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

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