Statistical Mechanics of a Discrete Nonlinear System

Abstract

Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of T = ∞, we identify a phase transition, through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breather-like localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.The pioneering studies of Fermi, Pasta and Ulam [1] (FPU) showed that energy exchange between coupled systems may be suppressed in the presence of nonlinearity; instead a type of behavior that severely contrasts equipartition among the linear modes is observed. The question of whether equipartition of excitation energy always appears is a contemporary issue in various fields of physics. Many manifestations of nonequilibrium and non-equipartition phenomena equivalent to the dynamical behavior of systems with few degrees of freedom contrasting statistical mechanics expectations have been observed. Some of these phenomena, and therefore the absence of immediate equipartition expressed in terms of self-trapping of energy, play an important role for optical storage patterns in nonlinear fibers, condensed matter physics, and biophysics [2].A particularity of discrete nonlinear systems is their ability to sustain strong localization of energy [3]. This is accomplished via intrinsic localized modes (breathers) which are modes that remain stable for extremely long times. So far it is a largely unaddressed problem how to handle and describe these excitations in a statistical mechanics framework although it has been argued that breathers may act as virtual bottlenecks [4] delaying the thermalization process.In this work, we develop a statistical understanding of the dynamics, including the breathers, in a discrete nonlinear Schrödinger (DNLS) equation. The DNLS equation plays a significant role in several branches nonlinear physics as a simple physical model because it may approximate many of the above mentioned nonlinear systems. We study analytically and numerically the thermalization of the lattice for T ≥ 0. We identify the regime in phase space wherein regular statistical mechanics considerations apply, and hence, thermalization is observed numerically and explored analytically using regular, grand-canonical, Gibbsian equilibrium measures.However, the nonlinear dynamics of the problem renders permissible the realization of regimes of phase space which would formally correspond to "negative temperatures" [5] in the sense of statistical mechanics. The novel feature of these states is that the energy tends to be localized in certain lattice sites forming breather-like excitations. Returning to statistical mechanics, such realizations, which would formally correspond to negative temperatures, are not possible (since the Hamil...

Boundary-Induced Instabilities in Coupled Oscillators

“…Here below, we show that a similar scenario can be observed for the DNLS equation, iż n = −2|z n | 2 z n − z n+1 − z n−1 , where z n = (p n + iq n )/ √ 2 is a complex variable. The DNLS Hamiltonian has two conserved quantities, the mass/norm a and the energy density h [29,30], so that it is a natural candidate for describing coupled transport [10,31]. We have numerically studied a DNLS chain interacting with two Langevin thermostats at T = 0 and different chemical potentials µ 1 and µ N imposed at the boundaries (see Ref.…”

Localization and coherence in nonintegrable systems

“…(12), (13) can be solved analytically for the low energy case E/N ≪ 1. The solution (Fig.12) is qualitatively different in the 'oversaturated phase' with M < M eq and in the 'overheated phase' M eq < M < M π with M eq = N − E/ √ 1 + 4J: (12) and (13). The energy is fixed as E w = 0.1 in (c).…”

“…These peaks can merge into stronger ones while radiating low-amplitude waves. The irreversible character of the system becomes apparent and its behavior is driven by statistical mechanics [12], [13], [14] as the solution's trajectory tries to explore more of the available phase space. As it does this, it must, at the same time, respect the preservation of conserved quantities.…”

Discrete nonlinear Schrödinger breathers in a phonon bath

“…The thermalization of the DNLS systems has been studied previously [20,21] and these studies have shown that the dynamics exhibits two very different characteristics depending on the initial energy density h = H/N and norm density n = N /N (for definition of Hamiltonian H and norm N see, Eq. (4) and below Eq.…”

“…(4), respectively), where N is the system size. At low densities (exact relationship is given in [21]) a thermalized state appears after a relatively short time, where all the correlations can be obtained from the partition function…”