Statistical Mechanics and Systems of Large Numbers of Elements with Random Interaction

Abstract

In this work, the evident analogy between problems of statistical mechanics and large systems with chance interaction is developed into a one-to-one correspondence. As an element of a system, a stochastic automaton described by an ergodic Markov chain is taken, so that we can receive “gas of automata”, “solid state of automata”, etc. The formula linking the physical notions of energy and temperature with the matrix of transient probabilities of Markov chain is given. The limits of implication of this formula are determined by the existence of a “detailed balance” in the ergodic finite Markov chain. Detailed balance by itself gives the opportunity to solve some complicated problems in Markov chain theory which have been unsolved for rather a long time.