Modelling anomalous diffusion in semi-infinite disordered systems and porous media
Ralf Metzler, Ashish Rajyaguru, Brian Berkowitz
Abstract
Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> in a semi-infinite space for the boundary condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> , 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.