Statistical mechanics of general discrete nonlinear Schrödinger models: Localization transition and its relevance for Klein-Gordon lattices

Abstract

We extend earlier work [Phys. Rev. Lett. 84, 3740 (2000)]] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrödinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physically motivated by the desire to describe situations with an excitation threshold for creation of localized excitations, as well as by recent work suggesting noncubic DNLS models to describe Bose-Einstein condensates in deep optical lattices, taking into account the effective condensate dimensionality. Considering ensembles of initial conditions with given values of the two conserved quantities, norm and Hamiltonian, we calculate analytically the boundary of the "normal" Gibbsian regime corresponding to infinite temperature, and perform numerical simulations to illuminate the nature of the localization dynamics outside this regime for various cases. Furthermore, we show quantitatively how this DNLS localization transition manifests itself for small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled anharmonic oscillators (in which energy is the only conserved quantity), and determine conditions for the existence of persistent energy localization over large time scales.

Condensation transition in joint large deviations of linear statistics

“…As we discussed above, the condition (39) leads to condensation; the rest of L − 1 random variables behave as if they were mutually independent and distributed with w(v; r) [15,16]. In other words, the distribution of background fluid that co-exists with the condensate is given by the grand canonical distribution with maximal fugacity.…”

“…It now remains to estimate δ( (39). As we discussed above, the condition (39) leads to condensation; the rest of L − 1 random variables behave as if they were mutually independent and distributed with w(v; r) [15,16].…”

“…, v L are iid random variables distributed by w(v; r); we will use this fact later. Using (39) and (40), the expression for Z L (M, V ) in (38) can be written as…”

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

“…At variance to its space-continuous counterparts, the disorder-free translationally invariant lattice model with ε l = 0 shows a non-Gibbsean phase [7,8]. It is separated from the Gibbsean phase by states of infinite temperature.…”

“…It is separated from the Gibbsean phase by states of infinite temperature. While average densities in the non-Gibbsean phase can be formally described by Gibbs distributions with negative temperature, in truth the dynamics shows a separation of the complex field ψ l into a two-component one -a first component of high density localized spots and a second component of delocalized wave excitations with infinite temperature [7,8,9,10,11]. The high density localized spots are conceptually very similar to selftrapping and discrete breathers [12].…”