Hydrodynamics of the classical Toda chain and random matrix theory
A talk in the Summer School Randomness in Physics & Mathematics series by Herbert Spohn
There has been a lot of interest, mostly on the quantum side, to
develop the hydrodynamics of integrable systems, thus involving an innite number
of conservation laws. Surprisingly a model-independent structure is claimed, except
for the specic two-particle phase shift. I will use the classical Toda chain as a road
map for the general picture. The Toda chain has a tridiagonal Lax matrix, which
under the generalised Gibbs ensemble becomes random. The corresponding density
of states can be determined through a tricky use of the Dumitriu-Edelman theorem.
Also a connection to Dyson's Brownian motion will be discussed.