Partial Absorption and “Virtual” Traps

The physical basis of Brownian motion in translationally invariant media are well understood, and models based on this type of transport have been applied to the interpretation of an enormous variety of physical phenomena. There is likewise a large literature on Brownian motion in the presence of absorbing boundaries. Perfectly absorbing boundaries are known to change the properties of Brownian motion in ways that are largely understood. However, for many practical applications, the trapping medium is partially absorbing, as in the attenuation and multiple scattering of light in biological media, in heat conduction processes, and in colloidal suspensions. For example, recent studies of photon migration in a turbid medium suggest the validity of Brownian motion as the description of photon transport, as well as Beer’s law of absorption [1,2]. The analysis of scattered laser light in biological tissue suggests, for example, the importance of determining the maximum penetration depth of a Brownian particle in a partially absorbing medium. Another useful parameter for interpreting experimental data is the time required for a diffusing particle to reach an absorbing surface at a given distance from the interface of the laser beam into the sample. Some aspects of this problem have been addressed for models in which the sample is a semi-infinite partially absorbing medium [1-4]. These situations motivate us to consider a theoretical analysis of the penetration of a Brownian particle in absorbing media. We are interested in developing a quantitative approach to describe how the probability distribution of the diffusing particles is influenced by partial absorption. For a one-dimensional composite that consists of an absorbing medium for x < 0 and a non-absorbing medium for x > 0, we will show that a continuum description can be given either by separate equations for the two media, or by a diffusion equation in the nonabsorbing medium with a radiation boundary condition at the interface. As a consequence, we will show that the concentration of particles inside the partially absorbing medium decays rather modestly in time, as t−1/2, leading to a concentration at the interface which also decays as t−1/2. The equivalence between the two descriptions is the basis for a physical construction in which a partially absorbing medium can be replaced by an equivalent perfectly absorbing medium of a smaller spatial extent, that is, a perfect “virtual” trap. This analogy can be easily extended to higher dimensions, both for steady state and time dependent problems, and provides a simple way to quantify the effects of partial absorption. In section II, we first determine the probability distribution for a one-dimensional composite system, which is described by a diffusion equation for x > 0 and a diffusion-absorption equation for x < 0. These results are exploited to obtain the time dependence of the maximal penetration of particles into the absorbing medium. In section III we show that for sufficiently weak absorption, an intermediate time regime exists, where the distance of the closest particle to the absorbing medium grows as t, before the asymptotic t growth sets in. An equivalence between the solution of the composite system, and that for diffusion in the half-space x > 0 with a radiation boundary condition at x = 0, is derived in section IV. This result is the basis for the correspondence between the partially absorbing medium in the range x ≤ 0 and a virtual perfect trap located at a position rT < 0. In section V, this virtual trap correspondence is extended to higher dimensions for both steady state and time dependent situations. As the strength of the partially absorbing trap decreases, the radius of the virtual trap vanishes, with a dependence that is strongly dimension dependent. Furthermore, in two dimensions there is an exponentially