We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for stationary transport of bosonic particles like photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e. transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.

We explore the statistical behaviour of the discrete nonlinear Schrödinger equation as a test bed for the observation of negative-temperature (i.e. above infinite temperature) states in Bose-Einstein condensates in optical lattices and arrays of optical waveguides. By monitoring the microcanonical temperature, we show that there exists a parameter region where the system evolves towards a state characterized by a finite density of discrete breathers and a negative temperature. Such a state persists over very long (astronomical) times since the convergence to equilibrium becomes increasingly slower as a consequence of a coarsening process. We also discuss two possible mechanisms for the generation of negative-temperature states in experimental setups, namely, the introduction of boundary dissipations and the free expansion of wavepackets initially in equilibrium at a positive temperature. Gesellschaft optics [1]. For instance, BEC in optical lattices are ideal benchmarks to investigate the role of nonlinearity and spatial discreteness in quantum transport phenomena [2,3]. The refined experimental techniques now available [4-8] enable investigations and applications of BEC in quantum coherence, quantum control, quantum information processing and the quantumclassical correspondence [9]. In addition, BEC in optical lattices can be considered as the 'atomic analogues' of light propagating in waveguide arrays [10].The discrete nonlinear Schrödinger equation (DNLSE) is a basic semiclassical model for the study of BEC in optical lattices [11] and light propagating in arrays of optical waveguides [10]. Numerical studies have revealed the important role played by localized solutions, notably discrete breathers [11,12]. Many aspects of the relationship between breathers and transport properties in BEC have recently been reviewed in [13]. The DNLSE has also been found to exhibit unusual thermodynamic features. A first statistical-mechanics study of the DNLSE identified a region in the parameter space characterized by the spontaneous formation of breathers and conjectured it to correspond to negative-temperature (NT) states [14]. NT states have attracted the curiosity of researchers since the pioneering work in systems of quantum nuclear spins [15]. From a thermodynamic point of view, the presence of NT states implies that the system's entropy is a decreasing function of the internal energy. In a series of recent papers [16][17][18][19], Rumpf provided a convincing theoretical argument that excludes the physical occurrence of NT equilibrium states in the DNLSE. In particular, he showed that even in the region of parameter space where breathers form spontaneously via a modulational instability, the DNSLE eventually reaches a maximum entropy (equilibrium) state formed by a background at infinite temperature superposed on a single breather that collects the 'excess' energy. It was also observed that the convergence to the equilibrium state predicted by Rumpf would need transients lasting over astronomical times [20]. Therefore, ...

We introduce suitable Langevin thermostats which are able to control both the temperature and the chemical potential of a one-dimensional lattice of nonlinear Schrödinger oscillators.The resulting non-equilibrium stationary states are then investigated in the limit of low temperatures and large particle densities, where the dynamics can be mapped onto that of a coupledrotor chain with an external torque. As a result, an effective kinetic definition of temperature can be introduced and compared with the general microcanonical (global) definition.

We present a detailed account of a first-order localization transition in the discrete nonlinear Schrödinger equation, where the localized phase is associated to the high energy region in parameter space. We show that, due to ensemble inequivalence, this phase is thermodynamically stable only in the microcanonical ensemble. In particular, we obtain an explicit expression of the microcanonical entropy close to the transition line, located at infinite temperature. This task is accomplished making use of large-deviation techniques, that allow us to compute, in the limit of large system size, also the subleading corrections to the microcanonical entropy. These subleading terms are crucial ingredients to account for the first-order mechanism of the transition, to compute its order parameter and to predict the existence of negative temperatures in the localized phase. All of these features can be viewed as signatures of a thermodynamic phase where the translational symmetry is broken spontaneously due to a condensation mechanism yielding energy fluctuations far away from equipartition: actually they prefer to participate in the formation of nonlinear localized excitations (breathers), typically containing a macroscopic fraction of the total energy.

We investigate the coarsening evolution occurring in a simplified stochastic model of the Discrete NonLinear Schrödinger (DNLS) equation in the so-called negative-temperature region. We provide an explanation of the coarsening exponent n = 1/3, by invoking an analogy with a suitable exclusion process. In spite of the equivalence with the exponent observed in other known universality classes, this model is certainly different, in that it refers to a dynamics with two conservation laws.

We provide evidence of an extremely slow thermalization occurring in the Discrete NonLinear Schrödinger (DNLS) model. At variance with many similar processes encountered in statistical mechanics -typically ascribed to the presence of (free) energy barriers -here the slowness has a purely dynamical origin: it is due to the presence of an adiabatic invariant, which freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events, where energy is suddenly released towards the background. We conjecture that this exponentially slow relaxation is a key ingredient contributing to the non-ergodic behavior recently observed in the negative temperature region of the DNLS equation.

We investigate numerically the magnetization dynamics of an array of nanodisks interacting through the magnetodipolar coupling. In the presence of a temperature gradient, the chain reaches a nonequilibrium steady state where energy and magnetization currents propagate. This effect can be described as the flow of energy and particle currents in an off-equilibrium discrete nonlinear Schrödinger (DNLS) equation. This model makes transparent the transport properties of the system and allows for a precise definition of temperature and chemical potential for a precessing spin. The present study proposes a setup for the spin-Seebeck effect, and shows that its qualitative features can be captured by a general oscillator-chain model.

We investigate thermal conduction in arrays of long-range interacting rotors and Fermi-Pasta-Ulam (FPU) oscillators coupled to two reservoirs at different temperatures. The strength of the interaction between two lattice sites decays as a power α of the inverse of their distance. We point out the necessity of distinguishing between energy flows towards or from the reservoirs and those within the system. We show that energy flow between the reservoirs occurs via a direct transfer induced by long-range couplings and a diffusive process through the chain. To this aim, we introduce a decomposition of the steady-state heat current that explicitly accounts for such direct transfer of energy between the reservoir. For 0≤α<1, the direct transfer term dominates, meaning that the system can be effectively described as a set of oscillators each interacting with the thermal baths. Also, the heat current exchanged with the reservoirs depends on the size of the thermalized regions: In the case in which such size is proportional to the system size N, the stationary current is independent on N. For α>1, heat transport mostly occurs through diffusion along the chain: For the rotors transport is normal, while for FPU the data are compatible with an anomalous diffusion, possibly with an α-dependent characteristic exponent.