Analytic study of the three-urn model for separation of sand

TL;DR

This paper provides an analytical investigation of the three-urn sand separation model, solving key equations and revealing how the system's stability and cluster lifetime behave near critical points.

Contribution

It analytically solves the master and first-passage equations for the model, elucidating the role of free energy and stability in sand separation dynamics.

Findings

Stationary distribution obeys detailed balance and is governed by free energy.

Cluster lifetime diverges algebraically with exponent 1/3 at stability limit.

Analytic solutions enhance understanding of sand separation processes.

Abstract

We present an analytic study of the three-urn model for separation of sand. We solve analytically the master equation and the first-passage problem. We find that the stationary probability distribution obeys the detailed balance and is governed by the {\it free energy}. We find that the characteristic lifetime of a cluster diverges algebraically with exponent 1/3 at the limit of stability.