F. Bonetto, J.L. Lebowitz and L. Rey-Bellet
Fourier's Law: a Challenge to Theorists
(274K, postscript)

ABSTRACT.  We present a selective overview of the current state of our knowledge 
(more precisely of ourignorance) regarding the derivation of Fourier's 
Law, ${\bf J}(\br) =-\kappa {\bf \nabla }T(\br)$; ${\bf J}$ the heat 
flux, $T$ the temperature and $\kappa$, the heat conductivity. This law is 
empirically well tested for both fluids and crystals, when the 
temperature varies slowly on the microscopic scale, with $\kappa$ 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 $\kappa$, involving integrals 
over equilibrium time correlations, for any system (or model) with a 
deterministic, \eg Hamiltonian, microscopic evolution.