Temporal Asymptotic Form of the Survival Probability in the Effective Medium Approximation for Trapping of Particles in
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Temporal Asymptotic Form of the Survival Probability in the Effective Medium Approximation for Trapping of Particles in Media with Anomalous Diffusion V. E. Arkhincheeva,b aLaboratory
of Applied Physics, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, 700000 Vietnam b Faculty of Applied Science, Ton Duc Thang University, Ho Chi Minh City, 700000 Vietnam e-mail: [email protected] Received January 28, 2020; revised January 28, 2020; accepted February 11, 2020
Abstract—The capture of particles diffusing anomalously in accordance with a power law in absorbing traps is analyzed in the effective medium approximation. A new slower power-law dependence of the asymptotic form of the particle survival probability over long times has been established. This result is due to the anomalous diffusion of particles in strongly anisotropic media. DOI: 10.1134/S1063776120060102
1. INTRIDUCTION The problem of diffusion of particles in media with absorbing traps was analyzed by many authors [1–3]. In the case of temporal capture of particles in traps followed by their release, it is transformed into the problem of continuous time random walk (CTRW) [4, 5]. Capture in traps is also of considerable interest for describing diffusion-controlled chemical reactions and has been studied in a large number of publications [6–8]. It was shown that in the case of fully absorbing traps and in the effective medium approximation presuming a uniform spatial distribution of absorbing traps, the survival probability of particles diffusing in conventional manner is given by [3] (1) W (t, c) ∝ W0 exp(−Dtc 2 ). Here, D is the diffusion coefficient and c is the concentration of traps in the 1D case. Accordingly, a characteristic diffusion time appears at distances on the order of the mean distance between traps, tc = 1/Dc2. It should be emphasized that result (1) given above corresponds to the case of so-called conventional diffusion. We interpret conventional diffusion as random walks in which the rms displacement is a linear function of time: (2) X (t ) ∝ Dt. In the effective medium approximation, we obtain result (1). However, there is a large number of stochastic processes of anomalous subdiffusion type (i.e., with a
power-law time dependence of the rms displacement) [9, 10]:
X 2(t ) ∝ t k ,
k < 1.
(3)
Accordingly, it would be interesting to find out how temporal dependence (1) changes in the case of anomalous diffusion processes. This study is devoted to analysis of this problem in the effective medium approximation. In Section 2, we briefly introduce the problem of capture of particle in traps. In Section 3, we consider the fractional-order generalized diffusion equation. In Section 4, the probability of particle survival in media with absorbing traps is investigated for anomalous subdiffusion processes. The results are summarized in Section 5. 2. PROBLEM OF CAPTURE OF DIFFUSING PARTICLEES IN MEDIA WITH TRAPS Let us briefly recall familiar results. According to the approach
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