Speaker: Angelo Vulpiani
Title: Equilibrium and thermalization in integrable systems:
the case of Toda model and harmonic systems
Abstract:
We provide explicit examples 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.
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.
A rather similar result is obtained for harmonic chains (trivially integrable systems): studying the relaxation from an atypical
condition given with respect to "random" modes, we observe that a thermal state with equilibrium properties is attained in short times.
Such a result is independent from the orthonormal base used to represent the chain state, provided it is random.
These results  suggest 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.