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 1n), and the total energy E=nepsilon, 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.
Benettin
Giancarlo
G
Università di Padua, Dipartimento di Matematica Pura e Applicata, Via G. Belzoni 7, 35131 Padua, Italy. benettin@math.unipd.it
D000465:Algorithms; D015233:Models, Statistical; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D010825:Physics / Q000379:methods*; D013816:Thermodynamics; D013997:Time Factors