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\def\pra #1 #2 #3 {{\sl Phys.\ Rev.\ A} {\bf #1}, #2 (#3)}
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\def\pre #1 #2 #3 {{\sl Phys.\ Rev.\ E} {\bf #1}, #2 (#3)}
\def\prl #1 #2 #3 {{\sl Phys.\ Rev.\ Lett.} {\bf #1}, #2 (#3)}
\def\prsl #1 #2 #3 {{\sl Proc.\ Roy.\ Soc.\ London Ser. A} {\bf #1}, #2 (#3)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\def\apr{{\it a priori}}\def\apo{{\it a posteriori}}
\def\pd#1#2{{\partial #1\over\partial #2}}      %partial derivative
\def\p2d#1#2{{\partial^2 #1\over\partial #2^2}} %second partial derivative
\def\td#1#2{{d #1\over d #2}}      %total derivative
\def\t2d#1#2{{d^2 #1\over d #2^2}} %second total derivative
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\pxt{P(x,t)}
\def\pyt{P(y,t)}
\def\pat{P(A,t)}
\def\pvt{P(V,t)}
\def\pxxt{P(x_1,x_2;t)}
\def\pix{P_0(x)}
\def\piy{P_0(y)}
\def\pixx{P_0(x_1,x_2)}
\def\pyyt{P(y_1,y_2;t)}
\def\msst{M(s_1,s_2;t)}
\def\mst{M(s,s;t)}
\def\mssi{M(s_1,s_2;0)}
\def\mss{M(s_1,s_2)}
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\def\mssst{M(s_1,\ldots,s_d;t)}
\def\dst{M(s,t)}
\def\ds1t{M(s+1,t)}
\def\dssst{M(s,\ldots,s;t)}
\def\ass{A(s_1,s_2)}
\def\ass1{A(s_1+1,s_2+1)}
\def\fiz{\Phi_1(z)}
\def\fz{\Phi_2(z)}
\def\Fz{\Phi_d(z)}
\def\fxt{\Phi_1(xt)}
\def\fAt{\Phi_2(At)}
\def\fVt{\Phi_d(Vt)}


\def\xniiav{\langle x_1^{n_1}\rangle}
\def\xniav{\langle x_1^n\rangle}
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\def\xnffav{\langle x_2^{n_2}\rangle}
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\def\Vav{\langle V\rangle}
\def\Aav{\langle A\rangle}
\def\lav{\langle l\rangle}
\def\lnav{\langle l^n\rangle}
\def\ARnav{\langle (x_1/x_2)^n \rangle}
\def\xxnnav{\langle x_1^{n_1} x_2^{n_2} \rangle}
\def\xxnav{\langle (x_1x_2)^n\rangle}

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\def\inxx{{\int\limits_{x_1}^{\infty} \int\limits_{x_2}^{\infty} dy_1 dy_2}}

\def\a{\alpha}
\def\b{\beta}
\def\G{\Gamma}
\def\g{\gamma}
\def\d{\delta}
\def\z{\zeta}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[graphicx,epsf,aps,multicol]{revtex}

\begin{document}

\title{Scaling and Multiscaling in Models of Fragmentation}
\author{P.~L.~Krapivsky and E.~Ben-Naim\footnote{Present address: 
The James Franck Institute, The University of Chicago, 
5640 South Ellis Avenue, Chicago, IL 60637}}
\address{Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\maketitle
\begin{abstract}
We introduce a simple
geometric model which describes the kinetics of fragmentation of
$d$-dimensional objects. In one dimension, our model coincides with the
random scission model and show a simple scaling behavior in the
long-time limit. For $d>1$, the volume of the fragments is characterized
by a single scale $1/t$, while other geometric properties such as
the length are characterized by an infinite number of length scales and
thus exhibit multiscaling.
\end{abstract}
\pacs{P. A. C. S. Numbers: 02.50.-r, 05.40.+j, 82.20.-w}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}
\section{Introduction}

The phenomenon of fragmentation which occurs in numerous physical, 
chemical, and geological processes, has attracted a considerable
recent interest. Fragmentation can be exemplified by polymer 
degradation, grinding of minerals, atomic collisions cascades, 
energy cascades in turbulence, multivalley structure of the phase space of
disorder systems, $\etc$ \cite{1,2,3,4,5,6,7}. In general, fragmentation is a 
kinetic process with scattering, breaking, or splitting of 
particular material into smaller fragments. With such wide-ranging 
applications it is natural to try to abstract the essential features 
of fragmentation and to model them as simple as possible. 
One characteristic feature of these cascade processes 
is that fragments continue splitting independently. 
This allows one to describe the evolution by linear rate equations. 
Another restriction which is used in almost all studies of 
fragmentation is the implicit assumption that fragments may 
be described by a single variable, say their mass or size. 
The simplest model satisfying these restrictions is the so-called 
random scission model \cite{5,8,9}. In this model,
the distribution of sizes is described by the integro-differential
equation, 
\begin{equation}
\pd \pxt t = -x\pxt + 2\inx \pyt,
\label{1}
\end{equation}
where $\pxt$ is the concentration of fragments of mass (size) $x$, 
$x$-mers, at time $t$. The loss term on the right-hand side represents
the decrease of $x$-mers due to binary breakups. The probability of
breaking at every point is assumed to be constant in the random 
scission model and hence the overall rate at which an $x$-mer breaks  
is equal to $x$. The gain term in Eq.~(\ref{1}) represents the increase of
$x$-mers due to breakups of longer fragments. The general solution  
\cite{5,8,9} to Eq.~(\ref{1}) is
\begin{equation}
\pxt=e^{-xt}\pix + e^{-xt}\inx \piy\left[2t+t^2(y-x)\right].
\label{2}
\end{equation}
In the long-time  limit, 
this exact solution  approaches the scaling form
\begin{equation}
\pxt = Ct^2e^{-xt}, \quad C=\ini y\piy,
\label{3}
\end{equation}
if we keep $xt$ finite while taking the limit $t \to \infty$ and $x\to 0$.


The random scission model is a representative  
example of ``one-dimensional'' fragmentation processes in which
fragments are described by a single variable.
The kinetics of such fragmentation processes is now well 
understood and numerous explicit and scaling solutions have
been found \cite{4,5,8,9,10,11,12,13,14}.  


The geometry of fragments clearly influences the fragmentation
processes. However, it was ignored in so far studied models.  
In this paper, we introduce simple kinetic models describing 
the splitting of two-dimensional and more generally 
$d$-dimensional objects. We find that multiscaling appears 
for dimensions larger than one. In section II, we present the
two-dimensional model and analyze the behavior of the moments of the
size distribution of the fragments. Furthermore, we investigate 
 the area distribution of the fragments an show that it
exhibits ordinary scaling. In Section III, we generalize the asymptotic
results to arbitrary dimensions and show that multiscaling occurs in
higher dimensions as well. In Section IV, we introduce a two-dimensional
isotropic fragmentation process. Numerical study of this process
suggests that it belongs to a different universality class. 

\begin{figure}
%\centerline{\epsfxsize=7cm \epsfbox{fig1.ps}}
\centerline{\includegraphics[width=7cm]{fig1}}
\caption{The fragmentation process.}
\end{figure}


\section{Fragmentation in two dimensions}





In close analogy with the one-dimensional fragmentation process 
we investigate the following process in two dimensions. 
A fragmentation event takes place at an arbitrary internal point 
of the rectangle and gives birth to four smaller rectangles 
as illustrated in Figure 1. The distribution function $\pxxt$
describing rectangles of size $x_1 \times x_2$, 
is governed by the following kinetic equation: 
\begin{equation}
\pd \pxxt t = -x_1x_2\pxt + 4\inxx \pyyt.
\label{4}
\end{equation}
Note that Eq.~(\ref{4}) implies the conservation of the total area, 
\begin{equation}
\inii x_1x_2\pxxt = const.
\label{5}
\end{equation}


To analyze Eq.~(\ref{4}) we introduce the double Mellin transform of the 
distribution function $\pxxt$,
\begin{equation}
\msst = \inii x_1^{s_1-1} x_2^{s_2-1}\pxxt.
\label{6}
\end{equation}
The functions $\msst$ at fixed $s_1$ and $s_2$ 
will be called the moments.
By combining Eqs.~(\ref{4}) and (\ref{6}) we arrive at the equation
\begin{equation}
\pd \msst t = \left({4 \over s_1s_2}-1\right)\mss1t.
\label{7}
\end{equation}


A surprising feature of Eq.~(\ref{7}) is that it implies the existence of
an infinite number of conservation laws. The moments $\msst$
with $s_1$ and $s_2$ satisfying the relation $s_1s_2=4$ are 
independent of time. Thus in addition to the conservation of 
the total area there is an infinite amount of 
hidden conserved integrals. These integrals are in fact responsible for 
the absence of scaling solutions to Eq.~(\ref{4}). 
Indeed, the scaling solution 
$\pxxt=t^wQ(t^zx_1,t^zx_2)$, implies an infinite amount of
scaling relations, $w=z(s_1+s_2)$ at $s_1s_2=4$, which cannot
be satisfied by the scaling exponents, $w$ and $z$.


We will solve Eq.~(\ref{7}) by Charlesby's method \cite{8} (for more
recent applications of this method see, $\eg$, \cite{10,15}).  For the
random scission of the unit square, $P(x_1,x_2;0)=\d(x_1-1)\d(x_2-1)$,
or equivalently $\mssi=1$.  By iterating Eq.~(\ref{7}) one can compute
all derivatives of $\msst$ at $t=0$ and then find $\msst$ from the
Taylor's series, $\msst=M(0)+tM'(0)+t^2M''(0)/2!+t^3M'''(0)/3!+\ldots$.
This gives a solution in terms of a generalized hypergeometric function
\cite{16},
\begin{equation}
\msst= {}_2F_2(a_+,a_-;s_1,s_2;-t),
\label{8}
\end{equation}
with
\begin{equation}
a_{\pm} \equiv a_{\pm}(s_1,s_2)
={s_1+s_2\over 2}\pm\sqrt{\left({s_1-s_2 \over 2}\right)^2+4}.
\label{9}
\end{equation}


Computation of first few moments gives $M(1,1;t)\equiv N(t)=1+3t$ 
for the total number of fragments, $N(t)$; $M(2,2;t) \equiv 1$ 
for the total area; and $M(3,3;t)={1\over t}+{1\over 3t^2}
+e^{-t}\left({1\over 6} -{2\over 3t}+{1\over 3t^2}\right)$ 
for the next diagonal moment. The first moment can be easily understood.
The rate of creation of rectangles is equal to 3 since every 
fragmentation event introduces 3 additional rectangles,
and hence, the total number of rectangles is $1+3t$.  These results
suggest the following power-law asymptotic behavior of the moments
$\msst$:
\begin{equation}
\msst \simeq A(s_1,s_2) t^{-\a(s_1,s_2)}.
\label{10}
\end{equation}
Substituting this asymptotic form into Eq.~(\ref{7}) yields the difference 
equations for the exponent $\a(s_1,s_2)$, and for the prefactor $A(s_1,s_2)$,
\begin{eqnarray}
\label{11}
\a(s_1,s_2)+1&=&\a(s_1+1,s_2+1),\\
\a(s_1,s_2)A(s_1,s_2)&=&\left(1-{4 \over s_1s_2}\right)A(s_1+1,s_2+1).
\nonumber
\end{eqnarray}
With the boundary conditions, $\a(s_1,s_2)=0$ and $A(s_1,s_2)=1$ at
$s_1s_2=4$, Eqs.~(11) are readily solved to give
\begin{eqnarray}
\label{12}
\a(s_1,s_2)&=&a_-(s_1,s_2)=
{s_1+s_2\over 2}-\sqrt{\left({s_1-s_2 \over 2}\right)^2+4},\nonumber\\
A(s_1,s_2)&=&{\G(s_1)\G(s_2)\G(a_+-a_-)
\over \G(a_+-s_1)\G(a_+-s_2)\G(a_+)}.
\end{eqnarray}
The  preceding formulas, Eqs.~(10) and (12), may be established 
rigorously from the asymptotic behavior of the generalized 
hypergeometric functions.  


For ordinary scaling distributions the exponent $\a(s_1,s_2)$, 
describing the asymptotic decay of the moments is linear in the  
variable $s_1+s_2$. However, for two-dimensional fragmentation 
this exponent depends also on the variable $s_1-s_2$. 
This manifests the non-trivial scaling properties of the
two-dimensional fragmentation process. One can also compare 
the average value of $x_1^{n_1}x_2^{n_2}, \xxnnav$, defined by
\begin{equation}
\xxnnav={\inii x_1^{n_1} x_2^{n_2}\pxxt \over \inii \pxxt}
\equiv {M(n_1+1,n_2+1;t) \over M(1,1;t)},
\label{13}
\end{equation} 
with the product $\xniiav \xnffav$. It turns out that the ratio of
these quantities {\it depends} asymptotically on time $t$, 
while for any scaling distribution $\pxxt$ such a ratio 
would be a constant. In particular,
\begin{equation}
{\xxnav \over \xniav \xnfav} \sim t^{-\left(\sqrt{n^2+16}-4\right)}.
\label{14}
\end{equation} 
Only in the limit $n\to 0$ this ratio reaches a constant, while for every
positive $n$ the ratio decays in time.  
By considering the case $n=1$
one sees that the average length, $\langle x_1\rangle\sim
t^{-(5-\sqrt{17})/2}\sim t^{-.438}$, decays slower than the square root
of the average area, $\sqrt{\langle x_1x_2\rangle} \sim t^{-1/2}$.  
This again confirms that the fragment distribution function $\pxxt$ 
in the two-dimensional random scission model does not approach 
a scaling form in the long-time limit. However, since all the moments 
still show a power-law behavior we conclude that the model exhibits 
a multiscaling asymptotic behavior. 


\begin{figure}
%\centerline{\epsfxsize=7cm \epsfbox{fig2.ps}}
\centerline{\includegraphics[width=7cm]{fig2}}
\caption{Realization of the fragmentation
process on a unit square at time $t=1000$.}  
\end{figure}


The moments provide an almost complete analytical description 
of the fragmentation process. However, a snapshot of the system 
at the later stages remains intriguing (see Fig.~2).  
This unexpectedly rich pattern arising in such a simple process 
can be viewed as a consequence of the fact that the process 
is not fully self-similar.  Instead, the pattern is formed of sets 
of different scales which are spatially interwoven. 
Fig.~2 also shows that a number of rectangles have large
aspect ratio. This qualitative observation is in agreement
with the asymptotic behavior of the \nth\ moment of aspect ratio,
\begin{eqnarray}
\label{15}
&&\ARnav \rightarrow B\,\, t^{\sqrt{n^2+4}-2} \\
&&B={\G(1+n)\G(1-n)\G(2\sqrt{n^2+4}) \over
\G(\sqrt{n^2+4}+n)\G(\sqrt{n^2+4}-n)\G(1+\sqrt{n^2+4})} ,\nonumber
\end{eqnarray} 
which is valid for $|n|<1$. [For $|n|\ge 1$, these moments do not exist 
as follows from a general feature of the $d$-dimensional 
random scission model - the moments $M(s_1,\ldots,s_d;t)$ do exist 
only if $s_j>0$ for all $j$]. The aspect ratio appears to be growing 
in time, in other words, perfect squares have a great
tendency of breaking into long and thin rectangles. 



Let us consider the {\it area} distribution function,
$\pat$, 
\begin{equation}
\pat = \inii \d(x_1x_2-A)\pxxt,
\label{16}
\end{equation} 
which provides a partial description of our system. We will show 
that $\pat$ approaches a scaling form similar to those found 
for a number of other one-dimensional fragmentation systems \cite{9,10,12}. 
Indeed,  the diagonal moments scale in time according to 
\begin{equation}
M(s,s;t) \simeq {6\G(s) \over s(s+1)}t^{2-s}, 
\label{17}
\end{equation}
or in other words, the normalized \nth\ moments of the area 
$\langle A^n\rangle^{1/n}$ are all proportional to $t^{-1}$.  
Hence,  the area distribution function follows the scaling form 
\begin{equation}
\pat \simeq t^2\fAt, 
\label{18}
\end{equation}
where the scaling function $\fz$ satisfies 
\begin{equation}
\int_0^{\infty}dz z^{s-1}\fz = {6\G(s) \over s(s+1)}.
\label{19}
\end{equation}
Performing the inverse Mellin transforms yields the
explicit expression for the scaling function \cite{17},
\begin{equation}
\fz = 6\int_0^1 d\z \left({1\over \z}-1\right)e^{-z/\z},
\label{20}
\end{equation}
with the limiting behavior
\begin{equation}
\fz \to  \cases {6 z^{-2}e^{-z},  &if $z \gg 1$,\cr 
                 6\ln(1/z),       &if $z \ll 1$.\cr}
\label{21}
\end{equation}


Notice that the scaling solution of Eq.~(18) is characterized
by the same exponents as the scaling solution (3) for the
one-dimensional random scission model, $\pxt \sim t^2\fxt$. 
However, the scaling functions are different: $\fiz=e^{-z}$
(see Eq.~(3)) is regular everywhere while $\fz$ diverges 
logarithmically near the origin. 



One can consider variations of this model for describing the kinetics
of fragmentation of multidimensional objects. For example,
one can change the governing rule of the fragmentation
events (see Fig.~1) and keep only two rectangles, 
say the rectangle in the bottom left corner and in the
upper right one. This rule implies that the total
length is conserved while the total area decays
to zero. Interestingly, this case has been partially
studied in connection with the problem of random sequential 
adsorption of needles \cite{18}. This model can be treated by 
applying our approach. One should just change in Eq.~(4) 
the factor 4, corresponding to creation of four rectangles,
by factor 2. Many results like Eqs.~(8), (10), and (12) remain
the same, with 
\begin{equation}
a_{\pm}={s_1+s_2\over 2}\pm\sqrt{\left({s_1-s_2 \over 2}\right)^2+2},
\label{22}
\end{equation}
instead of Eq.~(9). All the qualitative conclusions also
do not change: the model exhibits a multiscaling asymptotic
behavior and, $\eg$, 
\begin{equation}
{\xxnav \over \xniav \xnfav} \sim t^{-\left(\sqrt{n^2+8}-\sqrt{8}\right)}.
\label{23}
\end{equation}

The area distribution function again scales according to 
\begin{equation}
\pat \simeq t^{\sqrt{2}}\fAt.
\label{24}
\end{equation}
Here the scaling function $\fz$ is given by
\begin{equation}
\fz = C\int_0^1{d\z\over \z}(1-\z)^{\sqrt{2}-1}e^{-z/\z}, 
\label{25}
\end{equation}
where $C={\G(2\sqrt{2})/\G^2(\sqrt{2})} \cong 2.18482$.

In the limits of large and small area one has
\begin{equation}
\pat \to \cases {C A^{-2}e^{-At},            &if $At \gg 1$,\cr 
                 D t^{\sqrt{2}}\ln(1/At),    &if $At \ll 1$,\cr}
\label{26}
\end{equation}
with $D={\G(2\sqrt{2})/\G^3(\sqrt{2})} \cong 2.46432$.
Therefore the area distribution function again diverges 
logarithmically in the small-$A$ limit.


\section{Generalization to Higher Dimensions}


We turn now to the general $d$-dimensional random scission model.  
The asymptotic method presented for the two-dimensional case can be
generalized by using a simple geometric construction. 
We are interested in obtaining the moments $M({\bf s};t)$, where we
have used the notation ${\bf s}\equiv (s_1,\ldots,s_n)$. 
In analogy with the two-dimensional case, we will assume 
the power-law behavior: $M({\bf s};t)\sim t^{-\a(\bf s)}$. 
The exponents $\a$ satisfy the following difference equation
\begin{equation}
\a({\bf  s})+1=\a({\bf s}+{\bf 1}),\label{27}
\end{equation} 
with the notation ${\bf 1}=(1,\ldots,1)$. Meanwhile, 
the exponents should also reflect the hidden conserved integrals, 
\ie, $\a({\bf s}^*)=0$ on the hypersurface which is formed of
points ${\bf s}^*$ satisfying the relation $\Pi_j s^*_j=2^d$.  
The solution to Eq.~(\ref{27}) with these boundary conditions 
is given by the formal expression 
\begin{equation}
\a({\bf  s})=\a({\bf  s}^*+k{\bf 1})=k.  \label{28}
\end{equation} 
This solution clearly satisfies the boundary condition as well as
Eq.~(27).  Hence the problem is reduced to finding roots of 
the algebraic equation $\Pi_j (s_j-k)=2^d$. 
Since this equation is of degree $d$, 
a solution is  feasible only for $d\le4$. An alternative way 
of viewing the solution is geometric. In Eq.~(\ref{28}), 
${\bf  s}^*+k{\bf 1}$ represents a line along the
$(1,\ldots,1)$ direction originating at ${\bf s^*}$ and ending at 
${\bf s}={\bf  s}^*+k{\bf 1}$. The exponent $\a({\bf  s})$ equals the
projection of this line on an arbitrary axis, \eg\ , 
on the $s_1$ axis. Figure 3 illustrates this construction. 

\begin{figure}
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Fig. 3 The Geometric Solution. The hypersurface 
${\bf s}^*$ satisfies $\Pi_j s_j=2^d$.
\end{figure}







The main features found for the two-dimensional case such as
multiscaling occur for higher dimensions as well. As a manifestation of
the existence of multiple length scales in the system let us consider
the ratio of the average volume $\Vav$ to the \dth\ power of the average
length, $\lav$. We define the exponent $\beta_d$ by 
\begin{equation}
{\Vav\over\lav^d}\sim t^{-\beta_d},\label{29}
\end{equation}
or equivalently, $\beta_d=1-d\left[\alpha(2,1\ldots,1)+1\right]$. 
Using the construction of Eq.~(\ref{28}), 
we find $\beta_d=0,0.1231,0.1486$ for $d=1,2,3$ respectively, 
while in the limit $d\to\infty$ this exponent saturates 
at $1-2\log(3/2)\cong0.1891$.  Note that $\beta_d$ measures 
the deviation between the asymptotic behavior of the length
and the volume. As the dimension is increased, this discrepancy becomes
more pronounced, and hence multiscaling is stronger in higher
dimensions. Another consequence of the same phenomenon is the
nonuniversal behavior of the various moments of the length
distribution. We find that the \nth\ moment decays asymptotically
according to
\begin{equation}
\langle l^n\rangle\sim t^{-2\ln(1+n/2)/d},\label{30}
\end{equation}
indicating the presence of an infinite number of length scales.

One can also show that different directions behave independently to a
certain degree in the limit of infinite dimensions. Specifically, one
can show that 
\begin{equation}
\left\langle \Pi_j x_j^{n_j}\right\rangle=
\Pi_j \left\langle  x_j^{n_j}\right\rangle, \label{31}
\end{equation}
if $n_j=0$ for all indices $j$ except for a finite number. 
The average over a finite number of variables decouples into 
a product over single variable averages, while the average 
over an infinite number does not decouple. 


For completeness, we present the general dimension results for the 
diagonal moments, $\dssst$, which will be shortly denoted by $\dst$.
The governing equation for these moments reads
\begin{equation}
\pd \dst t = \left[(2/s)^d-1\right]\ds1t.
\label{32}
\end{equation}
We substitute the power-law asymptotic behavior,
$\dst \simeq A(s) t^{-\a(s)}$, into Eq.~(32) and  take into account 
the boundary conditions $\a(s=2)=0$ and $A(s=2)=1$. 
By solving the resulting difference equations we find 
\begin{equation}
\a(s)=s-2, \quad 
A(s)=\G^d(s)\prod_{j=1}^{d-1}{\G(2-2\cdot\z^j) \over\G(s-2\cdot\z^j)},
\label{33}
\end{equation} 
with $\z=\exp(2\pi i/d)$.


In the long-time limit the volume
distribution function, $\pvt$, approaches the scaling form
\begin{equation}
\pvt \simeq t^2\fVt,
\label{34}
\end{equation}
with $\Fz$ being the inverse Mellin transform of $A(s)$.
After a lengthy calculation one can find the asymptotic behavior
of the volume distribution function:
\begin{equation}
\pvt \to \cases {C_d V^{-2}e^{-Vt},            &if $Vt \gg 1$,\cr 
                 D_d t^2\ln^{d-1}(1/Vt),       &if $Vt \ll 1$,\cr}
\label{35}
\end{equation} 
where $C_d=\prod_{1 \le j \le d-1}\G(2-2\cdot\z^j)$ and
$D_d=2^{d-1}(2^d-1)/\G(d)$. Thus for all $d>1$ the volume distribution
function diverges logarithmically in the small-volume limit.  


To summarize, in the $d$-dimensional random scission model, 
the volume is characterized by only one scale, $V \sim t^{-1}$.  
However, other geometrical characteristics such as the average 
length and the surface area decay nonuniversally in time because 
of the existence of an infinite amount of length scales,
namely multiscaling. 


One can also consider a varying fragmentation rate and study 
the case where the overall rate depends on the volume as a power-law, 
$\ie$, as $V^{\lambda}$ (the case $\lambda=1$ corresponds 
to the random scission model).  
When $\lambda$ is positive, this generalization also results in
multiscaling of the fragments distribution.  
The total number of fragments $N(t)\equiv M({\bf 1};t)$ grows 
algebraically in time, $N(t)\sim t^{1/\lambda}$. 
Hence, the case $\lambda=0$ is a critical one and the
number of fragments grows exponentially in time.  
Finally, for $\lambda<0$ the shattering transition takes place: 
the total volume decreases monotonically and the total number 
of fragments reaches infinity within an infinitesimally small 
time interval. Moreover, a finite fraction of the volume breaks into 
zero-volume rectangles. This phenomenon is well known in the
context of one-dimensional fragmentation \cite{10,11} and has been 
examined in the context of two-dimensional fragmentation 
with length conservation in a very recent study \cite{19}.



\section{Isotropic Fragmentation}



\begin{figure}
%\centerline{\epsfxsize=7cm \epsfbox{fig4.ps}}
\centerline{\includegraphics[width=7cm]{fig4}}
\caption{Realization of the random orientation fragmentation on a unit square
at time $t=1000$.}
\end{figure}



Intrigued by the rich kinetics of the rectangular fragmentation 
problem, we also investigated numerically an isotropic 
fragmentation process. In situations such as shattering of a thin 
glass plate or in membrane crumpling, the fragments are polygons 
with a varying number of sides. Hence, we introduce a process 
where a randomly oriented crack appears with a uniform rate 
at a random point of the surface and propagates with an infinite 
speed until it meets an existing crack.  The original model can
be viewed as deposition of such perfectly oriented ``cross'' 
shaped cracks. The overall fragmentation rates in both processes 
are equal the volume of the fragment. For randomly oriented 
fragmentation, each fragmentation event creates an additional 
polygon and hence the total number of polygons grows linearly 
in time according to $N(t)=M(1,1;t)=1+t$. The average volume 
thus scales as $1/N(t)$ or $A\sim t^{-1}$.


A Monte-Carlo simulation study of isotropic fragmentation process
on a unit square with hard boundary conditions have been performed.
Numerical results suggest that unlike oriented fragmentation, 
only one length scale exists in the isotropic problem. 
The average length of a polygon side is plotted in Fig.~4 and
appears to decay as $t^{-1/2}$. Therefore, the length follows 
the same asymptotic behavior as does the square root of 
of the average area. A snapshot of a realization of the system 
at time $t=1000$ is shown in Fig.~5. This picture suggests that 
it may prove insightful to investigate various structure  
properties of the system such as the area distribution function 
and the side number probabilities of the polygons. 

\begin{figure}
%\centerline{\epsfxsize=9cm \epsfbox{fig5.ps}}
\centerline{\includegraphics[width=9cm]{fig5}}
\caption{The average length of a polygon side,
  for the random orientation fragmentation process.  Shown are $\langle
  l(t)\rangle$ \vs\ $t$ (diamonds) and a line of slope $-1/2$ (solid)
  for reference.}
\end{figure}



In conclusion, we have studied two fragmentation processes in 
spatial dimensions larger than one. For oriented fragmentation, 
where the fragments are always rectangular, 
multiscaling is found in the long-time limit. Specifically, 
the length distribution function has moments that scale 
algebraically in time with an infinite number of independent
length scales, while the area distribution function 
is characterized by a single length scale. 
The area distribution function also exhibits a weak logarithmic 
singularity near the origin. Multiscaling appears to 
depend strongly on the geometric nature of the process. 
For isotropic fragmentation, a single length scale 
describes the decay of the length as well as the area. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\section*{Acknowledgments}

After completing of this work we became aware of a closely related
study by D.~Boyer, G.~Tarjus and P.~Viot; we thank them for informing
us of their results.  We also want to thank S.~Cornell, E.~Kramer,
S.~Redner, and T.~Witten for interesting discussions.  We gratefully
acknowledge ARO grant \#DAAH04-93-G-0021, NSF grant \#DMR-9219845, and
the Donors of The Petroleum Research Fund, administered by the American
Chemical Society, for partial support of this research.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{multicols}
\end{document}



