Analytic study of the urn model for separation of sand

TL;DR

This paper provides an analytical solution to the urn model for sand separation, confirming previous numerical findings and revealing the governing free energy and critical distribution behavior.

Contribution

It analytically solves the master equation and first-passage problem for the urn model, advancing understanding of its stationary distribution and critical phenomena.

Findings

Analytical solution of the master equation and first-passage problem.

Stationary distribution and characteristic times governed by the same free energy.

Derived the form of the critical probability distribution on the critical line.

Abstract

We present an analytic study of the urn model for separation of sand recently introduced by Lipowski and Droz (Phys. Rev. E 65, 031307 (2002)). We solve analytically the master equation and the first-passage problem. The analytic results confirm the numerical results obtained by Lipowski and Droz. We find that the stationary probability distribution and the shortest one among the characteristic times are governed by the same free energy. We also analytically derive the form of the critical probability distribution on the critical line, which supports their results obtained by numerically calculating Binder cumulants (cond-mat/0201472).