We consider the steady state of an open system in which there is a flux of matter between two reservoirs at different chemical potentials. For a large system of size $N$, the probability of any macroscopic density profile $\rho \left(x\right)$ is $\exp [-NF(\{\rho \}\left)\right]$; $F$ thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for $F$ is a nonlocal functional of $\rho$, which yields the macroscopically long range correlations in the nonequilibrium steady state previously predicted by fluctuating hydrodynamics and observed experimentally.

• Received 30 April 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.150601