A study of the Fermi-Pasta-Ulam problem in dimension two

Continuing the previous work on the same subject, we study here different two-dimensional Fermi-Pasta-Ulam (FPU)-like models, namely, planar models with a triangular cell, molecular-type potential and different boundary conditions, and perform on them both traditional FPU-like numerical experiments, i.e., experiments in which energy is initially concentrated on a small subset of normal modes, and other experiments, in which we test the time scale for the decay of a large fluctuation when all modes are excited almost to the same extent. For each experiment, we observe the behavior of the different two-dimensional systems and also make an accurate comparison with the behavior of a one-dimensional model with an identical potential. We assume the thermodynamic point of view and try to understand the behavior of the system for large n (the number of degrees of freedom) at fixed specific energy epsilon=En. As a result, it turns out that: (i) The difference between dimension one and two, if n is large, is substantial. In particular (making reference to FPU-like initial conditions) the "one-dimensional scenario," in which the dynamics is dominated for a long time scale by a weakly chaotic metastable situation, in dimension two is absent; moreover in dimension two, for large n, the time scale for energy sharing among normal modes is drastically shorter than in dimension one. (ii) The boundary conditions in dimension two play a relevant role. Indeed, models with fixed or open boundary conditions give fast equipartition, on a rather short time scale of order epsilon(-1), while a periodic model gives longer equilibrium times (although much shorter than in dimension one).