On the statistical description of classical open systems with integer variables by the Lindblad equation

TL;DR

This paper introduces a statistical approach inspired by quantum open systems theory to analyze classical systems with integer variables, capturing effects of discreteness and fluctuations often ignored by mean field methods.

Contribution

It develops a novel method based on quantum theory concepts to compute distributions and moments of classical systems with integer states, accounting for discreteness and fluctuations.

Findings

Method matches dynamical results for large occupation numbers.

Reveals new effects and differences at small occupation numbers.

Applicable to models in physics, biology, and economics.

Abstract

We propose the consistent statistical approach to consider a wide class of classical open systems whose states are specified by a set of positive integers(occupation numbers).Such systems are often encountered in physics, chemistry, ecology, economics and other sciences.Our statistical method based on ideas of quantum theory of open systems takes into account both discreteness of the system variables and their time fluctuations - two effects which are ignored in usual mean field dynamical approach.The method let one to calculate the distribution function and (or)all moments of the system of interest at any instant.As descriptive examples illustrating the effectiveness of the method we consider some simple models:one relating to nonlinear mechanics,and others taken from population biology .In all this examples the results obtained by the method for large occupation numbers coincide with results of purely dynamical approach but for small numbers interesting differences and new effects arise.The possible observable effects connected with discreteness and fluctuations in such systems are discussed.