Generalised hydrodynamics of the classical Toda system

Abstract

We obtain the exact generalised hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorised scattering of Toda particles, we obtain the generalised free energy and exact average currents, and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta, and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and in particular the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorised scattering of Toda particles.

v1: 36 pages. v2: 37 pages, small adjustments

References in corpus (9)

• Hydrodynamic Diffusion in Integrable Systems
• Diffusion in generalized hydrodynamics and quasiparticle scattering
• Fluctuations in ballistic transport from Euler hydrodynamics
• Transport fluctuations in integrable models out of equilibrium
• Generalized Gibbs Ensembles of the classical Toda chain
• Current fluctuations for anharmonic chains in thermal equilibrium
• Long-Time Asymptotics of the Toda Lattice for Decaying Initial Data Revisited
• Equilibrium dynamical correlations in the Toda chain and other integrable models
• Kinetic theory of quantum and classical Toda lattices

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