Thermal Transport in Low Dimensions

Abstract

In this chapter we introduce some of the basic models and concepts that will be discussed throughout the volume. In particular we describe systems of nonlinear oscillators arranged on low-dimensional lattices and summarize the phenomenology of their transport properties.

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

“…In this review we will discuss results for both Hamiltonian and stochastic systems. There are two possible approaches for studying transport properties of a system [3,4]. A schematic of the two set-ups is shown in Fig.…”

“…(III B). A connection can be made between the superdiffusive scaling (σ 2 c (t) ∼ t β ) of the energy correlations and the power-law decay, ∼ t α−1 , of the current-current correlations [4,58,59], which can be seen as follows. Starting from the continuity equation for energy, one can obtain the relation [61,62] on the infinite line…”

“…In the previous section we discussed transport in the HCVE model which has two conserved quantities, namely volume and energy. In this section, we discuss heat transport in the harmonic chain momentum exchange (HCME) model which has three conserved quantities, namely volume, momentum and energy, that are the same as the ones in usual anharmonic chains with Hamiltonian dynamics [3,4]. The model consists of a harmonic chain of particles each of unit mass and described by the degrees of freedom q i , p i , with i ∈ Z, corresponding respectively to position and momentum.…”

“…The issue of deriving Fourier's law has been a long standing question and a very active field of research [1]. Several theoretical as well as large scale numerical studies have been performed on different mathematical model systems to understand the necessary and sufficient conditions needed in the microscopic description to validate Fourier's law at the macroscopic level [2][3][4]. Surprisingly, these studies suggest that Fourier's law is probably not valid in many one-dimensional systems and one finds that the thermal conductivity κ diverges with system size N as κ ∼ N α where 0 < α < 1 [2][3][4][5][6][7][8][9][10][11][12].…”

“…(2). The picture that has emerged from studies over the last few years [4,[25][26][27][28][29][30][31][32][33][34][35][36][37] is that Fourier's law gets replaced by a spatially non-local but linear equation wherein the current at a point gets contributions from temperature gradients in other parts of the system. This has the form j(x, t) = − dx K(x, x )…”