\documentstyle[graphicx,multicol,epsf,aps]{revtex}
\begin{document}
\title{Reaction Kinetics of Cluster Impurities}
\author{E.~Ben-Naim}
\address{The James Franck Institute, The University of Chicago, Chicago, IL 60637}
\maketitle
\begin{abstract}
We study the kinetics of
clustered immobile reactants in diffusion-controlled single-species
annihilation.  We consider the initial conditions where the immobile
reactants occupy a subspace of dimension $d_i$, while the rest of
the $d$-dimensional space is occupied by identical mobile particles.
The Smoluchowski rate theory suggests that the immobile reactants
concentration, $s(t)$, exhibits interesting behavior as a function of
the codimension, $\bar d\equiv d-d_i$.  This survival probability
undergoes a survival-to-extinction transition at $\bar d_c=2$.  For
$\bar d<\bar d_c$, a finite fraction of the immobile reactants
survives, while for $\bar d\ge \bar d_c$, $s(t)$ decays indefinitely.
The corresponding asymptotic properties of the concentration are
discussed. The theoretical predictions are verified by numerical
simulations in 2D and 3D.
\end{abstract}
\noindent P.A.C.S. Numbers: 02.50.-r, 05.40+j, 82.20.-w, 82.20.Wt

\begin{multicols}{2}
\section{Introduction}

The kinetics of diffusion-controlled  reactions have attracted
much interest recently. Despite their simplicity, the underlying
stochastic processes show complex nonequilibrium behavior.  For
homogeneous reaction processes, substantial theoretical knowledge is
available \cite{Smoluchowski-17,Chandra-43,Laider-65,Zeldo-78,Torney-83,
Racz-85,Kang-85,Lushnikov-86,Spouge-88,ben-Avraham-90}.  For reaction
processes such as single-species annihilation \cite{Kang-85} and
single-species aggregation \cite{ben-Avraham-90}, it is well known that
for $d\le 2$ spatial correlations are
important in the long time limit, while for $d>2$ such correlations
are negligible asymptotically.

A recent generalization of the annihilation process $A+A\to0$ to
situations where the reactants have different mobilities was shown to
exhibit a rich array of asymptotic behavior.  Such a process is well
suited for describing reactions that involve particles of different
sizes.  The case where a small number of particles move according
to one diffusion coefficient and the rest of the particles according
to another is especially interesting because of its simplicity. The
survival probability of a single ``impurity'' particle  in a
background of identical particles, $s(t)$, depends in a nonuniversal
fashion on the diffusivity of the impurity.  In one dimension, the
results are especially intriguing. While for the aggregation process,
a mapping to a random walk in three dimension enables an exact
solution \cite{Fisher-84}, for the annihilation process only an
approximate theory is available \cite{Krapivsky-94}.  The survival
probability of a single immobile impurity particle is equivalent to
the fraction of unvisited sites by annihilating random walkers
\cite{Cardy-94}. Additionally, as the annihilation process is
equivalent to the $T=0$ Ising model with Glauber dynamics
\cite{Glauber-63}, $s(t)$ equals the fraction of ``cold'' spins, {\it
i. e.}, the number of spins that did not flip up to time
$t$. Numerically, time-dependent and finite size simulations
\cite{Krapivsky-94,Derrida-94,Stauffer-94}, exact enumerations
\cite{Derrida-95}, as well as time power-series studies
\cite{Krapivsky-94} suggest that the impurity survival probability
depends algebraically on time, $s(t)\sim t^{-\alpha}$, with a
non-trivial exponent, $\alpha\cong0.375$, while the corresponding
exponent for the aggregation case, $A+A\to A$, is $\alpha=1$. A recent
theoretical work confirmed that the exact value is indeed $\alpha=3/8$
\cite{Derrida-pre}.  In addition, for trimolecular annihilation,
$A+A+A\to 0$, it was found that the impurity decays faster than a
power-law but slower than an exponential $s(t)\sim
\exp\big(-(\ln t)^{3/2}\big)$ \cite{Cardy-94}.  Hence, in
one-dimension, the impurity decay kinetics are highly sensitive to the
microscopic details of the reaction process.

In this study, we present a generalization of the isolated impurity
problem.  We restrict our attention to situations where the impurity
particles are immobile.  We investigate the collective behavior of
such immobile impurities by considering initial conditions in which
the immobiles are clustered. For simplicity, the immobiles occupy a
subspace of dimension $d_i$ embedded in a $d$-dimensional space.  For
convenience, we introduce the codimension $\bar d=d-d_i$. Note that
the case $\bar d=d$ corresponds to the single impurity problem.  In
the long time limit, neighboring immobile reactants ``shield'' each
other, and as a result, the immobile reactants decay significantly
slower than the background. The concentration inside the subspace is
larger than in the bulk, and consequently, the boundary between the
subspace and the bulk acts as an absorbing boundary. The geometry of
the system reduces to a $\bar d$-dimensional one.  Applying the
Smoluchowski rate theory to the mobile particles in dimension $d$ and
to the immobile particles in dimension $\bar d$, make theoretical
predictions concerning the asymptotic form of the survival probability
possible. The survival probability of an immobile reactant
undergoes a survival-extinction transition at $\bar d=\bar
d_c=2$. Below this critical codimension, a finite fraction of the
immobiles survive, while at a higher codimension their concentration decays
forever. For $\bar d=1$, the approach to the final concentration is an
algebraic one, $s(t)-s_{\infty}\sim t^{-1/2}$ for $d>2$, with a
logarithmic correction at $d=2$, $s(t)-s_{\infty}\sim
t^{-1/2}\ln t$.  At the critical codimension, $\bar d=2$, an unusual
logarithmic decay occurs $s(t)\sim(\ln t)^{-\alpha(d,\bar d\,)}$,
while for $\bar d>2$, a dimension dependent decay is found, $s(t) \sim
t^{-\alpha(d,\bar d\,)}$.

\section{The Single Impurity Problem}\smallskip

In this section we review the rate equation theory and apply it to the
single impurity problem. This approach truncates the infinite
hierarchy of equations describing the concentration at the first order.
Above the critical dimension, spatial fluctuations are asymptotically
irrelevant and thus, this theory is exact. Moreover, this theory can be
extended to arbitrary dimension, and is especially attractive due to
its simplicity.

In the lattice version of the homogeneous single-species annihilation
process, particles hop independently with rate $D$ to any one of
their nearest-neighbor sites.  An attempt to land on an occupied site
results in the removal of both particles from the system. Symbolically, the 
process can be written as 
\begin{equation}
A+A\to 0.
\label{Eq-0}
\end{equation}
Hence, the
concentration, $c(t)$, obeys the following rate equation, $dc/dt\propto
-c_{AA}$, where $c_{AA}$ is the concentration of pairs of neighboring
particles. This approach leads to an infinite hierarchy of 
equations and is of limited practical use. Alternatively, one assumes
that the annihilation rate is proportional to the flux experienced by
a particle, $dc/dt\propto -jc$ \cite{Smoluchowski-17}. This flux
can be evaluated by placing an absorbing particle in a diffusing
background of concentration $c$.  Since $j$ is proportional to $c$, the
concentration is described by the following rate equation,
\begin{equation}
{dc \over dt}=- k c^2, 
\label{Eq-1}
\end{equation}
with $k$ being the $d$-dimensional reaction rate. Above the critical
dimension $d_c=2$ the reaction rate aproaches a constant.
Otherwise, the reaction rate is time dependent since the
target particle is surrounded by a depletion zone the size of the
diffusion length $\sqrt{Dt}$. In appendix A, we detail a heuristic
derivation of the reaction rate in arbitrary dimension using the
quasistatic approximation.  Asymptotically, the reaction rate is given
by
\begin{equation}
k\sim\cases {t^{-1/2}&$d=1$;\cr
                (\ln t)^{-1}&$d=2$;\cr 
                 1&$d>2$.\cr}
\label{Eq-2}
\end{equation}
By introducing a modified time variable
\begin{equation}
z(t)=\int_0^t k(t')dt', 
\label{Eq-3}
\end{equation} 
the rate equation simplifies, $dc/dz=-c^2$.  Without loss of
generality, we set the initial concentration to unity and consequently,
the concentration $c=1/(1+z)$ is found.  
Evaluating the modified time variable, $z$, we arrive
at the following asymptotic behaviors of the homogeneous
single-species annihilation process,
\begin{equation}
c(t)\sim\cases    {t^{-1/2}&$d=1$;\cr
                   t^{-1}\ln t  &$d=2$;\cr 
                   t^{-1}       &$d>2$.\cr}
\label{Eq-4}
\end{equation}
Interestingly, these results are asymptotically exact 
\cite{Lushnikov-86,Spouge-88}, despite the assumptions 
involved with the rate theory. Moreover, a similar concentration is  obtained
for the aggregation process $A+A\to A$ as well.  In low dimensions,
the reaction proceeds with a slow rate, since particles are
effectively repelling each other. On the other hand, in high
dimensions spatial correlations are practically irrelevant, and the
concentration decays faster.  The critical dimension is characterized by
typical logarithmic corrections. 

The annihilation process can be generalized to heterogeneous
situations where the the mobilities of the reactants are not 
equal. Reactants with a diffusion coefficient $D_i$ are denoted by
$A_i$, and the annihilation process can be symbolically written as
\begin{equation}
A_i+A_j\to0. 
\label{Eq-4b}
\end{equation}
Let us consider the special case where a single impurity particle 
with diffusion coefficient $D_i$ is placed in a uniform background of 
particles with a diffusion coefficient $D$. The survival 
probability of such an impurity particle, denoted by $s(t)$, 
obeys the generalized rate equation 
\begin{equation}
{ds\over dt}=- k_i c s. 
\label{Eq-5}
\end{equation}
The reaction process between reactants of diffusivities $D$ and $D_i$
is characterized by the reaction rate $k_i $, given by (see
Appendix)
\begin{equation}
k_i\simeq\cases {K_1(D+D_i)^{-1/2}t^{-1/2}&$d=1$;\cr
            K_2(D+D_i)\left(\ln\left((D+D_i)t\right)\right)^{-1}&$d=2$;\cr
            K_d(D+D_i)&$d>2$.\cr}
\label{Eq-5b}
\end{equation}
Again, it is useful to
rewrite the rate equation in terms of the modified time variable $z$
defined by Eq.~(\ref{Eq-3}), \hbox{$ds/dz=-(k_i/k)cs$}.  Note
that in the long time limit the rate ratio approaches a constant that
depends only on the diffusion coefficient ratio.  By substituting the
asymptotic form of the concentration $c$, $c\sim 1/z$, one finds a purely
algebraic dependence of the impurity concentration $s\sim z^{k_i/k}$ for
$d\ne2$ \cite{Krapivsky-94}.  At the critical dimension, $d=2$, the
leading asymptotic correction to the rate ratio is important, and a
detailed calculation is necessary \cite{Krapivsky-94a}.  Using the
aforementioned forms of $z$ and $k_i$, the following 
 impurity concentration is found,
\begin{equation}
s(t)\sim\cases{t^{-\sqrt{\theta}/2}&$d=1$;\cr
            t^{-\theta}(\ln t)^{\theta(1-\ln \theta)}&$d=2$;\cr
            t^{-\theta} &$d\ge2$.\cr} 
\label{Eq-6}
\end{equation}
Hence, the ratio $\theta=(D+D_i)/2D$ governs the
long time kinetics.  For the case $D_i=0$, the decay exponent equals
$1/\sqrt{8}\cong0.353$ when $d=1$, and $1/2$ when $d>2$.  While the
latter value is exact, the former is only approximate.  Interestingly,
this approximate value is quite close to the observed numerical value
$0.375$ \cite{Krapivsky-94,Derrida-94,Stauffer-94,Derrida-95}.  
However, as this approximation is uncontrolled, its accuracy can
widely vary, and for the exactly soluble aggregation process, the
discrepancy in the exponents is  larger.  To summarize, both in
the supercritical regime and in the subcritical regime, the decay of
the impurities depends on the diffusivity ratio in a nongeneric
fashion.

\section{Kinetics of Cluster Impurities}


The following questions arise naturally from the above theory: Can the
presence of neighboring impurities increase the survival probability
of an impurity?  If yes, to what extent?  To answer these questions we
concentrate on the case of immobile reactants, $D_i=0$. We
introduce a generalization of the single immobile impurity problem.
Let us consider the initial configuration where the impurities occupy a
subspace of dimension $d_i$. For example, on a simple cubic
lattice, the subspace $\{ {\bf x}|x_1=\cdots=x_{d_i}=0\}$ is
occupied by impurities, with ${\bf x}\equiv x_1,\ldots,x_d$ the
Cartesian coordinates.  The rest of the space is occupied by particles
with an identical diffusion coefficient $D$.  The annihilation process is
the same as Eq.~(\ref{Eq-4b}).  The codimension $\bar d$ can be
conveniently defined as $\bar d=d-d_i$.  For example, when $\bar
d=1$, the impurities initially occupy a line in 2D, a plane in 3D,
{\it etc.}\ Figure 1 illustrates the initial configuration for the
case $\bar d=1$ in two spatial dimensions.  Note also that the single
impurity problem corresponds to the special case $\bar d=d$.

\begin{figure}[t]
\begin{center}
\begin{picture}(200,120)(40,640)
\thicklines
\put( 60,740){\circle{15}}
\put( 60,700){\circle{15}}
\put( 60,660){\circle{15}}
\put(100,660){\circle{15}}
\put(100,700){\circle{15}}
\put(100,740){\circle{15}}
\put(220,740){\circle{15}}
\put(220,700){\circle{15}}
\put(220,660){\circle{15}}
\put(180,660){\circle{15}}
\put(180,700){\circle{15}}
\put(180,740){\circle{15}}
\put(140,660){\circle*{15}}
\put(140,700){\circle*{15}}
\put(140,740){\circle*{15}}
\put(240,760){\line( 0,-1){120}}
\put(200,760){\line( 0,-1){120}}
\put(160,760){\line( 0,-1){120}}
\put(120,760){\line( 0,-1){120}}
\put( 80,760){\line( 0,-1){120}}
\put( 40,760){\line( 0,-1){120}}
\put( 40,640){\line( 1, 0){200}}
\put( 40,680){\line( 1, 0){200}}
\put( 40,720){\line( 1, 0){200}}
\put( 40,760){\line( 1, 0){200}}
\end{picture}
\end{center}
\label{Fig-1}
\caption{A line of impurities (bullets) in a two-dimensional 
background (circles)}
\end{figure}

Let us consider first a line of impurities in 2D.  The presence of nearby
static particles can only increase the survival probability of an
impurity, and thus, the survival probability is bounded by $s\sim
t^{-1/2}$, the corresponding result for the single impurity case for
$d\ge2$. However, the mobile particles decay $c\sim t^{-1}$ is
stronger and consequently, the boundary between the mobile reactants
and the immobile reactants is equivalent to an absorbing boundary for
the mobile reactants. This absorber is not a perfect one since not all
sites are occupied with impurities.  However, it is well known that in
the long time limit a partial absorber is equivalent to a perfect one
\cite{Placzek-47}.  A depletion layer of width $\sqrt{Dt}$ develops
around the subspace and the mobile reactant concentration profile is
strongly suppressed in this depletion zone.  As a result, the geometry
of the system is drastically altered.  A line in 2D reduces to a
one-dimensional geometry.  In other words, the codimension becomes the
relevant dimension.  This simple conclusion has a striking effect on
the long time kinetics of the impurities.


One possible way to tackle this problem is to describe this spatial
inhomogeneity by the reaction-diffusion equation with proper boundary
conditions.  However, the leading term in the reaction-diffusion
equation is the diffusion term, and this approach is equivalent in the
long time limit to the quasistatic approximation, or namely, the
Smoluchowsky rate theory. This theory, appealing in its simplicity,
leads to new and interesting behaviors for the cluster impurity
problem. As discussed above, the relevant dimension is the
codimension. Thus, we apply the $\bar d$-dimensional rate equation to
the impurity concentration 
\begin{equation}
{ds\over dt}=-k_i^{\bar d} c s,  
\label{Eq-7}
\end{equation}
where $k_i^{\bar d}$ is the reaction rate in $\bar d$ dimensions.  In
contrast with the previous analysis, introduction of a modified time
variable would not simplify the algebra, since the equations involve
different dimensions and consequently, different intrinsic time
scales.  Instead, the impurity survival probability, $s(t)/s(0)$, is
obtained by an integration of the above equation,
\begin{equation}
s(t)/s(0)=\exp\left(-\int_0^t dt' k_i^{\bar d}(t') c(t')\right). 
\label{Eq-8}
\end{equation}
Since the case $\bar d=d$ reduces to the single impurity case (see
Eq.~(\ref{Eq-6})), we concentrate on the case $\bar d<d$ only.  By
substituting the proper values of the concentration $c$ in
$d$-dimensions and the reaction rate $k_i^{\bar d}$ in $\bar d$-dimensions into
Eq.~(\ref{Eq-8}), we find the following asymptotic impurity densities,
\begin{equation}
s(t)\sim
\cases{
s_{\infty}+{\rm const}\times t^{-1/2}\ln t& $\bar d=1$ and $d=2$;\cr 
s_{\infty}+{\rm const}\times t^{-1/2}& $\bar d=1$ and $d>2$;\cr 
(\ln t)^{-K_{\bar d}/2K_d}&$\bar d=2$ and $d>\bar d$;\cr 
t^{-K_{\bar d}/2K_{d}}&$\bar d>2$ and $d>\bar d$.\cr 
}
\label{Eq-9}
\end{equation}
This rich behavior follows directly from the annihilation rate, {\it
i. e.}, the integrand in Eq.~(\ref{Eq-8}). If $k_i^{\bar d}(t)
c(t)$ decays faster than $1/t$, the integral remains finite in the
long time limit, and a finite fraction of the impurities survive
the annihilation process.  Otherwise, all of the impurities eventually
annihilate.  For $\bar d<2$, this integrand decays faster than $1/t$,
while for $\bar d\ge2$, the integrand is dominated by
$1/t$. Consequently, the system exhibits a survival-extinction
transition at $\bar d=2$.  Below this critical codimension, a fraction
of the impurities survive, while they decay indefinitely at a higher
codimension.

The approach to the final concentration, $s(t)-s_{\infty}\sim
t^{-1/2}\ln t$ for a line in 2D ($\bar d=1$) is reminiscent of the
single impurity decay in 2D. The two cases differ in that $s_{\infty}$
does not vanish for the cluster case.  They also differ in the
logarithmic correction.  The critical case is characterized by inverse
logarithmic decay since the annihilation rate is proportional to
$1/\left(t\ln t\right)=d\ln\left(\ln t\right)/dt$. Both the
critical case and the supercritical case follow a power-law with the
exponent $\alpha(d,\bar d\,)=K_{\bar d}/2K_{d}$. Thus, both decays
depend on the dimension as well as the codimension.  Only in the
extreme case, $\bar d=d$, corresponding to the single impurity case,
$K_d$ cancels out and the decay $s\sim t^{-1/2}$ is expected.
Generally, a detailed calculation for the dimensionless prefactors
$K_d$ is necessary in order to find the various decay exponents.
Since the decay in the cluster case $\bar d<d$ is slower than in the
single impurity case $\bar d=d$, we learn that the prefactor $K_d$ is
an increasing function of the spatial dimension $d$. This observation
is consistent with the fact that the initial reaction rate is given by
$k_i(t=0)=2(D+D_i)z_d$, with $z_d$ the number of neighboring sites in
$d$-dimensions. Indeed, $z_d$ is an increasing function of $d$. Above
the critical dimension $d=2$, the effective reaction rate depends
weakly on time.  Hence, the assumption $K_d\propto z_d$ leads to an
approximate value for the decay exponent $\alpha(d,\bar d\,)\cong
z_{\bar d}/2z_d$. For a simple square lattice one has the $z_d=2d$ or
$\alpha(d,\bar d\,)\cong \bar d/2d$. This approximation improves as
the dimension and the codimension increase.

The above results can be easily generalized to arbitrary dimensions.
Such a generalization is nontrivial only when the temporal nature of
the reaction rate is dimension dependent, or namely, below the
critical dimension. Using the reaction rates of Eq.~(A3), we evaluate
the immobile impurity densities,
\begin{equation}
s(t)\sim
\cases{
t^{-d/2^{1+d/2}}& $\bar d<2$ and $d=\bar d$;\cr s_{\infty}+{\rm
const}\times t^{-(d-\bar d\,)/2}& $\bar d<2$ and $\bar d<d<2$;\cr
s_{\infty}+{\rm const}\times t^{-(2-\bar d\,)/2}\ln t& $\bar d<2$
and $d=2$;\cr s_{\infty}+{\rm const}\times t^{-(2-\bar d\,)/2}& $\bar
d<2$ and $d>2$;\cr t^{-1/2}(\ln t)^{(1+\ln 2)/2}& $\bar d=2$ and
$d=2$\cr
(\ln t)^{-K_{\bar d}/2K_{d}}&$\bar d=2$ and $d>\bar d$;\cr 
t^{-K_{\bar d}/2K_{d}}&$\bar d>2$ and $d\ge\bar d$.\cr 
}
\label{Eq-10}
\end{equation}
To summarize, a fraction of the impurities survive only when $\bar d<2$ and
$d>\bar d$. The behavior for $d<2$ is influenced by the background
concentration behavior $c\sim t^{-d/2}$. As a result, the approach towards
the limiting concentration is algebraic with a vanishing decay exponent
$(d-\bar d\,)/2=d_i/2$ when $\bar d\,\,\,
{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\,\,\,d<2$.  It will be interesting to see how the above results
compare with Renormalization Group studies in the vicinity of the
critical codimension $\bar d=2-\epsilon$.



For completeness, we briefly discuss the early  behavior of the
system. We consider the case where all lattice sites are initially
occupied, such that $c(0)=s(0)=1$.  Following the above
discussion, the initial reaction rate is given
by $k(t=0)=2Dz_d$. From Eq.~(\ref{Eq-1}), the background
concentration is found
\begin{equation}
c(t)\cong \left(1+2Dz_d t\right)^{-1},\qquad t\to 0. 
\label{Eq-11}
\end{equation} 
On the other hand, only interfacial sites contribute to the
annihilation of impurities, and as a result $k_i (t=0)=D(z_d-z_{d-\bar
d})$. By substituting this rate and the early time mobile reactant
concentration into Eq.~(\ref{Eq-7}), the impurity concentration is
calculated in the small time limit,
\begin{equation}
s(t)\cong \left(1+2Dz_d t\right)^{-\beta(d,\bar d\,)},\qquad t\to 0. 
\label{Eq-12}
\end{equation} 
The above exponent, $\beta(d,\bar d\,)$, equals the reaction rate
ratio $\beta(d,\bar d\,)=(z_d-z_{d-\bar d})/2z_d$.  This exponent
should not be regarded as an asymptotic one, since it is relevant only
for the short time limit.  In addition to the dimension dependence,
the early time behavior depends on the lattice structure as well. As
the reaction process evolves, such details become irrelevant, and the
general asymptotic behavior is recovered. There is no sign of a
critical codimension in the early stages since the system is still
$d$-dimensional. After a sufficiently long time, the codimension
governs the kinetics.

\section{Simulation Results}

\begin{figure}
\narrowtext
%\centerline{\epsfxsize=7.6cm\epsfbox{fig2.ps}}
\centerline{\includegraphics[width=7.6cm]{fig2}}
\caption{The impurity concentration $s(t)$ versus $t$ in 2D and 3D. 
Shown are the case $\bar d=1$ in 2D (bullets) and 3D (squares), and the 
case $\bar d=2$ in 3D (triangles).}
\end{figure}


To test the theoretical predictions, we have performed Monte-Carlo
simulations for $d=2$ and 3.  The numerical implementation is
simple. Initially all $L^d$ sites of the cubic lattice are occupied
with reactants, of which $L^{d_i/d}$ are immobile and occupy the
subspace $\{ {\bf x}| x_1=\cdots=x_{d_i}=0\}$, with the Cartesian
coordinates ${\bf x}\equiv (x_1,\ldots,x_d)$. The rest of the lattice
is occupied by identical mobile particles.  Periodic boundary
conditions are imposed.  An elemental simulation step consists of
picking a mobile particle at random and moving it to a randomly chosen
neighboring site. If this site is occupied, both particles are removed
from the system.  Time is updated by the inverse of the total number
of mobile particles after each step. The linear dimension of the
lattice used in the simulation was $L=5\times10^3$, $2\times 10^2$ in
2D, and 3D respectively.  The total number of particles was thus
$25\times10^6$, and $8\times 10^6$ for 2D, and 3D.  In 2D the results
represent one realization of the process, while in 3D an average over
10 realizations is taken.  Typically, the simulation data are
reliable up to time $\propto L^2$ because due to finite size effects. 

\begin{figure}
\narrowtext
%\centerline{\epsfxsize=7.6cm\epsfbox{fig3.ps}}
\centerline{\includegraphics[width=7.6cm]{fig3}}
\caption{The quantity $s(t)-s_{\infty}$ versus time for the 
case $\bar d=1$. Shown are simulation results in 2D (bullets) and 3D (squares) 
versus the theoretical prediction of Eq.~(\ref{Eq-9}) (solid lines). }
\end{figure}
We have first verified that the average concentration of the mobile
reactants decays according to the well known concentration of
Eq.~(\ref{Eq-4}).  To verify the survival-extinction transition at
$\bar d=2$, we measured the impurity concentration versus time (see
figure 2). Indeed, for the case $\bar d=1$, $s(t)$ approaches a final
nonzero value, $s_{\infty}$, while the impurity concentration decays
indefinitely for the marginal case, $\bar d=2$.  Since the total
number of impurities initially present in the latter case is small, we
could not verify the inverse logarithmic decay of
Eq.~(\ref{Eq-9}). For the case $\bar d=1$, we performed a least square
fit of the data to the theoretical predictions of
Eq.~(\ref{Eq-9}). The optimal final concentration was found to be
$s_{\infty}\cong0.095$ and $0.335$ in 2D and 3D, respectively.  We
verified that similar results occured for smaller system sizes.  In
Figure 3, the concentration difference $s(t)-s_{\infty}$ is plotted
versus time. According to the rate theory, this quantity should decay
as $t^{-1/2}\ln t$ in 2D, and as $t^{-1/2}$ in 3D. It is seen that the
theoretical curves (solid lines) fit well the simulation data in the
long time limit. We conclude that the simulation results are
consistent with the theoretical predictions.


\section{Conclusions}

We have studied the kinetic behavior of clusters of immobile reactants
in the single-species annihilation process.  The codimension, $\bar
d$, plays an important role in the long time limit.  The Smoluchowsky
theory shows that a transition from survival to extinction takes place
at $\bar d=2$. For $\bar d<2$, a finite fraction of the impurities
survive, while for $\bar d \ge2$ all of the impurities eventually
vanish. Furthermore, the asymptotic behavior of the concentration of
impurities depends on the dimension as well as the codimension.  The
theoretical predictions are verified by Monte-Carlo simulations.

This study suggests that the rate theory can be extended to
heterogeneous situations. Moreover, a time-dependent reaction equation
is equivalent in the long time limit to the detailed
reaction-diffusion equation.  The success of this theory is remarkable
especially considering its simplicity.  It will be interesting to
apply the same mechanism to more complicated processes, such as
multispecies reactions.


\section{Acknowledgments}

I am thankful to P.~Krapivsky and S.~Redner for useful discussions.
This work was supported in part by the MRSEC Program of the National
Science Foundation under Award Number DMR-9400379 and by NSF under
Award Number 92-08527.





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\appendix
\section{The Quasistatic Approximation}\smallskip 

The reaction rate, $k$, can be found by evaluating the flux $j$
experienced by an absorbing test particle of radius $R$ due to a
diffusing background of concentration $c_0$. Hence, the diffusion
equation $\partial c/\partial t=D\nabla c$ is solved under the initial
conditions \hbox{$c|_{t=0}=c_0$} and the absorbing boundary condition
$c(r)|_{r=R}=0$.  Although an exact solution can be obtained, we
present a simpler approximate technique.  Since the length scale
governing the diffusion process is $\sqrt{Dt}$, one assumes that the
absorber does not affect the background concentration at distances larger
than $\sqrt{Dt}$. Inside this depletion zone the Laplacian term
dominates the spherically symmetric diffusion equation,
\renewcommand{\theequation}{A1}
\begin{equation}
Dr^{1-d}{\partial\over\partial r}r^{d-1}{\partial c(r)\over\partial r}=0,  
\qquad R<r<\sqrt{Dt}. 
\end{equation}
The quasistatic approximation imposes an additional time-dependent
boundary condition $c|_{r=\sqrt{Dt}}=c_0$ \cite{Redner-90,Burlatsky-92}. The
concentration profile is readily obtained for $R<r<\sqrt{Dt}$,
\renewcommand{\theequation}{A2}
\begin{equation}
c(r,t)\simeq\cases{
c_0\left((r/R)^{2-d}-1\right)\Big/\left((\sqrt{Dt}/R)^{2-d}-1\right)&$d<2$;\cr
c_0\ln(r/R)\big/\ln(\sqrt{Dt}/R)&$d=2$;\cr
c_0\left(1-(r/R)^{2-d}\right)\Big/\left(1-(\sqrt{Dt}/R)^{2-d}\right)&$d>2$.\cr
}
\end{equation}
Above the critical dimension $d_c=2$, $c(r,t)$ approaches a limiting
concentration profile. The reaction rate is given by calculating the
total flux seen by the test particle $j=DS_dR^{d-1}
\partial c/\partial r$, with $S_d$ the surface 
area of the $d$-dimensional unit sphere.  We quote the leading
asymptotic term of the reaction rate, $k=j/c_0$,
\renewcommand{\theequation}{A3}
\begin{equation}
k\propto\cases{
D^{d/2}t^{d/2-1}&$d<2$;\cr
D\left(\ln Dt\right)^{-1}&$d=2$;\cr
DR^{d-2}&$d>2$.\cr
}
\end{equation}
In the case where the target diffusivity $D_T$ is nonzero, the appropriate 
diffusion constant is \hbox{$D+D_T$}. We derived the continuum rates,
however, the lattice counterparts can be conveniently obtained by
setting $R\equiv 1$.

\end{multicols}
\end{document}
\clearpage

\centerline{\bf Figure Captions}\smallskip 
\begin{itemize}
\item[Figure 1:]A line of impurities (bullets) in a two-dimensional 
background (circles).

\item[Figure 2:]The impurity concentration $s(t)$ versus $t$ in 2D and 3D. 
Shown are the case $\bar d=1$ in 2D (bullets) and 3D (squares), and the 
case $\bar d=2$ in 3D (triangles).

\item[Figure 3:]The quantity $s(t)-s_{\infty}$ versus time for the 
case $\bar d=1$. Shown are simulation results in 2D (bullets) and 3D (squares) 
versus the theoretical prediction of Eq.~(\ref{Eq-9}) (solid lines). 
\end{itemize}
\clearpage


\end{document}
