Temporal Asymptotic Form of the Survival Probability in the Effective Medium Approximation for Trapping of Particles in Media with Anomalous Diffusion

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.

Derivation of the Fractional-Order Equation in the Comb Model

Let us briefly recall the comb model. This model was introduced for the first time for describing subdiffusion on percolation clusters [13]. This model includes a conducting axis (analog of the skeleton of percolation clusters) and reflects the main feature of random walks in inhomogeneous media (anomalous behavior).

Fig. 1.

Comb model: backbone axis and fingers fixed to it.

Diffusion along the x axis in the comb model is possible only for y = 0. This means that diffusion coefficient D_xx differs from zero only for y = 0 [14, 15]:

$${{D}_{{xx}}} = {{D}_{1}}\delta (y),$$

(A.1)

i.e., the x component of the diffusion current is given by

$${{J}_{x}} = - {{D}_{{xx}}}\frac{{\partial N}}{{\partial x}}.$$

(A.2)

Diffusion along the fingers of the comb structure is of conventional type (D_yy = D₂).

Consequently, random walks over the comb structure are described by diffusion tensor

$${{D}_{{ij}}} = \left( {\begin{array}{*{20}{c}} {{{D}_{1}}\delta (y)}&0 \\ 0&{{{D}_{2}}} \end{array}} \right).$$

(A.3)

Using Fick’s law J_d = –$\hat {D}$∇N for the diffusion current with diffusion tensor (A.3), we obtain the diffusion equation.

Further, we perform the Laplace transformation in time and the Fourier transformation in the x coordinate:

$$\left[ {s + {{D}_{1}}{{k}^{2}}\delta (y) - {{D}_{2}}\frac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}}} \right]G(s,k,y) = \delta (y).$$

(A.4)

Here, G(s, k, y) is the Green function with initial conditions in the form of point source δ(x)δ(y)δ(t). We will seek the expression for the Green function in form

$$G(s,k,y) = g(s,k)\exp ( - \lambda {\text{|}}y{\text{|}}).$$

(A.5)

Substituting this expression into Eq. (A.4), we obtain the regular equation and the equation with singular part δ(y):

$$[s - {{D}_{2}}{{\lambda }^{2}}]G(s,k,y) = 0,$$

(A.6)

$$[{{D}_{1}}{{k}^{2}} + 2\lambda {{D}_{2}}]\delta (y)g(s,k,y) = \delta (y).$$

(A.7)

This gives the fractional-order effective diffusion equation for random walks along the backbone axis of the comb structure:

$$\left( {\frac{{{{\partial }^{{1/2}}}}}{{\partial {{t}^{{1/2}}}}} - {{D}_{{{\text{eff}}}}}\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}}} \right)g(t,x) = 0.$$

(A.8)