Approach to hyperuniformity of steady states of facilitated exclusion processes.

We consider the fluctuations in the number of particles in a box of sizeLdinZd,d⩾1, in the (infinite volume) translation invariant stationary states of the facilitated exclusion process, also called the conserved lattice gas model. When started in a Bernoulli (product) measure at densityρ, these systems approach, ast→∞, a 'frozen' state forρ⩽ρc, withρc=1/2ford = 1 andρc<1/2ford⩾2. Atρ=ρcthe limiting state is, as observed by Hexner and Levine, hyperuniform, that is, the variance of the number of particles in the box grows slower thanLd. We give a general description of how the variances at different scales ofLbehave asρ↗ρc. On the largest scale,L≫L2, the fluctuations are normal (in fact the same as in the original product measure), while in a regionL1≪L≪L2, with bothL1andL2going to infinity asρ↗ρc, the variance grows faster than normal. For1≪L≪L1the variance is the same as in the hyperuniform system. (All results discussed are rigorous ford = 1 and based on simulations ford⩾2.).

Exact free energy functional for a driven diffusive open stationary nonequilibrium system.

We obtain the exact probability exp[-LF([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-->infinity. 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.

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[-NF([rho])]; 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.