The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics

Abstract

This paper is devoted to a numerical study of the familiar α+β FPU model. Precisely, we here discuss, revisit and combine together two main ideas on the subject: (i) In the system, at small specific energy ε = E/N, two well separated time-scales are present: in the former one a kind of metastable state is produced, while in the second much larger one, such an intermediate state evolves and reaches statistical equilibrium. (ii) FPU should be interpreted as a perturbed Toda model, rather than (as is typical) as a linear model per- turbed by nonlinear terms. In the view we here present and support, the former time scale is the one in which FPU is essentially integrable, its dynamics being almost indistinguishable from the Toda dynamics: the Toda actions stay constant for FPU too (while the usual linear normal modes do not), the angles fill their almost invariant torus, and nothing else happens. The second time scale is instead the one in which the Toda actions significantly evolve, and statistical equilibrium is possible. We study both FPU-like initial states, in which only a few degrees of freedom are excited, and generic initial states extracted randomly from an (ap- proximated) microcanonical distribution. The study is based on a close comparison between the behavior of FPU and Toda in various situations. The main technical novelty is the study of the correlation functions of the Toda constants of motion in the FPU dynamics; such a study allows us to provide a good definition of the equilibrium time τ , i.e. of the second time scale, for generic initial data. Our investigation shows that τ is stable in the thermodynamic limit, i.e. the limit of large N at fixed ε, and that by reducing ε (ideally, the temperature), τ approximately grows following a power law τ ∼ ε^{-a} , with a = 5/2

Route to thermalization in the α -Fermi–Pasta–Ulam system

“…In ref. 4, it has been shown that the α-FPU can be seen as a perturbation of the integrable Toda lattice (see also ref. 27).…”

“…More recently, very long numerical simulations have shown clear evidence of the phenomenon of equipartition (see, for instance, ref. 4 and references therein). However, despite substantial progress on the subject (5-10), to our knowledge no complete understanding of the original problem has been achieved so far, and the numerical results of the original α-FPU system remain largely unexplained from a theoretical point of view.…”

“…However, despite substantial progress on the subject (5-10), to our knowledge no complete understanding of the original problem has been achieved so far, and the numerical results of the original α-FPU system remain largely unexplained from a theoretical point of view. More precisely, the physical mechanism responsible for a first "metastable state" (4) and the observation of equipartition for very large times have not been understood.…”

Energy Transports in Toda Lattices with Quasiperiodic On-site Potentials

“…On the other hand, the energy transport behaviors of Toda model have been found very similar to the FPU model in the short time scale [21,22,24]. But the difference is also obvious.…”

“…Since then, extensive studies on the energy transport behaviors in nonlinear lattices have been carried out . For the FPU model, there is a general agreement that two well separated time scales are present for weak nonlinearity [15,[20][21][22]24]. That is, in a relative short time scale, a kind of metastable state is appeared.…”