Yang-Lee zeroes for an urn model for the separation of sand

TL;DR

This paper demonstrates that the Yang-Lee theory of phase transitions applies to a nonequilibrium urn model for sand separation, showing that zeros of the partition function indicate phase transition points.

Contribution

It extends the application of Yang-Lee theory to nonequilibrium systems by analyzing the zeros of an effective partition function in a sand separation model.

Findings

Zeros of the partition function lie on the unit circle in the thermodynamic limit.

Certain roots approach the transition point in the complex control parameter plane.

Yang-Lee theory is applicable to a broader class of nonequilibrium systems.

Abstract

We apply the Yang-Lee theory of phase transitions to an urn model of separation of sand. The effective partition function of this nonequilibrium system can be expressed as a polynomial of the size-dependent effective fugacity $z$. Numerical calculations show that in the thermodynamic limit, the zeros of the effective partition function are located on the unit circle in the complex $z$-plane. In the complex plane of the actual control parameter certain roots converge to the transition point of the model. Thus the Yang-Lee theory can be applied to a wider class of nonequilibrium systems than those considered previously.