Fractal-time approach to dispersive transport in single-species reaction-diffusion

The effect of dispersive transport in the single-species reaction-diffusion models of coagulation (A+A --> A) and annihilation (A+A --> 0) is considered. This transport is modelled through a fractal-time random walk, in which the stepping-times of the walker (a typical particle) follows a renewal process characterized by a pausing time distribution proportional to a stable law at long times, psi(t) similar to t(-1-gamma), with 0 < gamma < 1 (the fractal dimension of the time). This leads to a sublinear mean squared displacement for the particles: (r(2)(t)) similar to t(gamma). The decay of the concentration of particles, A(t), is obtained for all space dimensions d, and for the whole course of the reactions. The obtained results are exact for short and long times, with the long time asymptotics A(t) similar to t(-gamma/2) for d = 1, A(t) similar to ln(t)t(-gamma/2) for d = 2 and A(t) similar to t(-gamma) for d greater than or equal to 3. The effect of highly non-homogeneous space distributions of particles is also considered. It is found that a fractal segregation of dimension alpha (with 0 < alpha < d) in the initial distribution of particles in the space leads to A(t) similar to t(-gamma alpha/2) for d = 1, A(t) similar to ln(t)t(-gamma alpha/2) for d = 2 and A(t) similar to t(-gamma+gamma(d-alpha/2)) for d greater than or equal to 3, d - 2 < alpha < d and A(t) similar to cte > 0 for 0 < alpha < d - 2. This shows a subordination phenomenon in the combination of space-and time-fractal distributions.