Chemnitz UT: Natural Sciences: Physics: Comput...: Publications

Publications in 2010 - 2014

Optimal control in a quantum cooling problem
Salamon, Peter and Hoffmann, Karl Heinz and Tsirlin, Anatoly
Applied Mathematics Letters (in press) (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.

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, Gundula and Seeger, Steffen
it - Information Technology 53(2): 49--59 (2011) ; ISSN 1611-2776, 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, B. and Hoffmann, K. H. and Nulton, J. and Tsirlin, A. and Salamon, P.
European Journal of Physics 32(3): 827--843 (2011) ; ISSN 0143-0807, 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 $E_{ ext{T}}$ from a given initial state $(q_ ext{i}, p_ ext{i})$ by controlling its frequency $omega, omega_{ ext{min}} leq omega leq omega_{ ext{max}}$. 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, J. and Essex, C. and Hoffmann, K. H.
Physica A: Statistical Mechanics and its Applications 389(2): 214--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 R'enyi entropies in the space-fractional case. Complications arising due to the heavy tail solutions of space-fractional diffusion equations are discussed in detail.

Diffusion on fractals and space-fractional diffusion equations
Prehl, J.
PhD Thesis, Chemnitz University of Technology, 2010

The aim of this thesis is the examination of sub- and superdiffusive processes in fractal structures. The focus of the work concentrates on two separate approaches that are chosen and varied according to the corresponding regime. Thus, we obtain new insights about the underlying mechanisms and a more appropriate way of description for both regimes. In the first part subdiffusion is considered, which plays a crucial role for transport processes, as in living tissues. First, we model the fractal state space via finite Sierpinski carpets with absorbing boundary conditions and we solve the master equation to compute the time development of the probability distribution. To characterize the diffusion on regular as well as random carpets we determine the longest decay time of the probability distribution, the mean exit time and the Random walk dimension. Thus, we can verify the influence of random structures on the diffusive dynamics. In the second part of this thesis superdiffusive processes are studied by means of the diffusion equation. Its second order space derivative is extended to fractional order, which represents the fractal properties of the surrounding media. The resulting space-fractional diffusion equations span a linking regime from the irreversible diffusion equation to the reversible (half) wave equation. The corresponding solutions are analyzed by different entropies, as the Shannon, Tsallis or R'enyi entropies and their entropy production rates, which are natural measures of irreversibility. We find an entropy production paradox, i.e.~an unexpected increase of the entropy production rate by decreasing irreversibility of the processes. Due to an appropriate rescaling of the entropy we are able to resolve the paradox.

Simulating anomalous diffusion on graphics processing units
Hoffmann, K. H. and Hofmann, M. and Lang, J. and Rünger, G. and Seeger, S.
Proc.~of the 11th IEEE International Workshop on Parallel and Distributed Scientific and Engineering Computing (PDSEC-10) : 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, K. and Wilbrandt, S. and Stenzel, O. and Kaiser, N. and Hoffmann, K. H.
Applied Optics 49(16): 3150--3162 (2010)

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.

Anomalous diffusion and Random walks on random fractals
Anh, D. H. N.
PhD Thesis, Chemnitz University of Technology, 2010

The purpose of this research is to investigate properties of diffusion processes in porous media. Porous media are modelled by random Sierpinski carpets, each carpet is constructed by mixing two different generators with the same linear size. Diffusion on porous media is studied by performing random walks on random Sierpinski carpets and is characterized by the random walk dimension $d_w$. In the first part of this work we study $d_w$ as a function of the ratio of constituents in a mixture. The simulation results show that the resulting $d_w$ can be the same as, higher or lower than $d_w$ of carpets made by a single constituent generator. In the second part, we discuss the influence of static external fields on the behavior of diffusion. The biased random walk is used to model these phenomena and we report on many simulations with different field strengths and field directions. The results show that one structural feature of Sierpinski carpets called traps can have a strong influence on the observed diffusion properties. In the third part, we investigate the effect of diffusion under the influence of external fields which change direction back and forth after a certain duration. The results show a strong dependence on the period of oscillation, the field strength and structural properties of the carpet.

Publications in 2005 - 2009

Random Walks on random Koch curves
Seeger, S. and Hoffmann, K. H. and Essex, C.
Journal of Physics A: Mathematical and General 42(22): 22502 (2009) ; ISSN:1751-8121

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.

Funktionsorientierte Toleranzanalyse in der Motorenentwicklung
Schwalbe, Karsten
Bachelorarbeit an der TU Chemnitz, 2009

Maximum work in minimum time from a conservative quantum system
Salamon, P. and Hoffmann, K. H. and Rezek, Y. and Kosloff, R.
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, Y. and Salamon, P. and Hoffmann, K. H. and Kosloff, R.
Europhysics Letters 85: 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 $T_C ightarrow 0$. 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 $T_C^{3/2}$ 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, G. A. and Hoffmann, K. H.
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, J and Hoffmann, K. H.
Physical Review E 79: 046702-1--046702-8 (2009) ; DOI: 10.1103/PhysRevE.79.046702

$mathit{Genetic}$ $mathit{algorithms}$ 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 $mathit{threshold}$ $mathit{selection}$. 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, J. and Hoffmann, K. H.
Research and Development in Intelligent Systems XXVI : 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, K. H. and Salamon, P.
Applied Mathematics Letters 22: 1471--1475 (2009) ; ISSN: 0893-9659, 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.

A graphic based interface to Endoreversible Thermodynamics
Wagner, K.
Master Thesis, TU Chemnitz, 2008

The object of this thesis is a graphic based interface to endoreversible thermodynamics. It is meant to enable the user to visually create endoreversible systems and add the properties of the system by choosing features from a list and in form of equations. Then an equation system is built and the power output and efficiency of the endoreversible system is calculated and plotted. To illustrate the functions of the interface, some examples of heat and chemical engines are discussed.

Desiccation of a clay film: Cracking versus peeling
Sadhukhan, S. and Prehl, J. and Blaudeck, P. and Hoffmann, K. H. and Dutta, T. and Tarafdar, S.
The European 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 {it 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, J. and Hoffmann, K. H. and Hofmann, M. and Rünger, G. and Tarafdar, S.
Thermal Nonequilibrium - Lecture Notes of the 8th International Meeting on Thermodiffusion 3: 243--248 , 2008 ; ISSN: 1866-1807; ISBN: 978-3-89336-523-4

Sampling procedures for low temperature dynamics on complex energy landscapes
Nemnes, G. A.
PhD Thesis, Chemnitz University of Technology, 2008

The present work deals with relaxation dynamics on complex energy landscapes. The state space of a complex system possesses, as a hallmark, the multitude of local minima separated by higher states, called barrier states. This feature gives rise to a host of non-equilibrium phenomena. From case to case, for different complex systems, ranging from atomic clusters, spin glasses and proteins to neural networks or financial markets, the key quantities like energy and temperature may have different meanings, though their functionality is the same. The numerical handling of relaxational dynamics in such complex systems, even for relatively small sizes, poses a tough challenge if the entire state space is to be considered. Here, state space sampling procedures are introduced that provide an accurate enough description for the low temperature dynamics, using small subsets from the original state space. As test cases, short range Ising spin systems were considered. The samples - depending on the way they are constructed - provide either lower bounds for the largest relaxation timescales in a quasi-ergodic component of the state space or the isothermal relaxation of the mean energy, like in the proposed DRS method. Upon the latter procedure, a parallel heuristic is built which gives the possibility of handling large samples. The collected structural data provides information of the state space topology in systems with different levels of frustration, like disordered ferromagnets and spin glasses. It provides insights into the focusing/anti-focusing types of landscapes, which give rise to different ground state accessibilities. For the large samples, the domain formation and growth has been analysed and compared with existing experimental and numerical data in literature. The algorithms proposed here become more and more accurate as the temperature is decreased and therefore they can provide an alternative to the classical Monte Carlo approach for this temperature range.

Dynamically relevant structural properties of short-range spin glasses and disordered ferromagnets
Nemnes, G. A. and Hoffmann, K. H.
Physical Review B 77: 172410 (2008) ; ISSN: 1098-0121

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, J. and Hoffmann, K. H. and Enachescu, M.
Genetic and Evolutionary Computation Conference , 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.

Anomalous Transport in Disordered Fractals
Hoffmann, K. H. and Prehl, J.
Anomalous Transport - Foundations and Applications Wiley-VCH, Weinheim, 2008 ; ISBN: 978-3-527-40722-4

Numerical methods for density of states calculations
Haber, R.
Master Thesis, Chemnitz University of Technology, 2008

The parQ method, up to now only capable of calculating the density of states in the canonical ensemble, is extended to the grand canonical ensemble and compared to the Wang-Landau algorithm, a local-update flat-histogram method. Both algorithms have been implemented so that the performance and the respective benefits with increasing simulation time can be determined and compared.

An interlacing theorem for reversible Markov chains
Grone, R. and Hoffmann, K. H. and Salamon, P.
Journal of Physics A: Mathematical and General 41: 212002 (2008) ; ISSN: 1751-8113

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.

Modelling Complex Systems: Tree Structures
Fischer, A.
PhD Thesis, Chemnitz University of Technology, 2008

The state space is a very important and fundamental concept for the treatment of complex systems. All the system's properties can be understood by means of its structure. Due to the gigantic extent of a real system's state space, a coarse grained approach is inevitable for the analysis. In this work, based on the well established model of hierarchical trees, particular aspects of complex systems have been studied, while at the same time several extensions to the model have been made. In the first part of this research work the features of the probability flow are treated in detail at a single saddle point in the energy landscape. Influences of various parameters like energetic depth, density of states and connectivity are studied isolated and in their interaction. In the second part a whole system showing complex behavior is being considered, especially its energy exchange with the surroundings. It can be demonstrated that the hierarchical relaxation behavior observed in other realizations of complex systems is intrinsically covered by the tree model. Beside energy landscape based systems turbulent diffusion processes possess hierarchical structures, too. In the third part the tree structure has been used to model a turbulent superdiffusion process. The diffusion behavior observed there has been compared with four well known diffusion equation approaches. The results show that only one of the discussed continuum diffusion equations can model the turbulent transport based on the tree model in acceptable fashion.

Intermittent relaxation in hierarchical energy landscapes
Fischer, A. and Hoffmann, K. H. and Sibani, P.
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.

The Investigation of The parQ-Method for Continuous Systems
Schulz, R.
Master Thesis, 2007

To analyze the thermodynamical quantities of canonical systems the parQ-method has been introduced [1]. After testing this method for spin glasses [1, 2], this paper verifies that the parQ-method is also applicable to obtain the density of states of continuous systems, for example fluids, with the periodic boundary condition. Specifically, the energy transitions of a random walk through state space are observed. Therefore, the necessary methods and system configurations are introduced and examined. Furthermore, several potential error sources are investigated. The resulting data is compared to [3] where the density of states of a Lennard-Jones fluid is computed using the Wang-Landau scheme [4].

Numerically Optimized Diabatic Distillation Columns
Schaller, M.
PhD Thesis, Chemnitz University of Technology, 2007

Im Gegensatz zur konventionellen adiabatischen Destillation erfolgt bei der diabatischen Destillation Wärmeaustausch nicht nur am Kondensator und Verdampfer, sondern auch innerhalb der Kolonne an den einzelnen Siebböden, was die Entropieproduktion (=Exergieverlust) des Destillationsprozesses stark reduziert. In dieser Arbeit werden Modellsysteme zur diabatischen Destillation von idealen binären Gemischen mittels numerischer Optimierung untersucht. Das Ausgangsmodell beschränkt sich auf die Minimierung der Entropieproduktion verursacht durch Wärme- und Massentransport im Inneren der diabatischen Destillationskolonne. Im zweiten Modell wird das diabatische Modell um die Irreversibilität bedingt durch den Wärmeaustausch mit der Umgebung erweitert. Im dritten Modellsystem wird anstelle der bis dahin voneinander unabhängig geregelten Bodentemperaturen eine diabatische Implementierung mit seriellen Wärmetauschern untersucht, die nur mehr vier Kontrollvariablen besitzt und besonders zur praktischen Anwendung geeignet ist. Für alle diabatischen Modelle werden die minimale Entropieproduktion und optimalen Betriebsprofile numerisch ermittelt, und mit konventionellen Destillationskolonnen verglichen. Alle Ergebnisse zeigen eine deutlich Reduktion der Entropieproduktion für den diabatische Fall, besonders bei Kolonnen mit vielen Böden.

Quantifying Dissipation
Hoffmann, K. H.
Communications to SIMAI Congress 2: 1--12 (2007) ; ISSN: 1827-9015, 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, K. H. and Rodriguez-Brito, B. and Breitbart, M. and Bangor, D. and Angly, F. and Felts, B. and Nulton, J. and Rohwer, F. and Salamon, P.
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. Biologi- cally, 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
A. Fischer and S. Seeger and K. H. Hoffmann and C. Essex and M. Davison
Journal of Physics A: Mathematical and General 40(38): 11441-11452 (2007) ; ISSN 1751-8113/07/3811441+12$30.00

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
Do Hoang Ngoc Anh and P. Blaudeck and K. H. Hoffmann and J. Prehl and S. Tarafdar
Journal of Physics A: Mathematical and General 40: 11453-11465 (2007) ; ISSN 1751-8113/07/3811453+13$30.00

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, S. and Hoffmann, K. H. and Meyer, A.
Parallel algorithms and cluster computing : 335 Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

The structure of enumerated spin glass state spaces
Schubert, S. and Hoffmann, K. H.
Computer Physics Communications 174: 191--197 (2006) ; ISSN: 0010-4655

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
W. Muschik and K. H. Hoffmann
Journal of Non-Equilibrium Thermodynamics 31(3): 293--317 (2006) ; ISSN: 0340-0204

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, K. H. and Hofmann, M. and Rünger, G. and Seeger, S.
Proc. of the 12th International Euro-Par Conference 4128/2006: 1043--1025 , 2006 ; 978-3-540-37783-2

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 -
Hoffmann, K. H. and Meyer, A.
Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

Modelling aging experiments in spin glasses
Hoffmann, K. H. and Fischer, A. and Schubert, S. and Streibert, T.
Parallel algorithms and cluster computing : 281 Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

An Introduction to Endoreversible Thermodynamics
Hoffmann, K. H.
Atti dell'Accademia Peloritana dei Pericolanti, 2006

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 of internally reversible (endoreversible) subsystems exchanging energy in an irreversible fashion. This material provides an introduction to the subject.

Random walks on fractals
Franz, A. and Schulzky, C. and Do Hoang, N. A. and Seeger, S. and Balg, J. and Hoffmann, K. H.
Parallel algorithms and cluster computing : 303 Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

Optimizing simulated annealing schedules for amorphous carbons
Blaudeck, P. and Hoffmann, K. H.
Parallel algorithms and cluster computing : 227 Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

The coastline and lake shores of a fractal island
Blaudeck, P. and Seeger, S. and Schulzky, C. and Hoffmann, K. H. and Dutta, T. and Tarafdar, S.
Journal of Physics A: Mathematical and General 39: 1609--1618 (2006) ; ISSN: 0305-4470

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.

Simulation diskreter Markov-Prozesse zweiter Stufe
Bauer, M.
Bachelor Thesis, 2006

Simulationen von Markov-Prozessen erster Stufe sind in vielen Anwendungsbereichen bekannt, Erkenntnisse über die Eigenschaften von Prozessen zweiter Markov-Stufe sind jedoch rar. Wichtige Merkmale sollen durch die Simulation diskreter Markov-Prozesse zweiter Stufe herausgearbeitet werden. An erster Stelle steht dabei die Entwicklung eines einfachen Modells zur Simulation von Markov-Prozessen zweiter Stufe mittels eines Random Walkers, der über ein Gedächtnis verfügt. Das Modell bildet dabei zeit- und ortsdiskrete Prozesse ab und hält die Möglich- keit zur Simulation äußerer Kräfte offen. Im Weiteren geben Untersuchungen der zeitlichen Entwicklung der Ortsverteilung und des Entropieverlaufs an so simulierten Zufallswanderern Aufschluss über Ähnlichkeiten und Unterschiede zu Markov-Prozessen erster Stufe. Dabei werden analytische Abschätzun- gen zur Bestätigung der simulierten Daten genutzt und das Langzeitverhalten der Prozesse untersucht und mit dem von Markov-Prozessen erster Stufe verglichen. Weiterhin wird die Abhängigkeit der Stärke des Gedächtnisses auf das Verhalten der Zufallswanderer genauer betrachtet.

Diffusion on Fractals
Balg, J.
Master Thesis, TU Chemnitz, 2006

We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement $langle r^{2}(t) angle$. Applying different bias amplitudes on several regular {sc Sierpinski} carpets we obtain maximal drift velocities for weak field strengths. According to $langle r^{2}(t) angle sim t^{frac{2}{d_{ ext{w}}}}$, we determine random walk dimensions of $d_{ ext{w}}<2$~for applied external fields. These $d_{ ext{w}}$~corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus $d_{ ext{w}}<2$.

On symbolic derivation of the cumulant equations
Seeger, S. and Hoffmann, K. H.
Computer Physics Communications 168(3): 165--176 (2005) ; ISSN: 0010-4655, 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, S. and Hoffmann, K. H.
Journal of Statistical Physics 121(1--2): 75--90 (2005) ; ISSN: 0022-4715; 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, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 17(1): 51--60 (2005) ; ISSN: 0935-1175, 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.

Diffusion on Fractals
Schulz, R.
Bachelor Thesis, 2005

In this work a Sierpinski carpet with several randomly mixed generators is examined. A regular Sierpinski carpet is a self-similar fractal object fully determined by a recurrent pattern. In a random Sierpinski carpet, in order to model the irregularity of natural porous materials, several generators are mixed during the construction of the carpet. Furthermore it will be examined how the random walk dimension depends on the mixture ratio of two generators. This is explored with a numerical implementation of the master equation.

Simulation von Diffusion auf Fraktalen
Schönherr, M.
Bachelor Thesis, TU Chemnitz, 2005

Diffusion auf künstlichen oder natürlichen Strukturen kann durch Bewegung von Random Walkern auf Fraktalen untersucht werden. Diese Simulation wird fast nur numerisch durchgeführt. Um ungeordnete Strukturen zu untersuchen, kann das hier untersuchte auch zufällig zusammengesetzte Fraktale aus verschiedenen vorgegebenen Generatoren mit bestimmten Mischungsverhältnissen, in verschiedenen Iterationstiefen, betrachten. Das Verhalten der ausgegebenen Random-Walk-Dimension kann so über der Variation des Mischungsverhältnisses beobachtet werden. Verschiedene Ergebnisse wurden mit Werten aus der Theorie und anderen Arbeiten verglichen.

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: 084308 (2005) ; ISSN: 0021-8979

We present the nonlinear I-V characteristics of a nanoscale metal-oxide-semiconductor field-effect transistor in the Landauer-Büttiker 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 characteristic 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 et al., Intel Technol. J. 6, 42 (2002)].

Optimization by thermal cycling
Möbius, A. and Hoffmann, K. H. and Schön, C.
Complexity, Metastability and Nonextensitivity : 215-219 World Scientific, , 2005 ; ISBN: 981-256-525-6

Diffusion on Fractals
John, A.
Master thesis, TU Chemnitz, 2005

The structure of marine phage populations
Hoffmann, K. H. and Rodriguez-Brito, B. and Breitbart, M. and Bangor, D. and Angly, F. and Felts, B. and Nulton, J. and Rohwer, F. and Salamon, P.
Proceedings of ECOS 2005 : 1--5 , 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 $mathrm{CO_2}$ 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, K. H. and Schön, J. C.
Foundations of Physics Letters 18(2): 171--182 (2005) ; ISSN: 0894-9875, 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.

The State Space of Complex Systems
Heilmann, F.
PhD Thesis, Chemnitz University of Technology, 2005

In dieser Arbeit wird eine Beschreibung von Monte-Carlo-Verfahren zur Lösung komplexer Optimierungsaufgaben mit Hilfe von Markov-Ketten durchgefuhrt. Nach einer kurzen Einfuhrung werden Lösungsmenge solcher Aufgaben und der physikalische Zustandsraum komplexer Systeme identifiziert. Zunächst wird die Dynamik von Zufallswanderern im Zustandsraum mit Hilfe von Master-Gleichungen modelliert. Durch Einfuhrung von Performanzkriterien können verschiedene Optimierungsstrategien quantitativ miteinander verglichen werden. Insbesondere wird das Verfahren Extremal Optimization vorgestellt, das ebenfalls als Markov-Prozess verstanden werden kann. Es wird bewiesen, dass eine im Sinne der genannten Kriterien beste Implementierung existiert. Da diese von einem sogenannten Fitness Schedule abhängt, wird dieser fur kleine Beispielsysteme explizit berechnet. Daran anschließend wird die Zustandsdichte komplexer Systeme betrachtet. Nach einem kurzen Überblick über vorhandene Methoden folgt eine detaillierte Untersuchung des Verfahrens von Wang und Landau. Numerische und analytische Hinweise werden gegeben, nach denen dieser Algorithmus innerhalb seiner Klasse wahrscheinlich der Optimale ist. Eine neue Methode zur Approximation der Zustandsdichte wird vorgestellt, die insbesondere für die Untersuchung komplexer Systeme geeignet ist. Abschließend wird ein Ausblick auf zukünftige Arbeiten gegeben.

ParQ -- high-precision calculation of the density of states
Heilmann, F. and Hoffmann, K. H.
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 $O_2$ from arterioles
Bertuglia, S. and Limon, A. and Andresen, B. and Hoffmann, K. H. and Essex, C. and Salamon, P.
Journal of Non-Equilibrium Thermodynamics 30(2): 151--162 (2005) ; ISSN: 0340-0204, 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, D. H. N. and Hoffmann, K. H. and Seeger, S. and Tarafdar, S.
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 $d_w$ shows strong dependence on the mixture composition. For the mixed system it can be higher or lower than both the pure components. Further, $d_w$ may decrease on mixing, indicating faster diffusion in the disordered system.

Thermomechanical 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) ; ISSN: 0340-0204, 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) ; DOI: 10.1007/s00161-004-0183-3

Aging in enumerated spin glass state spaces
Schubert, S. and Hoffmann, K. H.
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, S. and Salamon, P. and Rivero, R. and Rendon, C. and Hoffmann, K. H. and Schaller, M. and Andresen, B.
Industrial & Engineering Chemical Research 43(23): 7566--7571 (2004) ; ISSN: 0888-5885(04)09593-4

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, E. S. and Salamon, P. and Rivero, R. and Rendon, C. and Hoffmann, K.H.
Proceedings of ECOS 2004 : 179--187 , 2004

Fitness Threshold Accepting over extremal optimization ranks
Hoffmann, K. H. and Heilmann, F. and Salamon, P.
Physical Review E 70(4): 046704-1 -- 046704-6 (2004) ; ISSN: 1539-3755/2004/70(4)/046704(6)

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, F. and Hoffmann, K. H. and Salamon, P.
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, A. and Hoffmann, K. H.
Journal of Non-Equilibrium Thermodynamics 29(1): 9--28 (2004) ; ISSN 0340-0204

Optimal Allocation of Heat Exchanger Investment
Burzler, J. M. and Amelkin, S. M. and Tsirlin, A.M. and Hoffmann, K. H.
Open Systems and Information Dynamics 11(3): 291--306 (2004) ; ISSN: 1301-9724

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.

Anwendung der Methode der gewichteten Residuen auf die eindimensionale Boltzmanngleichung
Balg, Janett
Bachelorarbeit an der TU-Chemnitz, 2004

Maximum power processes for multi-source endoreversible heat engines
Amelkin, S. A. and Andresen, B. and Burzler, J. M. and Hoffmann, K. H. and Tsirlin, A. M.
Journal of Physics D: Applied Physics 37(9): 1400--1404 (2004) ; PII: S0022-3727(04)73547-0

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.

The Cumulant Method
Seeger, S.
PhD Thesis, Chemnitz University of Technology, 2003

In this work, a new method to reduce Boltzmann equation to a system of partial differential equations is discussed. After a short introduction to kinetic theory of an inert mixture of gases an overview of the various moment methods known from literature is given. The cumulant method, presented in the following, is based on the assumption that due to the collision processes in a gas correlations of higher order decay more rapidly than correlations of lower order. Based on this assumption the eautions of motion for the cumulants are derived and the production terms of the resulting balance equations are calculated for a mixture of inter Maxwell gases are calculated. Examination of the relexation to an equilibrium state allows to relate these equations to models known from continuum mechanics and shows the validity of the assumption made for this case. In the second part results of numerical experiments are presented, where simulations with various boundary conditions are carried out for Couette and Poiseulle flows. Depending on the particular boundary conditions applied, both characteristic properties of rearefied gases but also of Navier-Stokes flows are observed. The last part of this work discusses moment equations as a particular form of the method of weighted residuals applied to the Boltzmann equation, giving an outlook to future work.

Fractional Diffusion, Irreversibility and Entropy
Li, X. and Essex, C. and Davison, M. and Hoffmann, K. H. and Schulzky, C.
Journal of Non-Equilibrium Thermodynamics 28(3): 279--291 (2003) ; ISSN: 0340-0204

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, K. H. and Burzler, J. and Fischer, A. and Schaller, M. and Schubert, S.
Journal of Non-Equilibrium Thermodynamics 28(3): 233--268 (2003) ; ISSN 0340-0204

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, A. and Hoffmann, K. H.
Applied Mathematics Letters 16(1): 27--31 (2003) ; ISSN: S0893-9659(02)

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 $qin R$, where for $q ightarrow 1$ 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 $q ightarrow -infty$.

Passive houses to reduce the utilization of classical fuels for space heating
Dragos, A. A. and Badescu, V. and Hoffmann, K. H. and Sicre, B.
Conferinta nationala pentru dezvoltare durabila : 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.

Performance Optima for Endoreversible Systems
Burzler, Josef Maximilian
Dissertation, TU-Chemnitz, 2003

Theoretical bounds for performance measures of thermodynamical systems are investigated under conditions of finite times and rates of processes using endoreversible models. These models consist of reversible operating sub-systems which exchange energy via generally irreversible interactions. Analytical and numerical calculations are performed to obtain performance optima and respective optimized process and design parameters for four model systems. A heat engine where the heat transfer between the working fluid and heat reservoirs is described by generalized, polytropic process is optimized for maximum work output. Thermal efficiencies, optimal values for temperatures and process times of the heat transfer processes are determined. A model of a generalized system suited to describe the operation of heat engines, refrigerators, and heat pumps is optimized with respect to thermal efficiency. Several examples illustrate how the results of the analysis are used to allocate financial resources to the heat exchanger inventory in an optimal way. A power-producing thermal system which exchanges heat with several heat reservoirs via irreversible heat transfer processes is analyzed to find the optimal contact times between the working fluid and each of the reservoirs. The piston motion of a Diesel engine is optimized to achieve maximum work for a given amount of fuel. The endoreversible model of the Diesel engine accounts for the temporal variations of the heat produced by the combustion process, the basic flow pattern within the engine's cylinder, the temperature dependence of the viscosity, thermal conductivity, and heat capacity of the working fluid and losses due to friction and heat leak through the cylinder walls.

Optimal Endoreversible Heat Engines with Polytropic Branches
Burzler, J. M. and Hoffmann, K. H. and Amelkin, S. A. and Tsirlin, A. S.
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, P. and Hoffmann, K. H.
Computer Physics Communications 150(3): 293--299 (2003) ; PII: 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.

Minimal Work for Separation Processes of Binary Mixtures
Amelkin, S. A. and Tsirlin, A. M. and Burzler, J. M. and Schubert, S. and Hoffmann, K. H.
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, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 14(2): 321--335 (2002) ; ISSN: 0935-1175

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, M. and Hoffmann, K.H. and Rivero, R. and Andresen, B. and Salamon, P.
Journal of Non-Equilibrium Thermodynamics 27(3): 257--256 (2002) ; ISSN: 0340-0204

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, G. M. and Kjelstrup, S. and Salamon, P. and Siragusa, G. and Schaller, M. and Hoffmann, K.H.
Industrial & Engineering Chemical Research 41(23): 5826--5834 (2002) ; DOI: 10.1021/ie010872p ISSN: 0888-5885

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, K. H. and Franz, A. and Salamon, P.
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, K. H.
Computational Statistical Physics : 57--76 Springer Verlag, Berlin, 2002 ; ISBN: 3-540-42160-2

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, K. H.
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
Hoffmann, K. H. and Schreiber, M.
Springer Verlag, Berlin, 2002 ; ISBN: 3-540-42160-2

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, A. and Schulzky, C. and Hoffmann, K. H.
SIGSAM Bulletin: Communications in Computer Algebra 36(2): 18--30 (2002) ; ISSN: 0163-5824

We present a new algorithm for calculating the chemical dimension $d_{ ext{l}}$ 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 $d_{ ext{min}}$ the minimum path dimension, which is related to the chemical dimension.

Optimal Annealing Schedules for a Modified Tsallis Statistics
Franz, A. and Hoffmann, K. H.
Journal of Computational Physics 176(1): 196--204 (2002) ; ISBN: 0021-9991/02

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 $q ightarrow infty$ 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, A. and Schulzky, C. and Seeger, S. and Hoffmann, K. H.
Fractal Geometry: Mathematical Methods, Algorithms, Applications IMA Conference Proceedings, : 52--67 Horwood Publishing Ltd., Chichester, West Sussex, , 2002 ; ISBN: 1-904275-00-1

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 $t^{2/d_w}$, where the random walk dimension $d_w$ 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, H. and Seeger, S. and Hoffmann, K.H.
Besondere Lernleistung, 2002

Modelling porous structures by repeated Sierpinski carpets
Tarafdar, S. and Franz, A. and Schulzky, S. and Hoffmann, K. H.
Physica A: Statistical Mechanics and its Applications 292(1-4): 1--8 (2001) ; PII: S0378-4371(00)00573-2, 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 $langle r^2 angle sim t^{2/d_w}$, where $langle r^2 angle $ is the mean square distance covered in time $t$ and $d_w>2$. 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 $d_w$ 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, S. and Franz, A. and Schulzky, C. and Hoffmann, K. H.
Computer Physics Communications 134(3): 307--316 (2001) ; PII: 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, M. and Hoffmann, K. H. and Siragusa, G. and Salamon, P. and Andresen, B.
Computers & Chemical Engineering 25(11--12): 1537--1548 (2001) ; ISSN: 0098-1354/01

Recently, the concept of equal thermodynamic distance (ETD) has been proposed to minimize entropy production in a distillation process using a diabatic column. mbox{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, P. and Hoffmann, K. H. and Schubert, S. and Berry, R. S. and Andresen, B.
Journal of Non-Equilibrium Thermodynamics 26(1): 73--83 (2001) ; ISSN: 0340-0204

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, G. and Kjelstrup, S. and Salamon, P. and Siragusa, G. and Schaller, M. and Hoffmann, K. H.
Proceedings of ECOS '01 : 667-677 , 2001 ; ISBN: 975-97568-2-2

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, U. and Seeger, S. and Petersen, K.
Praxis der Informationsverarbeitung und Kommunikation 24(2): 75--84 (2001) ; ISSN: 0930-5157

Durch die Massenfertigung moderner Personal-Computer sind heute sehr leistungsfähige Komponenten zu geringen Preisen zu erhalten -- auch hat bereits 1994 das emph{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
Hoffmann, K. H.
Annalen der Physik 10(1--2): 79--88 (2001) ; ISSN: 0003-3804

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
Hoffmann, K. H. and Schreiber, M.
Springer-Verlag, Berlin, 2001 ; ISBN: 7-03-008913-8/O 1296

The pore structure of Sierpinski carpets
Franz, A. and Schulzky, C. and Tarafdar, S. and Hoffmann, K. H.
Journal of Physics A: Mathematical and General 34(42): 8751--8765 (2001) ; PII: S0305-4470(01)19486-2

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, A. and Schulzky, C. and Hoffmann, K. H.
Nonlinearity 14(5): 1411--1418 (2001) ; PII: S0951-7715(01)16979-3

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, A. and Hoffmann, K. H. and Salamon, P.
Physical Review Letters 86(23): 5219--5222 (2001) ; ISSN: 0031-9007/01/86(23)/5219(4)

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.

A Comparison of Random Walks with Different Types of Acceptance Probabilities
Fachat, A.
Dissertation, Chemnitz University of Technology, 2001

In this thesis random walks similar to the Metropolis algorithm are investigated. Special emphasis is laid on different types of acceptance probabilities, namely Metropolis, Tsallis and Threshold Accepting. Equilibrium and relaxation properties as well as performance aspects in stochastic optimization are investigated. Analytical investigation of a simple system mimicking an harmonic oscillator yields that a variety of acceptance probabilities, including the abovementioned, result in an equilibrium distribution that is widely dominated by an exponential function. In the last chapter an optimal optimization schedule for the Tsallis acceptance probability for the idealized barrier is investigated.

The Differential Equation Describing Random Walks on the Koch Curve
Essex, C. and Davison, M. and Schulzky, C. and Franz, A. and Hoffmann, K. H.
Journal of Physics A: Mathematical and General 34(41): 8397-8406 (2001) ; PII: S0305-4470(01)27363-6

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, M. and Essex, C. and Schulzky, C. and Franz, A. and Hoffmann, K. H.
Journal of Physics A: Mathematical and General 34(20): L289--L296 (2001) ; PII: S0305-4470(01)15840-3

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, S. and Burzler, J. and Hoffmann, K. H. and Tsirlin, A. M.
Theoretical Foundations of Chemical Engineering 35(3): 217--223 (2001) ; ISSN: 0040-5795 (paper), ISSN: 1608-3431 (online), 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, S. and Burzler, J. and Hoffmann, K. H. and Tsirlin, A. M.
Theoretical Foundations of Chemical Engineering 35(3): 223--238 (2001) ; ISSN: 0040-5795 (paper), ISSN: 1608-3431 (online), DOI: 10.1023/A:1010485906403

Renewed Theory, Interfacing, and Visualization of Thermal Lattice Boltzmann Schemes
Späth, P.
Dissertation, TU Chemnitz, 2000

In this document the Lattice Boltzmann scheme, a heuristic method for the simulation of flows in complicated boundaries, is investigated. Its theory is renewed by emphasizing the entropy maximization principle, andnew means for the modeling of geometries (including moving boundaries) and the visual representation of evoluting flows are presented. An object oriented implementation is given with communication between objects realized by an interpreterobject and communication from outside realized via interprocess communication. Within the new theoretical approach the applicability of existing Lattice Boltzmann schemes to model thermal flows for arbitrary temperatures is reexamined.

The cumulant method for computational kinetic theory
Seeger, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 12: 403-421 (2000) ; ISSN: 0935-1175

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, C. and Essex, C. and Davison, M. and Franz, A. and Hoffmann, K. H.
Journal of Physics A: Mathematical and General 33(31): 5501-5511 (2000) ; ISSN: 0305-4470/00/315501+11

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, C. and Franz, A. and Hoffmann, K. H.
SIGSAM Bulletin: Communications in Computer Algebra 34(3): 1--8 (2000) ; ISSN: 0163-5824

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.

Anomalous Diffusion and Random Walks on Fractals
Schulzky, C.
PhD Thesis, University of Technology Chemnitz, 2000

In dieser Arbeit werden verschieden Ansätze diskutiert, die zum Verständnis und zur Beschreibung anomalen Diffusionsverhaltens beitragen, wobei insbesondere zwei unterschiedliche Aspekte hervorgehoben werden. Zum einen wird das Entropieproduktions-Paradoxon beschrieben, welches bei der Analyse der Entropieproduktion bei der anomalen Diffusion, beschrieben durch fraktionale Diffusionsgleichungen auftritt. Andererseits wird ein detaillierter Vergleich zwischen Lösungen verallgemeinerter Diffusionsgleichungen mit numerischen Daten präsentiert, die durch Iteration der Mastergleichung auf verschiedenen Fraktalen produziert worden sind. Die Entropieproduktionsrate für superdiffusive Prozesse wird berechnet und zeigt einen unerwarteten Anstieg beim Übergang von dissipativer Diffusion zur reversiblen Wellenausbreitung. Dieses Entropieproduktions-Paradoxon ist die direkte Konsequenz einer anwachsenden intrinsischen Rate bei Prozessen mit zunehmendem Wellencharakter. Nach Berücksichtigung dieser Rate zeigt die Entropie den erwarteten monotonen Abfall. Diese Überlegungen werden für generalisierte Entropiedefinitionen, wie die Tsallis- und Rényi-Entropien, fortgefhrt. Der zweite Aspekt bezieht sich auf die anomale Diffusion auf Fraktalen, im Besonderen auf Sierpinski-Dreiecke und -Teppiche. Die entsprechenden Mastergleichungen werden iteriert und die auf diese Weise numerisch gewonnenen Wahrscheinlichkeitsverteilungen werden mit den Lösungen vier verschiedener verallgemeinerter Diffusionsgleichungen verglichen.

Hausdorff dimension estimates for non-injective maps using the cardinality of the pre-image sets
Franz, A.
Nonlinearity 13(5): 1425-1438 (2000) ; PII: S0951-7715(00)02957-1

Upper bounds for the Hausdorff dimension of compact invariant sets of $C^1$ -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, A. and Schulzky, C. and Seeger, S. and Hoffmann, K. H.
Fractals - Complex Geometry, Patterns, and Scaling in Nature and Society 8(2): 155-161 (2000) ; ISSN: 0218-348X(00)

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, A. and Hoffmann, K.H. and Franz, A.
Computer Physics Communications 132(3): 232--240 (2000) ; ISSN: 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, C. and Schulzky, C. and Franz, A. and Hoffmann, K. H.
Physica A: Statistical Mechanics and its Applications 284(1-4): 299-308 (2000) ; ISSN: 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, C. and Davison, M. and Schulzky, C.
SIGSAM Bulletin: Communications in Computer Algebra 34(4): 16--32 (2000) ; ISSN: 0163-5824

When the results of certain computer calculations are shown to be not simply incorrect but {em dramatically} incorrect, we have a powerful reason to be cautious about {em 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, J. M. and Blaudeck, P. and Hoffmann, K. H.
Thermodynamics of Energy Conversion and Transport : 173--198 Springer, Berlin, 2000 ; ISBN: 0-387-98938-2

Extreme performance of heat exchangers of various hydrodynamic models of flows,
Amelkin, S. A. and Hoffmann, K. H. and Sicre, B. and Tsirlin, A. M.
Periodica Polytechnica Series Chemnical Engineering 44(1): 3--16 (2000) ; ISSN: 1587-3765 (online), ISSN: 0324-5853 (paper)

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, K. H.
Applied Mathematics Letters 12(5): 131--135 (1999) ; PII: 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.

Random Walks in Complex Systems -- Anomalous Relaxation
Schubert, S.
PhD thesis, TU Chemnitz, 1999

Goal of this work is the examination of the dynamics of complex systems. A central role play random walks, which are used to simulate anomalous relaxation in such systems. The complexity of the systems examined in this work is highly reflected in the state-space structure. After an introduction to different complex systems the algorithms are explained that play an important role in the exploration of the sometimes huge state-spaces. For the examination of the low-energy part of complex state-spaces a branch-and-bound algorithm is used. The simulation of dynamics is done by simulating random-walk processes and a statistical description by the master equation. A detailed description of different forms of the master equation and their solution is given. Important applications are simulations of random walks on fractals or hierarchical tree structures. Such simulations allow the comparison of experimental findings, e.g. for anomalous diffusion or non-equilibrium phenomena in spin-glasses. With such a modelling experimental results can be reproduced and understood. Another important contribution to the understanding of such processes is done by a newly developed algorithm for the coarse-graining of state-spaces.

Two physically motivated algorithms for combinatorial optimization: thermal cycling and iterative partial transcription
Möbius, A. and Diaz-Sanchez, A. and Freisleben, B. and Schreiber, M. and Fachat, A. and Hoffmann, K. H. and Merz, P. and Neklioudov, A.
Computer Physics Communications 121--122(1--3): 34--46 (1999) ; PII: 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, K. H.
Computer Physics Communications 121-122(1-3): 30-33 (1999) ; PII: 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, R. E. and Blaudeck, P. and Hoffmann, K. H. and Berry, S.
Journal of Chemical Physics 108(6): 2576--2582 (1998) ; PII: S0021-9606(98)50406-0

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, T. and Schubert, S. and Hoffmann, K. H.
The European Physical Journal B 2(3): 313--317 (1998) ; ISSN: 1434-6028

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.

Quantitative analysis of the state-space structure in a short-range Ising spin glass
Klotz, T. and Schubert, S. and Hoffmann, K. H.
Revista Mexicana de Fisica 44(S1): 81--84 (1998) ; ISSN: 0035-001X

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, T. and Schubert, S. and Hoffmann, K. H.
Journal of Physics: Condensed Matter 10(27): 6127-6134 (1998) ; ISSN: 0953-8984/98/276127+08

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, K. H. and Essex, C. and Schulzky, C.
Journal of Non-Equilibrium Thermodynamics 23(2): 166-175 (1998) ; ISSN: 0340-0204

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, A.
Nonlinearity 11(4): 1063-1074 (1998) ; PII: S0951-7715(98)87431-8

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
Fachat, André and Hoffmann, Karl Heinz
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, P. and Hoffmann, K. H.
Physica A: Statistical Mechanics and its Applications 234(3--4): 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, P. and Pedersen, J. M. and Sibani, P. and Hoffmann, K. H.
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, A. and Neklioudov, A. and Diaz-Sanchez, A. and Hoffmann, K. H. and Fachat, A. and Schreiber, M.
Physical Review Letters 79(22): 4297--4301 (1997) ; ISSN: 0031-9007/97/79(22)/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, K. H. and Burzler, J. M. and Schubert, S.
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, S. and Sibani, P.
Europhysics Letters 38(8): 613--618 (1997) ; ISSN: 0295-5075

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, A. and Hoffmann, K. H.
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, K.H.
Computational Physics : 44--63 Springer-Verlag, Berlin, Heidelberg, New-York, 1996

Computational Physics
Hoffmann, K.H. and Schreiber, M.
Springer-Verlag, Berlin, Heidelberg, New-York, 1996

Relaxation in Self Similar Hierarchies
Uhlig, C. and Hoffmann, K. H. and Sibani, P.
Zeitschrift fÃ¼r Physik B: Condensed Matter 96: 409--416 (1995)

Scaling features in complex optimization problems
R. Tafelmayer and K.H. Hoffmann
Computer Physics Communications 86: 81-90 (1995)

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, K. and Hoffmann, K. H. and Salamon, P.
Journal of Applied Physics 77(11): 5501--5508 (1995)

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, P. and Hoffmann, K. H.
Efficiency, Costs, Optimization and Environmental Impact of Energy Systems 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, K. H.
Periodica Polytechnica 2: 15 (1994) ; ISSN: 1216-0563

Optimizing Irreversible Thermodynamic Processes
Hoffmann, K. H.
Statistical physics and thermodynamics of nonlinear nonequilibrium systems : 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, K. H. and Meintrup, T. and Uhlig, C. and Sibani, P.
Europhysics Letters 22(8): 565--570 (1993)

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
Michael Christoph and Karl Heinz Hoffmann
Journal of Physics A: Mathematical and General 26(13): 3267-3277 (1993)

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
Claas de Groot and Diethelm W ü}rtz and Karl Heinz Hoffmann
Optimization 23: 369--384 (1992) ; ISSN: 0323-3898, ISSN: 0233-1934

Relaxation in Complex Systems: Local Minima and their Exponents
Sibani, P. and Hoffmann, K. H.
Europhysics Letters 16(5): 423 (1991)

Aging Phenomena in Complex Systems: A Hierarchical Model for Temperature Step Experiments
Schulze, C. and Hoffmann, K. H. and Sibani, P.
Europhysics Letters 15(3): 361--366 (1991)

Simulated Annealing for Single Minimum Optimization Problems
Karl Heinz Hoffmann and Peter Salamon
International Journal of Computer Mathematics 39: 193-204 (1991)

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
Karl Heinz Hoffmann and Michael Christoph and Martin Hanf
Parallel Problem Solving from Nature : 221--225 Springer-Verlag, Berlin, 1991 ; ISBN: 3-540-54148-9, ISBN: 0-387-54148-9

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, K. H. and Würtz, D. and de Groot, C. and Hanf, M.
Parallel and Distributed Optimization : 154--175 , 1991

Optimizing Complex Problems by Nature's Algorithms: Simulated Annealing and Evolution Strategy -- a Comparative Study
Claas de Groot and Diethelm Würtz and Karl Heinz Hoffmann
Parallel Problem Solving from Nature : 445-454 Springer-Verlag, Berlin, , 1991

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
Paolo Sibani and Jacob March Pedersen and Karl Heinz Hoffmann and Peter Salamon
Physical Review A 42(12): 7080-7086 (1990)

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, K. H. and Sibani, P.
Zeitschrift fÃ¼r Physik B: Condensed Matter 80: 429--438 (1990)

Optimal Ensemble Size for Parallel Implementations of Simulated Annealing
K.H. Hoffmann and P. Sibani and J.M. Pedersen and P. Salamon
Applied Mathematics Letters 3(3): 53-56 (1990)

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
K.H. Hoffmann and P. Salamon
Journal of Physics A: Mathematical and General 23: 3511--3523 (1990)

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 : 22 Taylor and Francis, New York, 1990

Implementation of a New Adaptive Simulated Annealing Schedule on a Multi Transputer System
M. Hanf and Y. Lehareinger and D. W ü}rtz and K.H. Hoffmann and C. de Groot and M. Anliker
IPS, ETH Z{ü}rich, CH-8092 Z{ü}rich, Switzerland; Technical Report 90-13, 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
Claas de Groot and Diethelm W ü}rtz and Karl Heinz Hoffmann
Helvetica Physica Acta 63: 843 (1990)

Publications in 1985 - 1989

Optimal Paths for a Bimolecular, Light-Driven Engine
S. J. Watowich and K. H. Hoffmann and R. S. Berry
Il Nuovo Cimento B 104(2): 131--147 (1989) ; ISSN: 1594-9982, ISSN: 0369-3554, ISSN: 0369-4100

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 $ m 2;SO_3Fleftrightarrow S_2O_6F_2$. 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, P. and Hoffmann, K. H.
Physical Review Letters 63(26): 2853-2856 (1989) ; ISSN: 1079-7114

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
Karl Heinz Hoffmann and Bjarne Andresen and P. Salamon
Physical Review A 39: 3618-3621 (1989)

The availability loss $-Delta A^u$ 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
K. H. Hoffmann and P. Sibani
Physical Review A 38(8): 4261-4270 (1988)

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, K. H.
Habilitationsschrift, Ruprecht-Karls-Universität, 1988

On lumped models for thermodynamic properties of simulated annealing problems
B. Andresen and K.H. Hoffmann and K. Mosegaard and J. Nulton and J.M. Pedersen and P. Salamon
Journal de Physique49: 1485--1492 (1988)

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
P. Sibani and K.H. Hoffmann
Europhysics Letters 4(9): 967-972 (1987)

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, K. H.
Solid State Communications 62(3): 135--139 (1987) ; ISSN: 0038-1098

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 $sigma_{xx}$ 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
Karl Heinz Hoffmann and Natalia Meshkov
Renewable Energy 9: 259--273 (1987) ; ISSN: 0165-0572

Lower Bounds on Dissipation in Composite Systems
Karl Heinz Hoffmann and Peter Salamon
Physical Review A 35: 369--373 (1987)

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
Karl Heinz Hoffmann and B. Neudecker
Zeitschrift fÃ¼r Physik B: Condensed Matter 62: 279--285 (1986) ; ISSN: 0722-3277, ISSN: 1431-584X

Intrinsically Irreversible Light-Driven Engine
Watowich, S. J. and Hoffmann, K. H. and Berry, S. R.
Journal of Applied Physics 58: 2893-2901 (1985)

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, K. H. and Grossmann, S. and Wegner, F.
Zeitschrift fÃ¼r Physik B: Condensed Matter 60: 401-414 (1985)

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

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, S. and Wegner, F. and Hoffmann, K. H.
Journal de Physique Lettres 46(13): L575-L583 (1985)

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 $sigma propto t^Theta$, $Theta$ larger than 2, is obtained as observed in turbulence.

Publications in 1980 - 1984

Die Hopfbifurkation unter dem Einfluß von weißem Rauschen
Karl Heinz Hoffmann
Dissertation, RWTH Aachen, 1982

The Hopf Bifurcation of Twodimensional Systems under the Influence of One External Noise Source
Karl Heinz Hoffmann
Zeitschrift fÃ¼r Physik B: Condensed Matter 49: 245--252 (1982) ; ISSN: 0722-3277, ISSN: 1431-584X

The Birkhoff Normalization Procedure and the Reductive Perturbation Approach: Two equivalent Methods to Discuss the Hopf Bifurcation
Hoffmann, K. H.
Physics Letters A 92(4): 163--164 (1982) ; ISSN: 0375-9601

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
Karl Heinz Hoffmann
Diplomarbeit, RWTH Aachen, 1979

K-Isomorphie und K-Äquivalenz von Ordnungen
Karl Heinz Hoffmann
Diplomarbeit, RWTH Aachen, 1978

Standard

Chemnitz UT: Natural Sciences: Physics: Computational Physics: Publications