Spectral solution of urn models for interacting particle systems

TL;DR

This paper classifies 2-Urn models for interacting particle systems using spectral methods, identifying solvable subclasses and analyzing their stationary distributions and entropy behavior.

Contribution

It provides a complete classification of solvable 2-Urn models, distinguishing Martingale and non-Martingale types, and links symmetry to entropy changes.

Findings

Non-Martingale models have Gaussian stationary distributions.

Symmetry conditions relate to entropy increase in the models.

Some social opinion models do not exhibit entropy increase.

Abstract

Using generating function methods for diagonalizing the transition matrix in 2-Urn models, we provide a complete classification into solvable and unsolvable subclasses, with further division of the solvable models into the Martingale and non-Martingale subcategories, and prove that the stationary distribution is a Gaussian function in the latter. We also give a natural condition related to the symmetry of the random walk in which the non-Martingale Urn models lead to an increase in entropy from Gaussian states. The condition also shows that universal symmetry in the macro-state is equivalent to increasing entropy. Certain models of social opinion dynamics, treated as Urn models, do not increase in entropy, unlike isolated mechanical systems.