Non-local linear response in anomalous transport

Abstract

The anomalous heat transport observed in low-dimensional classical systems is associated with super-diffusive spreading of the space–time correlation of the conserved fields in the system. This leads to a non-local linear response relation between the heat current and the local temperature gradient in the non-equilibrium steady state. This relation provides a generalization of Fourier’s law of heat transfer and is characterized by a non-local kernel operator related to the fractional operators describing super-diffusion. The kernel is essentially proportional, in an appropriate hydrodynamic scaling limit, to the time integral of the space–time correlations of local currents in equilibrium. In finite-size systems, the time integral of correlation of microscopic currents at different locations over an infinite duration is independent of the locations. On the other hand, the kernel operator is space-dependent. We demonstrate that the resolution of this apparent puzzle becomes evident when we consider an appropriate combination of the limits of a large system size and a long integration time. Our study shows the importance of properly handling these limits, even when dealing with (open) systems connected to reservoirs. In particular, we reveal how to extract the kernel operator from simulated microscopic current–current correlation data. For two model systems exhibiting anomalous transport, we provide a direct and detailed numerical verifications of the kernel operators.