  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.
