Dispersion in rectangular networks: effective diffusivity and large-deviation rate function

The dispersion of a diffusive scalar in a fluid flowing through a network has many applications including to biological flows, porous media, water supply and urban pollution. Motivated by this, we develop a large-deviation theory that predicts the evolution of the concentration of a scalar released in a rectangular network in the limit of large time $t \gg 1$. This theory provides an approximation for the concentration that remains valid for large distances from the centre of mass, specifically for distances up to $O(t)$ and thus much beyond the $O(t^{1/2})$ range where a standard Gaussian approximation holds. A byproduct of the approach is a closed-form expression for the effective diffusivity tensor that governs this Gaussian approximation. Monte Carlo simulations of Brownian particles confirm the large-deviation results and demonstrate their effectiveness in describing the scalar distribution when $t$ is only moderately large.