Rainer Feistel

Diffusion and reaction in random media and models of evolution processes

A diffusion equation including source terms, representing randomly distributed sources and sinks is considered. For quasilinear growth rates the eigenvalue problem is equivalent to that of the quantum mechanical motion of electrons in random fields. Correspondingly there exist localized and extended density distributions dependent on the statistics of the random field and on the dimension of the space. Besides applications in physics (nonequilibrium processes in pumped disordered solid materials) a new evolution model is discussed which considers evolution as hill climbing in a random landscape.

Models of Darwinian processes and evolutionary principles

Evolutionary processes are described as stochastic motions in a genotype space (set of sequences with a Hamming distance) and a phenotype space (vector space of phenotypic properties). Real value functions are introduced which form a landscape over these spaces; smoothness postulates are formulated. Evolution is considered as a kind of hill climbing on these adaptive landscapes. A rather simple diffusion approximation for the phenotypic processes is proposed which leads to similar mathematical problems as the Schrödinger equation for disordered potential distributions.

Entropy and the Self-Organization of Information and Value

Adam Smith, Charles Darwin, Rudolf Clausius, and Leon Brillouin considered certain “values” as key quantities in their descriptions of market competition, natural selection, thermodynamic processes, and information exchange, respectively. None of those values can be computed from elementary properties of the particular object they are attributed to, but rather values represent emergent, irreducible properties. In this paper, such values are jointly understood as information values in certain contexts. For this aim, structural information is distinguished from symbolic information. While the first can be associated with arbitrary physical processes or structures, the latter requires conventions which govern encoding and decoding of the symbols which form a message. As a value of energy, Clausius’ entropy is a universal measure of the structural information contained in a thermodynamic system. The structural information of a message, in contrast to its meaning, can be evaluated by Shannon’s entropy of communication. Symbolic information is found only in the realm of life, such as in animal behavior, human sociology, science, or technology, and is often cooperatively valuated by competition. Ritualization is described here as a universal scenario for the self-organization of symbols by which symbolic information emerges from structural information in the course of evolution processes. Emergent symbolic information exhibits the novel fundamental code symmetry which prevents the meaning of a message from being reducible to the physical structure of its carrier. While symbols turn arbitrary during the ritualization transition, their structures preserve information about their evolution history.

Dynamics of Interfaces in Random Media

One of the essential consequences of non-linear equations of motion is the possibility of several stable stationary states /1/. As a result, for a d-dimension system many configurations exist, where regions in which the system is in one of its stable states are separated by thin transition layers. As the non-linearity increases, the widths of these layers decrease, so that they can be described as (d-1)-dimensional hypersurfaces. The time evolution of the system is then determined by the dynamics of these phase-separating interfaces. Situations of this type arise in a variety of physical systems. Well-known examples are equilibrium phase transitions of first order as, e.g., liquid-vapour systems/2/. Of great practical importance are furthermore systems quenched into the phase region of distinct multistability /3–6/. After a very quick local relaxation process, a complicated pattern of domains separated by interfaces emerge. These domains coarsen with time, as can be verified experimentally by scattering techniques. Since the coarsening process is determined by the dynamics of the existing interfaces, comparison between theory and experiment becomes possible /5/. Of interest for practical purpose are in particular metallurgical systems as, e.g., binary alloys /3/.