Algorithmic complexity and statistical mechanics

Abstract

We apply the algorithmic complexity theory to statistical mechanics; in particular, we consider the maximum entropy principle and the entropy concentration theorem for non-ordered data in a non-probabilistic setting. The main goal of this paper is to deduce asymptotic relations for the frequencies of energy levels in a non-ordered collection ω^N = [ω₁, ..., ω_N] from the assumption of maximality of the Kolmogorov complexity K(ω^N) given a constraint , where E is a number and f is a numerical function; f(ω_i) is an energy level. We also consider a combinatorial model of the securities market and give some applications of the entropy concentration theorem to finance.