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

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Abstract
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 n0 of low frequency modes, with n0 proportional to n. We study numerically the time evolution of the so-called spectral entropy and the related “effective number” neff 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) ε, the time scale Tn(ε) for energy equipartition. Numerical results indicate that in the thermodynamic limit the equipartition times are short: more precisely, for large n at fixed ε we find a limit curve T∞(ε), and T∞ grows only as ε−1 for small ε. Larger equipartition times are obtained by lowering ε, at fixed n, below a crossover value εc(n). However, εc appears to vanish by increasing n (faster than 1∕n), and the total energy E=nε, rather than ε, appears to be the relevant variable when n is large and ε<εc. 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.