Fermi-Pasta-Ulam phenomenon for generic initial data

Abstract

The well known FPU phenomenon (lack of attainment of equipartition of the mode-energies at low energies, for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Montecarlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode-energies, and it is found that the high-frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing obtains within time-scales much longer than the presently available ones. PACS numbers: Valid PACS appear hereBy the "standard" FPU phenomenon we mean the celebrated one observed for the first time in the year 1955 [1]. Namely, in numerical integrations of the equations of motion for a chain of particles coupled by weakly nonlinear springs, equilibrium is not attained within the available computational time, and a kind of anomalous pseudoequilibrium does instead show up. This is observed at low energies, for initial data very far from equilibrium. The quantities studied were the energies E j (t) of the normal modes of the linearized system, and their time-averages E j (t) were found to relax each to a different value rather than to a common one, against the equipartition principle (see especially the last figure of the original FPU report).It was later found by Izrailev and Chirikov [2] that the phenomenon disappears, i.e., energy equipartition is quickly attained, if energy is large enough. A long debate then followed [3,4,5,6,7] concerning the questions (still unanswered) whether the phenomenon persists in the "thermodynamic limit" (i.e., when the number N of particles and the energy E both grow to infinity with a finite value of the specific energy ǫ = E/N ), and whether it can be interpreted in a metastability perspective [8,9]. Another still open problem is whether the phenomenon persists when the dimensions are increased, passing from a chain of particles to a 2-or a 3-dimensional lattice [10].In the present letter we address a further problem, namely whether some analog of the FPU phenomenon occurs if generic initial data are taken rather than some very special ones (see also [11,12,13]). More generally, we would like to look at the problem of the approach to equilibrium from the viewpoint of ergodic theory, in which one considers in principle all initial data, weighted * Electronic address: carati@mat.unimi.it † Electronic address: galgani@mat.unimi.it ‡ Electronic address: giorgilli@mat.unimi.it § Electronic address: paleari@mat.unimi.itwith an invariant measure such as the microcanonical one. Now, in ergodic theory it is well known that an approach to equilibrium is guaranteed if a system is proven to be mixing. Let us recall this. For a function f on p...

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