Exact Free Energy Functional for a Driven Diffusive Open Stationary Nonequilibrium System

Abstract

We obtain the exact probability $\exp [-LF(\{\rho \left(x\right)\}\left)\right]$ of finding a macroscopic density profile $\rho \left(x\right)$ in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system $L\to \infty$. $F$, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, $F$ is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of $F$, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.