Anharmonic Chain with Lennard-Jones Interaction

Abstract

The classical equations of motion of a one-dimensional, finite, anharmonic lattice, with nearest-neighbor interaction of the Lennard-Jones type, are investigated numerically.The results indicate that when the vibrational energy per particle is equal to or greater than 2-3% of the depth of the potential well, one has, in time average, equipartition of the energy among the normal modes, thus giving a hint toward ergodicity of the system at sufficiently high energy.For lower energies one finds recurrent motions if initially only one normal mode is excited in analogy with a famous result due to Fermi, Pasta, and Ulam. In this case the numerical results are consistent both with the existence of a long relaxation time and with a lack of ergodicity for low energies.

q-breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability

“…Finally we stress an even closer relation between the localization of q-breathers within the used perturbation theory and the Fourier series convergence of analytic periodic functions (11), (12), (13). One arrives from the QB problem to these equations by simply assuming ω q = const.…”

“…It aimed mainly at (FPU-3). Several other analytical [8,9] and numerical [10], [11], [12] threshold estimates have been published since, and will be discussed below. Note that similar effects have been observed in many other nonlinear discrete chain or field equations on a finite spatial domain, see e.g.…”

The Fermi–Pasta–Ulam problem: Fifty years of progress

“…This lattice is known to exhibit the stochasticity transition in a more clear way, compared with the FPU model (see, e.g., Ref. [77]). These data show that the standard procedure of computing the dynamical entropy due to the average h ∼< ln |d(t)/d(0)| > along the trajectories is quite stable with respect to different kinds of computational errors, and gives reliable results.…”

On the Definition of Temperature in FPU Systems

“…It is well known (see [1] and [2]) that, for energies below a certain threshold E c , if the energy is initially given to a few low frequency modes, equipartition of energy among the modes is eventually attained only after an extremely long time, while at intermediate times a kind of metaequilibrium state is attained, in which the energy is shared essentially within a packet of low frequency modes.…”