Exact Solutions for Small Systems: Urns Models

TL;DR

This paper develops an exact analytical method for solving urn models in small systems by expanding in moments, providing precise expressions for mean and variance, and applies it to Bernoulli-Laplace and Ehrenfest urns.

Contribution

It introduces a moment expansion approach as an alternative to traditional methods for deriving exact solutions in small stochastic systems.

Findings

Derived exact mean and variance expressions for urn models.

Applied method successfully to Bernoulli-Laplace and Ehrenfest urns.

Enhanced understanding of stochastic behavior in small systems.

Abstract

In this study, we analyzed urn models by solving the discrete-time master equation using an expansion in moments. This approach is a viable alternative to conventional methods, such as system-size expansion, allowing for the determination of analytical expressions for the mean and variance in an exact form and thus valid for any system size. In particular, this approach was used to study Bernoulli-Laplace and Ehrenfest urns, for which analytic expressions describing their evolution were found. This approach and the results will contribute to a more comprehensive understanding of stochastic systems and statistical physics for small-sized systems, where the thermodynamic limit cannot be assumed.