Time--scale for energy equipartition in a two-dimensional FPU model

The FPU problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in dimension two, more precisely in a model with triangular cell and nearest-neighbors Lennard-Jones interaction. The number n of degrees of freedom ranges from 182 to 6338. Energy is initially equidistributed among a small number n(0) of low frequency modes, with n(0) proportional to n. We study numerically the time evolution of the so-called spectral entropy and the related "effective number" n(eff) of degrees of freedom involved in the dynamics; in this (rather typical) way we can estimate, for each n and each specific energy (energy per degree of freedom) epsilon, the time scale T-n(epsilon) for energy equipartition. Numerical results indicate that in the thermodynamic limit the equipartition times are short: more precisely, for large n at fixed epsilon we find a limit curve T-infinity(epsilon), and T-infinity grows only as epsilon(-1) for small epsilon. Larger equipartition times are obtained by lowering epsilon, at fixed n, below a crossover value epsilon(c)(n). However, epsilon(c) appears to vanish by increasing n (faster than 1/n), and the total energy E=n epsilon, rather than epsilon, appears to be the relevant variable when n is large and epsilon <epsilon(c). In conclusion, it seems that in the thermodynamic limit, for this model and this kind of initial conditions, the FPU phenomenon, namely the lack of energy equipartition in physically reasonable times, practically disappears.