Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities

Abstract

The localization of energy in the discrete nonlinear Schrödinger equation is explained with statistical methods. The partition function and the entropy of the system are computed for low-amplitude initial conditions. Detailed predictions for the long-time solution are derived. Localized high-amplitude excitations absorb a surplus of energy when they emerge as a by-product of the production of entropy in the small fluctuations. The thermodynamic interpretation of this process applies to many dynamical systems with two conserved quantities.

Discrete breathers — Advances in theory and applications

“…In the right panel of Fig. 34 the results of another study of Rumpf [338] are shown, where the negative or positive temperature situations are achieved by initial conditions of extended waves with different wave numbers. The formation of the binary mixture is clearly observed for the former one (see also [339]).…”

Discrete breathers and negative-temperature states

“…We are indeed able to show that a broad range of initial conditions (IC) converges towards a well-defined thermodynamic state characterized by an NT and a finite density of breathers. Such a state does not contradict the theoretical arguments of [16,17], although the dynamical freezing of the high-amplitude breathers slows down the evolution so much as to render the convergence to equilibrium unobservable. Altogether, this phenomenon is reminiscent of ageing in glasses, although a more detailed analysis will be required to frame the analogy on more firm grounds.…”

“…From a thermodynamic point of view, the presence of NT states implies that the system's entropy is a decreasing function of the internal energy. In a series of recent papers [16][17][18][19], Rumpf provided a convincing theoretical argument that excludes the physical occurrence of NT equilibrium states in the DNLSE. In particular, he showed that even in the region of parameter space where breathers form spontaneously via a modulational instability, the DNSLE eventually reaches a maximum entropy (equilibrium) state formed by a background at infinite temperature superposed on a single breather that collects the 'excess' energy.…”

Quantum Gibbs distribution from dynamical thermalization in classical nonlinear lattices

“…The norm conservation can also be taken into account using the standard approach of statistical mechanics with the chemical potential and conservation of number of particles (or norm) [3,4] that is equivalent to the normalization used in (10). We note that possibilities of thermalization has been discussed in nonlinear chains starting from the FPU problem [9][10][11][12] and continuing even for nonlinear breathers [42][43][44]. However, here we consider the case of weak or moderate nonlinearity when the nonlinear terms are relatively small comparing to linear quadratic terms.…”

“…It is important to note that the above quantum Gibbs relations can also be obtained from the condition that the entropy S takes the maximal value at variation of probabilities ρ m . In fact, the quantum Gibbs ansatz was introduced in [33] for the DANSE and it was shown that it works at moderate nonlinearity β and not very strong disorder W (see also discussions in [44]). However, in [33] the striking paradox of the quantum Gibbs ansatz was not pointed out directly.…”