The spatial arrangement of elements pertaining to the GF solution, which depends on the two distances r and r 0. The initial location of the particle, r = r 0, can be either inside or outside the reactive sphere, r = σ. This divides space, in two different ways, into three regions (indicated by 1, 2, and 3). The particle can react only within the colored sphere, where r ⩽ σ.
The distance dependence of the Laplace transformed GF for an initially unbound particle at three values of r 0: inside (r 0 = 0.5), on the surface (r 0 = 1), or outside (r 0 = 2) the reaction sphere. Lines demonstrate Eqs. (3.5) and (3.6) , or (3.9) and (3.10) , for s = 1 (full line), 5 (dashed line), and 10 (dashed–dot line). All the figures in this publication use the same parameters set: σ = 1, D = 1, k d = 0.5, and k r = 3.
The distance dependence of the GF in the time-domain, for an initially unbound particle starting outside the reaction sphere, at r 0 = 2 > σ. Black lines demonstrate the Laplace inverse of Eqs. (3.5) and (3.6) for p(r, t|r 0), as well as that of Eq. (2.10) for the probability density of the bound state, q(r, t|r 0).
Same as Fig. 3 for r 0 = 1 = σ.
The time dependence of the GF for a particle that is initially bound at r 0 = 0.5. The LT of the GF for the unbound state is given by Eq. (3.9) multiplied by k d /(s + k d ) and its Laplace inverse is depicted by the blue lines. The LT of the GF for the bound state, , is calculated from using Eq. (2.13) and its Laplace inverse is depicted by the red lines. The numerical solution of the PDE (2.1) for these initial conditions is shown by the dotted green lines.
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