Boundary-Induced Instabilities in Coupled Oscillators

Abstract

A novel class of nonequilibrium phase-transitions at zero temperature is found in chains of nonlinear oscillators. For two paradigmatic systems, the Hamiltonian XY model and the discrete nonlinear Schrödinger equation, we find that the application of boundary forces induces two synchronized phases, separated by a non-trivial interfacial region where the kinetic temperature is finite. Dynamics in such supercritical state displays anomalous chaotic properties whereby some observables are non-extensive and transport is superdiffusive. At finite temperatures, the transition is smoothed, but the temperature profile is still non-monotonous. The characterization of steady-states is a widely investigated problem within non-equilibrium statistical mechanics [1], since it provides the basis for understanding a large variety of phenomena, including transport processes, pattern formation and the dynamics of living systems. In a nutshell, the simplest setup amounts to determining the currents that emerge as a result of the application of an external force, either across the system, as for electric currents, or at the boundaries, as in heat conduction [2][3][4]. Anyway, it is quite a nontrivial task to be accomplished, even when the departure from equilibrium is minimal and one can rely on the Green-Kubo formalism for establishing a connection between the microscopic and the hydrodynamic descriptions. For instance, this is testified by the discrepancy that still persists, after many years of careful studies, between the most advanced theories of heat conduction and some numerical simulations. The level of difficulty typically increases when one considers coupled transport [5-11] (i.e. when two or more currents coexist, such as heat and electric ones in thermo-electric effects) or, even worse, far-from-equilibrium. This is why most of the theoretical studies concentrate on stochastic models, where fluctuations can be easily controlled, although they lack a truly microscopic justification. This approach proved, nevertheless, very effective, since it has allowed discovering non-equilibrium transitions, such as those exhibited by TASEP-like models, that have been used to describe translation of proteins, or traffic flows [12].In this Letter we describe a novel class of boundaryinduced transitions for two models that are typically used as test beds for a wide range of physical phenomena: the so-called Hamiltonian XY (or rotor) model [13][14][15][16] sub- * Electronic address: stefano.lepri@isc.cnr.it ject to an applied mechanical torque and the Discrete NonLinear Schrödinger (DNLS) equation [17][18][19] under a gradient of the chemical potential. This type of qualitative change of the dynamics results from the joint effect of thermal and mechanical forces. It can be interpreted as a desynchronization phenomenon in a spatially-extended dynamical system, whereby mutual entrainment of oscillators' phases is abruptly destroyed. As a result of such unlocking, a regime characterized by phase-coexistence sets in where, although the chain...

Heat Transport in Low Dimensions: Introduction and Phenomenology

“…where F ± and T ± denote the torque applied to the chain boundary and the corresponding temperature, respectively; γ is the coupling strength with the external baths and η ± is a Gaussian white noise with unit variance. The effect of external forces on the Hamiltonian XY model has been preliminarly addressed in [32,47,49].…”

Coupled transport in rotor models

“…In the latter one, the boundary particles are assumed to exchange their velocity with equal-mass particles from an external heat bath, in equilibrium at some given temperature T. Both schemes can be easily generalized to account for an exchange of angular momentum, as well. In [33], the following Langevin scheme was proposed (here we just refer to the last particle)…”

“…Non stationary (time dependent) heat exchange processes have also been shown to be peculiar [31]. The effect of external forces has been previously addressed only in [32] and boundaryinduced transitions have also been discovered [33] (see also [34]). The important extension to 2D is characterized by the presence of a Kosterlitz-Thouless-Berezinskii phase transition between a disordered hightemperature phase and a low-temperature one, displaying anomalous and normal transport respectively [35].…”

Thermal Transport in Low Dimensions

“…The effect of external forces on the Hamiltonian XY model has been preliminarly addressed in [32,47,49].…”