Modelling anomalous diffusion in semi-infinite disordered systems and porous media

For an effectively one-dimensional, semi-infinite disordered system connected to a reservoir of tracer particles kept at constant concentration, we provide the dynamics of the concentration profile. Technically, we start with the Montroll-Weiss equation of a continuous time random walk with a scale-free waiting time density. From this we pass to a formulation in terms of the fractional diffusion equation for the concentration profile C(x, t) in a semi-infinite space for the boundary condition C(0, t) = C-0, using a subordination approach. From this we deduce the tracer flux and the so-called breakthrough curve (BTC) at a given distance from the tracer source. In particular, BTCs are routinely measured in geophysical contexts but are also of interest in single-particle tracking experiments. For the "residual' BTCs, given by 1- P(x, t), we demonstrate a long-time power-law behaviour that can be compared conveniently to experimental measurements. For completeness we also derive expressions for the moments in this constant-concentration boundary condition.