3 research outputs found
The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: an approach through Lyapunov Exponents
FPU models, in dimension one, are perturbations either of the linear model or
of the Toda model; perturbations of the linear model include the usual
-model, perturbations of Toda include the usual model. In
this paper we explore and compare two families, or hierarchies, of FPU models,
closer and closer to either the linear or the Toda model, by computing
numerically, for each model, the maximal Lyapunov exponent . We study the
asymptotics of for large (the number of particles) and small
(the specific energy ), and find, for all models, asymptotic
power laws , and depending on the model. The
asymptotics turns out to be, in general, rather slow, and producing accurate
results requires a great computational effort. We also revisit and extend the
analytic computation of introduced by Casetti, Livi and Pettini,
originally formulated for the -model. With great evidence the theory
extends successfully to all models of the linear hierarchy, but not to models
close to Toda
Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral
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 εc∼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