Normal approximation for a random elliptic equation

We consider solutions of an elliptic partial differential equation in (mathbb{R }^d) with a stationary, random conductivity coefficient that is also periodic with period (L) . Boundary conditions on a square domain of width (L) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit (L rightarrow infty ) , this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size (L) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.