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Physical review. B./Physical review. B

Equilibrium temperatures of discrete nonlinear systems

Uri Levy, Yaron R. Silberberg

Published Aug 30, 2018• 12 citations
Publisher

Abstract

During the evolution of discrete nonlinear systems with dynamics dictated by the discrete nonlinear Schr\"odinger equation, two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). It is then possible to analyze system evolution in relation to an energy-density phase diagram. Previous works have identified a ``thermalization zone'' on the phase diagram where regular statistical mechanics methods apply. Based on these statistical mechanics methods we have now assigned a specific equilibrium temperature to every point of the thermalization zone. Temperatures were derived in the grand canonical picture through an entropy-temperature relation, modified to suit the nonlinear lattice systems. Generally, everywhere in the thermalization zone of the phase diagram, temperatures along a fixed system-density line, grow monotonously from zero to infinity. Isotherms on the phase diagram are concave.

DOI
10.1103/physrevb.98.060303

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During the evolution of discrete nonlinear systems with dynamics dictated by the discrete nonlinear Schr"odinger equation, two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). It is then possible to analyze system evolution in relation to an energy-density phase diagram. Previous works have identified a ``thermalization zone'' on the phase diagram where regular statistical mechanics methods apply. Based on these statistical mechanics methods we have now assigned a specific equilibrium temperature to every point of the thermalization zone. Temperatures were derived in the grand canonical picture through an entropy-temperature relation, modified to suit the nonlinear lattice systems. Generally, everywhere in the thermalization zone of the phase diagram, temperatures along a fixed system-density line, grow monotonously from zero to infinity. Isotherms on the phase diagram are concave.
During the evolution of coupled nonlinear oscillators on a lattice, with dynamics dictated by the discrete nonlinear Schrödinger equation (DNLSE systems), two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). If the number of system oscillators is large enough, a significant portion of the array can be considered to be an "open system", in intimate energy and density contact with a "bath" - the rest of the array. Thus, as indicated in previous works, the grand canonical formulation can be exploited in order to determine equilibrium statistical properties of thermalized DNLSE systems. In this work, given the values of the two conserved quantities, we have calculated the necessary values of the two Lagrange parameters (typically designated β,μβ,μβ,μ) associated with the grand canonical partition function in two different ways. One is numerical and the other is analytic, based on a published approximate entropy expression. In addition we have accessed a purposely-derived approximate PDF expression of site-densities. Applying these mathematical tools we have generated maps of temperatures, chemical potentials, and field correlations for DNLSE systems over the entire thermalization zone of the DNLSE phase diagram, subjected to all system-nonlinearity levels. The end result is a rather complete picture, characterizing equilibrated large DNLSE systems.
We show that the one-dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultralow-, low-, and high-temperature regimes). This has been established via (i) one-point macroscopic thermodynamic observables (temperature T, energy density ε, and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference), and (iii) classical out-of-time-ordered correlators and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures T_{l-ul} (between low- and ultra-low-temperature regimes) and T_{h-l} (between the high- and low-temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward toward the understanding of DNLS which is ubiquitous in many fields and has a nonseparable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many-body systems.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
Abstract The discrete nonlinear Schrödinger (DNLS) model, owing to the existence of two conserved quantities, exhibits an equilibrium transition from a homogeneous phase at positive absolute temperature to a localized phase at negative absolute temperature. Here, we provide a mean-field (MF) theory of DNLS through a suitable approximation of the grand canonical partition function, which makes it factorizable and can be used to describe the equilibrium state at positive temperatures and the metastable state at negative temperatures. Comparison of our MF results with numerically exact calculations shows that this approximation is good to excellent in the entire grand canonical phase diagram. Explicit approximate expressions for equilibrium observables are provided in the high-temperature limit. Our theory represents a clear advancement over the model that neglects the interaction between sites.
We study equilibrium time correlations for the discrete nonlinear Schrödinger equation on a one-dimensional lattice and unravel three dynamical regimes. There is a high temperature regime with density and energy as the only two conserved fields. Their correlations have zero velocity and spread diffusively. In the low temperature regime umklapp processes are rare with the consequence that phase differences appear as an additional (almost) conserved field. In an approximation where all umklapp is suppressed, while the equilibrium state remains untouched, one arrives at an anharmonic chain. Using the method of nonlinear fluctuating hydrodynamics we establish that the DNLS equilibrium time correlations have the same signature as a generic anharmonic chain, in particular KPZ broadening for the sound peaks and Lévy 5/3 broadening for the heat peak. In the, so far not sharply defined, ultra-low temperature regime the integrability of the dynamics becomes visible. As an illustration we simulate the completely integrable Ablowitz–Ladik model and confirm ballistic broadening of the time correlations.