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\documentstyle[aps,multicol,epsf]{revtex}
\begin{document}
\title{Partial Absorption and ``Virtual'' Traps} 
\author{E.~Ben-Naim and S.~Redner}
\address{Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\author{G.~H.~Weiss}
\address{PSL, DCRT, National Institutes of Health, Bethesda, MD 20205}
\maketitle
\begin{abstract}
The spatial probability distribution associated with diffusion
and attenuation in partially absorbing media is studied.  An equivalence
is established between a system with free diffusion for $x>0$ and
partial absorption for $x<0$, and a semi-infinite system ($x>0$) with a
radiation boundary condition at $x=0$.  By exploiting this equivalence,
it is shown that the effect of a partially absorbing medium in the
long-time limit is equivalent to that of a perfect, ``virtual'' trap
whose size is smaller than the original absorbing medium.  For short
times, however, there is substantial penetration of diffusing particles
into the absorber.  The virtual trap approach is readily generalized to
higher dimensions.  This allows one to obtain the density profile of
diffusing particles around a partially absorbing spherical trap.  An
unusual crossover between short-time penetration and long-time trapping
occurs in two dimensions; the size of the virtual trap is
exponentially small in the case of weak absorption, corresponding to an
absorption time which is exponentially large.
\end{abstract}
\vspace{.1in}
\noindent Key Words: partial absorption, virtual trap, radiation
boundary condition
\begin{multicols}{2}
\section{Introduction}
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 {\it 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
non-absorbing 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 $t^{1/2}$, before the asymptotic $t^{1/4}$ 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 $x\leq0$ and a virtual perfect trap located at a position $r_T<0$.
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
long time regime of negligible absorption before the virtual trap
``turns on'', in the limit of weak trapping.
\section{Diffusion-Absorption Equation}
Consider a one-dimensional medium which is partially absorbing for
$x<0$, with a finite absorption rate $q$, and non-absorbing for $x>0$
(Fig.~1).  The time evolution of the density outside the absorber obeys
the diffusion equation, while the density inside obeys a diffusion
absorption equation: 
\begin{eqnarray}
{\partial c(x,t)\over \partial t}&=
D\displaystyle{{\partial^2 c(x,t)\over\partial x^2}}-qc(x,t) \hspace{.2in}
&x<0
\\ \nonumber\\ 
\vspace{-.1in}\\ \vspace{-.1in}
{\partial c(x,t)\over \partial t}&=
\displaystyle{D{\partial^2 c(x,t)\over \partial x^2}} \hspace{.7in}&x>0\nonumber
\end{eqnarray}
where $D$ is the diffusion coefficient.  At the origin, the
concentration and the flux ($-D\partial c/\partial x$) 
must be continuous.  Once the
initial condition is specified, one can then solve for basic quantities,
such as the concentration at the interface, and the total number of
particles inside the absorber, $S_-(t)=\int_{-\infty}^0c(x,t) dx$.  This
latter quantity is obtained by spatial integration of the first line of
Eq.~(1) to yield
\begin{equation}
{\partial S_-(t)\over\partial t} =D{\partial c(x,t)\over\partial x}
\Bigl |_{x=0}-qS_-(t).
\end{equation}
This reflects the decrease in $S_-(t)$ because of absorption, and its
increase because of flux entering at $x=0$.  We now present the solution
to Eq.~(1) for two simple initial conditions, although the analysis can
be carried out for an arbitrary initial condition.

\subsection{Uniform initial density for $x>0$}

For $c(x,t=0)=c_0H(x)$, with $H(x)$ the Heaviside step function, we
first introduce the Laplace transform, $c(x,s)=\int_0^\infty
c(x,t)\exp(-st)\,dt$, to reduce Eq.~(1) to the ordinary
differential equations 
\begin{eqnarray}
&\displaystyle{D{\partial^2c(x,s)\over\partial x^2}}-(s+q)c(x,t)=0 
\hspace{.2in}&x<0\nonumber\\ 
\vspace{-.2in}\\ \vspace{-.2in}
&\displaystyle{D{\partial^2c(x,s)\over\partial x^2}}-s c(x,t)=-c_0 
\hspace{.4in}&x<0\nonumber
\end{eqnarray}
subject to continuity of both $c(x,s)$ and its spatial derivative at the
origin.  The solution in Laplace space is
\begin{equation}
c(x,s)=\cases{
{\displaystyle{c_0\over s}}
\Bigl({\displaystyle{{1\over 1+\alpha(s)}}}\exp(x\sqrt{(s+q)/D})\Bigr)
&$x<0$,\cr
\cr
{\displaystyle{c_0\over s}}
\Bigl(1-{\displaystyle{{1\over 1+\alpha^{-1}(s)}}}\exp(-x\sqrt{s/D})\Bigr)
&$x>0$,
\cr}
\end{equation}
where $\alpha(s)=\sqrt{(s+q)/s}$.

We are primarily interested in extracting quantities which characterize
the penetration of particles into the absorber, such as the
concentration at the interface, $c(x=0,s)=c_0/s\big(1+\alpha(s)\big)$,
and the number of particles inside the absorber
$S_-(s)=c_0\sqrt{D/(s+q)}/s(1+\alpha(s))$.  Inverting the above Laplace
transforms yields the behavior of these two quantities in the time
domain[5],
\begin{equation} c(x=0,t)={c_0\over
2}\Bigl(I_0(t/2t_q)+I_1(t/2t_q)\Bigr)\exp(-t/2t_q),
\end{equation} 
\begin{equation} S_-(t)=c_0l_q{\sqrt{1\over\pi
t/t_q}}\Bigl(1-\exp(-t/t_q)\Bigr),
\end{equation} 
where $I_n(x)$ is the $n$th order modified Bessel function.  Here we
have introduced dimensionless variables by introducing the
characteristic absorption time, and the typical distance that a particle
travels in the absorber before absorption 
\begin{equation} t_q=1/q\qquad {\rm and}\qquad
l_q=\sqrt{D/q}\,(=\sqrt{Dt_q}).
 \end{equation} 
Interestingly, the concentration at the origin is independent of $D$.
In contrast, the behavior of $S_-(t)$ does depend on $D$, since the flux
entering the absorber at a fixed time depends on $D$.

For short times, $t\ll t_q$, absorption is negligible, and one recovers
purely diffusive behavior, namely $S_-(t)\sim c_0\sqrt{Dt/\pi}$ and
$c(x,t)\cong c_0/2$.  In this ``diffusion controlled'' time regime,
therefore, the absorption term in the diffusion-absorption equation can
be neglected.  In the long time limit, we use [5]
\begin{equation} 
I_n(z)\cong
{\displaystyle{e^z\over \sqrt{2\pi z}}}
\left(1-{n-1\over 8z}+\ldots\right)\qquad z\gg 1,
\end{equation}
to obtain 
\begin{equation}
c(x=0,t)\sim {c_0\over\sqrt{\pi t/t_q}},\qquad t\gg t_q.
\end{equation}
Comparison with Eq.~(6) shows that the concentration at the origin
and the number of particles inside the absorber are proportional to each
other in the long-time limit,
\begin{equation}
c(x=0,t)\sim {1\over l_q}S_-(t) \qquad t\gg t_q.
\end{equation}
Since
$S_-(t)\sim t^{-1/2}$, the time derivative will be a subdominant
contribution in Eq.~(2).  Neglecting this term and using Eq.~(6) gives,
\begin{equation}
D{\partial c(x,t)\over\partial x}\Big |_{x=0}\sim qS_-(t)\sim
Dc_0/\sqrt{\pi Dt},
\end{equation}
{\it i.e.}, the flux is the same as that entering a perfect trap.  There also
is a depletion zone (the extent over which the concentration is
spatially varying) which is of the order of $\sqrt{Dt}$.  In the
long-time regime, therefore, the solution to Eq.~(1) is essentially
determined by the absorption term, since the time derivative in the
diffusion-absorption equation is negligible.  Thus the solution can be
characterized as being ``absorption controlled''.

A useful way to characterize the penetration of particles into the
absorber is to monitor the time dependence of the average position of
the particles which are in the absorbing medium,
$\langle{\ell(t)}\rangle\equiv-\int_{-\infty}^0 x\,c(x,t)\, 
dx/S_-(t)$.  From the
first line of Eq.~(4), the Laplace transform of
the numerator of $\langle{\ell(t)}\rangle$ is 
$Dc_0/s(s+q)\bigl(1+\alpha(s)\bigr)$.
Laplace inversion gives
\begin{equation}
S_-(t)\,\langle{\ell(t)}\rangle=
c_0\,l_q^2\,\Bigl(I_0(t/2t_q)\exp(-t/2t_q)-\exp(-t/t_q)\Bigr).
\end{equation}
The penetration depth thus has the asymptotic behaviors
\begin{equation}
\langle{\ell(t)}\rangle\cong\cases{\sqrt{\pi Dt},&\qquad$t\ll t_q$\cr
                   l_q,&\qquad$t\gg t_q$.\cr} 
\end{equation} 
The short-time behavior merely recovers the behavior of freely diffusing
particles.  On the other hand, the long-time behavior can be understood
in a simple way by applying a quasistatic approximation to the
diffusion-absorption equation.  This involves neglecting the time
derivative term in this equation, and then solving subject to the
time-dependent boundary condition $c(0,t)=c_0\sqrt{t_q/\pi t}$ 
(Eq.~(9)).  Inside the absorber, this yields 
\begin{equation}
c(x,t)\sim {c_0\over \sqrt{\pi t/t_q}}\exp(x/l_q),\qquad x<0,
\end{equation}
which immediately gives the above long-time behavior of
$\langle{\ell(t)}\rangle$.

\subsection{Single particle initial condition}
For the complementary problem where one particle is initially
at the origin, $c(x,t=0)=\delta(x)$, the
solution to Eq.~(1) in Laplace space is 
\begin{equation}
c(x,s)=\cases{{\displaystyle{1\over\sqrt{Ds}}{1\over 1+\alpha(s)}}
\exp\big(x\sqrt{(s+q)/D}\big)&$x<0$,\cr\cr
{\displaystyle{1\over\sqrt{Ds}}{1\over 1+\alpha(s)}}
\exp\big(-x\sqrt{s/D}\big) &$x>0$.\cr}
\end{equation}
Thus the survival probability, $S(t)=\int_{-\infty}^{\infty}c(x,t) dx$, equals
\begin{equation}
S(t)=I_0(t/2t_q)\exp(-t/2t_q)\sim\left(t_q/\pi t\right)^{1/2}
\end{equation}
for $t/t_q\to\infty$,
while the survival probability in the absorbing medium is
\begin{eqnarray}
S_-(t)&&={1\over 2}\Bigl(I_0(t/2t_q)-I_1(t/2t_q)\Bigr)\exp(-t/2t_q)\nonumber \\
&&\sim{1\over{2\sqrt{\pi}}}(t_q/t)^{3/2}, 
\end{eqnarray} 
when $t/t_q\to\infty$. 
These quantities are independent of the diffusion coefficient, in
contrast to the corresponding behavior for the uniform initial
condition.  By applying standard Bessel function identities, it follows
that $\dot S(t)=-qS_-(t)$, {\it i.e.}, the loss of particles is simply related
to the concentration in the absorbing medium, as required by integration
of Eq.~(1) over all space.  Similarly, the density at the origin is 
\begin{equation}
c(x=0,t)={t_q\over \sqrt{4\pi Dt^3}}\Bigl(1-\exp(-t/t_q)\Bigr). 
\end{equation}
For short times, this density is diffusion controlled $\big(\sim (4\pi
Dt)^{-1/2}\big)$, while for later times $c(0,t)$ approaches the value
$S_-(t)/l_q$.  Other properties such as the concentration profile inside
the absorber and the average penetration are similar to the one obtained
in the many particle case.

Analogous results can be obtained within a discrete random walk
formulation, where absorption is described by a random walk
which is annihilated with probability $0\leq p\leq 1$ when it lands on
an absorbing site.  The dominant contribution to the survival probability at
the $n$th step arises from walks which remain in the region $x>0$ up to
this time, and this latter have probability which is proportional 
to $n^{-1/2}$.
Similarly, the survival probability inside the absorber is dominated by
walks which enter the absorbing region for the first time at the
$n$th step, which have probability proportional to $\sim n^{-3/2}$.
Formally, the full probability distribution in the composite can be
found in terms of $P_n(x,r)$, the probability that an $n$ step random
walk at position $x$ has visited the region $x<0$ exactly $r$ times [6,7].


\section{The Nearest-Neighbor Distance}


One useful way to quantify the distribution of diffusing particles near
a trap is through the position of the nearest particle to the trap,
$x_m$,[8-11] which may be defined by $\int_0^{x_m} c(x,t)dx=1$.  In
one dimension with a perfect absorber at the origin, $x_m\propto
(Dt/c_0^2)^{1/4}$.  For a sufficiently weak trap, however, we shall
demonstrate that there is an intermediate time regime where the closest
distance grows as $t^{1/2}$.  

From our previous results for the
concentration at the origin and its first derivative, the behavior of
the concentration near the origin is
\begin{equation}
c(x,t)\sim {c_0\over \sqrt{\pi t/t_q}}+{c_0\over\sqrt{\pi Dt}}x,\qquad t\gg 
t_q.
\end{equation}
Using this form for the concentration, we obtain for $x_m$,
\begin{equation}
x_m\cong l_q\Bigl(\sqrt{1+2\sqrt{\pi t/t_0}}-1\Bigr),
\end{equation}
with $t_0=Dc_0^2/q^2$.  Eq.~(20) is valid for $t\gg t_q=1/q$, and
thus $\theta=t_0/t_q=Dc_0^2/q$, which is
a dimensionless measure of the strength of the absorber, is a relevant
parameter.  For a sufficiently weak absorber, $\theta\gg 1$, and the
limiting behaviors of Eq.~(20) are,
\begin{equation}
x_m\sim\cases{(\pi t/t_qc_0^2)^{1/2}&$t_q\ll t\ll t_0$,\cr\cr
          (4\pi Dt/c_0^2)^{1/4}&$t_0\ll t$.\cr}
\end{equation}
The different limiting behaviors arise from the two terms in Eq.~(19);
the constant term dominates at early times, while the linear term
dominates at later times.  Note that the intermediate-time behavior
depends on the absorption rate, while the long-time behavior depends
essentially on the diffusion coefficient.  For $\theta < 1$, there is no
intermediate time regime, and the asymptotic behavior of a perfect trap
is recovered.  The physical meaning of $t_0$ can be seen from the
expression for the total number of particles in the absorber (Eq.~(6)).
Namely, for $t\gg t_q$, $S_-(t)\sim\sqrt{t/\pi t_0}$, so that $t_0$ gives the
time at which the total number of particles in the absorber falls below
unity.

\section{Equivalence to the Radiation Boundary Condition}

We now establish an equivalence between the partially absorbing
one-dimensional composite and the free medium for $x>0$ with a radiation
boundary condition at $x=0$.  Such a boundary condition arises naturally
in heat conduction between two media when the heat transfer is
proportional to the temperature difference.  As we shall see, this
equivalence provides a useful tool for describing the physical
manifestations of the partially absorbing medium.

Combining Eqs.~(10) and (11), which are
valid at long times, immediately gives the radiation boundary condition [12]
\begin{equation} 
D{\partial c(x,t)\over\partial x}\Big |_{x=0}=\sqrt{Dq}\, 
c(x,t)\Big |_{x=0}.
\end{equation} 
Eq.~(22) reduces to an absorbing boundary condition for $q=\infty$
and to a reflecting boundary condition for $q=0$.  However, for $q=0$,
the long-time limit $t\gg t_q$ is never reached, and
Eq.~(22) does not hold for this case.

To appreciate the consequences of the above equivalence, we solve
the diffusion equation
\begin{equation}
{\partial c(x,t)\over\partial t} =D{\partial^2 c(x,t)\over\partial x^2}\qquad x>0,
\end{equation}
with the radiation boundary condition of Eq.~(22) and the initial condition
$c(x,t=0)=c_0H(x)$.  In Laplace space, the solution is
\begin{equation}
c(x,s)={c_0\over s}\Bigl (1-{1\over 1+\beta(s)}\exp(-\sqrt{s/D}\,x)\Bigr),
\end{equation}
where $\beta(s)=\sqrt{st_q}$, while the solution in the time domain is
\begin{eqnarray}
c(x,t)&=&c_0\,{\rm erf}\left({x\over\sqrt{4Dt}}\right)\\
&+&c_0\Biggl(\exp\Bigl({x\over l_q}+{t\over t_q}\Bigr)
{\rm erfc}\left({x\over\sqrt{4Dt}}+\sqrt{{t\over t_q}}\right)\Biggl),\nonumber
\end{eqnarray} 
with the notations of Eq.~(7).  In the long time limit,
the behavior given by the solution to the diffusion-absorption
equation is recovered.  For example, the concentration at the interface is 
\begin{equation}
c(x=0,t)=c_0\,{\rm erfc}\left(\sqrt{{t/t_q}}\right)\,\exp(t/t_q) \sim
c_0\sqrt{t_q/\pi t}, 
\end{equation}
for $t\gg t_q$. 
Similarly, the intermediate time behavior for the distance of the
nearest particle from the interface varies as $t^{1/2}$,
provided that the trap is sufficiently weak ($\theta\gg 1$).

An essential aspect of the radiation boundary condition is that it
provides, in conjunction with the image method, an alternative and
useful way to quantify the effect of a partially absorbing medium.  For
the radiation boundary condition, the concentration profile given in
Eq.~(24), when evaluated at $t=0$ and for $x<0$, leads to an initial
distribution of ``image'' concentration 
\begin{equation}
c_I(x)=c_0\bigl(2\exp(x/l_q)-1\bigr)\qquad x<0. 
\end{equation} 
This is a uniform distribution of antiparticles at concentration $-c_0$
and an exponentially decaying distribution of particles (Fig.~2(a)).
The superposition of these two components gives rise to a change in sign
of this initial image distribution at $x=-l_q\ln 2$.
Owing to the equivalence between the solutions to the diffusion-absorption
equation and to the diffusion equation with the radiation boundary condition,
the effect of a partially absorbing medium is equivalent to a free
medium with an initial distribution of images given in Eq.~(27).

This equivalence motivates our introduction of the virtual trap, which is the
position where the time dependent concentration vanishes in the solution
to the diffusion equation with the radiation boundary condition.
Equating $c(r_T,t)=0$ in Eq.~(25) and keeping only the leading behavior
for $t\gg t_q$, gives $r_T\sim -l_q$.  Thus the time dependent solution
to a perfect trap located at $-l_q$ provides an excellent approximation
to the concentration obtained by imposing the radiation boundary
condition (Fig.~2(b)).  The position of this virtual trap is at the
interface in the limit of a perfectly absorbing medium, and is
infinitely far away in the no absorption limit.

\section{Partial Absorption in Higher Dimensions}

The notion of a virtual trap can be readily extended to higher
dimensions.  In (a) below, we use the steady-state solution to the
diffusion equation to find the location of the virtual trap for a
radially symmetric geometry in arbitrary spatial dimension.  An
application to a time dependent problem is presented in (b) below, where
the virtual trap method is combined with the quasistatic approximation.

\subsection{Steady state solution}
To illustrate the utility of the virtual trap method, consider radially
symmetric flow of particles (or heat) into an imperfect trap.  The
steady state concentration exterior to an absorber of radius $a$ obeys
the $d$-dimensional Laplace's equation
\begin{equation}
Dr^{1-d}{\partial\over\partial r}r^{d-1}{\partial c(r)\over\partial r}=0
\qquad r>a,
\end{equation}
with the radiation boundary condition
\begin{equation}
D{\partial c(r)\over\partial r}\bigl |_{r=a}=\sqrt{Dq}c(r)\bigl |_{r=a}.  
\end{equation}
This equation describes, for example, the flow of particles to the trap
from a continuous source of particles at a large outer sphere (in order to
reach a steady state in one and two dimensions the system must be finite
in extent).  We are primarily interested in the solution for $r>a$
where the density takes the following forms
\begin{equation}
c(r)\propto\cases{r-r_T &$r_T=a-l_q$\cr\cr
            \ln(r/r_T) &$r_T=a\exp(-l_q/a)$\cr\cr
            r_T^{-1}-r^{-1} &$r_T=a/(1+l_q/a)$\cr}
\end{equation} 

We interpret the point where the concentration vanishes as the location
of the virtual trap.  For $d=1$, this location agrees
with the form suggested in the previous section.  For strong absorption or
for a very large trap, {\it i.e.,} $l_q/a=\sqrt{Dt_q}/a\ll 1$, the
one-dimensional expression $r_T=a-l_q$ holds for any dimension.  In the
opposite limit of weak absorption, the position of the virtual trap is
\begin{equation}
r_T\cong\cases{a\exp(-l_q/a)&$d=2$\cr\cr a/(l_q/a)
&$d=3$\cr}\qquad\qquad l_q/a\gg 1 
\end{equation} 
Thus, as expected, the size of the virtual trap vanishes in the limit of
weak absorption.  The exponential behavior occurs only in two
dimensions, so that there is a significant difference between the 
dependence of the virtual trap size on the rate $q$ for two and three
dimensions.
\subsection{Time dependent solution}
The determination of the location of the virtual trap for transient
problems is based on solving the time dependent diffusion equation with
a perfect trap at $r_T$ given by Eq.~(30), and considering this solution
for $r\ge a$.  Although the exact solution to this problem is
straightforward, it is much simpler to solve the problem in the
quasistatic approximation to find the asymptotic behavior of the
concentration.  In this approximation, the time derivative in the
diffusion equation is neglected and the time dependence is introduced by
imposing the appropriate moving boundary conditions [9]
\begin{equation}
c(r)\bigl\vert_{r=\sqrt{4Dt}}\,=c_0\qquad{\rm and}\qquad
c(r)\bigl\vert_{r=r_T}=0.
\end{equation}
These conditions reflect the fact that the depletion zone width is
controlled by diffusion, and outside this zone the concentration equals
its initial value $c_0$.  Solving Laplace's equation within the
depletion zone, $a\leq t\leq\sqrt{4DT}$, and according to the above boundary
conditions gives,
\begin{equation}
c(r,t)=\cases{c_0(r-r_T)/\sqrt{4Dt}& $ d=1$\cr\cr
         c_0\big(\ln(r/r_T)/\ln(\sqrt{4Dt}/r_T)\big)&$d=2$\cr\cr
 c_0\big(r_T^{-1}-r^{-1}\big)/\big(r_T^{-1}-\sqrt{4Dt}^{-1}\big)&$d=3$\cr}
%\qquad a\le r\ll\sqrt{4Dt}.  
\end{equation}
The concentration at the interface $r=a$ can now be evaluated by using
the expressions for $r_T$ from .Eq.~(30), 
\begin{equation}
c(a,t)=\cases{c_0\sqrt{t_q/t} &$d=1$\cr\cr
                c_0/\big((a/l_q)\ln(\sqrt{4Dt}/a)+1\big)&$d=2$\cr\cr
                c_0/(1+a/l_q)&$d=3$.\cr}
\end{equation}

The expression for $d=1$ differs by a factor of $\sqrt{\pi}/2$ from the
exact result, a byproduct of the quasistatic approximation.  For three
dimensions, in contrast, the concentration reaches a constant value,
since a steady state is ultimately reached.  The strength of the trap
does not affect the time necessary to reach a steady concentration.  In
the marginal case of two dimensions on the other hand, the time for the
concentration to become appreciably less than the initial concentration
occurs when the two terms in the denominator of Eq.~(34) are comparable.
We define this as the ``initiation'' time of the trap,
\begin{equation}
\tau\equiv{a^2\over D}\exp(2l_q/a),\qquad l_q/a\gg 1.  
\end{equation}
In the weak absorption limit, $\tau$ is exponentially long, since the
virtual trap is exponentially small.  Amusingly, $\tau$ can be rewritten
as $t_0=(a^2/D)(a/r_T)^2$, {\it i.e.}, the initiation time equals a diffusion
time multiplied by the 
ratio of areas of the imperfect trap to the the perfect trap.
The initiation time is relevant in calculating the nearest neighbor
distance, since for $t<\tau$, the nearest distance varies roughly
as $c_0^{-1/2}$, and for $t\gg\tau$, the distance grows as
$\sqrt{\ln t}$ [9-11].

\section{Summary}

We investigated the properties of diffusion in the presence of an
imperfect absorber.  Our primary result is that the effect of a partial
absorber is equivalent to a virtual, perfect trap whose spatial extent
is smaller than the partially absorbing medium.  By exploiting this
analogy, the survival probability of a diffusing particle in the
presence of a semi-infinite one-dimensional absorber was found to decay
as $\sim t^{-1/2}$, just as in the case of a perfect trap.  This result
follows because the survival probability is dominated by particles in
the non-absorbing region ($x>0$) that reach the edge of the trapping
region for the first time at time $t$.  It is this observation that
accounts for the success of the virtual trap analogy in reproducing the
principal features of the kinetics of the system.  However, in the case
of weak absorption, there is an intermediate time regime where there is
substantial penetration of particles into the absorbing medium.  The
temporal range for which this penetration occurs can be described as the
time domain before the virtual trap turns on, in the virtual trap
description.

For higher dimensions, the virtual trap size tends to zero if the
absorber is weak.  In two dimensions, this size decays exponentially and
that gives rise to an exponentially large crossover time, compared with
a power law dependence in one dimension.  Although we used terminology
appropriate for diffusing particles, the temperature obeys the same
diffusion equation and thus, this analogy is relevant to heat conduction
problems as well.  The virtual trap method is relatively versatile for treating
more general geometries and may be applied to situations such as partial
absorption of an anisotropic absorber.

\section{Acknowledgements}
Two of us (EBN and SR) thank the ARO and NSF for partial support of this
research. 
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\end{thebibliography} 

\section*{Figure Captions}

Fig2 The basic geometry in one dimension in which the region $x<0$ 
absorbs at a rate $q$.

Fig1 Illustration of the virtual trap method.  (a) For a radiation
boundary condition, an initial uniform concentration of particles $c_0$
gives rise to an image distribution which consists of a uniform
concentration $c_0$ of antiparticles and an exponential distribution of
particles.  This image distribution vanishes at $x=-l_q\ln 2$ (dots).
(b) The time dependent concentration in the case of the radiation
boundary condition (solid) is well approximated by the concentration in
the presence of a perfect virtual trap at $x=-l_q$, namely $c(x,t)=c_0\,{\rm
erf}\bigl((x+l_q)/\sqrt{4Dt}\bigr)$ (dots).  For both plots
$l_q=t_q=c_0=1$, and for (b) $t=16$.

\end{multicols}
\end{document}

\beginsection\centerline{\bf Appendix}

\smallskip
1. Error functions
\begin{equation}
\eqalign{\hbox{The error function}\qquad
         &{\rm erf}(z)={2\over\sqrt{\pi}}\int_0^z \exp(-u^2)du\cr
         \hbox{The complementary error function}\qquad
         &{\rm erfc}(z)=1-{\rm erf}(z)\qquad\cr
         \hbox{Special values}\qquad
         &{\rm erf}(-\infty,0,\infty)=(-1,0,1)\cr
         \hbox{Asymptotic expansion}\qquad
         &{\rm erf}(z)\sim 1-{\exp(-z^2)\over\sqrt{\pi}z}
          \Bigl(1-{1\over 2z^2}+\cdots\Bigr)\qquad z\gg 1\cr
         \hbox{Power series expansion}\qquad
         &{\rm erf}(z)\cong {2z\over \sqrt{\pi}}\qquad z\ll 1\cr}
\end{equation}
\smallskip
2. Modified Bessel functions of the first kind $I_n(x)$.  
\begin{equation}
\eqalign{\hbox{Asymptotic expansion}\qquad

         \hbox{Power series expansion}\qquad
            &I_0(z)\cong 1+z^2/4+\cdots\qquad z\ll 1\cr                  
            &I_1(z)\cong z/2+\cdots\cr\qquad
         \hbox{Special identity}\qquad
         &I_0'(z)=I_1(z)\cr}
\end{equation}
\smallskip

3. Special cases solutions to the Diffusion Absorption Equation
(Eq.~(1)).  

\begin{equation}
\eqalign{\hbox{Perfect Absorber }(q=\infty)\qquad
         &c(x,t)=c_0{\rm erf}\Bigl({x\over\sqrt{4Dt}}\Bigr)\cr
         \hbox{No absorber }(q=0)\qquad
         &c(x,t)={c_0\over2}\Bigl(1+{\rm erfc}\Bigl({x\over\sqrt{4Dt}}\Bigr)
           \Bigr)\cr}
\end{equation}
\end