Fourier's Law: A Challenge to Theorists

Abstract

We present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J(r) = −κ∇T (r); J the heat flux, T the temperature and κ, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with κ an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for κ, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian, microscopic evolution.

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

“…(1,2) These results are in sharp contrast to those found earlier by Rieder, Lebowitz, and Lieb, (5) who studied a system with the same Hamiltonian dynamics, but with heat baths acting only at the boundaries. They found that the system had an infinite conductivity and a constant temperature profile away from the ends, results later generalized to the higher dimensional case by Nakazawa.…”

“…where T (1,0) is a solution of the problem with T L =1 and T R =0. The set of all such T (1,0) form a convex set, i.e., any non-uniqueness in the solution of the self-consistency condition would imply the existence of a whole continuum of solutions.…”

“…The set of all such T (1,0) form a convex set, i.e., any non-uniqueness in the solution of the self-consistency condition would imply the existence of a whole continuum of solutions. We shall now prove that for our model the solution to the self-consistency problem is unique.…”

“…Clearly, f is a rational function, analytic everywhere but at the zeroes of G in (2.20). On the other hand, 1], and since we have assumed that c > 0, there are no…”

Аналитический Подход К (Ан)гармоническим Кристаллическим Цепочкам С Самосогласованными Стохастическими Резервуарами

“…В частности, даже в одномерном контексте нет строгого вывода закона теплопроводности Фурье, согласно которому поток тепла пропорционален градиенту температуры [1]. Многие работы, почти все с использованием компьютерных вычислений [2] и с результатами, противоречащи-ми другу другу, были посвящены этой теме со времени первых строгих исследо-ваний по гармоническим цепочкам осцилляторов с термостатами на границах [3] -модели, которая не подчиняется закону Фурье.…”