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.