On the derivation of Fourier's law for coupled anharmonic oscillators

Details On the derivation of Fourier's law for coupled anharmonic oscillators On the derivation of Fourier's law for coupled anharmonic oscillators

2006-05-22

arxiv.org/abs/math-ph/0605062v1

Physics

Mathematical Physics

69 pages

Scientific paper

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing mimicking heat baths of temperatures T_1 and T_2 on the hyperplanes at x_1=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.

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