Thermodynamic Bethe Ansatz for the Toda chain from inverse scattering transform

Abstract

The Thermodynamic Bethe Ansatz (TBA) procedure developed by Yang-Yang provides the free energy functional to be minimized via a variational problem. Their result not only gives the correct thermodynamics, but it also sets the framework for Generalized Hydrodynamics. Originally developed for quantum systems, this procedure has been recently successfully applied to (integrable) classical particle systems. However, this has never been done for classical field theories: no classical derivation has consistently accounted for solitons, breathers, radiation (dispersive modes), and their mutual interactions. We here derive the TBA equations for the Toda chain directly from its inverse scattering transform. This results in solitonic modes interacting with radiation, extending the soliton gas picture. We provide numerical verification of our equations against well established TBA equations in terms of particle modes. We expect the same procedure should apply to KdV, and pure radiative models as defocusing NLS and Sinh-Gordon, while we discuss current limitations in other models such as focusing NLS (and Ablowitz-Ladik) and Sine-Gordon.