Luigi Galgani

We give a review of the Fermi–Pasta–Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light.

A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi–Pasta–Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence.