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Structure and Dynamics of Complex Systems

Many physical systems are characterized by their relaxation behavior. For such systems the concept of energy “landscapes” leads to a unified understanding of variety of different phenomena. All these systems are characterized by an energy function which possesses many local minima separated by barriers. An often-used picture for complex state spaces is that of a mountainous landscape, where the heights of the mountains represent the energy and the two horizontal axis have to mimic two of the many dimensions of the physical system. Typical examples of such complex systems are spin glasses, which show a wealth of interesting relaxation phenomena, cluster of molecules with their thermal relaxation behavior, or proteins with their folding dynamics. In such systems slow relaxation dynamics is intimately connected to metastable states, which are due to the many local minima in their energy function. An escape from those minima over the surrounding barriers can take a very long time. Such systems do not reach their thermal equilibrium easily. Studying the dynamics of complex systems is confronted with a major problem: the enormous number of states. This number can be reduced considerably by coarse graining the state space. Often the resulting structure has a tree topology. The result is that Marcov processes on tree structures are good modeling tools for the thermal relaxation of complex systems.

While in the systems described above the relaxation behavior is due to the crossing of energy barriers, there are also entropic and kinetic effects due to the state space topology which as well lead to a slowed down relaxation. Such topological barriers also exist in the anomalous diffusion observed for instance in porous media. Here the spatial features of the pores have a decisive effect on the particle movement. A good model system to study the observed features are fractal structures which capture the self-similarity of natural materials as well as of other complex state spaces.

Recent Publications

Tsallis Relative Entropy and Anomalous Diffusion
Prehl, Janett and Essex, Christopher and Hoffmann, Karl Heinz
Entropy 14: 701--716 (2012) ; ISSN:1099-4300

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

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): 215--224 (2010)

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

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 Rényi 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

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.

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.

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.

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)

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.

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

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.

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