  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. 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 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, still having no conflict with the established
validity of the Generalized Gibbs Ensemble for the Toda model. 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.
