Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

Abstract

It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.

Hydrodynamics and transport in the long-range-interacting $φ^4$ chain

“…The data are reported under the same rescalings as in the single-well case. They confirm that the dynamical scaling of is the same: relation (11) holds with z h as given in (12), for all the considered energies. Remarkably, this remains true even close to the critical point (see fig.…”

“…Anomalous transport in such many-body systems can be effectively described by a random Lévy walk [5] of the energy carriers, as demonstrated extensively in the literature [6,7,8]. This leads naturally to consider hydrodynamic equations where the standard Laplacian operator is replaced by a fractional one [9,10,11,12]. Nonlinear coupling among hydrodynamic fields yields slow algebraic decay of current correlations at equilibrium and effective non-local equations.…”

Finite-temperature transport in one-dimensional quantum lattice models

“…Localization corresponds to γ → ∞. See (Dhar et al, 2019) for a review of anomalous transport in classical systems. The type of transport can also be recognized from the NESS profile of a conserved density.…”

Anomalous Heat Transport in Classical Many-Body Systems: Overview and Perspectives

“…A related distinctive feature of anomalous transport is that the temperature profiles in non-equilibrium steady states are non-linear, even for vanishing applied temperature gradients [32,41]. There is indeed a close connection with the fractional heat equation, which has been demonstrated and discussed in recent literature [10,42].…”

“…Far from being a purely academic exercise, this research has unveiled the possibility of observing such peculiar effects in nanomaterials, such as nanotubes, nanowires, or graphene [5,6]. Extended review articles on this problem have existed for many years [7,8], while a collection of works about more recent achievements can be found in Lepri [9] and a review article [10] in the present issue. Here it is useful to first provide a short summary of the state of the art in the field, while the main part of the paper will focus on recent achievements that point to promising and challenging directions for future investigations.…”