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\documentstyle[aps,multicol]{revtex}

\begin{document}

\title{Aggregation with Multiple Conservation Laws} 

\author{P.~L.~Krapivsky$^1$ and E.~Ben-Naim$^2$}
\address{$^1$Center for Polymer Studies and Department of Physics, 
 Boston University, Boston, MA 02215, USA}
\address{$^2$The James Franck Institute, The University of Chicago, 
Chicago, IL 60637, USA}
\maketitle
\begin{abstract}
Aggregation processes with an arbitrary number of conserved quantities
are investigated. On the mean-field level, an exact solution for the
size distribution is obtained.  The asymptotic form of this solution
exhibits nontrivial ``double'' scaling. While processes with one
conserved quantity are governed by a single scale, processes with
multiple conservation laws exhibit an additional diffusion-like
scale. The theory is applied to ballistic aggregation with mass and
momentum conserving collisions and to diffusive aggregation with
multiple species.
\end{abstract}


\pacs{PACS numbers: 02.50.-r, 05.40.+j, 82.20.-w}


\begin{multicols}{2}


\section{Introduction}

Irreversible aggregation processes underly many natural phenomena
including, {\it e.\ g.}, polymerization [1], gelation [2], island
growth [3-4] and aerosols [5].  The classical rate theory of
Smoluchowski describes the kinetics of such processes [6-9]. Recently,
scaling [10-16] and exact [17-23] theoretical studies showed that
spatial correlations play a crucial role in low dimensions.  While the
above examples are diffusive driven, there are physical situations
such as formation of large-scale structure of the universe [24] and
clustering in traffic flows [25], where the aggregates move
ballistically.  So far, theories of ballistic aggregation [26] have
been restricted to scaling arguments [26-30].

In the ballistic aggregation process both the mass and the momentum
are conserved. In polymerization processes involving copolymers, each
monomer species mass is a conserved quantity. Hence, we study
aggregation processes with multiple conservation laws. The simplest
example for such a system is aggregation with $k$ distinct species.
Both the multivariate distribution and the single variable
distributions are of interest. We present exact solutions to the time
dependent and the steady state mean-field rate equations.  Although
they are straightforward generalizations to the well known solutions
they exhibit interesting behaviors.  An asymptotic analysis shows that
fluctuations associated with a single conserved quantity are Gaussian
in nature.  As a result, an additional ``diffusive'' size scale
emerges.

We apply the above theory to ballistic aggregation as well as
diffusive aggregation. In the case of ballistic aggregation, we use an
approximate collision rate to obtain a solution to the Boltzmann
equation in arbitrary dimension. While this approach agrees with the
scaling arguments, it suggests that for a given mass, the momentum
distribution is Gaussian.  We compare these predictions with one and
two dimensional simulations.  Furthermore, we consider steady state
properties of the aggregation process by introducing input of
particles.  For homogeneous input, a novel time scale describing the
density relaxation is found.  In the case of a localized input,
clustering occurs only for $d\le 2$.  Additionally, we apply the
theory to two-species aggregation with diffusing particles. Using the
density dependent reaction rate, we obtain the leading scaling
behavior of the two relevant mass scales.


This paper is organized as follows. In section II, we present exact
solutions of the rate equation theory. We investigate time dependent
as well as steady state properties of the process.  We then apply the
theory to ballistic aggregation with momentum conserving collisions
(section III) and to diffusive driven aggregation (section IV). We
conclude with a discussion and suggestions for further research in
section V.


\section{Theory}

Following the above discussion, there are aggregation processes where
several physical quantities are conserved.  In such processes, it is
natural to label the aggregates by a ``mass'' vector ${\bf
m}\equiv(m_1,\cdots,m_k)$, where every component represents a
conservation law.  Let us denote the probability distribution function
for particles of mass ${\bf m}$ at time $t$ by ${{P({\bf m},t)}}$.
Mean field theory of the binary reaction process assumes that 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 $d_c$ [10, 14-16].  The rate equations [6] are
written as
\begin{eqnarray}
{d{{P({\bf m},t)}}\over dt}&=&
{\sum_{\{m_i'\}}} P({\bf m'},t)P({\bf m}-{\bf m'},t)\nonumber\\
&&- 2{{P({\bf m},t)}}{\sum_{\{m_i'\}}} P({\bf m'},t),
\end{eqnarray}
where the sum is carried over all $k$ variables $m'_i$, $1\le i \le
k$.  The loss term prefactor reflects the fact that two particles are
lost in each collision.  One can verify that the rate equations
conserve each mass separately, {\it i.\ e.}, $\sum m_i{{P({\bf
m},t)}}\equiv\sum m_i P_0({\bf m})$, where $P_0({\bf m})$ is the
initial distribution. In writing Eq.~(1) we have implicitly assumed
that the rate $K({\bf m},{\bf n})$ at which the reaction $({\bf
m})+({\bf n})\to ({\bf m+n})$ proceeds does not depend on masses of
the reactants, $K({\bf m},{\bf n})={\rm const}$.  This constant can be
set to unity without loss of generality.  For an arbitrary reaction
rate kernel $K({\bf m},{\bf n})$, Eq.~(1) is easily generalized so
that, {\it e.\ g.}, the gain term is replaced by $\sum K({\bf m'},{\bf
m-m'})P({\bf m'},t)P({\bf m}-{\bf m'},t)$.

To solve Eq.~(1) we introduce the generating function, ${F({\bf
z},t)}$, defined by
\begin{equation}
{F({\bf z},t)}={\sum_{\{m_i\}}} {\bf z}^{{\bf m}}{{P({\bf m},t)}}.
\end{equation}
In the above equation we have used the shorthand notations ${\bf
z}\equiv(z_1,\cdots,z_k)$ and ${\bf z}^{{\bf m}}\equiv z_1^{m_1}\cdots
z_k^{m_k}$.  Moments of the distribution function are readily obtained
by evaluating the generating function and its derivatives in the
vicinity of the point ${\bf z}={\bf 1}\equiv (1,\ldots,1)$. For
example, the total cluster density is given by $N(t)=F({\bf
1},t)$. The equation describing the temporal evolution of the
generating function ${F({\bf z},t)}$ can be evaluated by a proper
summation of the rate equation. This yields
\begin{equation}
{dF\over dt}=F^2-2FN. 
\end{equation}
As a preliminary step, we evaluate the time dependence of the
density. The corresponding rate equation is obtained by evaluating
Eq.~(3) at ${\bf z}={\bf 1}$,
\begin{equation}
{dN\over dt}=-N^2. 
\end{equation}
Without loss of generality we set the initial density to unity and
therefore we have $N(t)=1/(1+t)$. We also note that by subtracting
Eq.~(4) from Eq.~(3), a simple differential equation for the quantity
$F-N$, emerges, $d(F-N)/dt=(F-N)^2$. This equation is readily solved
to find
\begin{equation}
{F({\bf z},t)}={F_0({\bf z})\over(1+t)\left(1+t-tF_0({\bf z})\right)}, 
\end{equation}
where the notation $F_0({\bf z})\equiv F({\bf z},t=0)$ has been used.
Eq.~(5) represents the general solution to Eq.~(1). Indeed, it is a
simple generalization of the well known solution [6] for the single
mass aggregation process.



We consider the simplest multivariate case,  
\begin{equation}
P_0({\bf m})=k^{-1}\sum_{i=1}^k \delta(m_i-1)\prod_{j\ne i}\delta(m_j).
\end{equation}
These initial conditions imply $F_0\equiv{\bar z}=(z_1+\cdots+z_k)/k$
and from Eq.~(5) we have
\begin{equation}
{F({\bf z},t)}={{\bar z}\over (1+t)\left(1+t-t{\bar z}\right)}.
\end{equation}
A possible application of these initial conditions is to aggregation
processes involving $k$ distinct species with equal initial densities.
Although this symmetric situation appears to be too simple at first,
it contains the necessary ingredients for exploring the long time
kinetics.

Before we investigate the multivariate distribution, let us first
study properties of the following distribution function
\begin{equation}
P(m_+,t)={\sum_{\{m_i\}}} {{P({\bf m},t)}} \delta \left(m_+-(m_1+\cdots+m_k)\right).
\end{equation}
This distribution corresponds to the sum variable
$m_+=m_1+\cdots+m_k$.  It is useful to introduce the generating
function $F(z,t)=\sum z^{m_+}P(m_+,t)$.  This generating function can
be obtained from ${F({\bf z},t)}$ by replacing ${\bar z}$ with
$z$. Expansion of $F(z,t)=z/(1+t)(1+t-tz)$ in powers of $z$ yields the
sum distribution
\begin{equation}
P(m_+,t)={t^{m_+-1}\over(1+t)^{m_+ +1}}.
\end{equation}
The variable $m_+$ ignores the ``identity'' of the different conserved
variables and thus, the problem reduces to ordinary aggregation.  In
the long time limit, $m_+\sim t$, and the corresponding distribution
is given by
\begin{equation}
P(m_+,t)\simeq t^{-2}\exp(-m_+/t).
\end{equation}
Using the scaling variable, $M_+=m_+/t$, this distribution can be also
written in terms of a scaling function $P(m_+,t)\simeq
t^{-2}\Psi(M_+)$, with $\Psi(x)=\exp(-x)$.

The multivariate probability distribution function can be now found by
expanding ${F({\bf z},t)}$ and comparing with Eq.~(2),
\begin{equation}
{{P({\bf m},t)}}=P(m_+,t)g({\bf m})\quad{\rm with}\quad {g({\bf m})}=
{k^{-m_+}(m_+)!\over m_1!\cdots m_k!}.
\end{equation}
Note that ${\sum_{\{m_i\}}}{g({\bf m})}=1$. The above expression is
the explicit solution to the mean field equations. However, its long
time nature is of particular interest and we proceed with an
asymptotic analysis.


To study the asymptotic properties of ${{P({\bf m},t)}}$, we
concentrate on the case $k=2$ and then generalize the results to
arbitrary $k$. The time independent geometric factor reads,
\hbox{$g(m_1,m_2)=2^{-m_+}(m_1+m_2)!/m_1!m_2!\sim
m_+^{-1/2}\exp\big(-(m_1-m_2)^2/2m_+\big)$}. The limit of large
masses, $m_1,m_2\gg1$, is the relevant one since the masses grow
indefinitely. Using the limiting form of $P(m_+,t)$, we arrive at the
asymptotic form of the mass density,
\begin{equation}
P(m_1,m_2,t)\sim t^{-2}m_+^{-1/2}\exp(-m_+/t)\exp(-m_-^2/2m_+),
\end{equation}
with $m_{\pm}=m_1\pm m_2$. Two mass scales govern the mass
distribution, $m_+\sim t$ and $|m_-|\sim \sqrt{m_+} \sim \sqrt{t}$.
Furthermore, introduction of the scaling variables, $M_+=m_+/t$ and
$M_-=m_-/\sqrt{t}$, enables us to write the solution in a convenient
scaling form
\begin{equation}
P(m_1,m_2,t)\sim t^{-5/2} \Phi(M_+,M_-).
\end{equation} 
The scaling function $\Phi$ depends on the scaling variables only, 
\begin{equation}
\Phi(x,y)=x^{-1/2}\exp(-x)\exp(-y^2/2x).
\end{equation}

Interestingly, each of the variables, $m_1$ and $m_2$, exhibit simple
scaling: $m_1,m_2\sim t$. Additionally, the moments ${M_{\bf n}(t)}$
of the distribution function, ${M_{\bf n}(t)}={\sum_{\{m_i\}}} {\bf
m}^{{\bf n}}{{P({\bf m},t)}}$, are described asymptotically by linear
exponents. By taking derivatives of the generating function, one can
show that ${M_{\bf n}}(t)\sim t^{\alpha(\bf n)}$ with $\alpha({\bf
n})=n_+-1$ .  This behavior agrees with ordinary scaling and therefore
one could naively expect single-size scaling to hold.  However, from
the complete form of the mass distribution we learn that it is {\it
impossible} to write the scaling solution in terms of a simple scaling
function, $P(m_1,m_2)\sim t^{-2}\Phi(m_1/t,m_2/t)$.  Such a scaling
solution would imply that the process has only one intrinsic size
scale. Instead, this problem exhibits ``double-scaling'', as the
process is governed by two distinct size scales. The second scale
$|m_-| \sim \sqrt{t}$ is hidden, {\it e.\ g.}, it does not clearly
appear in the moments of the distribution function.  The mass
difference is reminiscent of a diffusive scale for the following
reason. The mass difference, $m_-=m_1-m_2$, is also a conserved
variable, since each of its components is conserved.  Hence, for an
aggregate of total mass $m_+$, the mass difference is a sum of $m_+$
random variables. According to the central limit theorem this variable
is Gaussian and thus, $m_-\sim\sqrt{m_+}$.


The above results can be generalized to arbitrary $k$.  The
generalized mass difference variable is $m_-^2=k^{-1}\sum_{i,j}
(m_i-m_j)^2$ and the mass distribution follows the scaling form
\begin{equation}
{{P({\bf m},t)}}\sim t^{-(k+3)/2}\Phi(M_+,M_-),
\end{equation}
with the scaling function 
\begin{equation}
\Phi(x,y)=x^{-(k-1)/2}\exp(-x)\exp(-y^2/2x).
\end{equation}


The above results can be also generalized to situations with
asymmetric initial conditions, {\it i.\ e.}, the initial conditions
are not invariant with respect to a permutation of the mass variables,
$\{m_i\}$. Denoting by $M_i={\sum_{\{m_i\}}} m_i P_0({\bf m})$ the
$i^{\rm th}$ conserved mass, one can easily show that with the
transformation $m_i\to m_i/M_i$ and ${{P({\bf m})}}\to M_1\cdots
M_k{{P({\bf m})}}$, the system reduces to the symmetric case. For this
transformation to be valid, the process must be truly multivariate,
$M_i\ne 0$. In this case, the prefactor ${g({\bf m})}$ equals
probabilities associated with a biased random walk in a
$k$-dimensional ``mass'' space.


Previous results have been established for a constant rate kernel,
$K({\bf m},{\bf n})={\rm const}$.  Although the most general situation
is hardly tractable, in the case of sum-variable dependent reaction
rate, $K({\bf m},{\bf n})\equiv K(m_+, n_+)$, the sum-variable
distribution function $P(m_+, t)$ satisfies a simpler single-variable
Smoluchowski equation. In this case, the complete multivariate
distribution function $P({\bf m}, t)$ is related to $P(m_+, t)$ by
Eq.~(11). Therefore solvable variants of the single-variable
Smoluchowski equation provide exact solutions for multivariate
Smoluchowski equation with the corresponding reaction rates. For the
usual Smoluchowski equation, exact solutions have been found for three
types of reaction rates: $K(m,n)={\rm const}, ~K(m,n)=m+n,
~K(m,n)=mn$, and for their linear combinations [6,5,31]. In
particular, for the sum-kernel $K({\bf m},{\bf n})={1\over 2}(m_+ +
n_+)$, subject to the monodisperse initial conditions of Eq.~(6), we
again arrive at the exact solution of Eq.~(11) with
\begin{equation}
P(m_+,t)=\exp\bigl[-t-m_+(1-e^{-t})\bigr]~
{\bigl[m_+(1-e^{-t})\bigr]^{m_+-1}\over (m_+)!}.
\end{equation} 
Similarly, one can find an exact solution for the product kernel.


We now investigate the steady state properties of the aggregation
process by introducing input of particles into the system.  For
simplicity, we consider homogeneous input with rate $h$ and restrict
ourselves to the constant rate kernel.  The governing equations are
modified by adding an input term,

\begin{eqnarray}
{d{{P({\bf m},t)}}\over dt}&=&{\sum_{\{m_i'\}}} P({\bf m'},t)P({\bf m}-{\bf
m'},t)\nonumber\\&-& 2{{P({\bf m},t)}}{\sum_{\{m_i'\}}} P({\bf m'},t)+h{R({\bf m})}.
\end{eqnarray}

The input function, ${R({\bf m})}$, satisfies the normalization
condition ${\sum_{\{m_i\}}}{R({\bf m})}=1$ since the total input
$h{\sum_{\{m_i\}}}{R({\bf m})}$ is equal to $h$.

The solution to the above equation parallels the solution to the time
dependent problem.  We denote the steady state distribution by
${{P({\bf m})}}$ and the steady state generating function by ${F({\bf
z})}$. This generating function is obtained by eliminating the time
derivative of Eq.~(18), $F^2-2FN+hR=0$, where $R\equiv{R({\bf
z})}={\sum_{\{m_i\}}} {\bf z}^{{\bf m}} {R({\bf m})}$.  The
normalization condition $R({\bf 1})=1$ is satisfied by the input
function, and consequently the solution to the generating function
reads
\begin{equation}
{F({\bf z})}=\sqrt{h}\left(1-\sqrt{1-{R({\bf z})}}\right).
\end{equation}
The total monomer density is given by $N=F({\bf 1})$ and thus, 
\begin{equation}
N(h)=\sqrt{h}.
\end{equation}

In analogy with the initial conditions of Eq.~(6), we consider the
input ${R({\bf m})}=P_0({\bf m})$, which in turn implies ${R({\bf
z})}={\bar z}$.  In this case, the steady state generating function is
simply \hbox{${F({\bf z})}=\sqrt{h}\big(1-\sqrt{1-{\bar z}}\big)$}.
Expanding in powers of the variables $z_i$, and comparing with the
definition of the generating function, one finds the steady state
distribution
\begin{equation}
{{P({\bf m})}}=P(m_+){g({\bf m})},\quad
P(m_+)={(2m_+)!\over(2m_+-1)(2^{m_+} m_+!)^2}. 
\end{equation}
There is a strong similarity with the time dependent counterpart,
${{P({\bf m},t)}}$.  The distribution among aggregates with the same
total mass, $m_+$, is given by the same combinatorial factor ${g({\bf
m})}$ which has already appeared in solution to the time dependent
problem.  For large masses, the distribution describing the sum
variable, $m_+$, is an algebraic one, $N^{-1}P(m_+)\sim m_+^{-3/2}$.
We can write now the asymptotic form of the complete steady state
distribution,
\begin{equation}
P({\bf m})\equiv P(m_+,m_-)\sim \sqrt{h}\,\,m_+^{-(k+2)/2} 
\exp(-m_-^2/2m_+).
\end{equation}
For the time dependent problem, an additional size scale emerges for
the mass difference variable, $m_-\sim\sqrt{m_+}$. In the steady
state, on the other hand, the total mass diverges and thus, the
central limit theorem does not apply.  However, the similarity between
the two cases is still strong as for a fixed $m_+$, the mass
difference distribution function is Gaussian.


For completeness, we briefly discuss the relaxation properties of the
density towards the steady state. It is possible to obtain these
properties by incorporating the time dependent density, $N\sim
t^{-\alpha}$, and the steady state solution, $N\sim h^{\gamma}$, into
a single expression. Let us assume the scaling form [17]
\begin{equation}
N(h,t)\sim t^{-\alpha}\psi(t/\tau),
\end{equation}
with the relaxation time $\tau\sim h^{-\beta}$. In the limit of a
vanishing input rate the time dependent solution must be recovered and
thus, $\psi(x)\to 1$ as $x\to 0$. In the long time limit steady state
is approached and thus, $\psi(x)\sim x^{\alpha}$ for $x\gg1$.
Comparing with the definition of the relaxation time the exponent
relation
\begin{equation}
\gamma=\alpha\beta
\end{equation}
is found. The exponent $\beta=1/2$, is obtained from the decay
exponent, $\alpha=1$, and the steady state exponent, $\gamma=1/2$.
Hence, the relaxation time diverges in the limit of a vanishing input
rate, $\tau \sim h^{-1/2}$. This result can be obtained directly by
solving the equation $dN/dt=-N^2+h$ which is readily performed to find
\hbox{$N(h,t)=\sqrt{h}\tanh\big((t+t_0)\sqrt{h}\big)$}.  The time
shift $t_0$ is determined by the initial density,
$N(t=0)=\sqrt{h}\tanh(\sqrt{h}t_0)$. We prefer the above scaling
argument since it is applicable to a wide class of problems.




\section{Ballistic Aggregation}

Ballistic aggregation is a natural and important example of a process
with multiple conservation laws.  Both the mass and the momentum are
conserved quantities and thus, there are $d+1$ conservation laws in
$d$ dimension [26].  A collision between two particles results in an
aggregate whose mass as well as momentum are given by a sum over its
components. One can view this system as a gas of sticky particles,
{\it i.\ e.}, an inelastic gas with a vanishing restitution
coefficient. Heuristic arguments predict certain scaling properties of
the system.  However, other aspects of this problem, such as the
mass-momentum distribution function remain unsolved. Our theory is
well suited for this problem and yields an approximate form for the
particle distribution function.

We start by discussing the one dimensional case and then generalize to
higher dimensions. We denote the probability distribution function for
particles of mass $m$ and momentum $p$ at time $t$ by ${{P(m,p)}}$ (we
suppress the time variable).  The {\it stosszahlansatz} Boltzmann
equation, which describes the temporal evolution of this density, must
conserve mass and momentum. Additionally, the collision rate between
two particles is given by their velocity difference and hence, the
Boltzmann equation reads
\begin{eqnarray}
{d{{{P(m,p)}}} \over dt} &=& 
\sum_{m',p'}\left|v'-v''\right|P(m',p')P(m-m',p-p')\nonumber\\
&&-2{{P(m,p)}}\sum_{m',p'} \left|v-v'\right|P(m',p'),
\end{eqnarray}
where $v'-v''=p'/m'-(p-p')/(m-m')$ and $v-v'=p/m-p'/m'$. This
approximation ignores possible spatial correlations and is generally
uncontrolled.  By replacing the kernel terms $|v-v'|$ and $|v'-v''|$
with their average value ${\langle v\rangle}$, an approximate
Boltzmann equation is written,
\begin{eqnarray}
&&{d{{P(m,p)}}\over dt}={\langle v\rangle}\\
&&\left(\sum P(m',p')P(m-m',p-p')-2{{P(m,p)}} \sum P(m',p')\right).\nonumber 
\end{eqnarray}
Although this approximation is generally unjustified, it enables a
solution for the mass-momentum distribution.  The time-dependent
factor ${\langle v\rangle}$ can be absorbed into a novel time
variable, $T$, defined by $dT/dt={\langle v\rangle}$. The resulting
Boltzmann equation is equivalent to Eq.~(1).

For simplicity we consider an initial system of identical particles
with zero average momentum. Without loss of generality we assume that
typical initial quantities such as the mass and the momentum equal
unity. Hence, the initial distribution function, $P_0(m,p)$, is given
by
\begin{equation}
P_0(m,p)=\delta(m-1)\big(\delta(p-1)+\delta(p+1)\big)/2.
\end{equation}
In principle, the general solution of Eq.~(26) can be obtained using
the generating function method. However, with these specific initial
conditions, the process reduces to the two-mass process discussed in
the preceding section. Indeed, by identifying $m$ with the total mass
$m_+=m_1+m_2$ and $p$ with the mass difference $m_-=m_1-m_2$, the
initial conditions Eq.~(6) and Eq.~(27) are the same.  Moreover,
identical rate equations describe the time evolution of both
$P(m_1,m_2)$ and ${{P(m,p)}}$. From the solution of Eq.~(11), the
mass-momentum distribution function is found in terms of the time
variable $T$,
\begin{equation}
P(p,m)={T^{m-1}\over (1+T)^{m+1}}\,
\,{2^{-m}m!\over \left((m+p)/2\right)!\left((m-p)/2\right)!}. 
\end{equation}


Using the asymptotic scaling properties of the solution $m\sim T$ and
$p\sim\sqrt{T}$, we can rewrite the solution in terms of the
observable time $t$. Since $dT/dt\sim v\sim p/m\sim T^{-1/2}$, one has
$T\sim t^{2/3}$.  Hence, the well known the scaling laws
\begin{equation}
m\sim t^{2/3}\qquad{\rm and}\qquad p\sim t^{1/3},
\end{equation}
are recovered. Asymptotically, the mass-momentum distribution is given 
by the following form 
\begin{equation}
P(p,m)\sim 
{\langle m\rangle}^{-2}m^{-1/2}\exp
\Bigl(-{m\over {\langle m\rangle}}-{p^2\over 2m}\Bigr), 
\end{equation}
with the average mass ${\langle m\rangle}\sim t^{2/3}$.  As discussed
in the previous section, the momentum $p$ is a sum of $m$ independent
variables. As a result, for a fixed mass $m$ the momentum distribution
is Gaussian and $p\sim\sqrt{m}$. From Eq.~(10), the mass distribution
is [27] $P(m)={\langle m\rangle}^{-2}\exp\big(-m/{\langle
m\rangle}\big)$.  Direct integration of Eq.~(30) shows that the
momentum distribution is also purely exponential, $P(p)=\big({\langle
m\rangle}{\langle |p|\rangle}\big)^{-1}\exp(-|p|/{\langle
|p|\rangle})$, with the typical momentum, ${\langle
|p|\rangle}=\sqrt{{\langle m\rangle}/2}$.  On the other hand, the
velocity distribution is algebraic for large velocities $p(v)\sim
|v|^{-3}$.

It is interesting to compare these predictions with numerical
simulations.  Carnevale {\it et al.}\ [26] established a scaling
behavior of Eq.~(29) heuristically and confirmed it numerically.  They
also reported a distribution that is reminiscent of Eq.~(30).  Their
numeric form resembles Eq.~(30) in that the mass distribution is
exponential and in that for a fixed mass the velocity distribution is
Gaussian. However, there is a significant difference between the two
forms as the simulation data suggests that the velocity distribution
is independent of mass. This observation is an intriguing one, since
for a fixed mass $m$ the typical velocity is mass dependent $v(m)\sim
m^{-1/2}$. Jiang and Leyvraz [28] also studied the mass distribution
and found that it is singular near the origin, $P(m)\sim m^{-1/2}$ for
$m\ll{\langle m\rangle}$, in contradiction with our approximate
theory. Curiously, the mass-momentum distribution contains an
identical singularity. However, this singularity disappears when the
momentum is integrated over.  We conclude that further numeric
investigation of quantities such as velocity correlations between
neighboring aggregates and moments of the mass-velocity distribution
are needed to better the understanding of one-dimensional ballistic
aggregation.

A similar line of reasoning applies in $d$ dimensions.  We assume that
the density of an aggregate is constant, and thus, as the aggregation
process evolves, the size of an aggregate grows
indefinitely. Initially, only monomers with unit momentum occupy the
system.  Since the collision rate is also proportional to the surface
area of an aggregate, the typical collision rate is $va^{d-1}$, with
the radius $a\sim m^{1/d}$. The density satisfies the approximate
Boltzmann equation
\begin{eqnarray}
&&{d{P(m,{\bf p})}\over dt}=\langle va^{d-1}\rangle\\
&&\left(\sum P(m',{\bf p}')P(m-m',{\bf p}-{\bf p}')
-2P(m,{\bf p}) \sum P(m',{\bf p}')\right), \nonumber
\end{eqnarray}
where ${\bf p}$ is the $d$-dimensional momentum.  Repeating the above
analysis yield the leading scaling behavior for the mass and the
momentum,
\begin{equation}
m\sim t^{2d/(d+2)}\qquad{\rm and}\qquad |{\bf p}|\sim t^{d/(d+2)}.
\end{equation}
The distribution function is a simple generalization of Eq.~(31), 
\begin{equation}
P(m,{\bf p})\sim {\langle m\rangle}^{-2}m^{-d/2}\exp\Bigl(-{m\over
{\langle m\rangle}}-{d|{\bf p}|^2\over 2m}\Bigr),
\end{equation}
with the average mass ${\langle m\rangle}\sim t^{2d/(d+2)}$.  Eq.~(32)
suggests that ballistic aggregation has no upper critical dimension.
Hence, it is not clear whether our approximation holds in sufficiently
large dimension, as is the case for diffusive aggregation.  In a
recent study of the two-dimensional gas of sticky particles [30], some
deviations from the mean-field predictions have been observed. The
simulations [30] revealed that the growth exponent varies up to 10\%
as the initial density was varried.  The growth exponent of Eq.~(32)
was recovered for sufficiently high initial densities. In this limit,
multiple coalescence events dominate and mean-field thoery is
appropriate.  In general, an exponential mass distribution and a
Boltzmann energy distribution were found, in agreement with mean-field
theory. We conclude that despite the crude nature of the
approximation, it provides good estimates for the leading asymptotic
behavior as well as the various distribution functions.


Steady state can be achieved by adding particles to the system with
rate $h$. We consider homogeneous and isotropic input of particles
with unit mass and unit momentum.  From Eq.~(21), the distribution
function reads,
\begin{equation}
P(m,p)=\sqrt{h}\,\,m^{-(d+3)/2}
\exp\Bigl(-{d|{\bf p}|^2\over 2m}\Bigr).  
\end{equation}
Furthermore, the mass distribution is given by $N^{-1}P(m)\sim
m^{-3/2}$, with the density $N\sim \sqrt{h}$. Note that the velocity
kernel ${\langle v\rangle}$ is not important in the steady state since
${\langle v\rangle}\propto \sum P(m)v(m)\sim \int m^{-3/2}m^{-1/2}$ is
finite. One can also study the relaxation properties of the
system. The relaxation towards the steady state becomes slower and
slower as the input rate vanishes.  Following the analysis of
Eq.~(23), the corresponding relaxation exponent is obtained,
\begin{equation}
\tau(h)\sim h^{-\beta},\qquad{\rm with}\qquad \beta={d+2\over 4d}.
\end{equation}
However, only in the low input rate limit, $h\ll 1$, steady state is
achieved. Indeed, the previous description is valid in the low
coverage limit, {\it i.\ e.}, when $t\ll t_c$ with $t_c\sim h^{-1}$,
while for $t>t_c$ the space is covered by a single ``superparticle''.
Since the relaxation exponent $\beta$ is less than one, the time
scales $\tau$ and $t_c$ are well separated in the low input rate
limit.  Thus, the steady state distribution of Eq.~(34) provides an
intermediate asymptotics valid for $\tau\ll t\ll t_c$.


In contrast, the nature of steady state caused by a spatially {\it
localized} particle input is a truly asymptotic one. The spatial
dependence of the density distribution is of particular interest in
this problem.  We consider a spherically symmetric point-like source
of ballistic particles with unit mass and unit momentum.  Let
$P(m,{\bf p},r)=P(m,{\bf p},r,t=\infty)$ be the steady-state radial
mass-momentum density.  For $d>2$ the reaction is in fact irrelevant
away from the source [20].  The density satisfies the convection
equation, $d\bigl(r^{d-1}P(m,{\bf p},r)\bigr)/dr=0$, and thus, the
concentration decays as $r^{-(d-1)}$. So in the {\it inhomogeneous}
ballistic problem, $d=2$ is a critical dimension above which the
reaction merely leads to the renormalization of the strength of the
source.

For $d\le2$, away from the source, particles have typically collided
many times and they move with a velocity close to unity in the radial
direction.  Thus, collisions occur with rate proportional to the rms
velocity $v$. The steady-state radial distribution $P(m,{\bf p},r)$
satisfies
\begin{eqnarray}
&&r^{1-d}{d\over dr} \bigl(r^{d-1}P(m,{\bf p})\bigr)=\langle va^{d-1}\rangle\\
&&\left(\sum P(m',{\bf p}')P(m-m',{\bf p}-{\bf p}')
-2 P(m,{\bf p}) \sum P(m',{\bf p}')\right).\nonumber 
\end{eqnarray}
It is not difficult to verify that with the transformation
$r^{d-1}P(m,{\bf p})\to P(m,{\bf p})$ and $R=\int^r r'^{(1-d)}dr'\to
t$, the steady-state equation reduces to the time-dependent equation
(31). Therefore when $d<2$, the variable $R\sim r^{2-d}$ plays the
role of time and from Eq.~(32), $m(r)\sim R^{2d/(2+d)}\sim
r^{2d(2-d)/(2+d)}$.  As $d\to 2$, the exponent describing the mass
growth vanishes, indicating that $d=2$ is indeed the critical
dimension, above which the typical mass far from the source is
constant.  At the critical dimension $d=2$, logarithmic behavior
occurs, $R\sim\log(r)$, and consequently, $m(r)\sim R \sim \log(r)$.
Hence, for $d\le 2$ clustering is significant and the typical mass is
a growing function of the distance from the source.


To determine the mass-momentum distribution, we tacitly impose
boundary conditions similar to the initial conditions of the time
dependent problem, $r_0^{d-1}P(m,{\bf p},r_0)=\delta(m-1)\delta(|{\bf
p}|-1)$.  The steady-state density far from the source reads
\begin{eqnarray}
P(m,{\bf p},r)&\sim &r^{-(d-1)}{\langle m(r)\rangle}^{-2} m^{-d/2}\nonumber\\
&&\exp\left(-{m \over {\langle m(r)\rangle}}-{d(|{\bf p}|-m)^2\over 2m}
\right)\quad d\le 2,
\end{eqnarray}
with ${\langle m(r)\rangle}\sim r^{2d(2-d)/(2+d)}$ for $d<2$ and
${\langle m(r)\rangle}\sim\log(r)$ for $d=2$.  For a fixed mass, the
momentum distribution is Gaussian and as a result, the rms momentum is
characterized by $\sqrt{m(r)}$. By integration of the mass-momentum
density, one finds the concentration, $N(r)\sim 1/(r^{(d-1)}{\langle
m(r)\rangle})$.  As a result, $N(r)\sim r^{(2-5d+d^2)/(2+d)}$ for
$d<2$, and $N(r)\sim 1/\big(r\log(r)\big)$ for $d=2$.


Returning to the time dependent problem, the steady state solution
holds for $r<t$ only, while for a larger the space is essentially
empty. It is useful to estimate the total number of clusters ${{\cal
N}(t)}\sim \int_0^t r^{d-1}N(r)$.  For $d<2$ one finds ${{\cal
N}(t)}\sim t^{(2d^2-3d+2)/(d+2)}$, and for $d=2$ one has ${{\cal
N}(t)}\sim t/\log(t)$. In the ballistic regime, $d>2$, collisions do
not cause a significant reduction in the number of clusters, and the
total number of clusters grows linearly in time.


To summarize, we write the leading asymptotic behaviors of the mass 
\begin{equation}
m(r)\sim\cases{r^{2d(2-d)/(2+d)}&$d<2$;\cr
               \log(r) &$d=2$;\cr
               1  &$d>2$,\cr}
\end{equation}
the  density 
\begin{equation}
N(r)\sim\cases{r^{-(5d-d^2-2)/(2+d)}&$d<2$;\cr
               r^{-1}[\log(r)]^{-1} &$d=2$;\cr
                r^{-(d-1)}  &$d>2$,\cr}
\end{equation}
and the total number of clusters
\begin{equation}
{{\cal N}(t)} \sim\cases{t^{(2d^2-3d+2)/(2+d)}&$d<2$;\cr
                   t/\log(t) &$d=2$;\cr
                   t  &$d>2$.\cr}
\end{equation}
The typical rms momentum behaves as $\sqrt{m(r)}$. Numerical
simulations agree with the above in $d=1$ [26]. It will be interesting
to test these predictions in higher dimensions.


\section{Diffusive Aggregation}

In this section, we consider the diffusive driven aggregation
processes involving $k$ distinct species.  Each of the species masses
is conserved and thus there are $k$ conservation laws.  We apply the
rate theory described in Sec.~II and also investigate low dimensional
systems where the rate equation description is expected to fail
[15]. In all dimensions we find that in addition to the typical mass
scale, there exists an additional mass scale.


In spatial dimensions larger than the critical dimension, $d_c=2$,
Smoluchowski rate theory is exact.  In dimensions lower than the
critical dimension, spatial correlations are significant
asymptotically.  Particles are repelling each other, and the effective
reaction rate, $\kappa$, depends on the density [20]
\begin{equation}
\kappa\sim\cases{ N^{2/d-1}&$d<2$;\cr
                1/|\log(N)| &$d=2$;\cr
                1     &$d>2$.\cr}
\end{equation}
This reaction rate can be absorbed into a suitably defined time
variable $T$,  
\begin{equation}
{dT\over dt} = \kappa . 
\end{equation}
With this novel time, Smoluchowski rate equation reduces to Eq.~(1).


For $d\le d_c$, this approximation yields erroneous results for the
mass distribution. Nevertheless, it produces correct asymptotic
scaling behavior for quantities such as the typical mass. Thus for the
general case $d\le d_c$ we just quote the leading asymptotic behaviors
while in one dimension we provide exact results.  Let us consider the
case where there are two species, say $A$ and $B$.  A cluster is
characterized by the respective masses of its components, $m_A$ and
$m_B$.  The typical total mass is given by $m_A+m_B\sim T$, or
equivalently,
\begin{equation}
m_A+m_B\sim\cases{ t^{d/2}&$d<2$;\cr
                   t/\log(t) &$d=2$;\cr
                   t  &$d>2$.\cr}
\end{equation}
On the other hand, since the mass difference is a Gaussian variable,
for a fixed total mass, one has $|m_A-m_B|\sim\sqrt{m_A+m_B}$. This
additional mass scale can be also expressed in terms of time,
\begin{equation}
|m_A-m_B|\sim\cases{ t^{d/4}&$d<2$;\cr
                   t^{1/2}[\log(t)]^{-1/2} &$d=2$;\cr
                   t^{1/2}  &$d>2$.\cr}
\end{equation}
Note also that the present two-mass aggregation process can be mapped
onto a two-species aggregation-annihilation process with two
conservation laws [23].  Indeed, the above results agree with
analytical and numerical findings of Refs.~[23,32].

In one dimension, it is possible to derive a complete analytical
solution. We consider a linear lattice on which point clusters hop
randomly from site to nearest neighbor sites. We assume that the
diffusion coefficient $D$ does not depend on cluster's mass.  We also
assume that initially each site is occupied by some monomer, $A$ or
$B$ with equal probability. The multivariate distribution function can
be expressed in the form of Eq.~(11), with the sum variable given by
the Spouge's solution [19] of the ordinary one dimensional
diffusion-controlled aggregation problem,
\begin{equation}
P(m_+,t)=e^{-4Dt}\bigl[I_{m_+-1}(4Dt)-I_{m_++1}(4Dt)\bigr],
\end{equation}
where $I_n$ denotes a modified Bessel function. Introducing now the
scaling variables, $M_+=m_+/\sqrt{8Dt}$ and $M_-=m_-/(8Dt)^{1/4}$, one
can rewrite the solution in a convenient scaling form
\begin{equation}
P(m_1,m_2,t)\sim t^{-5/4} \Phi(M_+,M_-),
\end{equation} 
with the scaling function $\Phi$ 
\begin{equation}
\Phi(x,y)=\sqrt{x}\exp\Bigl(-x^2-{y^2\over 2x}\Bigr).
\end{equation}

Finally, we consider diffusive aggregation in a system with a steady
spatially localized monomer input.  We assume that monomers of all
types are added at random with an equal rate. Making use of exact and
scaling results for the corresponding ordinary aggregation [20], we
solve for our case. This inhomogeneous system is characterized by two
critical dimensions, the usual ``homogeneous'' critical dimension
$d_c=2$ and an additional critical dimension $d^c=4$ which demarcates
the pure diffusion regime $d>4$ (particles do not affect each other
far away from the source) and the diffusion-reaction regime $d\le
4$. The system approaches steady state as $t\to \infty$. In the
diffusion-reaction regime, the density distribution function
approaches a power-law in $r$, where $r$ is the distance from the
source. The borderline cases $d=d_c=2$ and $d=d^c=4$ should be treated
more carefully since logarithmic factors appear. We write the final
results for the typical total mass
\begin{equation}
m_A+m_B\sim\cases{ r^2&$d<2$;\cr
                   r^2/\log(r) &$d=2$;\cr
                   r^{4-d}  &$2<d<4$;\cr
                   \log(r) &$d=4$;\cr
                   1       &$d>4$.\cr}
\end{equation}
For a fixed total mass $m_A+m_B$, the mass difference is again Gaussian 
and consequently, $|m_A-m_B|\sim\sqrt{m_A+m_B}$. 


\section{Conclusions}

In summary, we have investigated irreversible aggregation with many
conservation laws.  The solution to this process is characterized by
the Gaussian statistics of the fluctuations in a given conserved
quantity. The process is governed asymptotically by two size scales. A
typical aggregation-induced scale characterizes the total mass, while
a novel diffusive scale characterizes individual masses. The latter
scale is hidden, {\it i.\ e.}, it is not present in the moments of the
multivariate distribution.  Application to a ``sticky gas'' suggests a
Boltzmann velocity distribution for a fixed mass. In addition, the
mass distribution is exponential. By comparing our predictions with
available numerical results we have found that the present approximate
theory gives a good description of the sticky gas. We have also
investigated steady-state properties by introducing a localized
source. We have observed two different behaviors, the
ballistic-reaction regime for $d\le 2$, and the pure ballistic regime
for $d>2$. Application to diffusive aggregation with more than one
species also exhibit a novel ``diffusive'' size scale.

This study suggests different avenues for further investigation.  The
theory might be applicable to problems such as catalysis and chemical
reactions with many species. It will be also interesting to analyze
the rate equations with more realistic reaction rates.  An important
question to be addressed is how robust is the Gaussian nature of the
fluctuations statistics. It is plausible that spatial correlations
introduce nontrivial internal arrangement of the clusters, leading to
more complicated statistics. It is also plausible that even on the
rate equation level but with the reaction rate not expressible as a
function of the sum-variables only, different asymptotic behavior
emerges. This could explain why for a dual fragmentation process the
multivariate generalization produces an infinite set of scales [33-35]
compared to the two scales in the models of multivariate aggregation
we have examined in this study.




We thank Y.~Du, L.~P.~Kadanoff, S.~Redner, and W.~R.~Young for useful
discussions.  P.L.K. was supported by ARO grant \#DAAH04-93-G-0021 and
NSF grant \#DMR-9219845.  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.

%\vfill\eject

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