Generalized Gibbs Ensembles of the Classical Toda Chain

The Toda chain is the prime example of a classical integrable system with strictly local conservation laws. Relying on the Dumitriu–Edelman matrix model, we obtain the generalized free energy of the Toda chain and thereby establish a mapping to the one-dimensional log-gas with an interaction strength of order 1 / N. The (deterministic) local density of states of the Lax matrix is identified as the object, which should evolve according to generalized hydrodynamics.