Exact solutions for small systems: urns models

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 its 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.