Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

Abstract

We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g. the 'spectral entropy' or various types of 'correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as ε c ∼ N −b , with b ∈ [1.5, 2.5]. For 'generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

“…As the J (k) 's are conserved along the Toda flow, and the FPUT chain is a perturbation of the Toda one, the Toda integrals are good candidates to be adiabatic invariants when computed along the FPUT flow. This intuition is supported by several numerical simulations, the first by Ferguson-Flaschka-McLaughlin [12] and more recently by other authors [4,6,10,20,35]. Such simulations show that the variation of the Toda integrals along the FPUT flow is very small on long times for initial data of small specific energy.…”

“…We are left to estimate (6.6) for FPUT, but this is exactly the quantity bounded in Proposition 5.1. We conclude that 10) for some constant C j > 0, j = 1, 2, 3 and for β > β 0 and N > N 0 . Combing Proposition 5.3 with (6.10) we obtain…”

Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit

“…As the J pkq 's are conserved along the Toda flow, and the FPUT chain is a perturbation of the Toda one, the Toda integrals are good candidates to be adiabatic invariants when computed along the FPUT flow. This intuition is supported by several numerical simulations, the first by Ferguson-Flaschka-McLaughlin [12] and more recently by other authors [4,6,10,19,34]. Such simulations show that the variation of the Toda integrals along the FPUT flow is very small on long times for initial data of small specific energy.…”

Complex Dynamics and Statistics in Hamiltonian 1-Dimensional Lattices

“…A more complete interpretation of the FPU paradox was provided by our group [6], where we introduced the concept of q-tori, reconciling q-breathers with the metastable packets of lowfrequency modes. Now we shall use the GALI indices to study the stability of these q-tori and the breakdown of the associated FPU recurrences.…”

“…The interesting question that arises, therefore, is whether a statistical analysis of this orbit also shows that this orbit can also be characterized as weakly chaotic, by plotting the probability distributions of averaged sums of its coordinates, as was done above for the orbit shown in Figure 16. for a total integration time t=10 6 . Right: Final integration time t =10 10 .…”