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

Kinetic theory of quantum and classical Toda lattices