Exact Thermal Distributions in Integrable Classical and Quantum Gases

We consider one-dimensional, integrable many-body classical and quantum systems in thermal equilibrium. In the classical case, we use the classical limit of the Bethe equations to obtain a self-consistent integral equation whose solution gives the distribution of asymptotic Bethe momenta, or rapidities, as well as the classical partition function in the canonical ensemble, and the thermal energy dispersion. For quantum gases, we obtain a similar integral equation, albeit in the grand canonical ensemble, with completely analogous results. We apply our theory to the classical and quantum Tonks and Calogero-Sutherland models, and our results are in perfect agreement with standard calculations using Yang-Yang thermodynamics. Remarkably, we show in a straightforward manner that the thermodynamics of the quantum Calogero-Sutherland model is in one-to-one correspondence with the ideal Fermi gas upon simple rescalings of chemical potential and density.