Analytic study of the urn model for separation of sand

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 (A. Lipowski and M. Droz, e-print cond-mat/0201472).