# Generalized Gibbs Ensembles of the classical Toda chain

## TL;DR

This paper derives the generalized free energy of the classical Toda chain using matrix models, linking it to log-gas systems, and discusses the evolution of local density of states via generalized hydrodynamics.

## Contribution

It introduces a novel approach to compute the free energy of the Toda chain through the Dumitriu-Edelman matrix model, connecting integrable systems with log-gas models.

## Key findings

- Derived the generalized free energy of the Toda chain.
- Mapped the Toda chain to a one-dimensional log-gas system.
- Identified the local density of states as an object evolving under generalized hydrodynamics.

## Abstract

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. In the current version missing factors of 2 are corrected.