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Publications since 2015

Thermodynamical estimation of the bounds on performance of irreversible binary distillation

Tsirlin, A. M. and Grigorevskiy, I. N. and Schwalbe, K.
International Journal of Heat and Mass Transfer 118: 289—296 (2018); DOI:10.1016/j.ijheatmasstransfer.2017.10.119

The limits of the ability of a distillation system to seperate a binary mixture are considered for two different cases. In the first case the heat supply solely takes place at the column bottom and the heat removal at the condenser. In the second case the heat supply and removal is distributed over the column height. For both cases, the limiting column capacity and the minimum heat consumption are related to the external stream compositions and to the heat and mass transfer coefficients.

Stochastic Novikov engine with time dependent temperature fluctuations

In this article a Stochastic Novikov engine is used to model a solar power plant. This heat engine is characterized by a fluctuating hot heat bath temperature. The distribution function of these fluctuations is derived based on a stochastic solar irradiation model. Using this distribution the control problem of determining the maximum of the expected work output is solved. This work output as well as the corresponding efficiency is deduced in dependence of the chosen control type. Considering this work output and this efficiency, which can be considered as performance measures of the solar power plant, the influence of the system parameters is discussed. It turns out that these performance measures increase monotonously with the parameter of the hot temperature's distribution function and that more control leads to higher efficiencies.

Performance Features of a Stationary Stochastic Novikov Engine

Karsten Schwalbe and Karl Heinz Hoffmann
Entropy 20: 52-1—13 (2018); DOI:10.3390/e20010052

In this article a Novikov engine with fluctuating hot heat bath temperature is presented. Based on this model, the performance measure maximum expected power as well as the corresponding efficiency and entropy production rate is investigated for four different stationary distributions: continuous uniform, normal, triangle, quadratic, and Pareto. It is found that the performance measures increase monotonously with increasing expectation value and increasing standard deviation of the distributions. Additionally, we show that the distribution has only little influence on the performance measures for small standard deviations. For larger values of the standard deviation, the performance measures in the case of the Pareto distribution are significantly different compared to the other distributions. These observations are explained by a comparison of the Taylor expansions in terms of the distributions’ standard deviations. For the considered symmetric distributions, an extension of the well known Curzon-Ahlborn efficiency to a stochastic Novikov engine is given.

Optimal Control of an Endoreversible Solar Power Plant

Karsten Schwalbe and Karl Heinz Hoffmann
Journal of Non-Equilibrium Thermodynamics 43(3): 255—271 (2018); DOI:10.1515/jnet-2018-0021

While in the classic Curzon–Ahlborn and Novikov engines the temperatures of the heat baths are kept fixed or follow a deterministic time function, it is the aim of this work to study the impact of fluctuating heat bath temperatures. As an example serves a solar power plant, where the stochastically varying cloud cover leads to fluctuations in the temperature of the hot heat bath. This solar thermal power plant is modeled as a stochastic endoreversible system. On the basis of this model the maximum expected work output of the power plant and the corresponding optimal control policy is derived. For the considered system it is found that the maximum expected work output changes with the reversion speed of the hot temperature depending on the relation of the starting hot temperature and the temperature of the power plant’s receiver. Additionally, it is found that the maximum expected work output increases with the hot temperature’s fluctuation strength.

Novikov engine with fluctuating heat bath temperature

Karsten Schwalbe and Karl Heinz Hoffmann
Journal of Non-Equilibrium Thermodynamics 43(2): 141—150 (2018); DOI:10.1515/jnet-2018-0003

The Novikov engine is a model for heat engines that takes the irreversible character of heat fluxes into account. Using this model, the maximum power output as well as the corresponding efficiency of the heat engine can be deduced, leading to the well known Curzon-Ahlborn efficiency. The classical model assumes constant heat bath temperatures, which is not a reasonable assumption in case of fluctuating heat sources. Therefore, in this article the influence of stochastic fluctuations of the hot heat bath’s temperature on the optimal performance measures is investigated. For this purpose, a Novikov engine with fluctuating heat bath temperature is considered. Doing so, an generalization of the Curzon-Ahlborn efficiency is found. The results can help to quantify how the distribution of fluctuating quantities effects the performance measures of power plants.

Modeling reaction kinetics of twin polymerization via differential scanning calorimetry

Prehl, Janett and Masser, Robin and Salamon, Peter and Hoffmann, Karl Heinz
Journal of Non-Equilibrium Thermodynamics 43(4): 347—357 (2018); DOI:10.1515/jnet-2018-0057

We present a kinetic model for the reaction mechanism of acid-catalyzed twin polymerization. Our model characterizes the reaction mechanism not by the reactants, intermediate structures, and products, but via reaction-relevant moieties. We apply our model for three different derivatives of 2,2’-Spirobi[4 extit{H}-1,3,2-benzodioxasiline] and determine activation energies, reaction enthalpies, and reaction rate constants for the reaction steps in our mechanism. We compare our findings to previously reported values obtained from density functional theory calculations. Furthermore, with this approach we are also able to follow the time development of the concentrations of the reaction-relevant moieties.

Modeling and simulation of nanostructure formation of TP

Prehl, Janett and Fiedler, Benjamin and Friedrich, Joachim
"Twin Polymerization: New Strategy for Hybrid Materials Synthesis" Eds: Spange, Stefan and Mehring, Michael: 135—166 DeGruyter, Berlin, Germany, 2018; ISBN: 978-3110500677

Twin polymerization is a novel approach where two distinct polymers are produced from a single source monomer, thus being an excellent tool for the synthesis of hybrid materials. The author introduces the principles of various twin polymerization processes, their classification and practical use. The book is supplied with numerous individual examples, demonstrating the potential of this strategy in materials synthesis.

A reactive bond fluctuation model (rBFM) for twin polymerization: Comparison of simulated morphologies with experimental data

Huster, Constantin and Nagel, Kevin and Spange, Stefan and Prehl, Janett
Chemical Physics Letters 713: 145—148 (2018); DOI:10.1016/j.cplett.2018.10.016

With twin polymerization (TP), nanostructured organic-inorganic hybrid materials are produced, which serve as intermediates for nano- and microporous materials. To manipulate the synthesis process appropriately for applications a detailed understanding of the emerging structures is desired.Here, we present a reactive bond fluctuation model (rBFM), that bases on a 3D lattice-based Monte-Carlo method, to model the full structure formation process of the complex TP process. Thus, we can analyze the final structures and compare them with experimental data. We show that the rBFM can capture the full TP process and that we find a good agreement between simulation and experiment.

Modeling the structure formation process of twin polymerization

Hoffmann, Karl Heinz and Prehl, Janett
Reaction Kinetics, Mechanisms and Catalysis 123: 367—383 (2018); DOI:10.1007/s11144-017-1303-y

Twin polymerization is an elegant technique to synthesize nanostructured organic—inorganic polymers with defined domain sizes of 0.5—3 nm. Although various classes of chemical structures undergoing twin polymerization have been found, the theoretical understanding of the overall twin polymerization process and especially of structure formation of the composite material is still at the beginning. Here, we introduce three different theoretical modeling approaches to investigate and analyze the structure formation process of twin polymerization on different levels of detail. We develop new methods and extend existing ones that range from reactive molecular dynamics simulations to reaction kinetics to reactive bond fluctuation models. We compare the obtained simulation results with experimental and quantum chemical data and find very good to reasonable agreement between theoretical modeling and experimental results. In doing so we achieve detailed insights to the structure formation process of twin polymerization on different length scales.zation on different length scales.

A Dual Power Law Distribution for the Stellar Initial Mass Function

Hoffmann, Karl Heinz and Essex, Christopher and Basu, Shantanu and Prehl, Janett
Monthly Notices of the Royal Astronomical Society 478(2): 2113—2118 (2018); DOI:10.1093/mnras/sty1251

We introduce a new dual power-law (DPL) probability distribution function for the mass distribution of stellar and substellar objects at birth, otherwise known as the initial mass function (IMF). The model contains both deterministic and stochastic elements, and provides a unified framework within which to view the formation of brown dwarfs and stars resulting from an accretion process that starts from extremely low-mass seeds. It does not depend upon a top down scenario of collapsing (Jeans) masses or an initial lognormal or otherwise IMF-like distribution of seed masses. Like the modified lognormal power-law (MLP) distribution, the DPL distribution has a power law at the high-mass end, as a result of exponential growth of mass coupled with equally likely stopping of accretion at any time interval. Unlike the MLP, a power-law decay also appears at the low-mass end of the IMF. This feature is closely connected to the accretion stopping probability rising from an initially low value up to a high value. This might be associated with physical effects of ejections sometimes (i.e. rarely) stopping accretion at early times followed by outflow driven accretion stopping at later times, with the transition happening at a critical time (therefore mass). Comparing the DPL to empirical data, the critical mass is close to the substellar mass limit, suggesting that the onset of nuclear fusion plays an important role in the subsequent accretion history of a young stellar object.

Between Waves and Diffusion: Paradoxical Entropy Production in an Exceptional Regime

Hoffmann, Karl Heinz and Kulmus, Kathrin and Essex, Christopher, and Prehl, Janett
Entropy 20(11): 881-1—15 (2018); DOI:10.3390/e20110881

The entropy production rate is a well established measure for the extent of irreversibility in a process. For irreversible processes, one thus usually expects that the entropy production rate approaches zero in the reversible limit. Fractional diffusion equations provide a fascinating testbed for that intuition in that they build a bridge connecting the fully irreversible diffusion equation with the fully reversible wave equation by a one-parameter family of processes. The entropy production paradox describes the very non-intuitive increase of the entropy production rate as that bridge is passed from irreversible diffusion to reversible waves. This paradox has been established for time- and space-fractional diffusion equations on one-dimensional continuous space and for the Shannon, Tsallis and Renyi entropies. After a brief review of the known results, we generalize it to time-fractional diffusion on a finite chain of points described by a fractional master equation.

A performance comparison of density-of-states methods

Haber, René and Hoffmann, Karl Heinz
Communications in Computational Physics 24(2): 383—407 (2018); DOI:10.4208/cicp.OA-2017-0058

Nowadays equilibrium thermodynamic properties of materials can be ob- tained very efficiently by numerical simulations. If the properties are needed over a range of temperatures it is highly efficient to determine the density of states first. For this purpose histogram- and matrix-based methods have been developed. Here we present a performance comparison of a number of those algorithms. The comparison is based on three different benchmarks, which cover systems with discrete and con- tinuous state spaces. For the benchmarks the exact density of states is known, for one benchmark — the FAB system — the exact infinite temperature transition matrix Q is also known. In particular the Wang-Landau algorithm in its standard and 1/t variant are compared to Q-methods, where estimates of the infinite temperature transition matrix are obtained by random walks with different acceptance criteria. Overall the Q-matrix methods perform better or at least as good as the histogram methods. In addition, dif- ferent methods to obtain the density of states from the Q-matrix and their efficiencies are presented.

The Rule of Temperature Coefficients for Selection of Optimal Separation Sequence for Multicomponent Mixtures in Thermal Systems

Tsirlin, Anatoly and Sukin, Ivan Andreevich and Balunov, Alexander and Schwalbe, Karsten
Journal of Non-Equilibrium Thermodynamics 42(4): 359—369 (2017); DOI:10.1515/jnet-2017-0024

In this paper an estimate for the reversible molar heat supply needed for fully separating a certain mixture is given on the basis of thermodynamic balance equations. It is shown that in order to estimate this heat supply one should solve the problem of selecting the optimal separation sequence. The algorithm solving this task is given. This algorithm allows to select the separation sequence on the basis of preliminary calculations, knowing only the properties of the component that one wants to separate. The solution algorithm is demonstrated for an exemplary system: a gas-fractionation plant.

Site Dependent Atom Type ReaxFF for the Proton-Catalyzed Twin Polymerization

Prehl, Janett and Schönfelder, Thomas and Friedrich, Joachim and Hoffmann, Karl Heinz
Journal of Physical Chemistry C 121(29): 15984—15992 (2017); DOI:10.1021/acs.jpcc.7b03219

ReaxFF is an efficient member of reactive molecular dynamics approaches to model chemical reactions for different chemical environments. Here it is applied to the structure formation process of twin polymerization, a newly developed method to obtain nanostructured functional materials. To achieve this, a site dependent atom type (SDAT) generalization of the classical ReaxFF approach is presented, which employs more then one atom type per chemical element. The efficacy of this SDAT-ReaxFF approach is demonstrated for two different cases: a benzene—benzyl reaction as well as for the twin polymerization.

Energy Recuperation System for Skip Trucks

Masser, Robin and Schwalbe, Karsten and Hoffmann, Karl Heinz
"Proceedings of ECOS 2017 - The 30th Environmental Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy systems" Eds: Beyene, Asfaw and MacPhee, David and Hernandez-Guerrero, Abel1: 650 , 2017

The reduction of the energy consumption by means of recuperation systems has been in the focus of research during the last decades. Following this goal, our work aims to reduce the fuel consumption of commercial vehicles, in particular of skip trucks. This is done by the installation of a module, consisting of a bladder accumulator and a hydraulic pump driven by the cardan shaft that is installed to store energy. The energy may then be used as additional propulsion, to support the thermal management and to operate the skipping mechanism. The latter application offers the advantage that the skipping mechanism can be operated without the need for running the combustion engine. Compared to other recuperation systems, hydraulic systems offer high energy and power density with relatively low weight. The hydraulic recuperation system is modeled applying endoreversible thermodynamics taking heat losses into account. Based on this model, system parameters and control strategies can be optimized in terms of power and efficiency. The resulting fuel and operational cost savings are estimated evaluating recorded urban driving data.

An embarrassingly parallel algorithm for random walk simulations on random fractal structures

Lang, Jens and Prehl, Janett
Journal of Computational Science 19: 1—10 (2017); DOI:10.1016/j.jocs.2016.11.014

Anomalous diffusion is often simulated by random walks on random fractal structures. As existing simulation methods either lack a high degree of parallelism or impose restrictions on the choice of fractal structures, a new approach is proposed here.  We present a parallel algorithm for simulating random walks on fractal structures that is suitable for a wide variety of hardware architectures. The degree of parallelism of the algorithm equals the number of random walkers, which is achieved by its communication-avoiding design. In contrast to other approaches, the random fractal structure is not pre-computed at whole. Instead, only the surrounding of each random walker is calculated by the parallel threads while the random walker moves around on the fractal structure.

Combining pressure and temperature control in dynamics on energy landscapes

Hoffmann, Karl Heinz and Schön, J. Christian
European Journal of Physics 90: 84-1—12 (2017); DOI:10.1140/epjb/e2017-70510-5

Complex systems from science, technology or mathematics usually appear to be very different in their specific dynamical evolution. However, the concept of an energy landscape with its basins corresponding to locally ergodic regions separated by energy barriers provides a unifying approach to the description of complex systems dynamics. In such systems one is often confronted with the task to control the dynamics such that a certain basin is reached with the highest possible probability. Typically one aims for the global minimum, e.g. when dealing with global optimization problems, but frequently other local minima such as the metastable compounds in materials science are of primary interest. Here we show how this task can be solved by applying control theory using magnesium fluoride as an example system, where different modifications of $MgF_2$ are considered as targets. In particular, we generalize previous work restricted to temperature controls only and present controls which simultaneously adjust temperature and pressure in an optimal fashion.

Chemical reactions in endoreversible thermodynamics

Wagner, Katharina and Hoffmann, Karl Heinz
European Journal of Physics 37: 015101 (2016); DOI:10.1088/0143-0807/37/1/015101

Endoreversible thermodynamics is a theory for the (approximate) description of thermodynamic non-equilibrium systems, which allows us to capture the ever present irreversibilities of real processes. For instance in heat engines the dissipation due to finite heat transport capabilities, as well as the resulting limitations in the energy fluxes, can be incorporated into the theory. It has thus been very successful in closing the gap between observed and theoretically predicted efficiencies. Here an extension of the theory is provided, with which chemical reactions can be included in the formalism. This opens up a wide field of applications for endoreversible modeling and the investigation of dissipative processes, for instance in fuel cells or batteries.

Rate constants, timescales, and free energy barriers

Salamon, Peter and Segall, Anca and Schön, J. Christian and Hoffmann, Karl Heinz and Andresen, Bjarne
Journal of Non-Equilibrium Thermodynamics 41(1): 13—18 (2016); DOI:10.1515/jnet-2015-0038

The traditional connection between rate constants and free energy landscapes is extended to define effective free energy landscapes relevant on any chosen timescale. Although the Eyring—Polanyi transition state theory specifies a fixed timescale of τ = h/kB T, we introduce instead the timescale of interest for the system in question, e.g. the observation time. The utility of drawing such landscapes using a variety of timescales is illustrated by the example of Holliday junction resolution. The resulting free energy landscapes are easier to interpret, clearly reveal observation time dependent effects like coalescence of short-lived states, and reveal features of interest for the specific system more clearly.

Symmetric Fractional Diffusion and Entropy Production

Prehl, Janett and Boldt, Frank and Hoffmann, Karl Heinz and Essex, Christopher
Entropy 18(7): 275-1—11 (2016); DOI:10.3390/e18070275

The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Extending the parQ transition matrix method to grand canonical ensembles

Haber, René and Hoffmann, Karl Heinz
Physical Review E 93: 063314-1—8 (2016); DOI:10.1103/PhysRevE.93.063314

Phase coexistence properties as well as other thermodynamic features of fluids can be effectively determined from the grand canonical density of states (DOS). We present an extension of the parQ transition matrix method in combination with the efasTM method as a very fast approach for determining the grand canonical DOS from the transition matrix. The efasTM method minimizes the deviation from detailed balance in the transition matrix using a fast Krylov-based equation solver. The method allows a very effective use of state space transition data obtained by different exploration schemes. An application to a Lennard-Jones system produces phase coexistence properties of the same quality as reference data.

Fastest Effectively Adiabatic Transitions for a Collection of Harmonic Oscillators

Boldt, Frank and Salamon, Peter and Hoffmann, Karl Heinz
The Journal of Physical Chemistry A 120(19): 3218—3224 (2016); DOI:10.1021/acs.jpca.5b11698

We discuss fastest effectively adiabatic transitions (FEATs) for a collection of noninteracting harmonic oscillators with shared controllable real frequencies. The construction of such transitions is presented for given initial and final equilibrium states, and the dependence of the minimum time control on the interval of achievable frequencies is discussed. While the FEAT times and associated FEAT processes are important in their own right as optimal controls, the FEAT time is an added feature which provides a measure of the quality of a shortcut to adiabaticity (STA). The FEAT time is evaluated for a previously reported experiment, wherein a cloud of Rb atoms is cooled following a STA recipe that took about twice as long as the FEAT speed limit, a time efficiency of 50%.

ETA-Graphics — an interface to endoreversible thermodynamics

Wagner, Katharina and Hoffmann, Karl Heinz
European Journal of Physics 36: 025010-1—11 (2015); DOI:10.1088/0143-0807/36/2/025010

Endoreversible thermodynamics is a theory for the description of irreversible thermodynamic processes. In this theory a non-equilibrium system is divided into a set of reversible subsystems which interact irreversibly with one another by exchanging energy and extensive quantities. These extensities act as carriers for the energy. ETA-Graphics is a graphics-based interface to endoreversible thermodynamics that can be used as an educational aid. It enables students to visually design endoreversible systems by drawing reversible subsystems and connecting them with irreversible (or reversible) interactions. Through special dialogs users specify the properties of the system, e.g., in form of transport laws for energy and extensive quantities. Based on the input ETA-Graphics allows students to analyse the endoreversible systems and their properties. Therefore, performance measures, i.e., efficiency and total power output, are calculated. Additionally, graphical representations of the results are shown.

Endoreversible modeling of a PEM fuel cell

Wagner, Katharina and Hoffmann, Karl Heinz
Journal of Non-Equilibrium Thermodynamics 40(4): 283—294 (2015); DOI:10.1515/jnet-2015-0061

Fuel cells are known for high efficiencies in converting chemical energy into electrical energy. Nonetheless, the processes taking place in a fuel cell still possess a number of irreversibilities that limit the power output to values below the reversible limit. To analyze these, we developed a model that captures the main irreversibilities occurring inside a proton exchange membrane or polymer electrolyte membrane fuel cell. We used the methods of endoreversible thermodynamics, which enable us to study the entropy production of the different sources of irreversibility in detail. Additionally, performance measures like efficiency and power output can be calculated with such a model, and the influence of different parameters, such as temperature and pressure, can be easily investigated. The comparison of the model predictions with realistic fuel cell data shows that the functional dependencies of the fuel cell characteristics can be captured quite well.

Quantum chemical investigation of the counter anion in the acid catalyzed initiation of 2,2'-spirobi[4H-1,3,2-benzodioxasiline] polymerization

Tchernook, Ivan and Prehl, Janett and Friedrich, Joachim
Polymer 60: 241—251 (2015); DOI:10.1016/j.polymer.2015.01.042

The recently discovered twin polymerization offers a remarkable access to new versatile hybrid materials. In order to develop an advanced model system to understand the reaction mechanism in detail we present an exemplary quantum chemical study on the role of the counter anion in the acid (trifluoroacetatic acid) catalyzed twin polymerization of 2,2'-spirobi[4H-1,3,2-benzodioxasiline]. Using three different model systems the reaction mechanism is investigated with respect to the influence of the anion (trifluoroacetate) in the reaction system. The extended model system with two monomers and two acid molecules exhibits lowest activation barriers for the first step of the organic network formation. The formation of the electrophile and the electrophilic substitution are supported by the present counter anion. The reaction mechanism of this initiating process is also influenced by sterical effects, since the size of the acid molecule is quite comparable to the size of other molecular structure units. The obtained sophisticated data base on the reaction mechanism and the energetics provides an important basis for structure formation modeling in future.

Free Energies of Staging a Scenario and Perpetual Motion Machines of the Third Kind

Salamon, Peter and Andresen, Bjarne and Hoffmann, Karl Heinz and Nulton, James D. and Segall, Anca M. and Rohwer, Forest L.
"Proceedings of the 240 Conference: Science's Great Challenges" Eds: Dinner, Aaron R. and Stuart, Rice A.157: 43—55 , 2015; ISBN: 978-1-118-95959-6; DOI:10.1002/9781118959602.ch4

Standard thermodynamics allows spontaneous reactions if their change of free energy ∆G=0. Following the tradition of thermodynamics, the author proposes to put the finite-time impossibility principle into the form of the nonexistence of Perpetual Motion Machines of the Third Kind (PM3). PM3 formalizes the statement that site-specific recombination of DNA strands cannot proceed in both directions spontaneously without some other input of free energy that is dissipated in the process. This chapter uses the term staging cost or staging free energy of the scenario to mean the free energetic investment that is required to make the scenario spontaneously reach its objective, that is, proceed as planned in the script of the scenario with high probability. Many biological processes seem to operate near the PM3 limit. The chapter presents a number of processes for which the energy dissipation is surprisingly low.

Quantum finite time availability for parametric oscillators

Hoffmann, Karl Heinz and Schmidt, Kim and Salamon, Peter
Journal of Non-Equilibrium Thermodynamics 40(2): 121—129 (2015); DOI:10.1515/jnet-2015-0025

The availability of a thermodynamic system out of equilibrium with its environment describes its ability to perform work in a reversible process which brings it to equilibrium with this environment. Processes in finite time can usually not be performed reversibly thus leading to unavoidable losses. In order to account for these losses the concept of finite time availability was introduced. We here add a new feature through the introduction of quantum finite time availability for an ensemble of parametric oscillators. For such systems there exists a certain critical time, the FEAT time. Quantum finite time availability quantifies the available work from processes which are shorter than the FEAT time of the oscillator ensemble.

Finite-Time Thermodynamics Tools to Analyze Dissipative Processes

Hoffmann, Karl Heinz and Andresen, Bjarne and Salamon, Peter
"Proceedings of the 240 Conference: Science's Great Challenges" Eds: Dinner, Aaron R. and Rice, Stuart A.157: 57—67 , 2015; ISBN: 978-1-118-95959-6; DOI:10.1002/9781118959602.ch5

The field of finite-time thermodynamics considers questions such as: does dissipation necessarily occur if a thermodynamic process takes place in finite time. This chapter presents four concepts: the tricycle, thermodynamic length, work deficiency, and network thermodynamics. These concepts highlight the basic features of finite-time thermodynamics and shed some light on the staging free energy problem. The concepts are also promising candidates for further development and application in biological systems. Beyond the realm of macroscopic systems, these concepts have been extended during the past years also to the realm of quantum systems. While network thermodynamics has proven its usefulness in macroscopic applications like the analysis of internal combustion engines, the complexity of biological systems remains a challenge.

Finite-time availability in a quantum system

Hoffmann, Karl Heinz and Salamon, Peter
EPL 109(4): 40004-1—6 (2015); DOI:10.1209/0295-5075/109/40004

Classically, availability refers to the work available in any reversible process that brings about equilibrium between the system and its environment. Here we introduce an additional meaning of availability as the maximum work associated with the change of an external parameter in the Hamiltonian of a quantum system. This availability can be gained in a FEAT process and for times larger than or equal to the FEAT time, there exists an optimal control that achieves the available work. For shorter times, quantum friction effects are unavoidable and the available work is thereby lowered. This finite-time availability is quantified here as a function of the time available.

Publications in 2010 - 2014

Prediction of Driver Intended Path at Intersections

Streubel, Thomas and Hoffmann, Karl Heinz
"Intelligent Vehicles Symposium Proceedings" Eds: : 134—139 , 2014; ISBN: 978-1-4799-3639-7; DOI:10.1109/IVS.2014.6856508

The complexity of situations occurring at intersections is demanding on the cognitive abilities of drivers. Advanced Driver Assistance Systems (ADAS) are intended to assist particularly in those situations. However, for adequate system reaction strategies it is essential to develop situation assessment. Especially the driver's intention has to be estimated. So, the criticality can be inferred and efficient intervention strategies can take action. In this paper, we present a prediction framework based on Hidden Markov Models (HMMs) and analyze its performance using a large database of real driving data. Our focus is on the variation of the model parameters and the choice of the dataset for learning. The direction of travel while approaching a 4-way intersection is to be estimated. A solid prediction is accomplished with high prediction rates above 90% and mean prediction times up to 7 seconds before entering the intersection area.

Fahrverhaltenanalyse an Kreuzungen auf Basis von Fahrdaten

Streubel, Thomas and Hoffmann, Karl Heinz
"AmE 2014 (GMM-FB 78)" Eds: : 5-1—6 , 2014; ISBN: 978-3-8007-3580-8

Kreuzungen sind ein wesentlicher Bestandteil der urbanen Infrastruktur. Je nach ihrer Beschaffenheit und dem Verkehrsaufkommen können sie ein hohes Maß an Komplexität annehmen und so den Fahrer vor besondere Herausforderungen stellen. Fahrerassistenzsysteme zielen darauf ab in genau solchen Situationen den Fahrer zu unterstützen. Inwieweit dies notwendig ist, lässt sich am Fahrerverhalten abschätzen. Um ein besseres Verständnis für das natürliche Annäherungsverhalten an eine Kreuzung zu gewinnen, soll hier anhand von Fahrdaten der Prozess der Annäherung nachvollzogen werden. Die Fahrdaten stammen aus dem Projekt ''Sichere Intelligente Mobilität Testfeld Deutschland'' (simTD). Dies war ein großangelegter Feldtest für Car-to-Car (C2C) Kommunikation. Hierbei wurden Daten verschiedener Fahrzeuge und Fahrer ausgewertet, die bei freier Fahrt auf dem Testgelände in Friedberg aufgezeichnet wurden. Ziel ist die Analyse der einzelnen Handlungsschritte während der Annäherung an eine spezifische 4-Wege Kreuzung. Der Fokus liegt in einer qualitativen Charakterisierung des Handlungsablaufs bei Abbiegeszenarien. Dabei werden quantitative Aussagen über den zeitlichen und räumlichen Verlauf einzelner Phasen — Beginn des Blinkens, der Verzögerung und des Abbiegevorgang sowie Gangschaltzeitpunkte — getroffen. Abhängig von der Richtung, aus der sich der Kreuzung angenähert wird, muss man zwischen dem Annäherungsverhalten bei Vorfahrt und bei Vorfahrtachten unterscheiden. Innerhalb dieser beiden Gruppen sind jeweils beide Abbiegerichtungen möglich.

Applied endoreversible thermodynamics: Optimization of powertrains

Schwalbe, Karsten and Fischer, Andreas and Hoffmann, Karl Heinz and Mehnert, Jens
"Proceedings of ECOS 2014 - 27th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems" Eds: Zevenhouen, R.: 45—55 , 2014; ISBN: 9781634391344

For the last decades, the theory of Endoreversible Thermodynamics has proven to be an important tool to investigate non-equilibrium thermal systems. In this theory, systems are considered as a network of reversible subsystems with irreversible interactions. A remarkable benefit of the Endoreversible Thermodynamics is the adaptivity of the desired level of detail for the model so that the complexity of the system remains relatively easy to handle. In this way, simplified models of good quality can be deduced for various systems, e.g. energy transformation devices or energy storage devices. The applicability of Endoreversible Thermodynamics to practically relevant systems is demonstrated by the ''Powertrain'' example. Therefore, both a combustion engine and a hydraulic energy storage (for recuperation purposes) are considered. The complex processes inside the engine and the energy storage are mapped to a simplified Endoreversible model. Synthesizing these Endoreversible models, we derive an estimate for the systems' full range operational behavior which can be used to optimize design and process parameters of the powertrain.

Reactive force field for electrophilic substitution at an aromatic system in twin polymerization

Schönfelder, Thomas and Friedrich, Joachim and Prehl, Janett and Seeger, Steffen and Spange, Stefan and Hoffmann, Karl Heinz
Chemical Physics 440: 119—126 (2014); DOI:10.1016/j.chemphys.2014.06.003

Twin polymerization is a new synthesis concept, which enables the formation of two different macromolecular structures from organic–inorganic hybrid materials in one single process step. To gain insights into formation processes we implement a first-principles-based ReaxFF reactive force field for C/H/O/Si for the initial electrophilic substitution of an aromatic system. We show that established parametrizations that have been developed to model chemical reactions of (hydro) carbon or carbon nanotubes systems successfully cannot reproduce this reaction although they include the same chemical elements and in parts same reaction mechanisms. Thus, we develop a new parametrization being capable in reproducing this aromatic reaction properly and compare it to the established ones to identify the differences.

A mathematical model for predicting lane changes using the steering wheel angle

Schmidt, Kim and Beggiato, Matthias and Hoffmann, Karl Heinz and Krems, Josef F.
Journal of Safety Research 49: 85—90 (2014); DOI:10.1016/j.jsr.2014.02.014

Positive safety effects of advanced driver assistance systems can only become effective if drivers accept and use these systems. Early detection of driver's intentionwould allowfor selective systemactivation and therefore reduce false alarms. Method: This driving simulator study aims at exploring early predictors of lane changes. In total, 3111 lane changes of 51 participants on a simulated highway track were analyzed. Results: Results show that drivers stopped their engagement in a secondary task about 7 s before crossing the lane, which indicates a first planning phase of the maneuver. Subsequently, drivers start moving toward the lane, marking a mean steering wheel angle of 2.5°. Steering wheel angle as a directly measurable vehicle parameter appears as a promising early predictor of a lane change. A mathematical model of the steering wheel angle is presented, which is supposed to contribute for predicting lane change maneuvers. Practical applications: The mathematical model will be part of a further predictor of lane changes. This predictor can be a newadvanced driver assistance system able to recognize a driver's intention. With this knowledge, other systems can be activated or deactivated so drivers get no annoying and exhausting alarm signals. This is one way how we can increase the acceptance of assistance systems.

Random walks of oriented particles on fractals

Haber, René and Prehl, Janett and Hoffmann, Karl Heinz and Herrmann, Heiko
Journal of Physics A: Mathematical and General 47(17): 155001-1—14 (2014); DOI:10.1088/1751-8113/47/15/155001

Random walks of point particles on fractals exhibit subdiffusive behavior, where the anomalous diffusion exponent is smaller than one, and the corresponding random walk dimension is larger than two. This is due to the limited space available in fractal structures. Here, we endow the particles with an orientation and analyze their dynamics on fractal structures. In particular, we focus on the dynamical consequences of the interactions between the local surrounding fractal structure and the particle orientation, which are modeled using an appropriate move class. These interactions can lead to particles becoming temporarily or permanently stuck in parts of the structure. A surprising finding is that the random walk dimension is not affected by the orientation while the diffusion constant shows a variety of interesting and surprising features.

Generische Umfeldmodellierung — Autonome Fahrzeugsteuerung durch eine Risikokarte

Streubel, Thomas and Moebus, Marco and Hoffmann, Karl Heinz
"VDI Wissensforum: 16. Internationaler Kongress — Elektronik im Fahrzeug" Eds: : 651—661 , 2013; ISBN: 978-3-1809-2188-4

We introduce a generic concept for environment modeling with artificial potential fields and its utilization for a temporary vehicle control. This approach is known in robotics and is already used for autonomous robot control successfully. However, the requirements differ highly in a vehicle environment considering relative velocities, driving dynamics and path restrictions. Nevertheless, artificial potential fields exhibit some advantages in the automotive context such as the opportunity to model the environment generically across multiple scenarios, including both lateral and longitudinal aspects in an elegant way. Road edges and vehicles are modeled by potential hills. Thus, the potential field can be interpreted as a risk map. Therefore, the driving task simplifies to staying in low potential areas. Since the gradient of a field is directing towards these areas, it can be utilized for vehicle control. Here, we present artificial potential models for the road and dynamic objects and show a gradient based automated control. This has been tested in a simulation and will be implemented in a vehicle.

Time Evolution of Relative Entropies for Anomalous Diffusion

Prehl, Janett and Boldt, Frank and Essex, Christopher and Hoffmann, Karl Heinz
Entropy 15(8): 2989—3006 (2013); DOI:10.3390/e15082989

The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is ''nearer'' to pure diffusion than another, placing pure wave propagation, desirably, ''furthest'' from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.

Optimal control of a collection of parametric oscillators

Hoffmann, Karl Heinz and Andresen, Bjarne and Salamon, Peter
Physical Review E 87: 062106-1—9 (2013); DOI:10-1103/PhysRevE.87.062106

The problem of effectively-adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact that leads us to adopting the coordinates: energy, Lagrangian, correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. A new invariant, the Casimir companion, consisting of a combination of our coordinates is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using 1-jump controls and enumerate all possible switchings for 1-jump effectively-adiabatic controls connecting any initial to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively-adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.

Controlled dynamics on energy landscapes

Hoffmann, Karl Heinz and Schön, J. Christian
The European Physical Journal B 86: 220-1—10 (2013); DOI:10.1140/epjb/e2013-31042-4

In systems with complex multi-minima energy landscapes, it is often not only the global minimum which is of great importance. For example, in materials science, metastable compounds corresponding to local minima on the landscape play a crucial role in many technological applications. In order to reach such modifications, both in computational and real world situations, it is necessary to optimally control the dynamics of the system on the landscape. We present a general method, how to design optimal temperature schedules for reaching particular basins on a complex landscape, by constructing a coarse-grained transition probability matrix from stochastic global landscape explorations, and subsequently using optimal control techniques on the Master equation describing the dynamics on the simplified energy landscape. As a demonstration example, the landscape of MgF2 is considered.

Diffusion of oriented particles in porous media

Haber, René and Prehl, Janett and Hermann, Heiko and Hoffmann, Karl Heinz
Physics Letters A 377: 2840—2845 (2013); DOI:j.physleta2013.08.036

Diffusion of particles in porous media often shows subdiffusive behavior. Here, we analyze the dynamics of particles exhibiting an orientation. The features we focus on are geometrical restrictions and the dynamical consequences of the interactions between the local surrounding structure and the particle orientation. This interaction can lead to particles getting temporarily stuck in parts of the structure. Modeling this interaction by a particular random walk dynamics on fractal structures we find that the random walk dimension is not affected while the diffusion constant shows a variety of interesting and surprising features.

''OptiVent'' - A New Approach for Controlling Mass Air Flow and Combustion in Direct Injection SI-Engines

Getzlaff, Jörn and Lambert, Thomas and Hoffmann, Karl Heinz and Fischer, Andreas
SAE International, Technical Papers : 2013-01-0592 (2013); DOI:10.4271/2013-01-0592

Combustion concepts for future SI engines try to meet CO2-emission commitments and legislation all over the world. Where the Diesel engine has an advantage by principle, the efficiency of the SI engine has to be improved significantly, while of course the exhaust emissions must not become worse. An approach is to reduce the gas exchange losses using fully variable valve trains on the intake side of the combustion engine. OptiVent is a patented new way of controlling the mass air flow in the cylinder of a combustion engine using opening valves during the compression phase of a four stroke engine. This technology regards a wider range of variability on the valvetrain components of the engine especially for opening the valves more than one time during a cycle. On the other hand it is necessary to combine this technology with direct injection to avoid fuel losses in the exhaust system and raising the exhaust hydrocarbon emission of the engine. Chemnitz University of Technology and the West Saxon University of Applied Sciences in Zwickau had performed numerical investigations on the potential of the OptiVent engine control and combustion system, using a fully variable valve train on the exhaust valves of the engine. The paper presents results from numerical simulations of the gas exchange, the mechanical losses of an engine with cylinder deactivation using OptiVent and regarding the effort for the starting process of engines with this new technology. The simulations show the potential of the new OptiVent-way of air mass control, so that the research can progress toward developing a running engine and testing it on a test bench. The research is funded by government and industrial partners.

Casimir companion: An invariant of motion for Hamiltonian systems

Boldt, Frank and Nulton, James D. and Andresen, Bjarne and Salamon, Peter and Hoffmann, Karl Heinz
Physical Review A 87: 022116-1—4 (2013); DOI:10.1103/PhysRevA.87.022116

In this paper an invariant of motion for Hamiltonian systems is introduced: the Casimir companion. For systems with simple dynamical algebras (e.g., coupled spins, harmonic oscillators) our invariant is useful in problems that consider adiabatically varying the parameters in the Hamiltonian. In particular, it has proved useful in optimal control of changes in these parameters. The Casimir companion also allows simple calculation of the entropy of nonequilibrium ensembles.

Optimal control in a quantum cooling problem

Salamon, Peter and Hoffmann, Karl Heinz and Tsirlin, Anatoly
Applied Mathematics Letters 25(10): 1263—1266 (2012); DOI:10.1016/j.aml.2011.11.020

The optimal control for cooling a quantum harmonic oscillator by controlling its frequency is considered. It is shown that this singular problem may be transformed with the proper choice of coordinates to an equivalent problem which is no longer singular. The coordinates used are sufficiently simple that a graphical solution is possible and eliminates the need to use a Weierstrass-like approach to show optimality. The optimal control of this problem is of significance in connection with cooling physical systems to low temperatures. It is also mathematically significant in showing the power and limitations of coordinate transformations for attacking apparently singular problems.

Tsallis Relative Entropy and Anomalous Diffusion

Prehl, Janett and Essex, Christopher and Hoffmann, Karl Heinz
Entropy 14: 701—716 (2012); DOI:10.3390/e1404701

In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback–Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order α ∈ (1, 2). They represent a bridging regime, where for α = 2 one obtains the diffusion equation and for α = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.

Characterizing a new composite material: Effect of NaOH coating of variable thickness on the properties of a tungsten microemitter

Mussa, Khalil O. and Fischer, Andreas and Mousa, Marwan S.
The Jordanian Journal of Physics 5(1): 27—31 (2012)

Tungsten based microemitter tips have been prepared with various tip radii ranging from 30 to 100 nm. These tips were manufactured by electrochemical etching a 0.1 mm diameter high purity (99.95%) tungsten wire at the meniscus of two molar NaOH solution. Contrary to the standard procedure, the tips' surfaces have not been cleaned off NaOH solution by ultrasonic cleaning in distilled water. Only a coarse cleaning by dipping the electro-polished samples a few times in distilled water has been performed. Thus, a layer of NaOH remained on the surface, which acts like a coating. par The thickness of this coating layer left on the core material depends on the number of dips of the sample in water after etching. This procedure produced composite microemitters which consisted of a tungsten core with three different thicknesses of coating — thick, medium or thin — consecutively produced by dipping the etched samples in water for one, six or twelve time(s). A conventional field emission microscopes with a tip (cathode) — screen (anode) separation standardized at 10 mm was used to characterize the electron emitters. The system was evacuated down to a base pressure of  ∼ 10−8 mbar when baked at up to  ∼ 180 °C overnight. This allowed measurements of typical Field Electron Emission (FE) characteristics, namely the current — voltage (IV) characteristics and the emission images on a conductive phosphorus screen (the anode).

Metallic and Composite Micropoint Cathodes: Aging Effect & Electronic and Spatial Characteristics

Mousa, Marwan S. and Fischer, Andreas and Mussa, Khalil O.
The Jordanian Journal of Physics 5(1): 21-26 (2012)

Composite micro-emitters consisting of a tungsten core coated with different dielectric materials were prepared. Various properties of these emitters were measured including the current-voltage (IV) characteristics and spatial current distributions. We compared coated and uncoated tips and determined differences between both types. It could be proven that coated emitters are superior to uncoated ones in terms of the current stability and the emission current obtained for the same applied voltages. After these samples have been stored under atmospheric conditions for a period of 10 to 20 years from the first time being characterized, they were tested again. The IV characteristics and spatial current distributions in addition to stability measurements were recorded. Various similarities as well as some differences compared to the initial characterization have been found. It is interesting to note that after one and a half decades these composite emitters are still functioning effectively without being subjected to field desorption processes. The dielectric layers built on the tungsten cores were still in shape and stable. Some theoretical analysis of the tip properties and their change during storage time is included. Particular attention is paid to the deviations from the ideal Fowler-Nordheim (FN) behavior as well as the related slope and intercept correction factors.

A unified approach to resolving the entropy production paradox

Hoffmann, Karl Heinz and Essex, Christopher and Prehl, Janett
Journal of Non-Equilibrium Thermodynamics 37(4): 393—412 (2012); DOI:10.1515/jnetdy-2012-0008

Bridging the regime between fully irreversible and fully reversible dynamics as represented by the two paradigmatic evolution equations for diffusion and wave propagation became possible by the use of fractional diffusion equations based on time- or space-fractional differential operators. These bridges are each characterized by a one-parameter family of distribution functions. In both cases one encounters a counter-intuitive behavior: the closer one gets to the reversible case, the larger the entropy production becomes. This feature is known as the entropy production paradox, and could be partly resolved for the time-fractional case by using the distribution mean as a way to characterize the internal quickness of the process, while for the space-fractional case a special location parameter was used. Here we are able to present a unified approach based on the distribution modes as the appropriate measure for the internal quickness of the processes.

Time-optimal processes for interacting spin systems

Boldt, Frank and Hoffmann, Karl Heinz and Salamon, Peter and Kosloff, Ronnie
Europhysics Letters 99: 40002-1—5 (2012); DOI:10.1209/0295-5079/99/40002

Reversible adiabatic processes connecting thermal equilibrium states are usually considered to be infinitely slow. Recently fast reversible adiabatic processes for quantum systems have been discussed. Here we present time-optimal processes for a paradigmatic ensemble of two interacting spin-½-systems in an external magnetic field, which previously had been employed as working fluid in a quantum refrigerator. These processes are realized by appropriate bang-bang or quasi bang-bang controls of the external magnetic field. Explicit control protocols including the necessary times for a transition connecting thermal equilibrium states depending on the limiting conditions on the magnetic field strength are presented.

Change of state variables in the problems of parametric control of oscillators

Tsirlin, Anatoly M. and Salamon, Peter and Hoffmann, Karl Heinz
Avtomatika i Telemekhanika (8): 53—64 (2011)

Решены задачи оптимального параметрического управления одиночным осциллятором и ансамблем квантовых осцилляторов. На их примере продемонстрированы возможности метода перехода к новым переменным пространства состояний управляемой системы.

Change of State Variables in the Problems of Parametric Control of Oscillators

Tsirlin, Anatoly M. and Salamon, Peter and Hoffmann, Karl Heinz
Automation and Remote Control 72(8): 1627—1638 (2011); DOI:10.1134/S0005117911080030

The problems of optimal parametric control of a single oscillator and an assembly of quantum oscillators were solved and used by way of example to demonstrate the potentialities of the method of transition to the new variables of the state space of the controlled system.

Time-optimal controls for frictionless cooling in harmonic traps

Hoffmann, Karl Heinz and Salamon, Peter and Rezek, Yair and Kosloff, Ronnie
Europhysics Letters 96(6): 60015-1—6 (2011); DOI:10.1209/0295-5075/96/60015

Fast adiabatic cooling procedures have important implications for the attainability of absolute zero. While traditionally adiabatically cooling a system is associated with slow thermal processes, for the parametric quantum harmonic oscillator fast frictionless processes are known, which transfer a system from an initial thermal equilibrium at one temperature into thermal equilibrium at another temperature. This makes such systems special tools in analyzing the bounds on fast cooling procedures. Previous discussions of those systems used frictionless cooling assuming real frequencies of the oscillator. Using a control with imaginary frequencies (repulsive potential) revises previous implications for the possible operation of a quantum refrigerator. Here we discuss these requisite revisions in the context of the third law of thermodynamics. In addition to minimum time controls, which are always of the bang-bang form, fast frictionless processes with a continuous variation of the frequency have been presented previously in the literature. Such continuous variation controls have been experimentally verified by cooling a Bose-Einstein condensate, while minimum time controls still await verification. As some implementations may indeed not be able to implement the instantaneous jumps in frequency required by bang-bang controls, constraining the rate of change in the frequency calls for ramped bang-bang solutions. We present such solutions and compare their performance to the continuous controls used in the experiment.

Accuracy of coarse grained Markovian dynamics

Hoffmann, Karl Heinz and Salamon, Peter
Physica A: Statistical Mechanics and its Applications 390: 3086—3094 (2011); DOI:10.1016/j.physa.2011.04.027

Markov chain models on a mesoscopic level are a widely used description for complex systems. They are based on the assumption that certain sets of microstates can be coarse grained as their internal dynamics is faster than the time scales considered in the modeling. Here we analyze quantitatively the errors made by using lumping techniques and present the first rigorous proof for bounds on such errors. Our bounds express the deviations from a full microscopic description for all subsequent time steps in terms of the deviations in the first time step.

Accelerating Physical Simulations Using Graphics Processing Units

Hoffmann, Karl Heinz and Hofmann, Michael and Lang, Jens and Rünger, Gudula and Seeger, Steffen
it - Information Technology 53(2): 49—59 (2011); DOI:10.1524/itit.2011.0625

Graphics processors are used in many fields of applications that require high computational power. Especially in scientific computing, the programming of graphics processing units is an active field of research. Because of their hardware characteristics, graphics processors are well-suited for regular parallelism, however the implementation of irregular problems requires more advanced strategies. In this article, the hardware architecture of graphics porcessors and different framewords for graphics processor programming, such as CAL, Brook+, CUDA and OpgenCL with their specific properties, are presented. Additionally, an overview of different physical applications that have been implemented successfully on graphics processors is given. The parallel implementation of a specific irregular physical application on graphics processors is presented in more detail. This application simulates anomalous diffusion in porous media using random walk on Random Sierpinski Carpets.

Competitive trapping in complex state spaces

Fischer, Andreas and Hoffmann, Karl Heinz and Schön, J. Christian
Journal of Physics A: Mathematical and General 44(7): 1—15 (2011); DOI:10.1088/1751-8113/44/7/075101

In complex state space dynamics at finite time scales, the trapping in certain regions of state space is of great importance, e.g. in the field of protein folding or in the application of stochastic global optimization algorithms. Here, we analyze the influence of the density of states on the features of the trapping process. In particular, we compare the trapping power of a valley with a power-law density of states to one with an exponentially growing density of states. The outcome of this competition crucially depends on the annealing speed and shows that the clear difference between these two paradigmatic densities of states observed at very slow (near-equilibrium) annealing is lost for fast non-equilibrium processes, and that the outcome of the relaxation can strongly depend on the time scale of the process and subtle features of the density of states.

Optimal control of the parametric oscillator

Andresen, Bjarne and Hoffmann, Karl Heinz and Nulton, James and Tsirlin, Anatoly and Salamon, Peter
European Journal of Physics 32(3): 827—843 (2011); DOI:10.1088/0143-0807/32/3/018

We present a solution to the minimum time control problem for a classical harmonic oscillator to reach a target energy ET from a given initial state (qi, pi) by controlling its frequency ω, ωmin ≤ ω ≤ ωmax. A brief synopsis of optimal control theory is included and the solution for the harmonic oscillator problem is used to illustrate the theory.

The superdiffusion entropy production paradox in the space-fractional case for extended entropies

Prehl, Janett and Essex, Christopher and Hoffmann, Karl Heinz
Physica A: Statistical Mechanics and its Applications 389(2): 215—224 (2010); DOI:10.1016/j.physa.2009.09.009

Contrary to intuition, entropy production rates grow as reversible, wave-like behavior is approached. This paradox was discovered in time-fractional diffusion equations. It was found to persist for extended entropies and for space-fractional diffusion as well. This paper completes the possibilities by showing that the paradox persists for Tsallis and Rényi entropies in the space-fractional case. Complications arising due to the heavy tail solutions of space-fractional diffusion equations are discussed in detail.

Simulating anomalous diffusion on graphics processing units

Hoffmann, Karl Heinz and Hofmann, Michael and Lang, Jens and Rünger, Gudula and Seeger, Steffen
" Proc. of the 11th IEEE International Workshop on Parallel and Distributed Scientific and Engineering Computing (PDSEC-10) " Eds: : 1—8 , 2010; ISBN: 978-1-4244-6534-7; DOI:10.1109/IPDPSW.2010.5470767

The computational power of modern graphics processing units (GPUs) has become an interesting alternative in high performance computing. The specialized hardware of GPUs delivers a high degree of parallelism and performance. Various applications in scientific computing have been implemented such that computationally intensive parts are executed on GPUs. In this article, we present a GPU implementation of an application for the simulation of diffusion processes using random fractal structures. It is shown how the irregular computational structure that is inherent to the application can be implemented efficiently in the regular computing environment of a GPU. Performance results are shown to demonstrate the benefits of the chosen implementation approaches.

Computational manufacturing of optical interference coatings: method, simulation results, and comparison with experiment

Friedrich, Karen and Wilbrandt, Steffen and Stenzel, Olaf and Kaiser, Norbert and Hoffmann, Karl Heinz
Applied Optics 49(16): 3150—3162 (2010); DOI:10.1364/AO.49.003150

Virtual depostion runs have been performed to estimate the procuction yield of selected oxide optical interference coatings when plasma ion-assisted deposition with an advanced plasma source is applied. Therby, depostion of each layer can be terminated either by broadband optical monitoring or quertz crystal monitoring. Numerous deposition runs of single-layer coatings have been performed to investigate the reproducibility of coating properties and to quantifydeposition errors for the simulation. Variations of the following parameters are considered in the simulation: refractive index, extinction coefficient, and film thickness. The refractive index and the extinction coefficient are simulated in terms of the oscillator model. The parameters are varied using an apodized normal distribution with known mean value and standard strategy. Several depositon runs of the selected oxide interference coatings have been performed to verify the simulation results by experimental data.

Publications in 2005 - 2009

Random Walks on random Koch curves

Seeger, Steffen and Hoffmann, Karl Heinz and Essex, Christopher
Journal of Physics A: Mathematical and General 42(22): 225002-1—11 (2009); DOI:10.1088/1751-8113/42/22/225002

Diffusion processes in porous materials are often modeled as random walks on fractals. In order to capture the randomness of the materials random fractals are employed, which no longer show the deterministic self-similarity of regular fractals. Finding a continuum differential equation describing the diffusion on such fractals has been a long-standing goal, and we address the question of whether the concepts developed for regular fractals are still applicable. We use the random Koch curve as a convenient example as it provides certain technical advantages by its separation of time and space features. While some of the concepts developed for regular fractals can be used unaltered, others have to be modified. Based on the concept of fibers, we introduce ensemble-averaged density functions which produce a differentiable estimate of probability explicitly and compare it to random walk data.

Maximum work in minimum time from a conservative quantum system

Salamon, Peter and Hoffmann, Karl Heinz and Rezek, Yair and Kosloff, Ronnie
Physical Chemistry Chemical Physics11: 1027—1032 (2009); DOI:10.1039/B816102J

This paper considers the problem of obtaining maximum work from a conservative quantum system corresponding to a given change in an external parameter in the Hamiltonian. The example we present is a non-interacting collection of harmonic oscillators with a shared frequency o which changes from a given initial to a given final value. The example is interesting for its role in experiments at ultra-low temperatures and for probing finite-time versions of the third law of thermodynamics. It is also the simplest system displaying quantum friction, which represents loss mechanisms in any reversible prelude to a thermal process. The example leads to a new type of availability. It is also the first example of a minimum time for transitions between thermal states of a thermodynamic system.

The quantum refrigerator: The quest for absolute zero

Rezek, Yair and Salamon, Peter and Hoffmann, Karl Heinz and Kosloff, Ronnie
Europhysics Letters 85: 30008-1—5 (2009); DOI:10.1209/0295-5057/85/30008

The emergence of the laws of thermodynamics from the laws of quantum mechanics is an unresolved issue. The generation of the third law of thermodynamics from quantum dynamics is analysed. The scaling of the optimal cooling power of a reciprocating quantum refrigerator is sought as a function of the cold bath temperature as TC → 0. The working medium consists of noninteracting particles in a harmonic potential. Two closed-form solutions of the refrigeration cycle are analyzed, and compared to a numerical optimization scheme, focusing on cooling toward zero temperature. The optimal cycle is characterized by linear relations between the heat extracted from the cold bath, the energy level spacing of the working medium and the temperature. The scaling of the optimal cooling rate is found to be proportional to TC3/2 giving a dynamical interpretation to the third law of thermodynamics.

Spin-box algorithm for low temperature dynamics of short range disordered Ising spin systems

Nemnes, George Alexandru and Hoffmann, Karl Heinz
Computer Physics Communications 180(7): 1098—1103 (2009); DOI:10.1016/j.cpc.2008.12.038

An approximate parallel approach was developed to describe efficiently the low temperature dynamics in short range Ising spin systems, based on the dynamically relevant sequence technique. It relates the low temperature dynamics to the structural properties of the state space of spin glasses and disordered ferromagnets, which has been proved to give accurate results for low temperatures. Large samples can be handled, which allows the analysis of domain formation and the discussion of the growth laws. The results are consistent with existing numerical and experimental data.

Threshold-selecting strategy for best possible ground state detection with genetic algorithms

Lässig, Jörg and Hoffmann, Karl Heinz
Physical Review E 79(4): 046702-1—8 (2009); DOI:10.1103/PhysRevE.79.046702

Genetic algorithms are a standard heuristic to find states of low energy in complex state spaces as given by physical systems such as spin flasses but also in combinatorial optimization. The paper considers the problem of selecting individuals in the current population in Genetic Algorithms for crossover. Many schemes have been considered in literature as possible crossover selection strategies. We show for a large class of quality measures that the best possible probability distribution for selecting individuals in each generation of the algorithm execution is a rectangular distribution over the individuals sorted by their energy values. This means uniform probabilities have to be assigned to a group of the individuals with lowest energy in the population. The considered strategy is dubbed threshold selection. The proof applies basic arguments of Markov chains and linear optimization and makes only a few assumptions on the underlying principles and hence applies to a large class of algorithms.

On the Structure of a Best Possible Crossover Selection Strategy in Genetic Algorithms

Lässig, Jörg and Hoffmann, Karl Heinz
"Research and Development in Intelligent Systems XXVI" Eds: Bramer, M. and Ellis, R. and Petridis, M.: 263—276 , 2009; ISBN: 978-1-84882-982-4 (print), 978-1-84882-983-1 (online); DOI:10.1007/978-1-84882-983-1_19

The paper considers the problem of selecting individuals in the current population in genetic algorithms for crossover to find a solution with high fitnee for a given optimization problem. Many different schemes have been described in the literature as possible strategies for this task but so far comparisons have been predominantly empirical. It is shown that if one wishes to maximize any linear function of the final state probabilities, e.g. the fitness of the best individual in the final population of the algorithm, then a best probability distribution for selecting an individual in each generation is a rectangular distribution over the individuals sorted in descending sequence by their fitness values. This means uniform probabilities have to be assigned to a group of the best individuals of the population but probabilities equal to zero to individuals from the current population can be chosen independently for each iteration and each individual. This result is then generalized also to typical practically applied performance measures, such as maximizing the expected fitness value of the best individual seen in any generation.

Bounding the lumping error in Markov chain dynamics

Hoffmann, Karl Heinz and Salamon, Peter
Applied Mathematics Letters 22: 1471—1475 (2009); DOI:10.1016/j.aml.2009.03.016

Forming lumped states in a Markov chain is a very useful device leading to a coarser level of description. The Markov chain on these lumped states is often taken as an approximation for the time evolution of the unlumped chain. In the present work we derive a bound on the error in this approximation.

Desiccation of a clay film: Cracking versus peeling

Sadhukhan, Supti and Prehl, Janett and Blaudeck, Peter and Hoffmann, Karl Heinz and Dutta, Tapati and Tarafdar, Sujata
The European Physical Journal E 27(4): 391—295 (2008); DOI:10.1140/epje/i2008-10401-9

We report a simulation study on competition between cracking and peeling, in a layer of clay on desiccation and how this is affected by the rate of drying, as well as the roughness of the substrate. The system is based on a simple 2-dimensional spring model. A vertical section through the layer with finite thickness is represented by a rectangular array of nodes connected by linear springs on a square lattice. The effect of reduction of the natural length of the springs, which mimics the drying is studied. Varying the strength of adhesion between sample and substrate and the rate of penetration of the drying front produces an interesting phase diagram, showing cross-over from peeling to cracking behavior. Changes in the number and width of cracks on varying the layer thickness is observed to reproduce experimental reports.

Anomalous diffusion in porous media

Prehl, Janett and Hoffmann, Karl Heinz and Hofmann, Michael and Rünger, Gudula and Tarafdar, Sujata
"Thermal Nonequilibrium - Lecture Notes of the 8th International Meeting on Thermodiffusion" Eds: Wiegand, S. and Köhler, W. and Dhont, J. K. G.3: 243—248 , 2008; ISBN: 978-3-89336-523-4; ISSN: 1866-1807

We studied anomalous diffusion under the influence of an external force on finite regular Sierpinski carpets. In order to investigate the time development of the probability density p(r,t) we utilize the master equation approach. Thus, we are able to determine important quantities depending on their space direction e ∈ {x,y}, like the mean drift velocities 〈vdre〉, the mean square displacements 〈e2〉 and the random walk dimensions dwe. Applying different force strengths in x-direction we find a maximum 〈vdrx〉 for small to medium force strengths in x. According to 〈x2〉 ∼ t[2/(dwx)], we determine that dwx < 2 along the external force. So, diffusion seems to be superdiffusive, although diffusion is hindered by structure and delayed be waiting times. Finally, this seems to be the result of two competing effects. First, the particles get accelerated due to the external force. However, they get also trapped according to the complex structure which takes more time to escape caused by the external force. Thus, the distribution spreads faster with than without an external force and dwx < 2.

Dynamically relevant structural properties of short-range spin glasses and disordered ferromagnets

Nemnes, George Alexandru and Hoffmann, Karl Heinz
Physical Review B 77: 172410 (2008)

Structural properties relevant for the low-temperature dynamics of short-range Ising systems are comparatively analyzed for spin glasses and disordered ferromagnets. The key elements, disorder and frustration, induce different topologies in the state space, going from funnel-like landscapes in the case of disordered ferromagnets to trapping landscapes for spin glasses. An efficient tool, dynamically relevant sequence, is introduced, which directly extracts the low-temperature dynamics.

Threshold Selecting: Best Possible Probability Distribution for Crossover Selection in Genetic Algorithms

Lässig, Jörg and Hoffmann, Karl Heinz and Enachescu, Mihaela
"Genetic and Evolutionary Computation Conference" Eds: , 2008; ISBN: 978-1-60558-131-6; DOI:10.1145/1388969.1389044

The paper considers the problem of selecting individuals in the current population in genetic algorithms for crossover to find a solution of high fitness of a given combinatorial optimization problem. Many different schemes have been considered in literature as possible crossover selection strategies, such as windowing, exponential reduction, linear transformation or normalization and binary tournament selection. It is shown that if one wishes to maximize any linear function of the final state probabilities, e.g. the fitness of the best individual of the final population of the algorithm, then the best probability distribution for selecting individuals in each generation is a rectangular distribution over the individuals sorted by their fitness values. This means uniform probabilities have to be assigned to a group of the best individuals of the population but probabilities equal to zero to individuals with fitness ranks higher than a fixed cutoff, which is equal to a certain rank in the sorted fitness vector. The considered strategy is called threshold selecting. The proof applies basic arguments of Markov chains and linear optimization and requires only a few assumptions on the underlying principles and hence applies to a large class of genetic algorithms.

An introduction to endoreversible thermodynamics

Reversible thermodynamic processes are convenient abstractions of real processes, which are always irreversible. Approaching the reversible regime means to become more and more quasistatic, letting behind processes which achieve any kind of finite transformation rate for the quantities studied. On the other hand studying processes with finite transformation rates means to deal with irreversibilities and in many cases these irreversibilities must be included in a realistic description of such processes. Endoreversible thermodynamics is a non-equilibrium approach in this direction by viewing a system as a network or internally reversible (endoreversible) subsystems exchanging energy in an irreversible fashion. This material provides an introduction to the subject.

Anomalous Transport in Disordered Fractals

Hoffmann, Karl Heinz and Prehl, Janett
"Anomalous Transport - Foundations and Applications" Eds: Klages, R. and Radons, G. and Sokolov, I. M.: 397—427 Wiley-VCH, Weinheim, 2008; ISBN: 978-3-527-40722-4

This multi-author reference work provides a unique introduction to the currently emerging, highly interdisciplinary field of those transport processes that cannot be described by using standard methods of statistical mechanics. It comprehensively summarizes topics ranging from mathematical foundations of anomalous dynamics to the most recent experiments in this field. In so doing, this monograph extracts and emphasizes common principles and methods from many different disciplines while providing up-to-date coverage of this new field of research, considering such diverse applications as plasma physics, glassy material, cell science, and socio-economic aspects. The book will be of interest to both theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic processes. It also forms an ideal starting point for graduate students moving into this area. 18 chapters written by internationally recognized experts in this field provide in-depth introductions to the following fundamental aspects of anomalous transport: Fractional calculus and stochastic theory; Dynamical systems and deterministic transport; Anomalous transport in disordered systems; Applications to complex systems and experimental results.

An interlacing theorem for reversible Markov chains

Grone, Robert and Hoffmann, Karl Heinz and Salamon, Peter
Journal of Physics A: Mathematical and General 41: 1—7 (2008); DOI:10.1088/1751-8113/41/21/212002

Reversible Markov chains are an indispensable tool in the modeling of a vast class of physical, chemical, biological and statistical problems. Examples include the master equation descriptions of relaxing physical systems, stochastic optimization algorithms such as simulated annealing, chemical dynamics of protein folding and Markov chain Monte Carlo statistical estimation. Very often the large size of the state spaces requires the coarse graining or lumping of microstates into fewer mesoscopic states, and a question of utmost importance for the validity of the physical model is how the eigenvalues of the corresponding stochastic matrix change under this operation. In this paper we prove an interlacing theorem which gives explicit bounds on the eigenvalues of the lumped stochastic matrix.

Intermittent relaxation in hierarchical energy landscapes

Fischer, Andreas and Hoffmann, Karl Heinz and Sibani, Paolo
Physical Review E 77(4): 041120/1-5 (2008); DOI:10.1103/PhysRevE.77.041120

We numerically simulate a thermalization process in an energy landscape with hierarchically organized metastable states. The initial configuration is chosen to have a large energy excess relative to the thermal equilibrium value at the running temperature. We show that the initial energy surplus is dissipated in a series of intermittent bursts, or quakes, whose rate decreases as the inverse of the age of the system. In addition, one observes energy fluctuations with a zero-centered Gaussian distribution. These pertain to the pseudoequilibrium dynamics within a single metastable state and do not contribute to the energy dissipation. The derivative of the thermal energy with respect to the logarithm of time is asymptotically constant and comprises a temperature-independent part and a part with an Arrhenius temperature dependence. The findings closely mirror recent numerical simulation results obtained for microscopic glassy models. For these models, record-sized energy fluctuations have been claimed to trigger intermittent events during low-temperature thermalization. In the present model record-sized fluctuations are by construction needed to trigger changes from one metastable state to another. This property thus suffices to explain the statistical property of intermittent energy flow in complex metastable systems.

Quantifying Dissipation

Hoffmann, Karl Heinz
Communications to SIMAI Congress 2: 1—12 (2007); DOI:10.1685/CSC06171

Reversible thermodynamic processes are convenient abstractions of real processes, which are always irreversible. Approaching the reversible regime means to become more and more quasistatic, letting behind processes which achieve any kind of finite transformation rate for the quantities studied. On the other hand studying processes with finite transformation rates means to deal with irreversibilities and in many cases these irreversibilities must be included in a realistic description of such processes. There are various approaches how to not negelect finite times and rates while not being slain by the real worlds complexity. Endoreversible thermodynamics is a non-equilibrium approach in this direction by viewing a system as a network of internally reversible (endoreversible) subsystem exchanging energy in an irreversible fashion.

Power law rank-abundance models for marine phage communities

Hoffmann, Karl Heinz and Rodriguez-Brito, Beltran and Breitbart, Mya and Bangor, David and Angly, Florent and Felts, Ben and Nulton, James and Rohwer, Forest and Salamon, Peter
FEMS Mircobiology Letters 273: 224-228 (2007); DOI:10.1111/j.1574-6968.2007.00790.x

Metagenomic analyses suggest that the rank-abundance curve for marine phage communities follows a power law distribution. A new type of power law dependence based on a simple model in which a modified version of Lotka-Volterra predator-prey dynamics is sampled uniformly in time is presented. Biologically, the model embodies a kill the winner hypothesis and a neutral evolution hypothesis. The model can match observed power law distributions and uses very few parameters that are readily identifiable and characterize phage ecosystems. The model makes new untested predictions: (1) it is unlikely that the most abundant phage genotype will be the same at different time points and (2) the long-term decay of isolated phage populations follows a power law.

Modeling anomalous superdiffusion

Fischer, Andreas and Seeger, Steffen and Hoffmann, Karl Heinz and Essex, Chistopher and Davison, Matt
Journal of Physics A: Mathematical and General 40(38): 11441-11452 (2007); DOI:10.1088/1751-8113/40/38/001

Continuous models for anomalous diffusion have previously been tested in the subdiffusive case by making comparisons to diffusion on a Sierpinski gasket. This paper extends this discussion to the superdiffusive case by comparing performance to diffusion on a tree model. Although there is reasonable agreement within limited regimes for all four models, one model, due to Compte and Jou, stands out as being consistently sound over all regimes studied.

Anomalous diffusion on random fractal composites

Anh, Do Hoang Ngoc and Blaudeck, Peter and Hoffmann, Karl Heinz and Prehl, Janett and Tarafdar, Sujata
Journal of Physics A: Mathematical and General 40(38): 11453-11465 (2007); DOI:10.1088/1751-8113/40/38/002

Stochastic fractals, generated from combinations of deterministic fractals, have the advantage of being tractable to some extent, but also being closer to real materials, since they are partially disordered. In the present work, we focus our attention on the remarkable nonlinear mixing behavior exhibited by fractals generated as random combinations of two different Sierpinski carpet generators. When patterns with different anomalous diffusion exponents and the same or different fractal dimensions are combined together, the effective diffusion exponent cannot in general be expressed as a linear weighted average of the diffusion exponents of the constituents. The effective exponent may show a maximum or minimum for certain compositions. An explanation of this interesting phenomenon is offered on the basis of details of the carpet generator, particularly on the number and position of `connection points', which determine the connectivity of the `fractal composite'.

The cumulant method for gas dynamics

Seeger, Steffen and Hoffmann, Karl Heinz and Meyer, Arndt
"Parallel algorithms and cluster computing" Eds: Hoffmann, Karl Heinz and Meyer, Arndt: 335—360 Springer-Verlag, Berlin Heidelberg, 2006; ISBN: 3-540-33539-9; DOI:10.1007/3-540-33541-2_19

Characterizing fluid flow by the ratio of mean free path and a characteristic flow length (the Knudsen number Kn) we have two extremes: dense gases (Kn « 1) where modeling by Euler or Navier-Stokes equations is valid and rarefied gases (Kn » 1) for which modeling by the Boltzmann equation is necessary. Developing models for the intermediate transition regime is subject to active current research because despite the tremendously growing increase in computational and algorithmic computing performance, numerical simulation of flows in the transition regime remains a challenging problem. Thus there is a considerable gap in the ability to model flows where mean free path and characteristic flow lengths are comparable. However, efficient methods for simulating transition regime flows will be an important design tool for micro-scale machinery, where dense gas models become invalid.

The structure of enumerated spin glass state spaces

Schubert, Sven and Hoffmann, Karl Heinz
Computer Physics Communications 174: 191—197 (2006); DOI:10.1016/j.cpc.2004.02.019

We enumerate the low energy part of the state space of an Ising spin glass using an e cient branch-and-bound algorithm. A coarse graining algorithm (NB-clustering) is employed to condense the inherent information to a system size which is treatable in computer simulations. The reduced state space still incorporates all ingredients necessary to simulate aging e ects. We investigate its structure in detail and find that certain assumptions made in heuristical state space models which have been presented in the past to reproduce aging phenomena in spin glass experiments are indeed compatible with the data from the observed state spaces.

Endoreversible Thermodynamics: A Tool for Simulating and Comparing Processes of Discrete Systems

Muschik, Wolfgang and Hoffmann, Karl Heinz
Journal of Non-Equilibrium Thermodynamics 31(3): 293—317 (2006); DOI:10.1515/JNETDY.2006.013

Endoreversible thermodynamics is concerned with reversible sub-systems which are in irreversible interaction with each other. Consequently, endoreversible thermodynamics represents the analogue for discrete systems to the local equilibrium hypothesis in continuum thermodynamics. Here a real cyclic 2-reservoir process is simulated by endoreversible model processes. Simulation means, that the simulating process has the same net heat exchanges, cycle time, power, entropy production, and efficiency as the original one. By introducing process-independent heat conduction coefficients as a constraint for the irreversible interaction, a family of comparative endoreversible processes is generated including the simulation of the original process. This procedure allows to compare process parameters of the family of comparative processes to those of the original one. The fraction ''power of the real process over the maximal power inbetween the comparative family'' is introduced as a parameter describing the process excellence.

Task Pool Teams Implementation of the Master Equation Approach for Random Sierpinski Carpets

Hoffmann, Karl Heinz and Hofmann, Michael and Rünger, Gudula and Seeger, Steffen
"Proc. of the 12th International Euro-Par Conference" Eds: 4128/2006: 1043—1052 , 2006; ISBN: 978-3-540-37783-2; DOI:10.1007/11823285_110

We consider the use of task pool teams in implementation of the master equation on random Sierpinski carpets. Though the basic idea of dynamic storage of the probability density reported earlier applies straightforward to random carpets, the randomized construction breaks up most of the simplifications possible for regular carpets. In addition, parallel implementations show highly irregular communication patterns. We compare four implementations on three different Beowulf-Cluster architectures, mainly differing in throughput and latency of their interconnection networks. It appears that task pool teams provide a powerful programming paradigm for handling the irregular communication patterns that arise in our application and show a promising approach to efficiently handle the problems that appear with such randomized structures. This will allow for highly improved modelling of anomalous diffusion in porous media, taking the random structure of real materials into account.

Parallel algorithms and cluster computing - implementations, algorithms, and applications

Eds: Hoffmann, Karl Heinz and Meyer, Arnd
Springer-Verlag, Berlin Heidelberg, 2006; ISBN: 978-3-540-33539-9

Modelling aging experiments in spin glasses

Hoffmann, Karl Heinz and Fischer, Andreas and Schubert, Sven and Streibert, Thomas
"Parallel algorithms and cluster computing" Eds: Hoffmann, Karl Heinz and Meyer, A.: 281 Springer-Verlag, Berlin Heidelberg, 2006; ISBN: 978-3-540-33539-9; DOI:10.1007/3-540-33541-2_16

Spin glasses are a paradigm for complex systems. They show a wealth of different phenomena including metastability and aging. Especially in the low temperature regime they reveal a very complex dynamical behaviour. For temperatures below the spin glass transition temperature one finds a variety of features connected to the inability of the systems to attain thermodynamic equilibrium with the ambient conditions on the observation time scale: aging and memory effects have been observed in many experiments [1–11]. Spin glasses are good model systems as their magnetism provides an easy and very accurate experimental probe into their dynamic behavior. In order to investigate such features different experimental techniques have been applied. Complicated setups including temperature and field changes with subsequent relaxation phases lead to more interesting effects such as age reinitialization and freezing [12, 13].

Random walks on fractals

Franz, Astrid and Schulzky, Christian and Do Hoang, Ngoc Anh and Seeger, Steffen and Balg, Janett and Hoffmann, Karl Heinz
"Parallel algorithms and cluster computing" Eds: Hoffmann, Karl Heinz and Meyer, A.: 303 Springer-Verlag, Berlin Heidelberg, 2006; ISBN: 978-3-540-33539-9; DOI:10.1007/3-540-33541-2_17

Porous materials such as aerogel, porous rocks or cements exhibit a fractal structure for a range of length scales [1]. Diffusion processes in such disordered media are widely studied in the physical literature [2, 3]. They exhibit an anomalous behavior in terms of the asymptotic time scaling of the mean square displacement of the diffusive particles.

Optimizing simulated annealing schedules for amorphous carbons

Blaudeck, Peter and Hoffmann, Karl Heinz
"Parallel algorithms and cluster computing" Eds: Hoffmann, K. H. and Meyer, A.: 227 Springer-Verlag, Berlin Heidelberg, 2006; ISBN: 978-3-540-33539-9; DOI:10.1007/3-540-33541-2_12

Annealing, carried out in simulation, has taken on an existence of its own as a tool to solve optimization problems of many kinds [1–3]. One of many important applications is to find local minima for the potential energy of atomic structures, as in this paper, in particular structures of amorphous carbon at room temperature. Carbon is one of the most promising chemical elements for molecular structure design in nature. An infinite richness of different structures with an incredibly wide variety of physical properties can be produced. Apart from the huge variety of organic substances, even the two crystalline inorganic modifications, graphite and diamond, show diametrically opposite physical properties. Amorphous carbon continues to attract researchers for both the fundamental understanding of the microstructure and stability of the material and the increasing interest in various applications as a high performance coating material as well as in electronic devices.

The coastline and lake shores on a fractal island

Blaudeck, Peter and Seeger, Steffen and Schulzky, Christian and Hoffmann, Karl Heinz and Dutta, Tapati and Tarafdar, Sujata
Journal of Physics A: Mathematical and General 39: 1609—1618 (2006); DOI:10.1088/0305-4470/39/7/006

We compute the fractal dimensions of the ''hulls'' or external boundary and the boundaries of the internal cavities in several deterministic as well as random fractal structures. Our conclusion is that the two fractal dimensions are in fact identical. The deterministic fractals we study are Sierpinski carpets (SC) in a two-dimensional space and the random fractals are percolation clusters at criticality. As an intermediate case, we present results on some randomized SC. In the random structures, statistics of the area and perimeters of all internal cavities or holes are taken and the fractal dimension of the hull borderline is computed. Two different definitions of the borderline are used, considering nearest neighbours as well as nearest and second nearest neighbours as connected. The conclusion is valid for both cases.

Energy and cost assessment of micro-CHP plants in high-performance residential buildings

Sicre, Benoit and Bühring, Andreas and Platzer, Bernd and Hoffmann, Karl Heinz
"Proceedings of ECOS 2005, the 18th International Conference on Efficiency, Cost, Optimization, Simulation, and Environmental Impact of Energy Systems: Trondheim, Norway, June 20 - 22, 2005" Eds: Kjelstrup, S.2: 1063—1071 , 2005; ISBN: 82-519-2041-8

Lately, efforts were made to scale-down some Combined Heat and Power (CHP) technologies, sch as fuel cell, gas turbine or organic Rankine cycle power plants in order to suit well for residential applications. The driving force for this is the high overall thermal efficiency and the low associated GHG emmissions. Beside this, distributed power generation is exected to alleviate partially the issue related to the general rise of electricity demand. However, as micro-CHP habe only modest electrical conversion efficiencies, the effective exploitation of the thermal output is critical to realising high levels of energy efficiency and the associated environmental benefits, Heat to power ratio of currently available micro-CHP devices (in the best case approx. 3 to 1) fits poorly with the heat to power ratio of high-performance buildings (in an average 1 to 1 according to recent construction practises). The present paper considers the building and the HVAC system (including the micro-CHP) and the resident's heat and power patterns as a whole, taking into account their inherent interactions. It describes first new simulation tools which were developed to balance in terms of energy and of costs the production and the consumption of both heat and electricity. Computation based on SOFC fuel cells and Stirling engines are discussed. Second, these technologies such as condensing gas boilers. Third, the role of the apropriate level of thermal insulation for the building as a compromise between building heat losses and overall primary energy efficiency, including electricity supply, is discussed.

On symbolic derivation of the cumulant equations

Seeger, Steffen and Hoffmann, Karl Heinz
Computer Physics Communications 168(3): 165—176 (2005); DOI:10.1016/j.cpc.2005.03.106

We discuss the application of Mathematica for automated, symbolic calculation of the cumulant equations of arbitrary order. Like moment equations, these partial differential equations-describing fluid motion on a mesoscopic scale-may be considered an approximation to the Boltzmann equation, a highly nonlinear integro-differential equation that describes the motion of gases at a microscopic scale. Though the cumulant method provides a simple and compact presentation of the theory, actual calculation of very high order equations turns out to be a challenging task.

On the Domain of Hyperbolicity of the Cumulant Equations

Seeger, Steffen and Hoffmann, Karl Heinz
Journal of Statistical Physics 121(1—2): 75—90 (2005); DOI:10.1007/s10955-005-6969-2

In this article we consider the influence of non-equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations. Real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid. We find that this condition puts no constraints on velocity and shear stress values, but specific energy must be positive, normal stress must be bounded by specific energy and heat flux not be too large.

The cumulant method for the space-homogeneous Boltzmann equation

Seeger, Steffen and Hoffmann, Karl Heinz
Continuum Mechanics and Thermodynamics 17(1): 51—60 (2005); DOI:10.1007/s00161-004-0187-z

In this work we give a comparison of the exact Bobylev/Krook-Wu solution to the space-homogeneous Boltzmann equation and numerical results obtained by a implementation of the cumulant method for the space-homogeneous case. We find excellent agreement of the numerical solution to the cumulant equations with the exact solution of the space-homogeneous Boltzmann equation as long as the exact, non-linear production terms are used. If a linearized variant of the production terms is used, relaxation rates may be underestimated due to convergence to the solution of the linearized equations.

Nonlinear $I$-$V$ characteristics of nanotransistors in the Landauer—Büttiker formalism

Nemnes, G. A. and Wulf, U. and Racec, P. N.
Journal of Applied Physics 98: 1—8 (2005); DOI:10.1063/1.2113413

We present the nonlinear $I$-$V$ characteristics of a nanoscale metal-oxide-semiconductor field-effect transistor in the Landauer-Büttikker formalism. In our three-dimensional ballistic model the gate, source, and drain contacts are treated on an equal footing. As in the drift-diffusion regime for ballistic transport a saturation of the drain current results. We demonstrate the quantum mechanism for the ballistic drain current saturation. As a specific signature of ballistic transport we find a specific threshold characteristic with a close-to-linear dependence of the drain current on the drain voltage. This threshold characterisitc separates the ON-state regime from a quasi-OFF-state regime in which the device works as a tunneling transistor. Long- and short-channel effects are analyzed in both regimes and compared qualitatively with existing experimental data by Intel [B. Doyle {it et al.}, Intel Technol. J. $mathbf 6$, 42 (2002)].

Optimization by thermal cycling

Möbius, Arnulf and Hoffmann, Karl Heinz and Schön, Christian
"Complexity, Metastability and Nonextensitivity" Eds: Beck, C. and Benedek, G. and Rapisarda, A. and Tsallis, C.: 215-219 World Scientific, , 2005; ISBN: 981-256-525-6

Thermal cycling is an heuristic optimization algorithm which consists of cyclically heating and quenching by Metropolis and local search procedures, respectively, where the amplitude slowly decreases. In recent years, it has been successfully applied to two combinatorial optimization tasks, the traveling salesman problem and the search for low-energy states of the Coulomb glass. In these cases, the algorithm is far more efficient than usual simulated annealing. In its original form the algorithm was designed only for the case of discrete variables. Its basic ideas are applicable also to a problem with continuous variables, the search for low-energy states of Lennard-Jones clusters.

The structure of marine phage populations

Hoffmann, Karl Heinz and Rodriguez-Brito and Breitbart, Mya and Bangor, David and Angly, Florent and Felts, Ben and Nulton, James and Rohwer, Forest and Salamon, Peter
"Proceedings of ECOS 2005, the 18th International Conference on Efficiency, Cost, Optimization, Simulation, and Environmental Impact of Energy Systems: Trondheim, Norway, June 20 - 22, 2005" Eds: Kyelstrup, Signe and Forskningsr aa}d, Norges(2): 711—715 , 2005

Phage are the most abundant biological entities in the biosphere, with an estimated $10^31$ particles on the planet. They also play a major role in carbon cycling; at least 25% of fixed carbon passes through phage. Their roles as predators of bacteria have important implications for possible marine CO$_2$ sequestration. Metagenomic analyses show that the rank-abundance curve for marine phage communities follows a power law distribution. This distribution is consistent with a proposed, modified version of Lotka-Volterra predator-prey dynamics, where blooms of a specific microbial species lead to blooms of their corresponding phage and a subsequent decrease in abundance. The model predicts that the majority of phage genotypes in a population will be rare and it is unlikely that the most abundant phage genotype will be the same at different time points. The model is based on spatial-temporal heterogeneity and a power law phage decay, which are both supported by empirical data.

Kinetic Features of Preferential Trapping on Energy Landscapes

Hoffmann, Karl Heinz and Schön, J. Christian
Foundations of Physics Letters 18(2): 171—182 (2005); DOI:10.1007/s10702-005-3960-8

The dynamics of complex systems can be mapped onto trajectories on their energy landscape. The properties of such trajectories as a function of temperature, and thus the chances of the system to enter certain regions of the state space, can be understood in terms of such energy landscapes. Here we show that their kinetic features are of equal importance as the previously discussed energetic and entropic features. Especially for barrier-crossing movements on mountainous landscapes, we observe competing effects between these three aspects, which can lead to surprising inversions in the chances to find certain states such as local minima in the systems.

ParQ — high-precision calculation of the density of states

Heilmann, Frank and Hoffmann, Karl Heinz
Europhysics Letters 70(2): 155-161 (2005); DOI:10.1209/epl/i2004-10486-8

We present a highly effective, parallelized random-walk-based algorithm to calculate the density of states of complex physical systems. Random walkers' attempted moves from one energy level to another are represented in a stochastic matrix, giving estimates for the transition matrix at infinite temperature. The eigenvector corresponding to the largest eigenvalue is the density of states up to a normalization. We verify the performance on selected examples of Ising spin systems with random coupling constants drawn uniformly from [-1,1], of which the exact density of states have been calculated by a branch-and-bound approach.

Transport of O2 from arterioles

Bertuglia, Silvia and Limon, Alfonso and Andresen, Bjarne and Hoffmann, Karl Heinz and Essex, Chistopher and Salamon, Peter
Journal of Non-Equilibrium Thermodynamics 30(2): 151—162 (2005); DOI:10.1515/JNETDY.2005.011

Oxygen delivery to the tissues is crucial to survival but our understanding of the processes involved in the transport of oxygen from blood to tissue is incomplete. The aim of the present work is to illustrate a long-standing paradox regarding such transport by reporting new state-of-the-art measurements and by analyzing the results in several ways, thereby exploring possible resolutions of the paradox. Our model calculations show that slight extensions of system parameters are sufficient to overcome the apparent inconsistencies. Alternatively, so far unappreciated mild effects like flow-assisted diffusion in the interstitium will explain the supernormal diffusion of oxygen.

Diffusion in disordered Fractals

Anh, Do Hoang Ngoc and Hoffmann, Karl Heinz and Seeger, Steffen and Tarafdar, Sujata
Europhysics Letters 70(1): 109—115 (2005); DOI:10.1209/epl/i2005-10002-x

Diffusion in disordered media can be modelled by the anomalous diffusion in fractals. Up to now, usually regular fractals were used as models for such disordered systems. Here we study disordered fractals in an attempt to capture the random nature of the disordered material. In particular, we investigate the diffusion in fractals obtained by randomly mixing different sierpinski carpet generators. We find that the random-walk exponent dw shows strong dependence on the mixture composition. For the mixed system it can be higher or lower than both the pure components. Further, dw may decrease on mixing, indicating faster diffusion in the disordered system.

Thermo-mechanical systems with several heat reservoirs: maximum power processes

Amelkin, S. A. and Andresen, B. and Burzler, J. M. and Hoffmann, K. H. and Tsirlin, A. M.
Journal of Non-Equilibrium Thermodynamics 30(1): 67—80 (2005); DOI:10.1515/JNETDY.2005.005

While endoreversible heat-to-power conversion systems operating between two heat reservoirs have been intensely studied, systems with several reservoirs have attracted little attention. Here we analyse the maximum power processes of such systems with stationary temperature reservoirs. We nd that independent of the number of reservoirs the working uid uses only two isotherms and two in nitely fast isentropes/ adiabats. One surprising result is that there may be reservoirs that are never used. This feature is explained for a simple system with three heat reservoirs.

Publications in 2000 - 2004

Erratum: The cumulant method applied to a mixture of Maxwell gases

Seeger, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 16(5): 515 (2004)

Aging in enumerated spin glass state spaces

Schubert, Sven and Hoffmann, Karl Heinz
Europhysics Letters 66(1): 118—124 (2004); DOI:10.1209/epl/i2003-10142-y

Aging phenomena are observed in many spin class experiments. Heuristic state space models were presented in the past to reproduce these effects. We here start the investigation by considering the real state space of an Ising spin glass Hamiltonian. A branch-and-bound algorithm is used to find the low-energy part of the state space. We solve the problem of the still huge size of the state space by employing a special coarse graining algorithm. The system can be reduced to a computational treatable size. We demonstrate that these systems still contain all properties necessary for aging effects.

Optimization of a Diabatic Distillation Column with Sequential Heat Exchangers

Jimenez, Steven and Salamon, Peter and Rivero, Ricardo and Rendon, Consuelo and Hoffmann, Karl Heinz and Schaller, Markus and Andresen, Bjarne
Industrial & Engineering Chemistry Research 43(23): 7566—7571 (2004); DOI:10.1021/ie0495933

Diabatic distillation is a separation process in which heat is transferred on the trays inside the column as opposed to classical adiabatic columns where heat is only supplied to the reboiler and extracted from the condenser. Such diabatic columns dramatically reduce the exergy needed to perform the separation. One implementation, particularly suitable for retrofitting applications, uses a single heating fluid circulating in series from one tray to the next below the feed tray and a single cooling fluid circulating in series above the feed tray. The optimal design of these sequential heat exchangers, minimizing the overall rate of entropy production in the separation process, is a difficult optimization problem because traditional algorithms for optimization invariably get stuck. However, an algorithm based on physical intuition for adjusting the temperature profile can find the optimum. The resulting column operation is compared to the optimal operation with independent heat transfer to each tray (the completely controlled diabatic column) and to a conventional adiabatic column. In the former comparison, we find how much exergy is lost by circulating a fluid in series rather than using independently adjustable heat exchanges. In the latter, we find the possible savings available by retrofitting. The comparisons show that most of the potential exergy savings can be captured by diabatization using heat exchangers in series. The potential impact of this technology on the chemical and process industry is enormous because distillation is the single largest energy degrading unit operation worldwide.

Optimal allocation of Heat Exchanger Inventory in a Serial Type Diabatic Distillation Column

Jimenez, Edward Steven and Salamon, Peter and Rivero, Ricardo and Rendon, Consuelo and Hoffmann, Karl Heinz
"Proceedings of ECOS 2004" Eds: Rivero, R. and Monroy, L. and Pulido, R. and Tsatsaronis, G.: 179—187 , 2004

Diabatic distillation is a separation process in which heat is transferred on the trays inside the column . We have previously shown (Jimenez et al. 2003) that optimal operation of serial heat exchangers can capture most of the wasted exergy. In the present work we explore the effect of locating a fixed total heat exchanger area in different trays and calculate the optimal allocation of a given heat exchanger inventory.

Fitness Threshold Accepting over extremal optimization ranks

Hoffmann, Karl Heinz and Heilmann, Frank and Salamon, Peter
Physical Review E 70(4): 046704-1 — 046704-6 (2004); DOI:10.1103/PhysRevE.70.046704

We treat the problem of selecting the next degree of freedom for update in an extremal optimization algorithm designed to find the ground state of a system with a complex energy landscape. We show that there exists a best distribution for selecting the next degree of freedom in order to optimize any linear function of the state probabilities, e.g., the expected number of visits to the ground state. We dub the class of algorithms using this best distribution in conjunction with extremal optimization fitness threshold accepting. In addition, we construct an extended random walk and use it to show that fitness threshold accepting is optimal also for several other measures of algorithm performance, such as maximizing the expected probability of seeing the ground state and minimizing the expected value of the lowest energy seen.

Best possible probability distribution over Extremal Optimization ranks

Heilmann, Frank and Hoffmann, Karl Heinz and Salamon, Peter
Europhysics Letters 66(3): 305—310 (2004); DOI:10.1209/epl/i2004-10011-3

We consider the problem of selecting the next degree of freedom (DoF) for update in an Extremal Optimization algorithm designed to find the ground state of a sy stem with a complex energy landscape. We show that in order to minimize any linear function of the state probabilities, e.g., the expectation value of the final energy, there exists a best distribution for selecting the next DoF. We dub the algorithm using this best distribution Fitness Threshold Accepting.

Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?

Fischer, Andreas and Hoffmann, Karl Heinz
Journal of Non-Equilibrium Thermodynamics 29(1): 9—28 (2004); DOI:10.1515/JNETDY.2004.002

In this case study a complex Otto engine simulation provides data including, but not limited to, effects from losses due to heat conduction, exhaust losses and frictional losses. This data is used as a benchmark to test whether the Novikov engine with heat leak, a simple endoreversible model, can reproduce the complex engine behavior quantitatively by an appropriate choice of model parameters. The reproduction obtained proves to be of high quality.

Optimal Allocation of Heat Exchanger Investment

Burzler, Josef Maximilian and Amelkin, Sergej A. and Tsirlin, Anatoliy Mikhailovich and Hoffmann, Karl Heinz
Open Systems and Information Dynamics 11(3): 291—306 (2004); DOI:10.1023/B:OPSY.0000047572.63034.66

The optimal allocation of a given investment capital to the heat exchanging inventory is studied for heat engines, refrigerators and heat pumps. The study is based on an endoreversible model operating between two constant temperature heat reservoirs at optimal thermodynamic performance, which is either minimal entropy production or maximum power production. The analysis accounts for the fact that the actual costs of heat exchangers equipment is subject to the material, design and operating conditions of the heat exchangers so that the dependency between the costs and heat transfer coe cients generally needs to be considered as nonlinear and di erent for the hot and cold side of the system. Contrary to existing results showing no di erence between cyclic and stationary operation for Newtonian heat transfer we find a distinct di erence. This result also pertains to non-Newtonian heat transfer.

Maximum power processes for multi-source endoreversible heat engines

Amelkin, Sergei A. and Andresen, Bjarne and Burzler, Josef Maximilian and Hoffmann, Karl Heinz and Tsirlin, Anatoliy Mikhailovich
Journal of Physics D: Applied Physics 37(9): 1400—1404 (2004); DOI:10.1088/0022-3727/37/9/015

The maximum power processes of multi-source endoreversible engines with stationary temperature reservoirs are investigated. We prove that the optimal solution is always time independent with a single hot and a cold engine contact temperature. The heat reservoirs fall into three groups: The hot reservoirs which are connected at all times for heat delivery, the cold reservoi rs which are connected at all times for heat drain, and possibly a group of reservoirs at intermediate temperatures which are unused. This phenomenon is demonstrated for a three-source system. We find that for a commonly used class of heat transfer functions, including Newtonian, Fourier and radiative heat transport, the efficiencies at maximum power are the same as for two-reservoir engines with appropriately chosen properties.

Fractional Diffusion, Irreversibility and Entropy

Li, Xiaorang and Essex, Christopher and Davison, Matt and Hoffmann, Karl Heinz and Schulzky, Christian
Journal of Non-Equilibrium Thermodynamics 28(3): 279—291 (2003); DOI:10.1515/JNETDY.2003.017

Three types of equations linking the diffusion equation and the wave equation are studied: the time fractional diffusion equation, the space fractional diffusion equation and the telegrapher's equation. For each type, the entropy production is calculated and compared. It is found that the two fractional diffusions, considered as linking bridges between reversible and irreversible processes, possess counter-intuitive properties: as the equation becomes more reversible, the entropy production increases. The telegrapher's equation does not have the same counter-intuitive behavior. It is suggested that the different behaviors of these equations might be related to the velocities of the corresponding random walkers.

Optimal Process Paths for Endoreversible Systems

Hoffmann, Karl Heinz and Burzler, Josef Maximilian and Fischer, Andreas and Schaller, Markus and Schubert, Sven
Journal of Non-Equilibrium Thermodynamics 28(3): 233—268 (2003); DOI:10.1515/JNETDY.2003.015

All energy transformation processes occurring in reality are irreversible and in many cases these irreversibilities must be included in a realistic description of such processes. Endoreversible thermodynamics is a non-equilibrium approach in this direction by viewing a system as a network of internally reversible (endoreversible) subsystems exchanging energy in an irreversible fashion. All irreversibilities are confined to the interaction between the subsystems. This review is dedicated to the dynamical investigation of such endoreversible systems. First the general framework for the endoreversible description of a system is briefly introduced, and then the necessary mathematical tools to determine optimal process paths for such systems are presented. These are complemented by simple examples for the application of the different methods. Then the optimal paths for endoreversible processes of increasing complexity are discussed: first the processes between given equilibrium states, and then cyclic processes. These are followed by a review of internal combustion engines and by a number of further selected applications. We conclude with an outlook to other areas of irreversible thermodynamics where path optimization methods have been successfully used.

Threshold accepting as limit case for a modified Tsallis statistics

Franz, Astrid and Hoffmann, Karl Heinz
Applied Mathematics Letters 16(1): 27—31 (2003); DOI:10.1016/S0893-9659(02)00140-4

Simulated annealing with different types of acceptance probabilities is widely used in stochastic optimization. Based on the Metropolis algorithm describing thermal relaxation Threshold Accepting was developed to speed up the computation and Tsallis statistics generalizes the Metropolis acceptance probability by introducing a new parameter q ∈ R, where for q→ 1 the Metropolis statistics is recovered. In this paper we will show that not only the Metropolis acceptance probability is a limit case of Tsallis statistics, but Threshold Accepting can also be considered as limit case of a modified Tsallis acceptance probability for q→ −∞.

Passive houses to reduce the utilization of classical fuels for space heating

Dragos, A. A. and Badescu, Viorel and Hoffmann, Karl Heinz and Sicre, Benoit
"Conferinta nationala pentru dezvoltare durabila" Eds: : 227—232 , 2003

In the scope of this work, a passive house is a cost efficient building that can manage througout the heating period, due to its specific construction design, with more than ten times less heat energy that the same building designed to standards presently applicable accross Europe. Its extended thermal insulation and enhanced air tightness removes the need for temperatures higher than 50 degrees Celsius what makes renewable energy sources particularly suitable for heating, cooling and DHW. Description of the ventilation/heating system of an existing passive house is the topic of this paper.

Optimal Endoreversible Heat Engines with Polytropic Branches

Burzler, Josef Maximilian and Hoffmann, Karl Heinz and Amelkin, Sergey Anatolievich and Tsirlin, Anatoliy Mikhailovich
International Journal of Applied Thermodynamics 6(2): 69—78 (2003)

Endoreversible engine cycles with two adiabatic and two heat transfer branches are investigated and optimized for maximum work output. The heat transfer branches are described as general polytropic processes which include common standard branches, like isotherms, isobars and isometrics, as special cases. The study considers the finite heat capacity of the working fluid and the finite-time character of the heat transfer processes, determines the optimal allocation of branch times, and derives analytic expressions for the maximized work output. The efficiency at maximum work is found to coincide with the Curzon-Ahlborn efficiency for endoreversible Carnot engines and does not depend on design parameters of the engine if the degree of the polytropic processes is equal in both heat transfer branches.

Ground states for condensed amorphous systems: Optimizing annealing schemes

Blaudeck, Peter and Hoffmann, Karl Heinz
Computer Physics Communications 150: 293—299 (2003); DOI:10.1016/S0010-4655(02)00688-4

Using optimized Simulated Annealing allows finding distinctly lower minima for the potential energy of amorphous systems. A new scheme resulting in an optimal annealing schedule has been found that can be readily applied to the simulation of molecules, clusters and condensed systems with any atomic composition. The scheme remains applicable if, due to the complexity of the system and its interatomic potentials, the configuration space cannot be explored in more detail.

Renewable energy for passive house heating: Part II: Model

Badescu, Viorel and Sicre, Benoit
Energy and Buildings 35(11): 1085—1096 (2003); DOI:10.1016/j.enbuild.2003.09.004

The evaluation of renewable energy used to increase the environmental friendliness of passive houses (PH) is the topic of this paper. A time-dependent model of passive house thermal behavior is developed. The heat-transfer through the high thermal inertia elements is analyzed by using a 1D time-dependent conduction heat-transfer equation that is solved numerically by using a standard Netlib solver (PDECHEB). Appropriate models for the conduction through the low thermal inertia elements are used, as well as a simple approach of the solar radiation transmission through the windows. The model takes into account in a detailed fashion the internal heat sources. Also, the operation of ventilation/heating system is described and common practice control strategies are implemented. Three renewable energy sources are considered. First, there is the passive solar heating due to the large window on the façade oriented south. Second, the active solar collectors system provides thermal energy for space heating or hot domestic water preparation. Third, a ground heat exchanger (GHE) increases the fresh air temperature during the cold season. The model was applied to the Pirmasens Passive House (Rhineland Palatinate, Germany). The passive solar heating system provides most part of the heating energy during November, December, February and March while in January the ground heat exchanger is the most important renewable energy source. January and February require use of additional conventional energy sources. A clever use of the active solar heating system could avoid consuming classical fuels during November, December and March. The ground heat exchanger is a reliable renewable source of energy. It provides heat during all the day and its (rather small) heat flux is increasing when the weather becomes colder. The air temperature at heater exit is normally lower than 46°C. This is a good reason for the use of renewable energy to replace the classical fuel or the wood to be burn in the heater.

Renewable energy for passive house heating. Part I: Building Description

Badescu, Viorel and Sicre, Benoit
Energy and Buildings 35(11): 1077—1084 (2003); DOI:10.1016/j.enbuild.2003.10.001

A passive house is a cost-efficient building that can manage throughout the heating period, due to its specific construction design, with more than 10 times less heat energy than the same building designed to standards presently applicable across Europe. Its extended thermal insulation and enhanced air-tightness removes the need for temperatures higher than 50°C. This makes renewable energy sources particularly suitable for heating, cooling and domestic hot water production. Modeling of renewable energy usage for space heating requires as a preliminary stage the detailed description of the building structure, of the HVAC equipment and of the internal heat sources. This paper shows the main data used to model the thermal behavior of a passive house. Details about Pirmasens Passive House (Rhineland Palatinate, Germany) are given, as for example, the internal heat sources, including electric appliances, heat and humidity released by human bodies, thermal internal facilities as hot and cold water pipes. All these are quantified by using statistically derived data. A detailed time schedule for a standard German family with two adults and two children was prepared. It takes into account the national celebrations, vacation and weekends among others.

Minimal Work for Separation Processes of Binary Mixtures

Amelkin, Sergei A. and Tsirlin, Anatoliy Mikhailovich and Burzler, Josef Maximilian and Schubert, Sven and Hoffmann, Karl Heinz
Open Systems and Information Dynamics 10(4): 335—349 (2003); DOI:10.1023/B:OPSY.0000009555.63816.86

The work expenditures for both perfect and imperfect separation processes are well known for the reversible case; yet such a description is often far from reality. Real processes operate at finite times and non-zero rates leading to an additional, irreversible energy expenditure. This paper employs an idealized van t'Hoff chamber as a theoretical model to derive lower bounds for the irreversible work in real separation processes such as membrane separation. Methods of optimal control for open systems and nonlinear programming of averaged problems are used to calculate the optimal mass transfer kinetics for the finite-time separation of binary mixtures of ideal gases.

The cumulant method applied to a mixture of Maxwell gases

Seeger, Steffen and Hoffmann, Karl Heinz
Continuum Mechanics and Thermodynamics 14(2): 321—335 (2002); DOI:10.1007/s001610100067

We apply the recently proposed cumulant method to derive the production terms for a mixture of gases of Maxwell-molecules in two and three dimensions. For the single component Maxwell gas we introduce a linear approximation of the production terms and give an analytical solution for the (space-)homogeneous case. We find that the eigenvariables of the linearized productions appear in three different kinds and the first few can be related to classical thermodynamic quantities.

The Influence of Heat Transfer Irreversibilities on the Optimal Performance of Diabatic Distillation Columns

Schaller, Markus and Hoffmann, Karl Heinz and Rivero, R. and Andresen, Bjarne and Salamon, Peter
Journal of Non-Equilibrium Thermodynamics 27(3): 257—256 (2002); DOI:10.1515/JNETDY.2002.015

A distillation column with the possibility of heat exchange on every tray (a fully diabatic column) is optimized in the sense of minimizing its total entropy production. This entropy production counts the interior losses due to heat and mass flow as well as the entropy generated in the heat exchangers. It is observed that the optimal heating distribution, i.e. the heat exchange required on each tray, is essentially the same for all trays in the stripping and rectification sections, respectively. This makes a column design with consecutive interior heat exchanger and only one exterior supply for each of the two sections very appealing. The result is only slightly dependent on the heat transfer law considered. In the limit of an infinite number of trays even this column with resistance to transfer of heat becomes reversible.

Comparison of Entropy Production Rate Minimization Methods for Binary Diabatic Distillation

Koeijer, Gelein M. and Kjelstrup, Signe and Salamon, Peter and Siragusa, Gino and Schaller, Markus and Hoffmann, Karl Heinz
Industrial & Engineering Chemistry Research 41(23): 5826—5834 (2002); DOI:10.1021/ie010872p

The purpose of this study is to compare two analytical methods with two numerical methods for minimizing the entropy production rate in diabatic distillation columns (i.e., with heat exchangers on all trays). The first analytical method is the equal thermodynamic distance method. The second uses Lagrange minimization on a model derived from irreversible thermodynamics. The numerical methods use Powell s and a Monte Carlo algorithm and gave the same results. Both analytical methods agreed well with the numerical ones for two columns with low separation per tray, while they did not agree well for a column with large separation per tray.

Structure of best possible strategies for finding ground states

Hoffmann, Karl Heinz and Franz, Astrid and Salamon, Peter
Physical Review E 66(4): 046706/1—046706/7 (2002); DOI:10.1103/PhysRevE.66.046706

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that for a wide range of objective functions threshold accepting is the best possible strategy within a large class of algorithms that simulate random walks on the landscape. In particular, it can perform better than simulated annealing, Tsallis and Glauber statistics.

The Statistical Physics of Energy Landscapes: From Spin Glasses to Optimization

Hoffmann, Karl Heinz
"Computational Statistical Physics" Eds: Hoffmann, K. H. and Schreiber, M.: 57—76 Springer Verlag, Berlin, 2002; ISBN: 978-3-642-07571-1; DOI:10.1007/978-3-662-04804-7_4

The concept of energy ''landscapes'' leads to a unified understanding of phenomena in a number of different complex physical systems. All these systems are characterized by an energy function which possesses many local minima separated by barriers as a function of the state variables. If graphically depicted such energy function looks very much like a mountainous landscape. Typical examples of such complex systems are spin glasses which show a wealth of interesting relaxation phenomena, but also a number of industrially important minimization problems, which have a mountainous cost function landscape. These problems are intimately connected by the thermally activated relaxation dynamics on complex energy landscapes.

Recent Developments in Finite Time Thermodynamics

Hoffmann, Karl Heinz
Technische Mechanik 22(1): 14—25 (2002)

Finite time thermodynamics is a non-equilibrium theory. Its aim is to provide performance bounds and extremes for irreversible thermodynamics processes. Recent developments in different areas of this theory are presented. First it is shown how irreversible processes between reversible systems can be described by the endoreversible theory. Then maximum power an minimum entropy production processes are introduced. And finally the extensions of finite time thermodynamics to the realm of quantum theory is demonstrated.

Computational Statistical Physics

Eds: Hoffmann, Karl Heinz and Schreiber, Michael
Springer Verlag, Berlin, 2002; ISBN: 978-3-642-07571-1

In recent years statistical physics has made significant progress as a result of advances in numerical techniques. While good textbooks exist on the general aspects of statistical physics, the numerical methods and the new developments based on large-scale computing are not usually adequately presented. In this book 16 experts describe the application of methods of statistical physics to various areas in physics such as disordered materials, quasicrystals, semiconductors, and also to other areas beyond physics, such as financial markets, game theory, evolution, and traffic planning, in which statistical physics has recently become significant. In this way the universality of the underlying concepts and methods such as fractals, random matrix theory, time series, neural networks, evolutionary algorithms, becomes clear. The topics are covered by introductory, tutorial presentations.

Using Computer Algebra Methods to Determine the Chemical Dimension of Finitely Ramified Sierpinski Carpets

Franz, Astrid and Schulzky, Christian and Hoffmann, Karl Heinz
SIGSAM Bulletin: Communications in Computer Algebra 36(2): 18—30 (2002); DOI:10.1145/581316.581318

We present a new algorithm for calculating the chemical dimension dl of finitely ramified Sierpinski carpets. Using an algorithm of Dijkstra, we compute iteratively, using extsc{Mathematica}, the shortest paths through a carpet. The scaling exponent of the lengths of these shortest paths over the linear size of the carpet is dmin the minimum path dimension, which is related to the chemical dimension.

Optimal Annealing Schedules for a Modified Tsallis Statistics

Franz, Astrid and Hoffmann, Karl Heinz
Journal of Computational Physics 176(1): 196—204 (2002); DOI:10.1006/jcph.2001.6975

In this paper, for a number of example systems, optimal schedules for simulated annealing with a modified Tsallis statistics for various parameters q are analyzed. It turns out that in general depending on the objective function (minimizing the mean energy or maximizing the ground state probability), different schedules have to be chosen. Furthermore, the optimal objective function value, reached with the optimal schedule, shows a monotonic dependency on q, where better values are reached for smaller q. Thus, in stochastic optimization the limit case q → ∞ corresponding to threshold accepting should be chosen in order to get the best possible optimization results with as little effort as possible.

Diffusion on Fractals — Efficient algorithms to compute the random walk dimension

Franz, Astrid and Schulzky, Christian and Seeger, Steffen and Hoffmann, Karl Heinz
"Fractal Geometry: Mathematical Methods, Algorithms, Applications" IMA Conference Proceedings, Eds: Blackledge, Jonathan M. and Evans, Allan K. and Turner, Martin J.: 52—67 Horwood Publishing Ltd., Chichester, West Sussex, , 2002; ISBN: 1-904275-00-1; DOI:10.1533/9780857099594.52

Self-similar fractals are used as a simple model for porous media in order to describe diffusive processes. The diffusion or Brownian motion of particles on a fractal is approximated by random walks on pre-fractals. Since there are a lot of holes in the fractal, where a random walker is not allowed to move in, the mean square displacement scales with time t asymptotically as t2/dw, where the random walk dimension dw is usually greater than 2. This dimension is an important quantity to characterize diffusion properties. In this paper three efficient methods to calculate the random walk dimension of finitely ramified Sierpinski carpets are presented: First a simulation of random walks on pre-carpets, where an efficient storing scheme decreases the needed amount of memory and speeds up the computation. Secondly we iterate the master equation describing the time evolution of the probability distribution. Thirdly a resistance scaling algorithm is presented which yields a resistance scaling exponent. This exponent is related to the random walk dimension via the Einstein relation, using analogies between random walks on graphs and resistor networks.

Durch Zufall schneller ans Ziel: Anwendung eines stochastischen Optimierungsalgorithmus auf das Problem des Handlungsreisenden

Diering, Hermann and Seeger, Steffen and Hoffmann, Karl Heinz
Besondere Lernleistung, 2002

Modelling porous structures by repeated Sierpinski carpets

Tarafdar, Sujata and Franz, Astrid and Schulzky, Christian and Hoffmann, Karl Heinz
Physica A: Statistical Mechanics and its Applications 292(1-4): 1—8 (2001); DOI:10.1016/S0378-4371(00)00573-2

Porous materials such as sedimentary rocks often show a fractal character at certain length scales. Deterministic fractal generators, iterated upto several stages and then repeated periodically, provide a realistic model for such systems. On the fractal, diffusion is anomalous, and obeys the law 〈r2 〉 ∼ t2/dw, where 〈r2 〉 is the mean square distance covered in time t and dw > 2. The question is how is the macroscopic diffusivity related to the characteristics of the small scale fractal structure, which is hidden in the large scale homogeneous material? In particular do structures with same dw neccesarily lead to the same diffusion coefficient at same porosities? The present paper tries to shed some light on these questions.

Random Walks on Finitely Ramified Sierpinski Carpets

Seeger, Steffen and Franz, Astrid and Schulzky, Christian and Hoffmann, Karl Heinz
Computer Physics Communications 134(3): 307—316 (2001); DOI:10.1016/S0010-4655(00)00208-3

A new algorithm is presented that allows an efficient computer simulation of random walks on finitely ramified Sierpinski carpets. Instead of using a bitmap of the n-th iteration of the carpet to determine allowed neighbour sites, neighbourhood relations are stored in small lookup tables and a hierarchical coordinate notation is used to give the random walker position. The resulting algorithm has low memory requirements, shows no surface effects even for extremely long walks and is well suited for modern computer architectures.

Numerically optimized performance of diabatic distillation columns

Schaller, Markus and Hoffmann, Karl Heinz and Siragusa, Gino and Salamon, Peter and Andresen, Bjarne
Computers & Chemical Engineering 25(11—12): 1537—1548 (2001); DOI:10.1016/S0098-1354(01)00717-7

Recently, the concept of equal thermodynamic distance (ETD) has been proposed to minimize entropy production in a distillation process using a diabatic column. ETD gives the optimal temperature profile to first-order in N−1, where N is the number of trays. ETD however, does not in general give the true minimum for distillation columns with few trays. We therefore apply a fully numerical, multidimensional optimization routine to determine minimum entropy production. Since this method does not depend on an underlying theory we expect a true minimum to be revealed. We then compare the performance of ETD and numerical optimization by varying the number of trays and the purity requirements. Our results show surprisingly good agreement between the ETD results and the ones obtained numerically.

What conditions make minimum entropy production equivalent to maximum power production

Salamon, Peter and Hoffmann, Karl Heinz and Schubert, Sven and Berry, R. Stephen and Andresen, Bjarne
Journal of Non-Equilibrium Thermodynamics 26(1): 73—83 (2001); DOI:10.1515/JNETDY.2001.006

Optimization of processes can yield a variety of answers, depending not only on the objective of the optimization but also on the constraints that define the problem. Within the context of thermodynamic optimization, the role of the constraints is particularly important because, among other things, their choice can make some objectives either equivalent or inequivalent, and can limit or broaden the possible kinds of processes one might choose. After a general discussion of the principles, a specific example of a model power plant is analyzed to see how the constraints govern the possible solutions.

Comparison of Entropy Production Rate Minimization Methods for Binary Diabatic Tray Distillation

de Koeijer, Gelein and Kjelstrup, Signe and Salamon, Peter and Siragusa, Gino and Schaller, Markus and Hoffmann, Karl Heinz
"Proceedings of ECOS'01" Eds: Öztürk, A. and Gö g}= u}c s}, Y. A.: 667—677 , 2001; ISBN: 975-97568-2-2; DOI:10.1021/ie010872p

The purpose of this study is to compare two analytical methods with two numerical methods for minimizing the entropy production rate in diabatic distillation columns (i.e. with heat exchangers on all trays). The first analytical method is the Equal-Thermodynamic-Distance method. The second uses Lagrange minimization on a model derived from irreversible thermodynamics. The numerical methods use Powell's and a Monte-Carlo algorithm and gave the same results. Both analytical methods agreed well with the numerical ones for two columns with low separation per tray, while they did not agree well for a column with large separation per tray.

Das CLiC-Projekt — Planung und Inbetriebnahme eines PC-Clusters

Hübner, Uwe and Seeger, Steffen and Petersen, Karsten
Praxis der Informationsverarbeitung und Kommunikation 24(2): 75—84 (2001); DOI:10.1515/PIKO.2001.75

Durch die Massenfertigung moderner Personal-Computer sind heute sehr leistungsfähige Komponenten zu geringen Preisen zu erhalten — auch hat bereits 1994 das Beowulf-Projekt demonstriert, dass sich aus diesen Komponenten leistungsfähige Parallelrechner konstruieren lassen. Dieser Artikel zeigt nun am Beispiel des Chemnitzer Linux Clusters CLiC, wie ein solcher Supercomputer unter Beachtung der Anwendungen projektiert, realisiert und administriert werden kann.

Quantum thermodynamics

Quantum theory and thermodynamics are two important corner stones in our understanding of nature. In this paper we discuss a number of interesting topics where both fields interact starting from Max Planck's introduction of the energy quantum to todays open questions about the validity of the second law in the quantum regime.

Computational Physics

Eds: Hoffmann, Karl Heinz and Schreiber, Michael
Springer-Verlag, Berlin, 2001; ISBN: 7-03-008913-8/O 1296

The pore structure of Sierpinski carpets

Franz, Astrid and Schulzky, Christian and Tarafdar, Sujata and Hoffmann, Karl Heinz
Journal of Physics A: Mathematical and General 34(42): 8751—8765 (2001); DOI:10.1088/0305-4470/34/42/303

In this paper, a new method is developed to investigate the pore structure of finitely and even infinitely ramified Sierpinski carpets. The holes in every iteration stage of the carpet are described by a hole-counting polynomial. This polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters, from which dimensions governing the scaling of these quantities can be determined. Whereas the hole area is known to be two dimensional, the dimension of the hole perimeter may be related to the random walk dimension.

The Einstein relation for finitely ramified Sierpinski carpets

Franz, Astrid and Schulzky, Christian and Hoffmann, Karl Heinz
Nonlinearity 14(5): 1411—1418 (2001); DOI:10.1088/0951-7715/14/5/324

Based on an analogy consideration between random walks and resistor networks it is shown that for a wide class of random walks on graphs resulting from finitely ramified Sierpinski carpets the Einstein relation is satisfied, which is an important equation relating conductivity and diffusivity. On fractal graphs this relation means, for instance, that a wide class of random walk algorithms including the blind and myopic ant random walks have the same random walk dimension.

Best Possible Strategy for Finding Ground States

Franz, Astrid and Hoffmann, Karl Heinz and Salamon, Peter
Physical Review Letters 86(23): 5219—5222 (2001); DOI:10.1103/PhysRevLett.86.5219

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that, within the large class of algorithms that simulate a random walk on the landscape, threshold accepting is the best possible strategy. In particular, it can perform better than simulated annealing and Tsallis statistics. Our proof is the first example of a provably optimal strategy in this area.

The Differential Equation Describing Random Walks on the Koch Curve

Essex, Christopher and Davison, Matt and Schulzky, Christian and Franz, Astrid and Hoffmann, Karl Heinz
Journal of Physics A: Mathematical and General 34(41): 8397-8406 (2001); DOI:10.1088/0305-4470/34/41/301

Consider a particle which is released at some point on a fractal and which moves about the fractal at random. Along standing goal has been to determine a differential equation governing the probability density function which describes this walk. As well as being interesting in its own right, this problem is thought to provide an insight into the problem of anomalous diffusion. Many attempts to derive such an equation have been made, all with limited success, perhaps because of the tension between smoothness required by differential equation tools and the lack of smoothness inherent in fractals. Here we present, for the first time, the equation governing the random walk on a simple fractalthe Koch curve. We show that this equation makes computation of the probability density function for this problem a simple matter.

Clouds, fibres and echoes: a new approach to studying random walks on fractals

Davison, Matt and Essex, Christopher and Schulzky, Christian and Franz, Astrid and Hoffmann, Karl Heinz
Journal of Physics A: Mathematical and General 34(20): L289—L296 (2001); DOI:10.1088/0305-4470/34/20/101

Up to now the general approach of constructing evolution differential equations to describe random walks on fractals has not succeeded. Is this because the true probability density function is inherently fractal? When plotted in the appropriate similarity variable, we find a cloud which is not too smooth. Further investigation shows that this cloud has a structure that might be overlooked if one is looking for the usual single-valued probability density function. The cloud is composed of an infinite family of smooth fibres, each of which describes the behaviour of the walk on an infinite echo point class. The fibres are individually smooth and so are naturally amenable to analysis with differential equations.

Evaluating the Efficiency Frontier of Separation Processes

Amelkin, Sergei A. and Burzler, Josef Maximilian and Hoffmann, Karl Heinz and Tsirlin, Anatoliy Mikhailovich
Theoretical Foundations of Chemical Engineering 35(3): 217—223 (2001); DOI:10.1023/A:1010485906403

The problem of finding the minimum work to be done to separate a mixture at a fixed process duration or at a given process capacity is considered. The estimates of the work done in an irreversible process substantially exceed those of the work done in reversible separation, and the work done in irreversible separation of depleted mixtures is finite even when the concentration of the minor component is arbitrarily close to zero. A method is proposed for extending these estimates to separation processes consuming heat rather than mechanical energy.

Estimates of Limiting Possibilities of Separation Processes

Amelkin, Sergei A. and Burzler, Josef Maximilian and Hoffmann, Karl Heinz and Tsirlin, Anatoliy Mikhailovich
Theoretical Foundations of Chemical Engineering 35(3): 223—238 (2001); DOI:10.1023/A:1010485906403

The problem of finding the minimum work to be done to separate a mixture at a fixed process duration or at a given process capacity is considered. The estimates of the work done in an irreversible process substantially exceed those of the work done in reversible separation, and the work done in irreversible separation of depleted mixtures is finite even when the concentration of the minor component is arbitrarily close to zero. A method is proposed for extending these estimates to separation processes consuming heat rather than mechanical energy.

The cumulant method for computational kinetic theory

Seeger, Steffen and Hoffmann, Karl Heinz
Continuum Mechanics and Thermodynamics 12: 403-421 (2000); DOI:10.1007/s001610050145

We propose a new method for numerical simulation of gas dynamics based on kinetic theory. The method is based on a cumulant-expansion-ansatz for the phase space density, which leads to a set of quasi-linear, hyperbolic partial differential equations. The method is compared to the moment method of Grad. Both methods agree for low-order approximations but the method proposed shows additional non-linear terms for high order approximations. Boundary conditions on the cumulants for an ideally reflecting and an ideally rough boundary surface are derived from conditions on the phase space density. A Lax-method is used for numerical analysis of a 2d-BGK fluid, which results in an easy-to-implement algorithm well suited for implementation on massivly parallel computers. The results are found to agree qualitatively with predictions from moment theories.

The similarity group and anomalous diffusion equations

Schulzky, Christian and Essex, Christopher and Davison, Matt and Franz, Astrid and Hoffmann, Karl Heinz
Journal of Physics A: Mathematical and General 33(31): 5501-5511 (2000); DOI:10.1088/0305-4470/33/31/305

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.

Resistance Scaling and Random Walk Dimensions for Finitely Ramified Sierpinski Carpets

Schulzky, Christian and Franz, Astrid and Hoffmann, Karl Heinz
SIGSAM Bulletin: Communications in Computer Algebra 34(3): 1—8 (2000); DOI:10.1145/377604.377608

We present a new algorithm to calculate the random walk dimension of finitely ramified Sierpinski carpets. The fractal structure is interpreted as a resistor network for which the resistance scaling exponent is calculated using Mathematica. A fractal form of the Einstein relation, which connects diffusion with conductivity, is used to give a numerical value for the random walk dimension.

Hausdorff dimension estimates for non-injective maps using the cardinality of the pre-image sets

Franz, A.
Nonlinearity 13(5): 1425-1438 (2000); DOI:10.1088/0951-7715/13/5/301

Upper bounds for the Hausdorff dimension of compact invariant sets of C1 -maps on smooth Riemannian manifolds are given in terms of the singular values of the tangent map and the multiplicity function of the map, describing the number of pre-images of a certain point in a given set. For non-injective maps this improves previous results using only the singular values.

An Efficient Implementation of the Exact Enumeration Method for Random Walks on Sierpinski Carpets

Franz, Astrid and Schulzky, Christian and Seeger, Steffen and Hoffmann, Karl Heinz
Fractals - Complex Geometry, Patterns, and Scaling in Nature and Society 8(2): 155-161 (2000); DOI:10.1142/S0218348X00000172

In the following we present a highly efficient algorithm to iterate the master equation for random walks on effectively infinite Sierpinski carpets, i.e. without surface effects. The resulting probability distribution can, for instance, be used to get an estimate for the random walk dimension, which is determined by the scaling exponent of the mean square displacement versus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal with positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet has not to be fixed at the beginning of the algorithm. It is restricted only by the amount of available computer RAM. Furthermore all the information which is needed in every step to update the probability distribution is stored in tables. The lookup of this information is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number of time steps is decreased.

Simulated annealing with Threshold Accepting or Tsallis statistics

Fachat, André and Hoffmann, Karl Heinz and Franz, Astrid
Computer Physics Communications 132(3): 232—240 (2000); DOI:10.1016/S0010-4655(00)00153-3

Threshold Accepting and Tsallis statistics have shown good results when applied to optimization problems. In contrast to the Metropolis acceptance probability these two algorithms do not have detailed balance and also may have broken ergodicity. This makes it impossible to compute the equilibrium distribution analytically for general state spaces and neighborhood relations. In this paper we investigate the equilibrium properties of Threshold Accepting and Tsallis statistics numerically. For simple problems as a ladder of states both algorithms yield exponential functions as equilibrium probability distributions. However, as detailed balance does not hold, the neighborhood relation has an important influence on the resulting probability distribution. This is most obvious in systems with random energy values and random neighborhood structure.

Tsallis and Rényi Entropies in Fractional Diffusion and Entropy Production

Essex, Christopher and Schulzky, Christian and Franz, Astrid and Hoffmann, Karl Heinz
Physica A: Statistical Mechanics and its Applications 284(1-4): 299-308 (2000); DOI:10.1016/S0378-4371(00)00174-6

The entropy production rate for fractional diffusion processes using Shannon entropy was calculated previously, which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. Rényi and Tsallis entropies, which have an additional parameter q generalizing the Shannon case q=1), are shown here to have similar counter intuitive behaviours. However, the issue can be successfully treated in exactly the same manner as with Shannon entropy for q being not too large (i.e. generalizations near the Shannon case), whereas for larger q the Rényi and Tsallis entropies behave in a different way.

Numerical Monsters

Essex, Chistohper and Davison, Matt and Schulzky, Christian
SIGSAM Bulletin: Communications in Computer Algebra 34(4): 16—32 (2000); DOI:10.1145/377626.377635

When the results of certain computer calculations are shown to be not simply incorrect but dramatically incorrect, we have a powerful reason to be cautious about all computer-based calculations. In this paper we present a Rogue's Gallery of simple calculations whose correct solutions are obvious to humans but whose numerical solutions are incorrect in pathological ways. We call these calculations, which can be guaranteed to wreak numerical mayhem across both software packages and hardware platforms, Numerical Monsters. Our monsters can be used to provide deep insights into how computer calculations fail, and we use them to engender appreciation for the subject of numerical analysis in our students. Although these monsters are based on well-understood numerical pathologies, even experienced numerical analysts will find surprises in their behaviour and can use the lessons they bring to become even better masters of their tools.

Optimal Piston Paths for Diesel Engines

Burzler, Josef Maximilian and Blaudeck, Peter and Hoffmann, Karl Heinz
"Thermodynamics of Energy Conversion and Transport" Eds: Stanislaw Sieniutycz, S. and de Vos, A.: 173—198 Springer, Berlin, 2000; ISBN: 0-387-98938-2; DOI:10.1007/978-1-4612-1286-7_7

The performance of a Diesel engine is analysed for a model which includes losses due to mechanical friction and heat losses through the cylinder walls. Using the work output of the Diesel engine as an objective the optimal piston trajectories for the compression and power stroke are determined simultaneously. Results for a linear approximation of the heat leakage are compared to a more realistic, empirical heat transfer law due to Annand. Optimal operating conditions are found and discussed and significant improvements in the engine’s efficiency relative to conventionally designed engines are obtained.

Extreme performance of heat exchangers of various hydrodynamic models of flows

Amelkin, Sergey A. and Hoffmann, Karl Heinz and Sicre, Benoit and Tsirlin, Anatoly M.
Periodica Polytechnica Series Chemnical Engineering 44(1): 3—16 (2000)

The problem of minimization of entropy production is considered for one-pass heat exchangers of various types of description of hydrodynamic characteristics of the flows. Two models of the flows are considered, namely models of ideal mixing and ideal exclusion. The solution of the problem at issue allows one to construct a measure of thermodynamic perfectness of the heat exchanger taking into account the irreversibility of the heat exchange process.

Publications in 1995 - 1999

Adaptive Schedules for Ensemble-Based Threshold Accepting

Tafelmayer, R.  and Hoffmann, Karl Heinz
Applied Mathematics Letters 12(5): 131—135 (1999); DOI:10.1016/S0893-9659(99)00068-3

We study numerically an adaptive schedule in the ensemble approach of threshold accepting by considering a traveling salesman problem. We find that the probability for finding low lying minima is higher than in the widely used conventional exponential schedules. The algorithm is well suited for parallel implementations.

Two physically motivated algorithms for combinatorial optimization: thermal cycling and iterative partial transcription

Möbius, Arnulf and Diaz-Sanchez, Anastasio and Freisleben, Bernd and Schreiber, Michael and Fachat, André and Hoffmann, Karl Heinz and Merz, Peter and Neklioudov, A.
Computer Physics Communications 121—122(1—3): 34—46 (1999); DOI:10.1016/S0010-4655(99)00273-8

Among the various heuristic approaches to combinatorial optimization, local-search-based evolutionary algorithms have been particularly successful for the last years. We present two algorithms developed for jumping from local minimum to local minimum: Thermal cycling consists of cyclically heating and quenching by Metropolis and local search procedures, respectively, where the amplitude decreases during the process. Iterative partial transcription acts as a local search in the subspace spanned by the differing components of two approximate solutions corresponding to the relaxation of a spin glass by flipping clusters. The high efficiency of the proposed procedures is illustrated for the traveling salesman problem.

Slow relaxation dynamics — from spin glasses to stochastic optimization

Hoffmann, Karl Heinz
Computer Physics Communications 121-122(1-3): 30-33 (1999); DOI:10.1016/S0010-4655(99)00272-6

Metastable systems such as spin glasses show a wealth of interesting relaxation phenomena. Stochastic optimization procedures such as simulated annealing help to solve a number of industrially important minimization problems. Here we show that the two fields are intimately connected by the thermally activated relaxation dynamics of complex energy landscapes. The numerical as well as the analytical tools to analyze it are discussed. Finally two applications, aging phenomena in spin glasses and adaptive simulated annealing procedures, are presented.

Atomic clusters and nanoscale particles: From coarse-grained dynamics to optimized annealing schedules

Kunz, Ralph E. and Blaudeck, Peter and Hoffmann, Karl Heinz and Berry, R. Stephen
Journal of Chemical Physics 108(6): 2576—2582 (1998); DOI:10.1063/1.475642

An adaptive method is presented to optimize schedules for the simulated annealing of clusters and nanoscale particles. The method, based on both molecular-dynamics simulations and a set of master equations, is applied to a model configuration space for which the exact optimal schedule can also be found. The adaptive method is demonstrably suitable for optimizing larger and more realistic systems than can be treated by an exact method, even one based on a statistical-sample master equation.

The state space of short-range Ising spin glasses: the density of states

Klotz, Thomas and Schubert, Sven and Hoffmann, Karl Heinz
The European Physical Journal B 2(3): 313—317 (1998); DOI:10.1007/s100510050254

The state space of finite square and cubic Ising spin glass models is analysed in terms of the global and the local density of states. Systems with uniform and Gaussian probability distribution of interactions are compared. Different measures for the local state density are presented and discussed. In particular, the question whether the local density of states grows exponentially or not is considered. The direct comparison of global and local densities leads to consequences for the structure of the state space.

Qunatitative analysis of the state-space structure in a short-range Ising spin glas

Klotz, Thomas and Schubert, Sven and Hoffmann, Karl Heinz
Revista Mexicana de Fisica 44(S1): 81—84 (1998)

The state space structure of a finite cubic Ising spin glass model with a uniform distribution of short-range interactions is analysed in detail. The global and different measures for the local state density are presented and discussed quantitatively. The comparison of these densities gives an interesting insight in the structure of the state space. In addition the density of local minima and it's scaling behaviour is considered. The geometry of barriers in the system is investigated.

Coarse Graining of a spin-glass state space

Klotz, Thomas and Schubert, Sven and Hoffmann, Karl Heinz
Journal of Physics: Condensed Matter 10(27): 6127-6134 (1998); DOI:10.1088/0953-8984/10/27/013

The complex structure of a spin-glass state space can be simplified by a coarse-graining procedure, i.e. microscopic states being assembled into larger clusters. An algorithm for the coarse graining of the state space of a short-range Ising spin glass is provided, which is the basis of a coarse-grained dynamics. Different ways for modelling the transition rates in the coarse-grained state space are discussed. A comparison with the dynamics of the microscopic system shows that the dynamics in the coarse-grained state space gives an appropriate approximation.

Fractional Diffusion and Entropy Production

Hoffmann, Karl Heinz and Essex, Christopher and Schulzky, Christian
Journal of Non-Equilibrium Thermodynamics 23(2): 166—175 (1998); DOI:10.1515/jnet.1998.23.2.166

The entropy production rate for fractional diffusion processes is calculated and shows an apparently counter-intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. This is deduced directly from invariant and non-invariant factors of the (probability) density function, arising from a group method applied to the fractional differential equation which exists between the pure wave and diffusion equations. However, the counter-intuitive increase of the entropy production rate within the transition turns out to be a consequence of increasing quickness of processes as the wave case is approached. When this aspect is removed the entropy shows the expected decrease with the approach to the reversible wave limit.

Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting

Franz, Astrid
Nonlinearity 11(4): 1063-1074 (1998); DOI:10.1088/0951-7715/11/4/017

Upper bounds for the Hausdorff dimension of compact and invariant sets of diffeomorphisms are given using a singular value function of the tangent map and the topological entropy under the assumption, that there exists an equivariant splitting of the tangent bundle. This improves previous results for compact uniformly hyperbolic sets of diffeomorphisms satisfying an additional pinching condition. Furthermore it is shown that the results can be extended to a special class of non-injective maps.

Blocking vs. Non-blocking Communication under MPI on a Master-Worker Problem

André Fachat and Karl Heinz Hoffmann
Technische Universität Chemnitz; SFB393/98-18, 1998

In this report we describe the conversion of a simple Master-Worker parallel program from global blocking communications to non-blocking communications. The program is MPI-based and has been run on different computer architectures. By moving the communication to the background the processors can use the former waiting time for computation. However we find that the computing time increases by the time the communication time decreases in the used MPICH implementation on a cluster of workstations. Also using non-global communication instead of the global communication slows the algorithm down on computers with optimized global communication routines like the Cray T3D.

Aging and relaxation dynamics in free-energy landscapes with multiple minima

Sibani, Paolo and Hoffmann, Karl Heinz
Physica A: Statistical Mechanics and its Applications 234: 751—763 (1997); DOI:10.1016/S0378-4371(96)00312-3

We consider the stochastic dynamics of a system thermally relaxing in a free-energy landscape with multiple attractors, and show that lack of translational homogeneity in this landscape leads to aging effects, e.g. to the dependence of the susceptibilities on the time elapsed from a thermal quench to the imposition of the probing field. We then prove an inequality between response and correlation which generalizes the fluctuation dissipation theorem to a situation far from thermodynamical equilibrium. As an application and a check we specialize our formalism in a way which we suggest is appropriate for spin-glass systems: we assume a hierarchical organization of the landscape, and find aging behavior in the response curves in good agreement with relevant experimental data. We finally conclude with a summary and a brief discussion of different approaches to slow relaxation in complex systems.

Ensemble Implementations of Simulated Annealing: A Modelling Approach

Salamon, Peter and Pedersen, Jacob M. and Sibani, Paolo and Hoffmann, Karl Heinz
Mathematical Modelling and Scientific Computing 7(1): 28—37 (1997)

We address the problem of optimal ensemble size in simulated annealing algorithms by presenting some simple models which give insight into the dynamics of simulated annealing problems. We show that the presence of entrapment in these models leads to the conclusion that, for sufficiently large computational effort, the optimal ensemble size grows linearly with effort.

Optimization by thermal cycling

Möbius, Arnulf and Neklioudov, A. and Diaz-Sanchez, Anastasio and Hoffmann, Karl Heinz and Fachat, André and Schreiber, Michael
Physical Review Letters 79(22): 4297—4301 (1997); DOI:10.1103/PhysRevLett.79.4297

An optimization algorithm is presented which consists of cyclically heating and quenching by Metropolis and local search procedures, respectively. It works partially well when it is applied to an archive of samples instead of to a single one. We demonstrate for the travelling salesman problem that this algorithm is far more efficient than usual simulated annealing; our implementation can compete concerning speed with recent, very fast genetic local search algorithms, and exhibits good scaling properties.

Endoreversible Thermodynamics

Hoffmann, Karl Heinz and Burzler, Josef Maximilian and Schubert, Sven
Journal of Non-Equilibrium Thermodynamics 22(4): 311—355 (1997)

All energy transformation processes occurring in reality are irreversible and in many cases these irreversibilities must be included in a realistic description of such processes. Endoreversible thermodynamics is a non-equilibrium approach in this direction by viewing a system as a network of internally reversible (endoreversible) subsystems exchanging energy in an irreversible fashion. All irreversibilities are confined to the interaction between the subsystems. In this review a general framework for the endoreversible description of a system is presented, followed by a discussion of the performance of such systems. Thereafter the scope of the review is narrowed to time-independent stationary or cyclicly operating systems. We present the endoreversible theory of heat engines, and give an overview over the different heat transfer laws used in the entropy interactions between the subsystems. Also engine cycles different from the Carnot cycle and internal irreversibilities as well as the design optimization for such systems are discussed. These aspects are also important in the description of refrigerators and heat pumps which follows. Then combined and staged systems comprising several subsystems and their performance are reviewed and we conclude with a presentation of selected applications of endoreversible thermodynamics.

Age reinitialization in hierarchical relaxation models for spin-glass dynamics

Hoffmann, K. H. and Schubert, Sven and Sibani, Paolo
Europhysics Letters 38(8): 613—618 (1997); DOI:10.1209/epl/i1997-00292-4

We show that thermal relaxation on a tree structure is reinitialized by a temperature pulse, similarly to the experimental behavior of spin-glasses. The models' behavior originates from fast dynamical modes being excited — a mechanism which goes beyond the usual quasi-equilibrium description of slow relaxation based on the concept of a free-energy landscape. We demonstrate the excellent agreement of the model predictions with the thermoremanent magnetization experiments and discuss some of the implications of the results for the understanding of complex relaxation.

Implementation of Ensemble Based Simulated Annealing with Dynamic Load Balancing under MPI

Fachat, André and Hoffmann, Karl Heinz
Computer Physics Communications 107(1—3): 49-53 (1997); DOI:10.1016/S0010-4655(97)00096-9

This paper describes an implementation of Ensemble Based Simulated Annealing (EBSA) with dynamic load balancing. It is running under the MPI Message Passing Library allowing parallel execution on various types of computers. The load balancing is used to get maximum use of the available processing power, even on heterogeneous workstation clusters where the machines differ a lot in computing power.

Metastable Systems and Stochastic Optimization

Hoffmann, Karl Heinz
"Computational Physics" Eds: Hoffmann, Karl Heinz and Schreiber, Michael: 44—63 Springer-Verlag, Berlin, Heidelberg, New-York, 1996; ISBN: 978-3-642-85240-4; DOI:10.1007/978-3-642-85238-1_4

Metastable systems such as spin glasses show a wealth of interesting relaxation phenomena. Stochastic optimization procedures such as simulated annealing help to solve a number of industrially important minimization problems. Here we show that the two fields are intimately connected by the thermally activated relaxation dynamics of complex energy landscapes. The numerical as well as the analytical tools to analyse it are discussed. Finally two applications, aging phenomena in spin glasses and adaptive simulated annealing procedures, are presented.

Computational Physics

Eds: Hoffmann, Karl Heinz and Schreiber, Michael
Springer-Verlag, Berlin, Heidelberg, New-York, 1996; ISBN: 978-3-642-85240-4

Relaxation in Self Similar Hierarchies

Uhlig, C. and Hoffmann, Karl Heinz and Sibani, Paolo
Zeitschrift für Physik B: Condensed Matter 96: 409—416 (1995); DOI:10.1007/BF01313064

We investigate in some detail the relaxation process in self similar hierarchies. We find that the process can be divided in four different time regimes. After an initial phase in which the connectivity of the hierarchy determines the relaxation, the system enters a kind of stationary state, which can be accurately described by a simple analytical sink-picture. At longer times the behavior of the process is correctly described by the idea of quasiequilibrium. In this regime, propagators decay with power-laws. Finally, the global equilibrium state is reached, and the evolution stops.

Scaling features in complex optimization problems

Tafelmayer, R. and Hoffmann, Karl Heinz
Computer Physics Communications 86: 81-90 (1995); DOI:10.1016/0010-4655(95)00004-Y

We study the scaling behaviour in the ensemble approach of simulated annealing and threshold accepting considering two examples of complex optimization problems, namely the Grötschel's traveling salesman problem and a spin glass problem with Gaussian distribution of the couplings. If scaling is present it should allow for an estimation of the ground state energy. Our numerical results show a different qualitative behaviour for the two kinds of problems. Whereas scaling is present in the spin glass problem it is widely absent in the traveling salesman problem.

Optimal Simulated Annealing Schedules for Self Similar Systems

Ergenzinger, Klaus and Hoffmann, Karl Heinz and Salamon, Peter
Journal of Applied Physics 77(11): 5501—5508 (1995); DOI:10.1063/1.359253

The successful application of the stochastic optimization method known as simulated annealing can depend very much on the appropriate annealing schedule. While determining optimal schedules for arbitrary complex optimization problems is beyond the current scope, we here determine optimal schedules for a special class of systems with known properties. The state spaces of these special systems have the structure of self similar trees. Using methods of optimal control theory, we are able to predict the optimal schedule analytically for two distinct optimization criteria. These predictions are shown to be in good agreement with numerical results.

Optimization of the Power Output for the Compression and Power Stroke of the Diesel Engine

Blaudeck, Peter and Hoffmann, Karl Heinz
"Efficiency, Costs, Optimization and Environmental Impact of Energy Systems" Eds: Gö g}= u}c s}, Y. A. and Öztürk, A. and Tsatsaronis, G.2: 754 , 1995

For the Diesel engine the compression stroke and the power stroke are optimized to get the maximum power output. Contrary to previous papers the important heat transfer is completely taken into account. For both cases with and without constraints in piston acceleration a significant improvement of the efficiency in comparison with the conventional engine is found.

Publications in 1990 - 1994

Optimal control theory and irreversible thermodynamics

Hoffmann, Karl Heinz
Periodica Polytechnica 2: 15 (1994)

Optimizing Irreversible Thermodynamic Processes

Hoffmann, Karl Heinz
"Statistical physics and thermodynamics of nonlinear nonequilibrium systems" Eds: Ebeling, W. and Muschik, W.: 109—120 World-Scientific Publishing Co., Singapore, 1993; ISBN: 981-02-1134-1

Often ideal thermodynamic processes and the limits for process variables derived from them are compared to real industrial processes. But sometimes these are too far from equilibrium to be considered reversible, and thus the irreversibilities have to be taken into account to obtain a more realistic description. Then the question arises whether one can determine process limits and accompanying process paths for these irreversible processes. This paper addresses this question by means of two examples, one using a classical macroscopic thermodynamic description while the other uses statistical concepts: The first example deals with internal combustion engines and the second with simulated annealing. For both examples optimal process paths are determined.

Linear-Response Theory for Slowly Relaxing Systems

Hoffmann, Karl Heinz and Meintrup, Thomas and Uhlig, C. and Sibani, Paplo
Europhysics Letters 22(8): 565—570 (1993); DOI:10.1209/0295-5075/22/8/002

Slowly relaxing systems as spin glasses below the transition temperature are far from thermal equilibrium on experimental time scales. Nontheless experiments suggest the applicability of the fluctuation dissipation theorem out of equilibrium. To test this suggestion, we first derive the non-equilibrium response and correlation function for a large class of marcovian relaxation dynamics obeying detailed balance. We find that there exist no dynamics where the corrections to the FDT vanish exactly. Applying the formalism to a specific model, we then find that the corrections remain small in agreement with the experiments.

Scaling behaviour of optimal simulated annealing schedules

Christoph, Michael and Hoffmann, Karl Heinz
Journal of Physics A: Mathematical and General 26(13): 3267-3277 (1993); DOI:10.1088/0305-4470/26/13/028

The success of simulated annealing depends strongly upon the choice of a suitable annealing schedule. For a class of small sample systems the optimal annealing schedules are determined. They show distinct scaling behaviour as a function of the number of Metropolis steps carried out at each temperature of the schedule. This behaviour can be traced back to the influence of dominating barriers during cooling. Knowing the optimal schedule for a few different total annealing steps allows to predict the optimal annealing schedule for intermediate times.

Low Autocorrelation Binary Sequences: Exact Enumeration and Optimization by Evolution Strategies

De Groot, Claas and Würtz, Diethelm and Hoffmann, Karl Heinz
Optimization 23: 369—384 (1992); DOI:10.1080/02331939208843771

We investigate skew-symmetric sequences with chain lengths up to N =71, giving a complete table of all merit factors F≥7 and their associated configurations. We also calculate the exact thermodynamical properties of shorter chains (N≤55). We then introduce an evolutionary strategy, describing the properties of our search algorithm and comparing our results to those of other heuristic methods such as simulated annealing. We find the highest merit factors ever reached for chains of length 81≤N≤201.

Relaxation in Complex Systems: Local Minima and their Exponents

Sibani, Paolo and Hoffmann, Karl Heinz
Europhysics Letters 16(5): 423 (1991); DOI:10.1209/0295-5075/16/5/002

Existing models of random walks on regular trees are generalized by introducing nondegenerate local energy minima. We find that the long-time algebraic decay of the propagator is characterized by a multitude of exponents, rather than just by one as in the regular case. The relaxation to the lowest-lying minimum can be described by the Grossmann-Hoffmann-Wegner exponent for regular trees, but with an effective temperature-dependent branching ratio z(T),1 < z(T) < 2.

Aging Phenomena in Complex Systems: A Hierarchical Model for Temperature Step Experiments

Schulze, C. and Hoffmann, Karl Heinz and Sibani, Paolo
Europhysics Letters 15(3): 361—366 (1991); DOI:10.1209/0295-5075/15/3/022

We show that a previously introduced hierarchical model for relaxation in spin glasses and other complex systems, which described successfully the aging behaviour in the time domain, can also account for the effects of temperature changes on the magnetic response of the sample.

Simulated Annealing for Single Minimum Optimization Problems

Hoffmann, Karl Heinz and Salamon, Peter
International Journal of Computer Mathematics 39: 193-204 (1991); DOI:10.1080/00207169108803991

Two examples are presented which show that simulated annealing can perform better than quenching and steepest descent even on problems with a single minimum. An implication for real global optimization problems is that simulated annealing can be useful even on time scales which are short compared to the time required for a greedy algorithm to reach the nearest local minimum.

Optimizing Simulated Annealing

Hoffmann, Karl Heinz and Christoph, Michael and Hanf, Martin
"Parallel Problem Solving from Nature" Eds: H.-P. Schwefel and R. Maenner: 221—225 Springer-Verlag, Berlin, 1991; ISBN: 978-3-540-70652-6; DOI:10.1007/BFb0029756

This paper reviews efforts towards optimizing simulated annealing. In particular we address the question of the optimal schedule and of how estimates of system properties needed in optimizing simulated annealing can be obtained. We describe the ensemble approach to simulated annealing which lends itself readily to the implementation on parallel and vector computers and which thus leads to improved adaptive schedules.

Concepts in optimizing simulated annealing schedules: an adaptive approach for parallel and vector machines

Hoffmann, Karl Heinz and Würtz, Diethelm and de Groot, Class and Hanf, Martin
"Parallel and Distributed Optimization" Eds: Grauer, M. and Pressmar, D. B.: 154—175 , 1991; ISBN: 978-3-540-54434-0; DOI:10.1007/978-3-642-95665-2_10

Simulated Annealing (Čemy 1983, Kirkpatrick et al. 1983) is a technique which allows to find optimal or near optimal solutions to difficult optimization problems. It has been especially successful in applications to NP-complete or NP-hard problems, which occur in a variety of fields (Garey and Johnson 1979). These include mathematics with many graph problems (e.g. Brelaz 1979, Bonomi and Lutton 1987, Andresen et al. 1988), condensed matter physics, e.g. with the problem of finding the ground state of spin glasses (Ettelaie and Moore 1985), with the problem of solving the Ginzberg-Landau equations (Doria et al. 1989), engineering problems with the design of integrated circuits including the partitioning as well as the wiring problem (Vecchi and Kirkpatrick 1983, Sechen and Sangiovanni-Vincentelli 1985, Siarry et al. 1987), the design of binary sequences with low autocorrelation (Beenker et al. 1985, Bernasconi 1987, 1988), image processing (Carnevali et al. 1985), design of X-ray mirrors (Würtz and Schneider 1989), statistics with the application as a learning paradigm in neural network theory (Bernasconi 1990) and economics for instance with the travelling salesman problem (e.g. Bonomi and Lutton 1984, Kirkpatrick and Toulouse 1985, Hanf et al. 1990). Naturally, these are only some selected examples, since it is not possible here to give reference to the few hundred simulated annealing papers which appeared during the last years.

Optimizing Complex Problems by Nature's Algorithms: Simulated Annealing and Evolution Strategy — a Comparative Study

de Groot, Claas and Diethelm Würtz and Karl Heinz Hoffmann
"Parallel Problem Solving from Nature" Eds: H.-P. Schwefel and R. Maenner: 445-454 Springer-Verlag, Berlin, , 1991; ISBN: 978-3-540-54148-6; DOI:10.1007/BFb0029786

We compare two optimization algorithms which glean their heuristics from nature: simulated annealing and evolution strategy. These algorithms are applied to difficult optimization problems: finding binary sequences with low autocorrelation, calculating ground states of certain spin glass Hamiltonians, and giving the optimal tour in a traveling salesman problem. Our findings show a problem dependence of the quality of the results. Because of fundamental difficulties in the judgement of the algorithms' quality no final conclusions can be drawn, but the comparison gives valuable insight in the behaviour of the algorithms.

Monte Carlo dynamics of optimization problems: A scaling description

Sibani, Paolo and Pedersen, Jacob M. and Hoffmann, Karl Heinz and Salamon, Peter
Physical Review A 42(12): 7080-7086 (1990); DOI:10.1103/PhysRevA.42.7080

We show that some hard optimization problems studied by Monte Carlo methods, such as simulated annealing, have features that can be estimated by a statistical analysis of the data, well before being actually observed. This applies, for instance, to the estimation of the ground-state energy of the problem. We start by showing that the density of states and the distribution of extremes of energy seen in a given time interval in the Monte Carlo dynamics of combinatorial optimization problems are strongly related to each other through the first-passage-time distribution of the stochastic dynamics of the system. We then introduce a scaling ansatz for this last quantitiy, which allows an estimate of the ground state energy. Finally, we demonstrate the method on a travaling-salesman problem with know ground state energy and apply it to the simulated annealing of a graph-bipartitioning problem.

Relaxation and aging in spin glasses and other complex systems

Hoffmann, Karl Heinz and Sibani, Paolo
Zeitschrift für Physik B: Condensed Matter 80: 429—438 (1990); DOI:10.1007/BF01323526

We describe how a hierachical model of spin glass relaxation can display aging behaviour, similarly to what is found in spin glasses and other complex systems out of thermodynamic equilibrium. Since we deal with a nonequilibrium situation, the usualF(luctuation)D(issipation)T(heory) does not apply. We therefore derive a general relation between the linear response function and the non equilibrium propagator of the unperturbed system. The relation is shown to be very similar to the equilibrium FDT under certain conditions, which one can reasonably assume for spin glass systems. Having thus related the linear response of the system to a small external field to the autocorrelation function of the magnetization, we calculate the latter quantity by a master equation on a set of states which have the topology of a tree. The model can reproduce the main qualitative features of theZ(ero)F(ield)C(ooled) spin glass experiments, i.e. the maximum in the logarithmic time derivative of the magnetization, with only two free parameters.

Optimal Ensemble Size for Parallel Implementations of Simulated Annealing

Hoffmann, Karl Heinz and Sibani, Paolo and Pedersen, Jacob M. and Salamon, Peter
Applied Mathematics Letters 3(3): 53-56 (1990); DOI:10.1016/0893-9659(90)90136-Y

We determine the optimal ensemble size for a simulated annealing based on assumptions about scaling properties of the system dynamics and of the density of states in the low energy regime. The derivations indicate the optimal annealing time for any one ensemble member, thereby providing a stopping criterion and an explanation for the ''brick wall effect''.

The Optimal Simulated Annealing Schedule for a Simple Model

Hoffmann, Karl Heinz and Salamon, Peter
Journal of Physics A: Mathematical and General 23: 3511—3523 (1990); DOI:10.1088/0305-4470/23/15/023

Used as a tool for large scale global optimization, simulated annealing incurs heavy computational costs. Therefore, choosing an optimal cooling schedule is of great scientific and economic importance. For the first time an analytic as well as a numeric solution to this problem is presented, albeit only for a small example system. The example shows the role of optimal control theory for this problem.

Optima and Bounds for Irreversible Thermodynamic Processes

Karl Heinz Hoffmann
"Finite-Time Thermodynamics and Thermoeconomics, Advances in Thermodynamics 4" Eds: Stanislaw Sieniutycz and Peter Salamon: 22—65 Taylor and Francis, New York, 1990; ISBN: 0-8448-1668-X

In this paper bounds and optima for irreversible thermodynamic processes and their application in different fields are discussed. The tools of finite time thermodynamics are presented and especially optimal control theory is introduced. These methods are applied to heat engines, including models of the Diesel engine and a light-driven engine. Further bounds for irreversible processes are introduced, discussing work deficiency and its relation to thermodynamic length. Moreover the problem of dissipation in systems composed of several subsystems is studied. Finally, the methods of finite time thermodynamics are applied to thermodynamic processes described on a more microscopic level. The process used as an example is simulated annealing. It is shown how optimal control theory is applied to find the optimal cooling schedule for this important stochastic optimization method

Implementation of a New Adaptive Simulated Annealing Schedule on a Multi Transputer System

Hanf, Martin and Lehareinger, Yves and Würtz, Diethelm and Hoffmann, Karl Heinz and de Groot, Claas and Anliker, Max
IPS, ETH Zürich, CH-8092 Zürich, Switzerland; Technical Report 90-13—21, 1990

Simulated annealing is known to be a widely applicable optimization procedure from theoretical physics. We present an adaptive algorithm to optimize the annealing schedule using an ensemble approach. This approach is well-suited for a parallel computer. We give details of an implementation on a 32 transputer farm. We also compared this code to results obtained on other machines by applying them to the 532-city travelling salesman problem of Padberg and Rinaldi.

Simulated Annealing and Evolution Strategy — a Comparison

de Groot, Claas and Würtz, Diethelm and Hoffmann, Karl Heinz
Helvetica Physica Acta 63: 843 (1990)

We compare two optimization algorithms which glean their heuristic from nature: simulated annealing and evolution strategy. These algorithms are applied to two different difficult optimization problems which present themselves in a physical context as calculations of ground states with respect to certain Hamitonians. Our two Hamiltonians belong to short range two dimensional spin glasses and an autocorrelation funtion on a linear binary sequence which can be considered as a chain of long range interacting spins. Our results show a problem dependence of the behavior of the algorithms. While simulated annealing performs slightly better in the case of spin glasses, our evolution strategy for the autocorrelation function finds better results than ever obtained by other stochastic methods.

Publications in 1985 - 1989

Optimal Paths for a Bimolecular, Light-Driven Engine

Watowich, Stanley J. and Hoffmann, Karl Heinz and Berry, R. Stephen
Il Nuovo Cimento B 104(2): 131—147 (1989); DOI:10.1007/BF02906311

We examine a light-driven dissipative engine, which must necessarily operate far from equilibrium and at nonzero rate to be capable of providing power and work. The engine's working fluid consists of a buffer gas and the reacting system 2 SO3F↔ S2O6F2. we model the concentrations of the reacting system as function of both temperature and pressure. Piston trajectories maximizing work output and minimizing entropy production are determined for such an engine with the rate-dependent loss mechanisms of friction and heat conduction.

Hierarchical models for aging and relaxation of spin glasses

Sibani, Paolo and Hoffmann, Karl Heinz
Physical Review Letters 63(26): 2853—2856 (1989); DOI:10.1103/PhysRevLett.63.2853

We show that the aging phenomena found in spin glasses and other complex systems can be reproduced by a hierarchical model of relaxation.

Measures of Dissipation

Hoffmann, Karl Heinz and Andresen, Bjarne and Salamon, Peter
Physical Review A 39: 3618—3621 (1989); DOI:10.1103/PhysRevA.39.3618

The availability loss −∆Au in a process is equal to the flow of extensive thermodynamic quantities multiplied by the respective intensity differences only if the degraded work, the ''uncompensated heat'' of Clausius, is disposed of into the environment. We define work deficiency as the above product in all situations and relate it to the dissipation bound based on thermodynamic length.

Diffusion in Hierarchies

Hoffmann, Karl Heinz and Sibani, Paolo
Physical Review A 38(8): 4261—4270 (1988); DOI:10.1103/PhysRevA.38.4261

In this paper we show that diffusion processes in a 'complex' phase space with many local minima can be mapped into a random walk problem on a tree structure. We then rigorously solve the latter problem for regular trees, under the quite general assumption about the rates. Finally, we extend our results to the case of inhomogenious trees.

Bounds and Optima for Irreversible Thermodynamic Processes and their Application to Simulated Annealing

Hoffmann, Karl Heinz
Habilitationsschrift, Ruprecht-Karls-Universität, 1988

On lumped models for thermodynamic properties of simulated annealing problems

Andresen, Bjarne and Hoffmann, Karl Heinz and Mosegaard, Klaus and Nulton, Jim and Pedersen, Jacob M. and Salamon, Peter
Journal de Physique49: 1485—1492 (1988); DOI:10.1051/jphys:019880049090148500

The paper describes a new method for the estimation of thermodynamic properties for simulated annealing problems using data obtained during a simulated annealing run. The method works by estimating energy-to-energy transition probabilities and is well adapted to simulations such as simulated annealing, in which the system is never in equlibrium.

Random Walks on Cayley Trees: Temperature-Induced Transience-Recurrence Transition, Small Exponents and Logarithmic Relaxation

Sibani, Paolo and Hoffmann, Karl Heinz
Europhysics Letters 4(9): 967-972 (1987); DOI:10.1209/0295-5075/4/9/003

Random walks on tree structures are as useful tools in physics as they are interesting themselves. Here we show that for a certain class of models they can undergo a transition from being recurrent to being transient depending on the temperature. At the transition the relaxation is logarithmic. The significance of the pole in the relaxation exponent is also discussed.

Electrical Potential and current distribution for the quantized Hall effect

Neudecker, B. and Hoffmann, Karl Heinz
Solid State Communications 62(3): 135—139 (1987); DOI:10.1016/0038-1098(87)90177-3

The current distribution and the electrical potential is calculated for a system showing the Quantized Hall effect. The magnetic field strength enters the calculation in the form of the conductivity tensor of the sample. For the case of vanishing σxx the potential was calculated analytically, while for the other cases — representing different magnetic fields — current distribution and potential were computed numerically.

A Problem From Empirical Economics II: Determining Uncertainties Arising From Incomplete Data Using Information Theory

Hoffmann, Karl Heinz and Meshkov, Natalia
Renewable Energy 9: 259—273 (1987); DOI:10.1016/0165-0572(87)90005-3

Information theory is used to extract numerical values for quantities that are not directly available in existing data sources, but occur as linear combinations of other quantities for which data exist. The method uses known data to construct a distribution for the unknown quantities, which minimizes the Shannon information; hence, mean values and mean deviations for the unknowns can also be calculated. The results of an approach based on information theory are compared with the results obtained by another method.

Lower Bounds on Dissipation in Composite Systems

Hoffmann, Karl Heinz and Salamon, Peter
Physical Review A 35: 369—373 (1987); DOI:10.1103/PhysRevA.35.369

The dissipation inherent in the time evolution of a composite system consisting of a number of subsystems is characterized. We present two thermodynamic quantities which serve as general lower bounds to the minimum entropy production in a composite system. These lower bounds are easier to compute than the minimum entropy production and the difference between the bounds and the minimum entropy production quantifies a mismatch in the coevolution of subsystems. This mismatch has important implications for process control and design.

Fractional quantized Hall effect on a model with shortranged interaction in a sphere

Hoffmann, Karl Heinz and Neudecker, B.
Zeitschrift für Physik B: Condensed Matter 62: 279—285 (1986); DOI:10.1007/BF01313448

We consider a model forN electrons confined to the surface of a sphere in a strong radial magnetic field. The electron-electron interaction is modelled as a simple hardcore like repulsion. Numerical calculations have been carried out for up to N=6, neglecting contributions from higher Landau levels. Spectral properties of low-lying states are discussed within the quasihole-quasiparticle picture.

Intrinsically Irreversible Light-Driven Engine

Watowich, Stanley J. and Hoffmann, Karl Heinz and Berry, R. Stephen
Journal of Applied Physics 58: 2893-2901 (1985); DOI:10.1063/1.336281

We examine a reciprocating heat engine which necessarily operates far from equilibrium and about an unstable steady state. The piston of the engine is driven by the nonlinear coupling of the working fluid to an external light source which provides high quality heat and to the environment into which waste heat is dumped. We determine the piston trajectories that optimize two different criteria of process performance, the maximization of work output, and the minimization of entropy production. The trajectories optimizing different performance goals are qualitatively different. In engines not dominated by friction losses, the cycle optimizing work output requires that the expansion stroke begins with a slight compression and the temperature of the working fluid increases briefly.

Random Walk on a Fractal: Eigenvalue Analysis

Hoffmann, Karl Heinz and Grossmann, Siegfried and Wegner, Franz
Zeitschrift für Physik B: Condensed Matter 60(2-4): 401—414 (1985); DOI:10.1007/BF01304462

The eigenvalues of the master equation describing the motion on a nested hierarchy of $d$-dimensional intervals with selfsimilar scaling of spatial extension as well as of the level dependent transition rates are derived. Based on this spectrum the diffusion behaviour is obtained, which is anomalous, either exponential or obeying a power law with various exponents. Emphasis is put on the insight into the mechanism of the anomalous diffusion, in particular the geometrical structure of the decay rate spectrum.

Optimal Paths for Thermodynamic Systems: The Ideal Diesel Cycle

Hoffmann, Karl Heinz and Watowich, Stanley J. and Berry, R. Stephen
Journal of Applied Physics 58(6): 2125-2134 (1985); DOI:10.1063/1.335977

Optimal control theory is used to determine the piston trajectory which yields maximum power output for a model which incorporates the Diesel engine's major irreversibilities. Optimal trajectories were obtained for the cases of unconstrained piston acceleration. Optimizing the path four our standard engine increased both the net work output per cycle and the net efficiency by about 10%.

Anomalous Diffusion on a Selfsimilar Hierarchical Structure

Grossmann, Siegfried and Wegner, Franz and Hoffmann, Karl Heinz
Journal de Physique Lettres 46(13): L575-L583 (1985); DOI:10.1051/jphyslet:019850046013057500

The temporal increase of the moments in diffusion on a fractal with variable hopping range and lower cut-off is given. The essential parameters are the growth ratio, the length scaling and, as a new feature, the time scaling along the hierarchy. We find algebraical or exponential increase, logarithmic corrections, or trapping if the cut-off is removed. For the first time anomalous enhancement of the varaince increase σ ∝ tΘ, Θ larger than 2, is obtained as observed in turbulence.

Publications in 1980 - 1984

Die Hopfbifurkation unter dem Einfluß von weißem Rauschen

Hoffmann, Karl Heinz
PhD Thesis, RWTH Aachen, 1982

The Hopf Bifurcation of Twodimensional Systems under the Influence of One External Noise Source

Hoffmann, Karl Heinz
Zeitschrift für Physik B: Condensed Matter 49: 245—252 (1982); DOI:10.1007/BF01313033

The behaviour of the Hopf bifurcation under the influence of external noise is investigated by means of a twodimensional model which uses Gaussian white noise as input. The model includes the case of multiplicative and/or additive noise. Applying the Birkhoff transformation the model is transformed to the coordinates normally used to discuss the deterministic Hopf bifurcation. Then the stationary solution of the model is calculated as an expansion for weak noise: The Hopf bifurcation under the influence of noise exhibits a bifurcation interval with width and position depending on the noise power. Moreover, a class of the systems described by the model can perform noise driven bifurcations.

The Birkhoff Normalization Procedure and the Reductive Perturbation Approach: Two equivalent Methods to Discuss the Hopf Bifurcation

Hoffmann, Karl Heinz
Physics Letters A 92(4): 163—164 (1982); DOI:10.1016/0375-9601(82)90522-9

It is shown that the reductive perturbation approach yields exactly the same dynamical equations for systems with a small deviation of the control parameter from the onset of the Hopf bifurcation as the Birkhoff normalization procedure with an additional expansion with respect to this deviation.

Publications in 1975 - 1979

Stationäre Lösungen stochastischer Bewegungsgleichungen und ihr Zusammenhang mit der deterministischen Dynamik

Hoffmann, Karl Heinz
Diplomarbeit, RWTH Aachen, 1979

K-Isomorphie und K-Äquivalenz von Ordnungen

Hoffmann, Karl Heinz
Diplomarbeit, RWTH Aachen, 1978