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 $\alpha$ 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 $0\le \alpha <1$, 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 $\alpha >1$, 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 $\alpha$-dependent characteristic exponent.

We numerically investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schrödinger chain in contact with a heat reservoir (a bath) at temperature $T_L$ and a pure dissipator (a sink) acting on opposite edges. Long-time molecular-dynamics simulations are performed by evolving the equations of motion within a symplectic integration scheme. Mass and energy are steadily transported through the chain from the heat bath to the sink. We observe two different regimes. For small heat-bath temperatures $T_L$ and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures $T_L$ , the transport of energy is strongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death of breathers. The corresponding statistics exhibit the typical signature of rare events of processes with large deviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law.

The discrete nonlinear Schrödinger (DNLS) equation displays a parameter region characterized by the presence of localized excitations (breathers). While their formation is well understood and it is expected that the asymptotic configuration comprises a single breather on top of a background, it is not clear why the dynamics of a multi-breather configuration is essentially frozen. In order to investigate this question, we introduce simple stochastic models, characterized by suitable conservation laws. We focus on the role of the coupling strength between localized excitations and background. In the DNLS model, higher breathers interact more weakly, as a result of their faster rotation. In our stochastic models, the strength of the coupling is controlled directly by an amplitude-dependent parameter. In the case of a power-law decrease, the associated coarsening process undergoes a slowing down if the decay rate is larger than a critical value. In the case of an exponential decrease, a freezing effect is observed that is reminiscent of the scenario observed in the DNLS. This last regime arises spontaneously when direct energy diffusion between breathers and background is blocked below a certain threshold.

The Discrete NonLinear Schr\"odinger (DNLS) equation displays a parameter region characterized by the presence of localized excitations (breathers). While their formation is well understood and it is expected that the asymptotic configuration comprises a single breather on top of a background, it is not clear why the dynamics of a multi-breather configuration is essentially frozen. In order to investigate this question, we introduce simple stochastic models, characterized by suitable conservation laws. We focus on the role of the coupling strength between localized excitations and background. In the DNLS model, higher breathers interact more weakly, as a result of their faster rotation. In our stochastic models, the strength of the coupling is controlled directly by an amplitude-dependent parameter. In the case of a power-law decrease, the associated coarsening process undergoes a slowing down if the decay rate is larger than a critical value. In the case of an exponential decrease, a freezing effect is observed that is reminiscent of the scenario observed in the DNLS. This last regime arises spontaneously when direct energy diffusion between breathers and background is blocked below a certain threshold.

We investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schroedinger chain in contact with a heat reservoir (a bath) at temperature $T_L$ and a pure dissipator (a sink) acting on opposite edges. We observe two different regimes. For small heat-bath temperatures $T_L$ and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures $T_L$, the transport of energy is strongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death events of the breathers. The corresponding statistics exhibits the typical signature of rare events of processes with large deviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law.

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 $\alpha$ of the inverse of their distance. We point out the necessity of distinguishing between energy flows towards/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 $0\leq \alpha 1$, 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 $\alpha$ -dependent characteristic exponent.