Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case

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 $\rho(x)$ is $\exp[-N{\cal F}(\{\rho\})]$; ${\cal F}$ thus generalizes to nonequilibrium systems the notion of free energy density for equilibrium systems. Our exact expression for $\cal F$ 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.