Spectral solution of urn models for interacting particle systems

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.