# Kinetic theory of quantum and classical Toda lattices

## TL;DR

This paper develops a kinetic theory for quantum and classical Toda lattices, deriving a Bethe-Boltzmann type equation, analyzing quantum corrections, and validating results with numerical simulations.

## Contribution

It introduces a unified kinetic framework for quantum and classical Toda lattices and connects it with inverse scattering methods, highlighting quantum corrections.

## Key findings

- Classical kinetic theory agrees with numerical simulations.
- Quantum corrections depend smoothly on Planck's constant.
- The approach links Bethe-Boltzmann equations with inverse scattering spectrum.

## Abstract

We consider the kinetic theory of the quantum and classical Toda lattice models. A kinetic equation of Bethe-Boltzmann type is derived for the distribution function of conserved quasiparticles. Near the classical limit, we show that the kinetic theory depends smoothly on Planck's constant, and explicitly characterise the leading quantum corrections to classical behaviour. The classical kinetic theory is compared with direct numerical simulations and shows excellent agreement. Finally, we connect the Bethe-Boltzmann approach with the classical inverse scattering method, by identifying conserved quasiparticles with the spectrum of the Lax matrix.