Transition behavior of the discrete nonlinear Schrödinger equation

Abstract

Many nonlinear lattice systems exhibit high-amplitude localized structures, or discrete breathers. Such structures emerge in the discrete nonlinear Schrödinger equation when the energy is above a critical threshold. This paper studies the statistical mechanics at the transition and constructs the probability distribution in the regime where breathers emerge. The entropy as a function of the energy is nonanalytic at the transition. The entropy is independent of the energy in the regime of breathers above the transition.

Stable and metastable states and the formation and destruction of breathers in the discrete nonlinear Schrödinger equation

“…The transition behavior is due to the quadratic conservation law for the wave action, and the quartic conservation law for the energy. It is not caused by the interaction force, and it exists also for zero interaction [11]. Similar processes are also relevant in continuous nonequilibrium systems where a second conserved quantity is due to four-wave interactions [20].…”

“…The second is a two-phase state, where a localized mode coexists with low-amplitude waves. This explains the appearance of localized modes (breathers) in the domain II [11]. This state is stable, and it corresponds to the absolute maximum of entropy.…”

“…This distribution plus a small number of peaks achieves any energy E > A 2 /N, and has the same entropy as for E = A 2 /N in the thermodynamic limit [11].…”

“…de. system is high-dimensional and nonintegrable, and the trajectory may reach any point at the shells of fixed values of A = A and H = E. This has led to the question of whether breathers are associated with almost all microstates on the shell of fixed energy and waveaction, so that breathers should be considered a statistical phenomenon. A number of papers have studied the statistical mechanics of nonlinear Schrödinger models [7][8][9][10][11][12] and other lattices that support breathers [13][14][15][16][17][18]. The direct approach of computing the microcanonic ensemble would require us to solve integrals over the δ-functions of A and H. One alternative approach is to compute the grand canonical distribution function y(α, β, N) = exp(−αA − βH)dΓ , where the parameters α and β are inverse temperatures that control the content of the quantities A and H. dΓ is a phase space volume element.…”

Condensation transition in joint large deviations of linear statistics

“…In Section 5, we review several examples of condensation transition: jamming transition in exclusion process, condensation transition in polydisperse rods diffusing on a ring [9] and phase transition in a random pure state of a large bipartite quantum system [36,37]. We also mention a possibly related phenomenon of localised solutions (breathers) of the discrete nonlinear Schrödinger equation [38][39][40][41].…”