On the probability of violations of Fourier's law for heat flow in small systems observed for short times

We study the statistical mechanics of thermal conduction in a classical many-body system that is in contact with two thermal reservoirs maintained at different temperatures. The ratio of the probabilities, that when observed for a finite time, the time averaged heat flux flows in and against the direction required by Fourier's Law for heat flow, is derived from first principles. This result is obtained using the transient fluctuation theorem. We show that the argument of that theorem, namely, the dissipation function is, close to equilibrium, equal to a microscopic expression for the entropy production. We also prove that if transient time correlation functions of smooth zero mean variables decay to zero at long times, the system will relax to a unique nonequilibrium steady state, and for this state, the thermal conductivity must be positive. Our expressions are tested using nonequilibrium molecular dynamics simulations of heat flow between thermostated walls.