Transport with multiple-rate exchange in disordered media

We investigate transport of particles subject to exchange using the continuous-time random-walk framework. Transition is controlled by macroscale, and exchange by both macroscale and microscale disorder. A wide class of exchange mechanisms is represented using the multiple-rate exchange model. Particles are transported along random trajectories viewed as one-dimensional lattices. The solution of the transport problem is obtained in the form of the crossing-time density, h(t; L), at an exit surface L; h is dependent on two functions, g and f. g characterizes exchange controlled by microscale disorder. The joint density f is central for the solution as it relates the microscale and macroscale disorder along random trajectories. For the case of transition and exchange disorder, we show that power-law exponent eta (characterizing microscale disorder) and power-law exponents alpha(tau) and alpha(mu) (characterizing macroscale disorder), define two regions delimited by a line alpha(tau) = alpha(mu)(eta + 1): One in which the asymptotic transport is dominated by transition, and one in which it is dominated by the exchange. For the case of transition disorder with uniform exchange, both transition and exchange cart influence the late-time behavior of h(t). Microscale exchange processes will unconditionally influence the late-time behavior of h(t) only if eta<0. If eta>0, exchange will dominate at late time provided that transition is asymptotically Gaussian.