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

We present an analytic study of the three-urn model for separation of sand, which can be regarded as a zero-range process. 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 free energy. We find that the characteristic lifetime of a cluster diverges algebraically with exponent [Formula presented] at the limit of stability. We also give a general argument that the scaling behavior is robust with respect to different expressions of the flux.