How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems

The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there exists an ongoing controversy whether the notion of the maximum entropy principle can be extended in a meaningful way to non-extensive, non-ergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the (c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept for non-ergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.Comment: 6 pages, 1 figure. To appear in PNA

Topology without cooling: instantons and monopoles near to deconfinement

In an attempt to describe the change of topological structure of pure SU(2) gauge theory near deconfinement a renormalization group inspired method is tested. Instead of cooling, blocking and subsequent inverse blocking is applied to Monte Carlo configurations to capture topological features at a well-defined scale. We check that this procedure largely conserves long range physics like string tension. UV fluctuations and lattice artefacts are removed which otherwise spoil topological charge density and Abelian monopole currents. We report the behaviour of topological susceptibility and monopole current densities across the deconfinement transition and relate the two faces of topology to each other. First results of a cluster analysis are described.Comment: 6 pages, 8 figures, LaTeX with espcrc2.sty. Talk and poster presented at Lattice97, Edinburgh, 22-26 July 1997, to appear in Nucl. Phys. B (Proc.Suppl.

Towards a Topological Mechanism of Quark Confinement

We report on new analyses of the topological and chiral vacuum structure of four-dimensional QCD on the lattice. Correlation functions as well as visualization of monopole currents in the maximally Abelian gauge emphasize their topological origin and gauge invariant characterization. The (anti)selfdual character of strong vacuum fluctuations is reveiled by smoothing. In full QCD, (anti)instanton positions are also centers of the local chiral condensate and quark charge density. Most results turn out generically independent of the action and the cooling/smoothing method.Comment: 14 pages, Contribution to YKIS9

Scaling-violation phenomena and fractality in the human posture control systems

By analyzing the movements of quiet standing persons by means of wavelet statistics, we observe multiple scaling regions in the underlying body dynamics. The use of the wavelet-variance function opens the possibility to relate scaling violations to different modes of posture control. We show that scaling behavior becomes close to perfect, when correctional movements are dominated by the vestibular system.Comment: 12 pages, 4 figures, to appear in Phys. Rev.

Coexistence of monopoles and instantons for different topological charge definitions and lattice actions

We compute instanton sizes and study correlation functions between instantons and monopoles in maximum abelian projection within SU(2) lattice QCD at finite temperature. We compare several definitions of the topological charge, different lattice actions and methods of reducing quantum fluctuations. The average instanton size turns out to be $\sigma \approx 0.2$ fm. The correlation length between monopoles and instantons is $\zeta \approx 0.25$ fm and hardly affected by lattice artifacts as dislocations. We visualize several specific gauge field configurations and show directly that there is an enhanced probability for finding monopole loops in the vicinity of instantons. This feature is independent of the topological charge definition used.Comment: 10 pages, LaTeX, uses elsart.sty and elsart12.sty, 16 eps files, 4 figures, published, for corresponding movies (MPEG) see http://www.tuwien.ac.at/e142/Lat/qcd.htm