Statistical Mechanics of an Integrable System

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