Statistical fluctuations of the light emitted from amplifying random media are studied theoretically and numerically. The characteristic scales of the diffusive motion of light lead to Gaussian or power-law (Lévy) distributed fluctuations depending on external control parameters. In the Lévy regime, the output pulse is highly irregular leading to huge deviations from a mean-field description. Monte Carlo simulations of a simplified model which includes the population of the medium demonstrate the two statistical regimes and provide a comparison with dynamical rate equations. Different statistics of the fluctuations helps to explain recent experimental observations reported in the literature.

In the present Rapid Communication we present an analytical and numerical solution of the self-consistent mode-coupling equations for the problem of heat conductivity in one-dimensional systems. Such a solution leads us to propose a different scenario to accommodate the known results obtained so far for this problem. More precisely, we conjecture that the universality class is determined by the leading order of the nonlinear interaction potential. Moreover, our analysis allows us to determine the memory kernel, whose expression puts on a more firm basis the previously conjectured connection between anomalous heat conductivity and anomalous diffusion.

Energy transfer in small nano-sized systems can be very different from that in their macroscopic counterparts due to reduced dimensionality, interaction with surfaces, disorder, and large fluctuations. Those ingredients may induce non-diffusive heat transfer that requires to be taken into account on small scales. We provide an overview of the recent advances in this field from the points of view of nonequilibrium statistical mechanics and atomistic simulations. We summarize the underlying basic properties leading to violations of the standard diffusive picture of heat transport and its universal features, with some historical perspective. We complete this scenario by illustrating also the effects of long-range interaction and integrability on non-diffusive transport. Then we discuss how all of these features can be exploited for thermal management, rectification and to improve the efficiency of energy conversion. We conclude with a review on recent achievements in atomistic simulations of anomalous heat transport in single polymers, nanotubes and two-dimensional materials. A short account of the existing experimental literature is also given.

In this review paper we survey recent achievements in anomalous heat diffusion, while highlighting open problems and research perspectives. First, we briefly recall the main features of the phenomenon in low-dimensional classical anharmonic chains and outline some recent developments in the study of perturbed integrable systems and the effect of long-range forces and magnetic fields. Selected applications to heat transfer in material science at the nanoscale are described. In the second part, we discuss of the role of anomalous conduction in coupled transport and describe how systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-off for the conversion of thermal to particle current.

We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schr\"odinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, 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.