Abstract
We study numerically time evolution in classical lattices with weak or moderate nonlinearity which leads to interactions between linear modes. Our results show that in a certain strength range a moderate nonlinearity generates a dynamical thermalization process which drives the system to the quantum Gibbs distribution of probabilities, or average oscillation amplitudes. The effective dynamical temperature of the lattice varies from large positive to large negative values depending on the energy of the initially excited modes. This quantum Gibbs distribution is drastically different from the usually expected energy equipartition over linear modes corresponding to a regime of classical thermalization. Possible experimental observations of this dynamical thermalization are discussed for cold atoms in optical lattices, nonlinear photonic lattices and optical fiber arrays.
Highlights
The problem of thermal distribution for photons led to the invention of the Planck constant and Planck law [1]
In this work we studied numerically the time evolution in various types of classical lattices with moderate nonlinearities
The probability distribution is well described by the quantum Gibbs anzats (10) being drastically different from the classical energy equipartition over linear modes expected from classical thermalization picture
Summary
The problem of thermal distribution for photons led to the invention of the Planck constant and Planck law [1]. The first numerical investigations of onset of ergodicity and dynamical thermalization in a nonlinear lattice of coupled oscillators had been performed for the Fermi-Pasta-Ulam problem [9, 10, 11, 12] with an expectation to find energy equipartition over linear oscillator modes. The absence of ergodicity stimulated a great interest to the Fermi-Pasta-Ulam problem even if later it became clear that this model is rather close to the integrable Toda lattice and, it does not belong to a class of generic models (see discussions in [8, 11, 12]) Another approach to investigation of onset of ergodicity over linear oscillator modes in nonlinear lattices had been proposed in [13] by analyzing the effects of nonlinearity on the Anderson localization [14] in systems with disorder or systems of quantum chaos.
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