New Temporal Asymptotics of the Survival Probability in the Capture of Particles in Traps in Media with Anomalous Diffusion

Abstract

The problem of capture of particles experiencing normal diffusion as well as anomalous subdiffusion to absorbing traps has been investigated. It is shown that two characteristic diffusion times (and, accordingly, three time intervals) emerge in such a problem. It is established that new temporal (power-law and fractional–exponential) asymptotic forms of the particle survival probability appear in these intervals, which are determined by the character of diffusion of particles in strongly anisotropic media.

APPENDIX

1.1 Problem of Capture of Diffusing Particles in Media with Traps

Let us briefly recall the known results. According to the general approach [1–3], the solution to standard diffusion equation

$$\frac{{\partial W(t)}}{{\partial t}} = D\frac{{{{\partial }^{2}}W{{{(t)}}^{2}}}}{{\partial t}}$$

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with initial and boundary conditions

$$W(x,0) = \frac{{1 - c}}{L},\quad W({{x}_{i}},t) = W({{x}_{{i + 1}}},t) = 0$$

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is constructed. Here, L is the length of the one-dimensional chain and x_i, x_{i + 1} are the coordinates of absorbing traps along the 1D line. The solution has form

$$W(x,t) = \frac{4}{L}\sum\limits_{n = 0}^\infty {\exp \left( { - \frac{{Dk_{n}^{2}t}}{2}} \right)\frac{{\sin ({{k}_{n}}(x - {{x}_{i}}))}}{{{{k}_{n}}{{l}_{i}}}}.} $$

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Further, the obtained solution is averaged over the random distribution of absorbing traps:

$$W(t) = \sum\limits_i^{} {{{W}_{i}} = \sum\limits_i^{} {\int\limits_{{{x}_{i}}}^{{{x}_{{i + 1}}}} {W(x,t)\partial x.} } } $$

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The corresponding averaged solution obtained in this way describes the survival probability of particles after their capture in absorbing traps.