Diffusion in periodic porous media: the large deviation regime

We consider diffusion of a tracer released suddenly inside a porous medium composed of a periodic array of impenetrable circular obstacles. Classical homogenisation theory has for long established that at large times $t\gg 1$, the evolution of the concentration may be approximated by a Gaussian approximation parameterised by an effective diffusivity. This approximation is however only valid at $O(t^{1/2})$ distances from the centre of mass. We develop a large-deviation approximation for the concentration that remains valid at large distances $O(t)$ from the centre of mass. We provide reduced descriptions for the concentration in the limiting cases of dilute and densely packed arrays as a function of the ratio of the gap width to the period which agree with finite-element simulations. These demonstrate that a  Gaussian approximation is appropriate in the dilute limit. However, the difference between the two approximations is large in the densely packed limit.