\documentstyle[graphicx,multicol,epsf,aps]{revtex}
\begin{document}
\title{Kinetics of Aggregation-Annihilation Processes} 
\author{E.~Ben-Naim$^1$ and P.~L.~Krapivsky$^2$}
\address{$^1$The James Franck Institute, 
The University of Chicago, Chicago, IL 60637}
\address{$^2$Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\maketitle
\begin{abstract}
We investigate the kinetics of many-species systems with aggregation
of similar species clusters and annihilation of opposite species
clusters.  We find that the interplay between aggregation and
annihilation leads to rich kinetic behaviors and unusual conservation
laws.  On the mean-field level, an exact solution for the cluster-mass
distribution is obtained.  Asymptotically, this solution exhibits a
novel scaling form if the initial species densities are the same while
in the general case of unequal densities the process approaches single
species aggregation.  The theoretical predictions are compared with
numerical simulations in 1D, 2D, and 3D.  Nontrivial growth exponents
characterize the mass distribution in one dimension.
\end{abstract}
\pacs{PACS numbers: 02.50.-r, 05.40.+j, 82.20.-w}
\vspace{-.4in}
\begin{multicols}{2}

Irreversible aggregation and annihilation processes occur in many
natural phenomena \cite{smoke,meak}.  The kinetics of each of these
processes is well understood both on the mean-field level which
provides an accurate description when fluctuations in reactant
densities can be ignored \cite{smol,smoke,ernst} and in the
fluctuation-dominated regime which occurs in low-dimensional systems,
see \cite{zel}, and references therein.  It was found in particular
that the cluster-mass distribution in aggregating systems typically
approaches a universal scaling form \cite{meak,ernst}.


In this work we investigate the competition between aggregation and
annihilation, which gives rise to a surprisingly rich kinetic
behavior. Our model is the simplest implementation of the two
underlying processes, aggregation and annihilation: Similar species
clusters combine to form larger clusters while dissimilar clusters
combine to form an inert substitute.  It is worth noting that when 
the number of species $n$ diverges, $n\to\infty$, our model is 
equivalent to single-species
annihilation, while in the other extreme, $n=1$, single-species
aggregation is recovered. Hence, the process is well suited for
investigating the interplay between aggregation and annihilation.  We find
that the exponents characterizing the kinetics are nonuniversal in
that they may depend on the reaction rates as well as the initial
conditions. Still, there is a basic scaling form which describes the
process in the long-time limit.


Although our goal in this study was to understand how the competition
between aggregation and annihilation could change the kinetic behavior
we want to stress that our model does mimic some natural
processes. For example, in a two-component system with constituent
monomers $A$ and $B$, aggregation of similar species can produce {\it
open} linear polymers, $A\cdots A$ and $B\cdots B$, while {\it closed}
linear polymers can be energetically less favorable and hence never
appear.  On the other hand, when dissimilar polymers meet each other a
ring copolymer is formed.  While linear chains continue to participate
in the reaction process, ring copolymers lose their reactive edge and
stop participating in the reaction process.
 

We shall study our model both in the mean-field limit and in the 
diffusion-controlled limit for $d=1,2$ and 3. The mean-field approach
to the binary reaction process assumes that the reaction proceeds with
a rate proportional to the product of the reactants densities. Thus
the mean-field approximation neglects spatial correlations and
therefore typically holds in dimensions larger than some critical
dimension $d_c$.  For $n$-species pure annihilation processes, it was
suggested \cite{bAR} that $d_c=4(n-1)/(2n-3)$, the result which turns
into rigorously established values \cite{leb} of the critical
dimension for two- and single-species annihilation, $d_c=4$ and 2,
respectively (the latter case corresponds to $n=\infty$).  For pure
aggregation processes, the critical dimension depends on the details
of the reaction events and on the relation between the diffusion
coefficient and the mass of the cluster \cite{vD}.  Numerically, we
investigate the particle coalescence model (PCM) in which clusters
occupy single lattice sites and perform nearest neighbor hoping with
the diffusion coefficient independent of the mass.  For the PCM, it
is known that $d_c=2$ \cite{kang}. Since the aggregation-annihilation
model interpolates between single-species aggregation and
single-species annihilation processes, both of which belong to the
same universality class, it is natural to expect that the critical
dimension remains the same, $d_c=2$.  We confirm the mean-field
predictions above this critical dimension numerically.  In
one dimension, spatial correlations are relevant asymptotically. A
simple heuristic argument provides the exact asymptotic behavior of
the concentration and a good approximation for the growth of the
typical mass.
 

Following the above discussion, similar species aggregation is
described by the binary reaction scheme
\begin{equation}
A_i+A_j\to A_{i+j}, \quad
B_i+B_j\to B_{i+j}, 
\end{equation}
where $A_i$ denotes a cluster consisting of $i$ monomers of species
$A$, and similarly for $B_i$. The two-species case contains the
generic many-species behavior. Therefore, we shall focus on the two-species
situation and cite the many-species results if appropriate.
Annihilation between dissimilar clusters reads
\begin{equation}
A_i+B_j\to {\rm inert}.
\end{equation}
Thus, we assume that dissimilar clusters completely annihilate
independent of their masses. In some situations it may be more
reasonable to assume a partial annihilation, where the monomer
difference number $i-j$ is conserved \cite{paul1,paul2,blum}.

Denote by $a_k(t)$ and $b_k(t)$ the densities of $A$- and $B$-clusters, 
consisting of $k$ monomers. Then the mean-field rate equations 
for the two-species aggregation-annihilation process read
\begin{eqnarray}
\dot a_k&=&\sum_{i+j=k} a_ia_j-2a_k(a+b),\nonumber\\
\dot b_k&=&\sum_{i+j=k} b_ib_j-2b_k(a+b).
\end{eqnarray}
Here $a(t)$ and $b(t)$ denote the total densities of $A$- and
$B$-clusters, $a=\sum_{k\ge 1}a_k$ and $b=\sum_{k\ge 1}b_k$, and the
overdot denotes the time derivative.  In writing Eq.~(3) we have
assumed that the aggregation rate equals the annihilation rate. For
convenience, we set this rate to 2 such that the prefactor of the gain
term equals unity.

Summing up Eq.~(3) we obtain
\begin{equation}
\dot a=-a^2-2ab, \quad
\dot b=-b^2-2ab.
\end{equation}
For the symmetric initial conditions, $a(0)=b(0)$,
we get $\dot a=-3a^2$ which is immediately solved to find 
\begin{equation}
a(t)=b(t)={a(0)\over 1+3a(0)t}.
\end{equation}
For the asymmetric initial conditions, $a(0)>b(0)$,
it is helpful to rewrite Eq.~(4) in terms of $u\equiv (a+b)/(a-b)$
and $v\equiv a-b$. This gives
\begin{equation}
\dot u=-{1\over 2}v(u^2-1), \quad
\dot v=-uv^2.
\end{equation}
Expressing $v$ as a function of $u$ yields $dv/du=2uv/(u^2-1)$,
and as a result 
\begin{equation}
v=K(u^2-1), \quad
K={[a(0)-b(0)]^3\over 4a(0)b(0)}.
\end{equation}
Inserting Eq.~(7) into  Eq.~(6), we obtain a closed 
equation for $u\equiv u(t)$ with the solution
\begin{equation}
{2u\over u^2-1}-{2u(0)\over u^2(0)-1}
+\ln\left[{u-1\over u+1}~{u(0)+1\over u(0)-1}\right]=2Kt.
\end{equation}
The densities $a(t)$ and $b(t)$ of the total number of $A$- and 
$B$-clusters, $a=v(u+1)/2$ and $b=v(u-1)/2$, cannot be expressed
as explicit functions of $t$. However, asymptotically they exhibit
simple power-law behaviors
\begin{equation}
a\simeq t^{-1}, \quad 
b\simeq a(0)b(0)[a(0)-b(0)]^{-3}t^{-2}, 
\end{equation} 
for $t\to\infty$. 
This indicates that the majority and the minority species evolve very
differently. Similarly, in the many-species case the total density of
the majority species decays as $t^{-1}$ while the minorities densities
decay as $t^{-2}$. 

We turn now to the determination of the cluster densities.  
Introducing the generating functions,
\begin{equation}
A(z,t)=\sum_{j=1}^\infty z^ja_j(t), \quad
B(z,t)=\sum_{j=1}^\infty z^jb_j(t), 
\end{equation}
transforms the governing equations into
\begin{equation}
\dot A=A^2-2A(a+b), \quad
\dot B=B^2-2B(a+b).
\end{equation}
These Bernoulli equations are straightforwardly solved to get
\begin{eqnarray}
A(z,t)&=&{A_0(z)E(t)\over 1- A_0(z)\int_0^t dt'E(t')},\nonumber\\
B(z,t)&=&{B_0(z)E(t)\over 1- B_0(z)\int_0^t dt'E(t')},
\end{eqnarray}
with the shorthand notations $A_0(z)\equiv A(z,t=0), ~B_0(z)\equiv
B(z,t=0)$, and $E(t)\equiv\exp\left(-2\int_0^t
dt'[a(t')+b(t')]\right)$.

Eq.~(12) represents the general solution for arbitrary initial
conditions. Consider now the simplest but important case of
monodisperse, generally asymmetric, initial conditions
\begin{equation}
a_k(0)=\delta_{k1}, \quad 
b_k(0)=\lambda\delta_{k1}.
\end{equation}
These initial conditions imply $A_0(z)=z, B_0(z)=\lambda z$.
Expansion of the resulting generating functions 
$A(z,t)=zE(t)[1- z\int_0^t dt'E(t')]^{-1}$ and 
$B(z,t)=\lambda zE(t)[1- z\lambda\int_0^t dt'E(t')]^{-1}$ yields
\begin{eqnarray}
a_k(t)&=&E(t)\left[\int_0^t dt'E(t')\right]^{k-1}, \nonumber\\
b_k(t)&=&\lambda^k E(t)\left[\int_0^t dt'E(t')\right]^{k-1}.
\end{eqnarray}

In the symmetric case $\lambda=1$, we find $E(t)=(1+3t)^{-4/3}$ and
\begin{equation}
a_k(t)=b_k(t)=(1+3t)^{-4/3}\left[1-(1+3t)^{-1/3}\right]^{k-1}.
\end{equation}
Asymptotically, the cluster-mass distribution approaches the
scaling form
\begin{equation}
a_k(t) \sim t^{-4/3}\exp(-x), \quad x\sim k t^{-1/3}.
\end{equation}
The total mass densities, $m_a(t)=\sum_{k\geq 1}k a_k(t)$ and
$m_b(t)=\sum_{k\geq 1}k b_k(t)$, decrease with time, $m=m_a=m_b\sim
t^{-2/3}$.  However, there exists the quantity $I_1\equiv m a^{-2/3}$
which is conserved by the dynamics of the aggregation-annihilation
model subject to any {\it symmetric} initial conditions (not
necessarily the monodisperse one). The conservation of $I_1$ is
verified by a direct computation which makes use of the evolution
equations, $\dot a=-3a^2$ and $\dot m=-2a m$.  For pure aggregation
processes, the total mass is conserved.  Thus the quantity $I_1$ plays
the role of a ``mass'' in the aggregation-annihilation model.
Furthermore, the rate equations (3) actually admit an infinite set of
conservation laws which include higher moments, $M_p(t)=\sum_{k\geq
1}k^p a_k(t)$, of the distribution function $a_k(t)$. (The first two
moments are the number and mass densities, $M_0\equiv a$ and
$M_1\equiv m$).  These conservation laws are found consequently and
read $I_2=(M_2-2m^2/a)a^{-2/3}$, $I_3=(M_3-6M_2m/a+6m^2/a^2)a^{-2/3}$,
{\it etc.}\ We have constructed these integrals $I_j$ recursively.
Unfortunately, we  understand neither the physical meaning 
nor the algebraic structure of these integrals. 


In the asymmetric case we shall set $\lambda < 1$ thus forcing $B$-species
to be a minority. Making use of Eq.~(3) one can express 
$E(t)=\exp\left(-2\int_0^t dt'[a(t')+b(t')]\right)$ as an explicit 
function of $a$ and $b$, $E=(a(0)^{-1}-b(0)^{-1})/(a^{-1}-b^{-1})$. 
In the long-time limit we use the asymptotic values, 
$a\simeq t^{-1}$ and $b\simeq \lambda(1-\lambda)^{-3}t^{-2}$, to compute 
$E(t)\simeq \lambda^{-1}(1-\lambda)b\simeq (1-\lambda)^{-2}t^{-2}$. Therefore,
\begin{eqnarray}
a_k(t) &\simeq& (1-\lambda)^{-2}t^{-2}\exp(-y), \nonumber\\
b_k(t) &=& \lambda^k a_k(t), \qquad
y={k\over (1-\lambda)^2t}.
\end{eqnarray}
Thus, for arbitrary initial conditions the majority species
cluster-mass distribution can be written in the scaling form
\begin{equation}
a_k(t) \sim t^{-w}\Phi[k/S(t)], \quad
S(t)\sim t^z, 
\end{equation}
where $S(t)$ is the characteristic mass. The scaling function is
exponential, $\Phi(x)=\exp(-x)$, both for symmetric and asymmetric
initial conditions.  In the symmetric case, the governing exponents
are $w=4/3$ and $z=1/3$.  The asymmetric case is equivalent to 
single-species aggregation, and the exponents are $w=2$ and $z=1$.  
The minority, on the other hand, does not scale according to the usual
definition although it can be expressed in the modified scaling form,
$b_k(t) \sim \lambda^k t^{-w}\Phi[k/S(t)]$. The modified scaling form also
indicates that two different mass scales are associated with the
minority species.  A growing scale $S(t)\sim t$, which is forced by
the majority species, and a time independent scale $S_{\lambda}=1/(1-\lambda)$
which dominates in the long-time limit.  The latter scale diverges,
$S_{\lambda}\cong 1/(1-\lambda)$, in the limit $\lambda\to 1$.  It is also
instructive to compute the total mass densities. By summing Eq.~(17) we
find that as $t\to\infty$,
\begin{equation}
m_a(t)\to (1-\lambda)^2, \quad
m_b(t)\to \lambda(1-\lambda)^{-4}t^{-2}.
\end{equation}
Thus the final mass difference $\Delta m_{\infty} \equiv
m_a(\infty)-m_b(\infty)=m_a(\infty)$ may be expressed through the
initial mass difference $\Delta m_0 \equiv m_a(0)-m_b(0)=1-\lambda$ via a
surprisingly simple relation, $\Delta m_{\infty}=(\Delta m_0)^2$.  
In comparison, for the pure annihilation process, 
the final mass density is significantly larger, 
$\Delta m_{\infty}=\Delta m_0$.  More generally, the concentration 
difference, $c_-=a-b$, is a conserved variable in the pure 
annihilation process as it is obvious physically and can also 
be seen from the evolution equations $\dot a=\dot b=-2ab$. 
In the aggregation-annihilation process a related ``hidden'' 
conservation law exists. From the rate equations, 
one can verify that the quantity $J_1=(a-b)(ab)^{-1/3}$ is conserved. 
This unusual conservation law is trivially satisfied in the case 
of equal initial densities. There exists another hidden conservation law, 
$m_am_b (ab)^{-2/3}={\rm const}$. In the case of symmetric initial 
conditions this conservation law is transformed into the aforementioned 
aggregation-like conservation law $I_1=m_a a^{-2/3}={\rm const}$.  
Additionally, there is an infinite set of other integrals which 
generalize the integrals $I_2, ~I_3$, {\it etc.}\ found for symmetric 
initial conditions, to arbitrary initial conditions.
We conclude that the aggregation-annihilation process exhibits 
nontrivial conservation laws which are generalization 
to the usual conservation laws that underly annihilation 
and aggregation separately. Although the existence of an infinite
amount of hidden conservation laws is not customary for irreversible 
processes, it has been recently found in several such processes \cite{int}.


We turn now to the general $n$-species model. Denote by $a^i$ ($m^i$)
the concentration (mass) of the $i^{\rm th}$ species.  One can verify
that the quantities $J^{ij}=N(a^i-a^j)/a^ia^j$, with $N=(\prod_{1\leq
k\leq n}a^k)^{2/(2n-1)}$, are conserved for every $i\ne j$ independent
of the initial conditions. These integrals are reminiscent of the
integrals $L^{ij}=(\prod_{1\leq k\leq
n}a^k)^{1/(n-1)}(a^i-a^j)/a^ia^j$ which are the conservation laws for
the pure annihilation process \cite{bAR}. In both cases, there are
$n-1$ independent conservation laws. The aggregation-type conservation
laws can also be generalized to the $n$-specie case. The quantity
$M/N$ is conserved, with $M=(\prod_{1\leq k\leq n}m^k)^{1/(n-1)}$.  In
the case of the symmetric initial conditions, one recovers the
previously established integral $I_1=m^i (a^i)^{-(2n-2)/(2n-1)}$.
Similarly to the first conserved quantity, one can generalize the
$I_j$ integrals and it appears that there is again an infinite set of
conservation laws. 


For asymmetric initial conditions the majority species scales 
as in single-species aggregation, while the minorities scale 
only in a modified sense.  For the symmetric initial conditions, 
the cluster-mass distribution approaches the scaling form of Eq.~(18) with
\begin{equation}
w_n={2n\over2n-1},\qquad{\rm and}\qquad z_n={1\over2n-1}.
\end{equation}
Thus the exponents depend on the number of species $n$.  When $n=1$,
single-species aggregation is recovered, $w_1=2$ and $z_1=1$, while
the limit $n\to\infty$ corresponds to single-species annihilation,
$w=1$ and $z=0$.  


Generally, we expect that the exponents characterizing the scaling
behavior are universal, {\it i.\ e.}, they depend only on important
aspects of the kinetics. The fact that the exponents do depend on the
number of species does not contradict universality.  However, the
exponents in the aggregation-annihilation system may depend also on
the reaction rates. To demonstrate that we consider the general case
where the aggregation rate and the annihilation rate are different.
We set the aggregation rate to unity as previously and denote the
annihilation rate by $J$. In the $n$-species case with symmetric
monodisperse initial conditions we have
\begin{equation}
\dot a_k=\sum_{i+j=k} a_ia_j-2a_k[1+(n-1)J]a, 
\end{equation}
with $a_k(t=0)=\delta_{k1}$.
By employing the above technique  we solve Eq.~(21) 
and find
\begin{equation}
a_k(t)=(1+\nu t)^{-1-1/\nu}\left[1-(1+\nu t)^{-1/\nu}\right]^{k-1},
\end{equation}
where the notation $\nu=1+2(n-1)J$ has been used. The 
cluster mass distribution obeys the scaling form of Eq.~(18) with 
the exponents 
\begin{equation}
w_n=2{1+(n-1)J\over 1+2(n-1)J}, \quad
z_n={1\over 1+2(n-1)J}. 
\end{equation}
Interestingly, both exponents dependent continuously on the (relative)
magnitude of the annihilation rate, $J$. This nonuniversality
contrasts the bulk of previously investigated aggregation models
\cite{ernst} (see, however, \cite{lr,paul1}). Varying the rate ratio
represents another way of interpolating between aggregation and
annihilation. Indeed, independent of $n$, when $J=0$, single-species
aggregation is recovered, while the limit $J\to\infty$ corresponds to
single-species annihilation.


We now consider aggregation-annihilation in the diffusion-controlled
limit in low dimensions. We shall study the particle coalescence model
(PCM) in which clusters occupy single lattice sites, hop to 
nearest neighbor cites with a rate independent of the mass, and
aggregate or annihilate instantaneously whenever they meet. The PCM
allows one to focus on the kinetic aspects of the process and is well
suited for numerical implementation. Additionally, in the mean-field
approximation the PCM is governed by the same Eq.~(3) which has been
examined previously.


Let us ignore the mass and the identity of the clusters and denote an
arbitrary cluster by $C$.  Then the reduced reaction process can be
described by the reaction scheme $C+C\to C$ and $C+C\to 0$ for
aggregation and annihilation, respectively. For both processes, 
the total density $c=\sum_{k\geq 1} c_k$ is inversely proportional 
to $N(t)$, the number of {\it distinct} sites visited by a single 
random walk in $d$ dimensions \cite{zel}. 
$N(t)$ is well known in probability theory \cite{feller} 
and thus the density $c, ~c\sim N^{-1}$, behaves as 
\begin{equation}
c(t)\sim\cases{ t^{-1/2}&$d=1$;\cr
                t^{-1}\log(t) &$d=2$;\cr
                t^{-1}&$d>2$.\cr}
\end{equation}
In other words, the time variable $t$ is replaced with a modified
time variable $N(t)$.  We further assume that the rate equation theory
describes the process in low dimensions, with the time variable
$N(t)$.  Although this approach is a heuristic one, it provides a
good approximation for the subcritical behavior.  For the symmetric 
initial conditions, the characteristic mass for the case $n=2$ is given by 
\begin{equation}
S(t)\sim\cases{ t^{1/6}&$d=1$;\cr
                t^{1/3}\left(\log(t)\right)^{-1/3} &$d=2$;\cr
                t^{1/3}&$d>2$.\cr}
\end{equation}
Using the asymptotic forms of the concentration and the typical mass, 
a scaling form similar to the one of Eq.~(18) can be written. 
The same analysis can be repeated for the general multi-species case, 
and we quote the scaling exponents in one dimension only,  
\begin{equation}
w_n={n\over2n-1},\qquad{\rm and}\qquad z_n={1\over2(2n-1)}.
\end{equation} 
In the extreme cases of $n=1$ and $n\to\infty$, these exponents agree
with the solutions to single-species aggregation and single-species
annihilation, respectively \cite{spoug}. In the case of asymmetric 
initial conditions, the majority species concentration decays
according to single-species aggregation, or equivalently, according
to Eq.~(24).  The minority species, on the other hand, decays much
faster.  Since the total concentration of the minority species 
decays as the monomer density of the majority species, $b\sim a_1$, 
and the latter is known in one and two dimensions (see \cite{spoug,leb}),
we find that the minority species concentration decays as
$t^{-3/2}$ in 1D, and as $t^{-2}\log^2(t)$ in 2D.



It is interesting to compare the above theoretical predictions with
numerical simulations. We have performed simulations for $d=1, 2$, and 3.
The numerical implementation of the process is simple. Initially all
$L^d$ sites on the cubic lattice are occupied with monomers. An 
elemental simulation step consists of picking a cluster at random and
moving it to a randomly chosen neighboring site. If the site is
occupied, an aggregation or an annihilation event takes place,
depending on the identity of the two clusters. Time is updated by the
inverse of the total number of particles in the system after each
step. The linear dimension of the lattice used in the simulation was
$L=3\times10^7$, $4\times10^3$, $2\times 10^2$ in 1D, 2D, and 3D.




We discuss first results for the symmetric initial conditions
involving two species. To verify that the process belongs to the PCM
universality class we measured the density of particles as a function
of time. Indeed, Eq.~(24) appears to hold asymptotically (see Figure
1).  We also measured the average cluster size $\langle
k(t)\rangle=\sum k c_k/\sum c_k$. We expect that the average cluster
size is proportional to the characteristic size $S(t)$,
asymptotically.  For $d=3$, the mean-field prediction, $\langle
k(t)\rangle \sim t^{1/3}$ is verified (see Figure 2). Note that the
simulation results are reliable up to time $\propto L^2$ due to finite
size effects. At the critical dimension, $d_c=2$, the average mass
grows slightly slower, consistent with the logarithmic correction of 
Eq.~(25). In 1D, the growth exponent $z$ as determined by a least square
fit is $z=0.190\pm 0.002$. However, the corresponding value from the
approximate theory is lower $z=1/6$. The simulation was carried over a
relatively large temporal range, suggesting that the growth exponent
is indeed different than suggested by the heuristic theory.  A similar
trend is observed when more than two species are involved. We find
that the numerically determined exponents $z_3\cong0.12$ and
$z_4\cong0.09$ are slightly higher than their theoretical counterparts
$z_3=1/10$ and $z_4=1/14$. We have made some additional consistency
checks. For example, we verified that the cluster mass distribution
follows the scaling form of Eq.~(18).  Above the critical dimension the
scaling function is indeed a simple exponential, in agreement with the
rate equation predictions. Also, we verified numerically that both the
aggregation-type and the annihilation-type conservation laws are
satisfied by the PCM above the critical dimensions $d_c=2$. This
suggests that existense of hidden conserved quantities is an intrinsic
property of the model.


\begin{figure}
%\centerline{\epsfxsize=9cm \epsfbox{fig1.eps}}
\centerline{\includegraphics[width=7.8cm]{fig1}}
\caption{The total density versus time in 1D (top), 2D (middle) and
3D (bottom).  Shown are the simulation data (bullets) and the
theoretical prediction of Eq.~(24) (solid line). Time is given 
in terms of Monte-Carlo Steps (MCS).}
\end{figure}





In summary, we introduced an aggregation-annihilation model and
presented the solution to the governing rate equations. The mass
distribution follows a general scaling form with nontrivial growth
exponents, and the process is characterized by unusual conservation
laws.  The kinetic behavior depends both on the number of species and
on the specific rates of the annihilation and the aggregation
processes. Numerical simulations confirm the rate equation predictions
above the critical dimension. Below the critical dimension, a
heuristic argument provides a good estimate for the mass distribution.
Nontrivial exponents describe the process in one dimension and it
would be interesting to study this system rigorously.  It is plausible
that further investigation of the underlying conservation laws below
the critical dimension will pave the way for understanding the long
time kinetics.


\begin{figure}
%\centerline{\epsfxsize=9cm \epsfbox{fig2.eps}}
\centerline{\includegraphics[width=7.8cm]{fig1}}
\caption{The average cluster size $\langle k(t)\rangle$ versus $t$ in 1D
(bottom), 2D (middle), and 3D (top). Shown are simulation data
(bullets) and lines of slope $0.19$ (solid line) and $1/3$ (dashed
line) for reference.}
\end{figure}


We are thankful to S.~Redner for discussions.
E.B. 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. 
P.L.K. was supported by ARO grant \#DAAH04-93-G-0021 
and NSF grant \#DMR-9219845.  

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