Herbert Spohn

Generalised Gibbs Measures for the Classical Toda Lattice

According to statistical mechanics, Gibbs measures for a many-particle
system are constructed from number, momentum, and energy, which are
believed to be generically the only locally conserved fields. The
classical Toda lattice is an integrable system and thus possesses a
countable list of local conservation laws. Accordingly Gibbs measures
of the Toda chain are indexed by an infinite set of parameters, rather
than only three. This is the meaning of “generalised" in the title.
Specifically for the Toda lattice, we will discuss the structure of
generalised Gibbs measures and point out the connection to the
one-dimensional log gas. This information is a central building block
when writing down the appropriate Euler type equations for the Toda
lattice.