FM 2000-8       mp_arc@math.utexas.edu \# 00-89
                math-ph@xxx.lanl.gov   \# 0002052 

Authors: Federico Bonetto, Joel L. Lebowitz and Luc Rey-Bellet

Title: 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) = -k \nabla T(r); J the heat flux, T the
temperature and k, the heat conductivity.  This law is empirically
well tested for both fluids and crystals, when the temperature varies
slowly on the microscopic scale, with k 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 k, involving integrals over equilibrium time correlations,
for any system (or model) with a deterministic, e.g. Hamiltonian,
microscopic evolution.

Keywords: Fourier's law, heat conduction, steady state, 
          Green-Kubo formula

Addresses:
F.B.: IHES, 75 route de Chartres, 91440 Bures sur Yvette, France
J.L.L.: Department of Mathematics and Physics, Rutgers University,
        110 Frelinghuysen Road, Piscataway NJ 08854.
L.R.-B.: Department of Mathematics, University of Virginia, 
         Kerchof Hall, Charlottesville VA 22903

e-mail: bonetto@ihes.fr
        lebowitz@math.rutgers.edu
        lr7q@virginia.edu