Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere

We study the problem of a diffusing particle confined in a large sphere in the n-dimensional space being absorbed into a small sphere at the center. We first non-dimensionalize the problem using the radius of large confining sphere as the spatial scale and the square of the spatial scale divided by the diffusion coefficient as the time scale. The non-dimensional normalized absorption rate is the product of the physical absorption rate and the time scale. We derive asymptotic expansions for the normalized absorption rate using the inverse iteration method. The small parameter in the asymptotic expansions is the ratio of the small sphere radius to the large sphere radius. In particular, we observe that, to the leading order, the normalized absorption rate is proportional to the (n − 2)-th power of the small parameter for n 3 ≥ .


Introduction
Search theory represents the birth of operations analysis [1]- [4]. One of the classical search problems involves a searcher equipped with a cookie-cutter sensor looking for a single moving target. A cookie-cutter sensor can detect a target instantly when the target gets within distance R to the searcher and there is no deteciton when the target range is larger than R. One interesting mathematical challenge is to find the probability of a diffusing target avoiding detection by a stationary cookie-cutter sensor. This problem has been addressed by Eagle [5] where the search region is a two-dimensional disk. Recently we have revisited this problem and have derived a unified asymptotic expression for the decay-rate of the non-detection problability which is valid for the cases where the search region is either a disk or a square [6].
In this paper, we would like to extend our earlier work [6] to high dimensions. More specifically, we investigate the absorption rate into a small sphere such as a cookie-cutter sensor for a difusing particle (i.e. target) confined in a large sphere (i.e. search region).
From the next section, the paper is outlined as follows. We first present the mathematical formulation of the problem in Section 2. Then we consider the special case of the three dimensions in Section 3 and derive the exact solution for this case in Section 4. Section 5 and Section 6 describe the solutions for dimension four and dimension five, respectively. These asymptotic solutions are validated against the accurate numerical solutions of a Sturm-Liouville problem in Section 7. Finally, Section 8 summarizes the paper.

Mathematical Formulation
We consider a particle in the n-dimensional space n  , undergoing a Brownian diffusion with diffusion coefficient D. Let ( ) 0, B R denote the ball in n  , of radius R and centered at the origin S R denote the sphere in n  , of radius R and centered at the origin. ( ) 0, S R is the boundary of ( ) 0, B R . We consider the situation where the diffusing particle is confined from outside by a large sphere ( ) 2 0, S R and is absorbed near the origin by a a small sphere ( ) Figure 1 shows the geometry of the problem setpup in the three dimensional space ( 3 n = ).

Let ( )
, p x t be the probability of the particle being at position x at time t . ( , ) p x t is governed by the diffusion equation with boundary and initial conditions: where 2 ∇ denotes the Laplace operator and p p ∂ ≡ ⋅ ∇ ∂ n n represents the directional derivative of p along the normal vector n of ( ) 2 0, S R . We first perform non-dimensionalization to make the problem dimensionless. Let x t has the meaning of probability density with respect to new x . It satisfies the initial boundary value problem below (we drop the subscript " new " for simplicity): After non-dimensionalization, the outside confining sphere has radius 1 and the inside absorbing sphere has radius 1 ε  .
The solution of initial boundary value problem (2) can be expressed in terms of exponentially decays of eigenfunctions.
We consider the survival probability: Over long time, the decay of survival The normalized decay rate of survival probability is 1 λ , which is dimensionless. The physical decay rate (before non-dimensionalization) of survival probability is related to the normalized decay rate 1 λ as In the two-dimensional case ( 2 n = ), we showed that for small ε the smallest eigenvalue 1 λ has the expansion (Wang and Zhou, 2016) In this study, we derive asymptotic expansions for the smallest eigenvalue 1 λ in the cases of 3 n = , 4 n = and 5 n = . For simplicity, we drop the subscript " 1 ", and use λ to denote the smallest eigenvalue and ( ) u x to denote a corresponding eigenfunction. Since an eigenfunction for the smallest eigenvalue is axisymmetric, We use the inverse iteration method to derive an asymptotic expansion for λ , starting with an initial guess for eigenfunction: Specifically, we solve the linear differential equation with boundary conditions below to update the In the first iteration ( 0 k = ), the delta function on the right hand side can be conveniently incorporated into the boundary condition at 1 r = . For 0 k = , Equation (10) becomes An approximation to the smallest eigenvalue λ is calculated as In the subsequent sections, we show that 2 9 459 3 1 ) for 3 5 175 n λ ε ε ε

The Three Dimensional Case: n = 3
For the three dimensional case ( 3 n = ), the differential equation in (10) has the form ( ) We first solve for two independent solutions of (16) in the case of ( ) 0 g r ≡ without any boundary condition ( ) ( ) With these results, we start the inverse iteration. For the first iteration ( 0 k = ), the solution of (11) is a linear combination of two independent solutions 1 r and 1.
The corresponding approxomation for λ using (12) is In the third iteration ( 2 k = ), the right hand side of (10) is The solution of (10) is constructed using The corresponding approxomation for λ using (12) is Therefore, in the three dimensional case, λ has the expansion 2 9 459 3 1 for 3 5 175 n λ ε ε ε For the three dimensional case, the smallest eigenvalue λ can be written as the exact solution of a transcendental equation, which provides an alternative way of deriving the asymptotic expansion. This is carried out in the next section.

Exact Solution for the Special Case of n = 3
For the special case of 3 n = , we write ( ) u r as Substituting it into (9) for 3 n = , we derive the differential equation for ( ) The iterative formula gives us With these results, we start the inverse iteration. For the first iteration ( 0 k = ), the solution of (11) is a linear combination of two independent solutions 3 1 r and 1. The one-term asymptotic solution is not very good in this range of ε . Nevertheless, as ε is reduced, the one-term asymptotic solution converges slowly to the true solution, which is represented by the accurate numerical solution in Figure 2. The two-term asymptotic solution is better than the one-term solution. The three-term asymptotic solution is even better. For 0.2 ε ≤ , the three-term asymptotic solution is indistinguishable from the true solution. Figure 3 compares 2 asymptotic solutions and a very accurate numerical solution in the four dimensional case ( 4 n = ). The one-term asymptotic solution in the four dimensional case (Figure 3) is much more accurate than that in the three dimensional case (Figure 2). In Figure 3, the one-term solution is very close to the true solution for 0.1 ε ≤ . This indicates that as n is increased, the leading order asymptotic solution becomes more accurate. The two-term asymptotic solution in Figure 3  , confirming the trend that in higher dimensional space (larger n), the leading order asymptotic solution is more   accurate than that in lower dimensional space (smaller n). The two-term asymptotic solution in Figure 4 is indistinguishable from the true solution even at 0.4 ε = .
In each case ( 3 n = , 4 n = , or 5 n = ), the most accurate asymptotic solution coincides with the true solution, at least, for 0.2 ε ≤ .

Concluding Remarks
The focus of this paper was to calculate the absorption rate into a small sphere for a diffusing particle which was confined in a large sphere. Under the assumption that the ratio of the small sphere radius to the large sphere radius was small, we derived asymptotic expansions for the normalized absorption rate with the inverse iteration method.