Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis that explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasiparticles with a very large mean free path $\ell$. Upon increasing the system size $L$, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario.

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.

Steady nonequilibrium states are investigated in a one-dimensional setup in the presence of two thermodynamic currents. Two paradigmatic nonlinear oscillators models are investigated: an XY chain and the discrete nonlinear Schrödinger equation. Their distinctive feature is that the relevant variable is an angle in both cases. We point out the importance of clearly distinguishing between energy and heat flux. In fact, even in the presence of a vanishing Seebeck coefficient, a coupling between (angular) momentum and energy arises, mediated by the unavoidable presence of a coherent energy flux. Such a contribution is the result of the 'advection' induced by the position-dependent angular velocity. As a result, in the XY model, the knowledge of the two diagonal elements of the Onsager matrix suffices to reconstruct its transport properties. The analysis of the nonequilibrium steady states finally allows to strengthen the connection between the two models.

A class of non-local contact processes is introduced and studied using the mean field approximation and numerical simulations. In these processes particles are created at a rate which decays algebraically with the distance from the nearest particle. It is found that the transition into the absorbing state is continuous and is characterized by continuously varying critical exponents. This model differs from the previously studied non-local directed percolation model, where particles are created by unrestricted Levy flights. It is motivated by recent studies of non-equilibrium wetting indicating that such non-local processes play a role in the unbinding transition. Other non-local processes which have been suggested to exist within the context of wetting are considered as well.

Strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of Lyapunov exponents. This regime, denoted as 'stable chaos', has been so far mainly characterized by means of numerical studies. In this paper we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries. In order to clarify how linear stability can coexist with nonlinear instability, we construct a suitable stochastic model. In the absence of spatial coupling, the model does not reveal the existence of any self-sustained chaotic phase. Nevertheless, already this simple regime reveals peculiar differences between the behaviour of finite-size and that of infinitesimal perturbations. A mean-field analysis of the spatially extended case provides a semi-quantitative description of the onset of irregular behaviour. Possible relations with directed percolation as a synchronization transition are also briefly discussed.