Rigorous approximation of diffusion coefficients for expanding maps

We use Ulam's method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps. An algorithm is provided and its implementation is illustrated using Lanford's map.Comment: In this version Lanford's map has been used to illustrate the computer implementation of the algorithm. To appear in Journal of Statistical Physic

Annealed and quenched limit theorems for random expanding dynamical systems

In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erd\"os-R\'enyi laws and concentration inequalities.Comment: Appeared online in Probability Theory and Related Field

Quantum mechanics as an approximation of statistical mechanics for classical fields

We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties, we start with the finite dimensional quantum mechanics. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within the quantum formalism from experimental averages (which are supposed to be equal to classical averages given by our model).Comment: Talks at the conferences: "Quantum Theory: Reconsideration of Foundations-3", Vaxjo, Sweden, June-2005; "Processes in Physics", Askloster, Sweden, June-2005; "The nature of light: What is photon?", San-Diego, August-2005; "Nonlinear Physics. Theory and Experiment", Lece, Italy, July-200