Statistical Mechanics of an Integrable System

We provide here an explicit example of Khinchin's idea that the validity equilibrium statistical mechanics in high dimensional systems does not depend on details dynamics. This point view is supported by extensive numerical simulation one-dimensional Toda chain, integrable non-linear Hamiltonian system where all Lyapunov exponents are zero definition. study relaxation to starting from very atypical initial conditions and focusing energy equipartion among Fourier modes, as done original Fermi-Pasta-Ulam-Tsingou experiment. find evidence general case, i.e., perturbative regime modes close each other, there a fast reaching thermal terms single temperature. also fluctuations, particular behaviour specific heat function temperature, agreement with analytic predictions drawn ordinary Gibbs ensemble, still having no conflict established Generalized Ensemble for model. Our results suggest thus even reaches thermalization constant hypersurface, provided considered observables do strongly one or few conserved quantities. suggests dynamical chaos irrelevant large-$N$ limit, any macroscopic observable reads collective variable respect coordinate which diagonalize Hamiltonian. The possibility our be relevant problem generic quantum systems, non-integrable ones, commented at end.