Alex Little

Large deviations of the periodic Toda chain

The Toda chain was introduced by M. Toda in 1967 as a discrete analogue of the KdV equation and as a model of a nonlinear crystal. It is a classical integrable system that exhibits soliton-like solutions. Being a strongly interacting system, its statistical mechanics are very interesting. This means imposing a Gibbs-Boltzmann distribution on the system, or, because of the extensive number of conserved quantities, a Generalised Gibbs Distribution. In joint work with T. Grava, A. Guionnet, A. Its and K. Kozlowski, we prove a Large Deviation Principle for the periodic Toda chain subject to a Generalised Gibbs Ensemble. This quantifies the probability of observing the system far from its equilibrium point. This yields information about the free energy and entropy of the system.