Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators

Abstract

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice 2N × 2 , and subjected to a stochastic forcing mimicking heat baths of temperatures T 1 and T 2 on the hyperplanes at 0 and N . We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equlibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.

Non-Equilibrium Steady States for Chains of Four Rotors

“…Since the system is rapidly driven to small p 1 , p 4 , it is really the dynamics of ( p 2 , p 3 ) that plays the most important role. We will often argue in terms of the 8-dimensional dynamics projected onto the p 2 p 3 -plane.…”

“…We think in terms of the following fast-slow decomposition: the variables q 1 , q 3 , q 4 and p evolve slowly, while q 2 evolves rapidly, sinceq 2 = p 2 , and p 2 is large. In this regime, the variable q 2 swipes through T many times before any other variable changes significantly.…”

“…We will then find that Lp (3) 2 contains terms of order −3 and −4. To replace the terms of order −3 with their average (which is finally non-zero), we will introduce a function p (4) 2 . This will complete the averaging procedure.…”

“…See for example [2,3,7,13,15,19,21] for chains of rotors and [1][2][3][4]16,20,21] for chains of oscillators. From a rigorous point of view however, the mere existence of an invariant measure is not evident, and has been proved only in special cases.…”

“…In Sect. 4 we construct a Lyapunov function that is valid across all regimes, and in Sect. 5 we provide the technicalities necessary to obtain the main result.…”