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Computational Physics
Computational Physics

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.

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