3 research outputs found
Energy localization on q-tori, long term stability and the interpretation of FPU recurrences
We focus on two approaches that have been proposed in recent years for the
explanation of the so-called FPU paradox, i.e. the persistence of energy
localization in the `low-q' Fourier modes of Fermi-Pasta-Ulam nonlinear
lattices, preventing equipartition among all modes at low energies. In the
first approach, a low-frequency fraction of the spectrum is initially excited
leading to the formation of `natural packets' exhibiting exponential stability,
while in the second, emphasis is placed on the existence of `q-breathers', i.e
periodic continuations of the linear modes of the lattice, which are
exponentially localized in Fourier space. Following ideas of the latter, we
introduce in this paper the concept of `q-tori' representing exponentially
localized solutions on low-dimensional tori and use their stability properties
to reconcile these two approaches and provide a more complete explanation of
the FPU paradox.Comment: 38 pages, 7 figure
Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties
This is a continuation of our study concerning q-tori, i.e. tori of low
dimensionality in the phase space of nonlinear lattice models like the
Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU
system, and we showed that the dynamical features of the q-tori serve as an
interpretational tool to understand phenomena of energy localization in the FPU
space of linear normal modes. In the present paper i) we employ the method of
Poincare - Lindstedt series, for a fixed set of frequencies, in order to
compute an explicit quasi-periodic representation of the trajectories lying on
q-tori in the alpha model, and ii) we consider more general types of initial
excitations in both the alpha and beta models. Furthermore we turn into
questions of physical interest related to the dynamical features of the q-tori.
We focus on particular q-tori solutions describing low-frequency `packets' of
modes, and excitations of a small set of modes with an arbitrary distribution
in q-space. In the former case, we find formulae yielding an exponential
profile of energy localization, following an analysis of the size of the
leading order terms in the Poincare - Lindstedt series. In the latter case, we
explain the observed localization patterns on the basis of a rigorous result
concerning the propagation of non-zero terms in the Poincare - Lindstedt series
from zeroth to subsequent orders. Finally, we discuss the extensive (i.e.
independent of the number of degrees of freedom) properties of some q-tori
solutions.Comment: To appear in Physica D, 34 pages, 9 figure
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