Hans Zessin

Ergodic theorems for spatial processes

SummaryWe investigate the ergodic properties of spatial processes, i.e. stochastic processes with an index set of bounded Borel subsets in ℝv, and prove mean and individual ergodic theorems for them. As important consequences we get a generalization of McMillans theorem due to Fritz [4]; the existence of specific energy for a large class of interactions in the case of marked point processes in ℝv and the existence of the specific Minkowski Quermaßintegrals for Boolean models in ℝv with convex, compact grains.

Microcanonical distributions for lattice gases

In this article, a large deviation principle (cf. Theorem 1.3) for the empirical distribution functional is applied to prove a rather general version of Boltzmanns principle (cf. Theorem 3.5) for models with shift-invariant, finite range potentials. The final section contains an application of these considerations to the two dimensional Ising model at sub-critical temperature.

Large deviations and the maximum entropy principle for marked point random fields

SummaryWe establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ℝd. The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.

Punktprozesse mit Wechselwirkung

We consider classical, continuous systems of particles in r dimensions described by infinite system equilibrium states which have been defined by Dobrushin [5] and Lanford/Ruelle [24]. For a large class of potentials we prove the theorem of Lee/Yang [43] together with a variational characterizafor these equilibrium states. The main idea stems from Föllmer [9] who showed that in the case of lattice systems, the theorem of Lee/Yang is intimately related to Birkhoffs ergodic theorem and McMillans theorem (ergodic theorem of information theory). Following this idea we obtain as main results an r-dimensional ergodic theorem for random measures in ℝ r, limit theorems concerning energy and entropy and an r-dimensional version of Breimans theorem showing that there is almost sure convergence behind McMillans theorem.

The method of moments for random measures

SummaryFirst a general condition is derived which assures uniqueness of a random measure in terms of its moment measures. Then weak convergence of a sequence of random measures is established under the assumptions that the moment measures converge vaguely and the resulting limiting measures satisfy the uniqueness criterium. A partial converse is also given. This result is sharpened to a criterium on weak convergence of certain time evolutions of random measures to a Poisson process. As an application of this result in non-equilibrium statistical mechanics, a theorem of Lanford on the approach to equilibrium in the case of the free motion is derived.

Der Papangelou Prozess

Das Ziel dieser Note ist die Konstruktion von Punktprozessen in einem abstrakten Raum, die spezifiziert sind durch eine gewisse Klasse von Kernen, die für diese Prozesse Papangelou Kerne, d.h. bedingte Intensitäten sind. Wir bezeichnen sie aus diesem Grunde als Papangelousche Prozesse. Diese Klasse von Prozessen enthält viele Gibbssche Prozesse der klassischen statistischen Mechanik sowie neben dem Poissonschen Prozess die neue Klasse der hier so genannten Polyaschen Summenprozesse. Diese haben eine ebenso fundamentale Bedeutung wie Poissonsche Prozesse.

The Papangelou process. A concept for Gibbs, Fermi and Bose processes

This note is a revised and enlarged version of the german article [16] in a slightly different framework. We here correct a serious mistake in the first version and generalize the class of Polya sum processes considered there. (A corrected version of the same results can be found already in the thesis of Mathias Rafler [12].) Moreover, the class of Polya difference processes is constructed here for the first time. In analogy to classical statistical mechanics we propose a theory of interacting Bosons and Fermions. We consider Papangelou processes. These are point processes specified by some kernel which represents the conditional intensity of the process. The main result is a general construction of a large class of such processes which contains Cox, Gibbs processes of classical statistical mechanics, but also interacting Bose and Fermi processes.

Stochastic dynamics for an infinite system of random closed strings: A Gibbsian point of view

We consider the stochastic dynamics of infinitely many, interacting random closed strings, and show that the law of this process can be characterized as a Gibbs state for some Hamiltonian on the path level, which is represented in terms of the interaction. This is done by means of the stochastic calculus of variations, in particular an integration by parts formula in infinite dimensions. This Gibbsian point of view of the stochastic dynamics allows us to characterize the reversible states as the Gibbs states for the underlying interaction. Under supplementary monotonicity conditions, there is only one stationary distribution, and we prove that there is exactly one Gibbs state.

An integral characterization of random permutations. A point process approach

We consider random finite permutations and prove the following version of Thoma’s theorem in [8]: Random finite permutations which are class functions satisfy a new integration by parts formula if and only if they are given by a certain Ewens-Sütö process. The main source of inspiration for the results in this note is the fundamental work of Andras Sütö [7], from which some results are reestablished here again in the present point process approach.

On rigorous Hydrodynamics, Self-diffusion and the Green-Kubo formulae

We describe some thoughts and results revolving around the mathematically rigorous derivation of hydrodynamics from deterministic microscopic mechanical model. We recall first the theoretical physics standard on linearized hydrodynamics. In particular we point to the fact that the Green-Kubo formulae may serve as a starting point for a rigorous theory: One looks upon hydrodynamics only from the asymptotics of certain relevant two point equilibrium correlation functions. These describe fluctuations of the locally conserved hydrodynamic fields. We try first to catch the flavour in simple examples of mathematical nature, in particular we observe the well known fact that self-diffusion is part of the theory. Then we elaborate on the physical example of the hard rods system and show how one may understand the bulk diffusion in that system — of the particle density per velocity — as a phenomenon of self-diffusion of a velocity pulse.