 
Summary: Statistical mechanical theory for steady state systems.
V. Nonequilibrium probability density
Phil Attarda
School of Chemistry F11, University of Sydney, New South Wales 2006, Australia
Received 15 March 2006; accepted 17 April 2006; published online 12 June 2006
The phase space probability density for steady heat flow is given. This generalizes the Boltzmann
distribution to a nonequilibrium system. The expression includes the nonequilibrium partition
function, which is a generating function for statistical averages and which can be related to a
nonequilibrium free energy. The probability density is shown to give the GreenKubo formula in the
linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the
probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown
to be much more efficient for the thermal conductivity of a LennardJones fluid than the
GreenKubo equilibrium fluctuation method. The theory for heat flow is generalized to give the
generic nonequilibrium probability densities for hydrodynamic transport, for timedependent
mechanical work, and for nonequilibrium quantum statistical mechanics. © 2006 American Institute
of Physics. DOI: 10.1063/1.2203069
I. INTRODUCTION
This paper is the culmination of a series on the theory of
steady state systems.14
The goal of the research has been to
