Mean field variation in random media

We consider here the basic equation \[ \frac {\partial }{{\partial {x_i}}}\left [ { \epsilon \left ( \textrm {X} \right )\frac {{\partial \phi }}{{\partial {x_i}}}\left ( \textrm {X} \right )} \right ] = \rho \left ( \textrm {X} \right )\], where $\in \left ( \textrm {X} \right )$ is a random function of position and $\rho \left ( \textrm {X} \right )$ is a prescribed source term. A formal equation is derived that governs $\left \{ {\phi \left ( \textrm {X} \right )} \right \}$, where the braces indicate an ensemble average. The equation, which depends on the boundary conditions of the stochastic problem, is presented in terms of an infinite sequence of correlation functions associated with $\in \left ( \textrm {X} \right )$. The equation is investigated first for the case of an infinite dielectric where isotropy may be assumed. An impulse response function is obtained and an explicit form of this response function is presented for the limit of small perturbations. Further, it is shown that the equation governing $\left \{ {\phi \left ( x \right )} \right \}$ is greatly simplified for the case in which all characteristic lengths associated with $\left \{ {\phi \left ( x \right )} \right \}$ are large compared to all correlation lengths ${l_i}$ associated with the $\in \left ( \textrm {X} \right )$ field. The question of boundary conditions is next considered and as an example a spherical boundary (radius $R$) is studied. It is demonstrated, in this case, that if $R \gg {l_i}$ the effects of the boundary conditions on the equation governing $\left \{ {\phi \left ( x \right )} \right \}$ are negligible except at points within a thin layer near the boundary. The relationship between the ensemble average and the local volume average is also discussed.