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\documentstyle[graphicx,epsf,aps,multicol]{revtex}
\begin{document}
\title{Multiscaling in fragmentation}
\author{E.~Ben-Naim\dag\ and P.~L.~Krapivsky\ddag}
\address{\dag Theoretical Division and Center for Nonlinear Studies, 
Los Alamos National Laboratory, Los Alamos, NM 87545}
\address{\ddag Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\maketitle
\begin{abstract}
We review recent results on random fragmentation of multidimensional
objects.  In addition to the obvious volume conservation, these
processes exhibit an infinite number of hidden conservation laws. The
fragment size distribution is characterized by an infinite number of
scales and thus, exhibits multiscaling. Nevertheless, the volume
distribution function shows ordinary scaling, {\it i.e.}, it is
characterized by a single scale.
\end{abstract}
Keywords: Fragmentation, Multiscaling, Size Distribution, Mellin Transform, 
Moments.\\
{P. A. C. S. Numbers: 02.50.-r, 05.40.+j, 82.20.-w}
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\begin{multicols}{2}

Fragmentation underlies a number of physical, chemical, and geological
processes, such as polymer degradation, atomic collisions cascades,
energy cascades in turbulence, martensitic transformations,
multivalley structures of the phase space of disorder systems, meteor
impacts, {\it etc.} [1-8] The fragmented quantity in such processes
are diverse: mass, momentum, energy, or area.  A characteristic
feature of these cascade processes is that fragments continue
splitting independently. One simplifying assumption used in most
theoretical studies is that fragments may be described properly by a
single variable, say their mass or size [9-15]. However, in many
cases, more than one variable may be necessary. Below, we review
recent results concerning fragmentation with multiple variables
[16-22]. We generalize the random scission model to higher dimension
and solve it analytically.  Using the moments of the fragment size
distribution, we show that the process exhibits many interesting
properties such as multiscaling and the existence of an infinite
number of hidden conservation laws. We also discuss applications of
the model to arbitrary dimensions, homogeneous breakage kernels, and
stochastic fractals.



The random scission model [5,9,10] is the simplest realization of a
fragmentation process, where the distribution of fragments of length
$x$, $P(x,t)$, satisfies the integro-differential equation
\begin{equation} 
{\partial P(x,t) \over \partial t} = 
-xP(x,t) + 2\int_x^{\infty}dy P(y,t).
\end{equation}
The negative term on the right-hand side represents loss due to binary
breakups. Since fragmentation is uniform, the corresponding loss rate
is proportional to the fragment size.  The gain term accounts for the
increase of fragments of size $x$ due to breakups of longer
fragments. The asymptotic solution to equation (1) can be written in a
scaling form
\begin{equation}
P(x,t) \simeq {C\over \langle x \rangle^2} \Phi
\left({x\over \langle x \rangle }\right),
\end{equation}
with $\langle x \rangle \simeq t^{-1}$ and the scaling function
$\Phi(z)=e^{-z}$.  The constant $C=\int dx P(x,0)$ ensures
conservation of the total mass.  The random scission process is
equivalent to uniform deposition of points on a one-dimensional
interval. Indeed, every deposition of such a ``crack'' leads to 
fragmentation of the underlying segment. 


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

The following process is a natural generalization to two dimensions: a 
fragmentation event takes place at a random internal point of the
rectangle and gives birth to four smaller rectangles as illustrated in
Figure 1. The distribution function $P(x_1,x_2;t)$ describing
rectangles of size $x_1 \times x_2$ is governed by the following
linear rate equation [16-19] 
\begin{equation}
{\partial P(x_1,x_2;t)\over \partial t} = -x_1x_2P(x,t) +
4\int_{x_1}^{\infty}\int_{x_2}^{\infty}dy_1 dy_2 P(y_1,y_2;t).
\end{equation}
A simple integration of this equation shows that the total area is
conserved, $\int_0^{\infty}\int_0^{\infty} dx_1 dx_2
x_1x_2P(x_1,x_2;t)={\rm const.}$ The average number of fragments
$N(t)=N(0)+3t$ is readily found. The rate of creation of
rectangles reflects the fact that 3 additional rectangle are created
in each fragmentation event.  It is useful to introduce the double
Mellin transform (or alternatively the moments) of the distribution
function $P(x_1,x_2;t)$,
\begin{equation} 
M(s_1,s_2;t) = \int_0^{\infty}\int_0^{\infty} 
dx_1 dx_2 x_1^{s_1-1} x_2^{s_2-1}P(x_1,x_2;t) 
\end{equation}
The rate equation (3) implies the following rate equation for the 
moments,
\begin{equation}
{\partial M(s_1,s_2;t)\over \partial t} = \left({4 \over s_1s_2}-1\right)
M(s_1+1,s_2+1;t).
\label{momeq}
\end{equation}

A surprising feature of Eq.~(5) is that it implies the existence of an
infinite number of conservation laws. The moments $M(s_1^*,s_2^*;t)$
with $s_1^*$ and $s_2^*$ satisfying $s_1^*s_2^*=4$ are time
independent. Thus, in addition to the conservation of the total area
$A\equiv M(2,2;t)$, there are an infinite number of hidden conserved
integrals. These integrals are in fact responsible for the absence of
scaling solutions to Eq.~(3).  Indeed, the scaling solution
$P(x_1,x_2;t)=t^wQ(t^zx_1,t^zx_2)$, implies an infinite amount of
scaling relations, $w=z(s_1^*+s_2^*)$ at $s_1^*s_2^*=4$, which cannot be
satisfied by the scaling exponents, $w$ and $z$.

Although an exact solution for the moment in terms of generalized
hypergeometric functions is possible [17-19], we present an alternative
technique for obtaining the asymptotic behavior of the moments [17].
Asymptotically, the moments depend algebraically on time
\begin{equation}
M(s_1,s_2;t) \simeq A(s_1,s_2) t^{-\alpha(s_1,s_2)}. 
\end{equation}
Substituting this form into Eq.~(\ref{momeq}) gives the difference
equations $\alpha(s_1,s_2)+1=\alpha(s_1+1,s_2+1)$.  Additionally, the
conservation laws imply that $\alpha(s_1^*,s_2^*)=0$ when
$s_1^*s_2^*=4$. Solving the difference equation subject to the
boundary constraint gives $\alpha(s_1^*+k,s_2^*+k)=k$.  Thus, the
exponent $\alpha$ can be easily obtained by solving a quadratic
equation:  
\begin{equation}
\alpha(s_1,s_2)={1\over 2}\left[s_1+s_2-\sqrt{(s_1-s_2)^2+16}\right]. 
\label{exp}
\end{equation}

To see the nontrivial structure of the moments, consider the average 
value of $x_1^{n_1} x_2^{n_2}$ defined by 
\begin{equation}
{\langle x_1^{n_1} x_2^{n_2} \rangle}= {\int_0^{\infty}\int_0^{\infty}
dx_1 dx_2 x_1^{n_1} x_2^{n_2}P(x_1,x_2;t) \over
\int_0^{\infty}\int_0^{\infty} dx_1 dx_2 P(x_1,x_2;t)}
\end{equation} 
or equivalently, by $\langle x_1^{n_1}x_2^{n_2}\rangle
=M(n_1+1,n_2+1;t)/ M(1,1;t)$.  For example, the ratio $\langle
x_1^{n} x_2^{n} \rangle/\langle x_1^{n}\rangle \langle x_2^{n}
\rangle\sim t^{\sqrt{n^2+16}-4}$ is time dependent, while for a scaling
distribution such a ratio should approach a constant. Furthermore, the
moments of the length $\langle l^n\rangle\equiv \langle
x_1^n\rangle\equiv \langle x_2^n\rangle$ can be found as well
\begin{equation}
\langle l^n\rangle\sim t^{-(n+4-\sqrt{n^2+16})/2}.
\end{equation}
Only in the limit $n\to 0$, the leading behavior $\langle
l^n\rangle\sim t^{-n/2}$ follows an ordinary scaling behavior, as the
exponent $n/2$ is linear in $n$. The general $n$ dependence is more
complicated. For example, $\langle l\rangle\sim
t^{-(5-\sqrt{17})/2}\sim t^{-.438}$, and $\langle l^2\rangle\sim
t^{-(3-\sqrt{5})}\sim t^{-.764}$.  Note that the average length 
decays slower than the square root of the average area, $\sqrt{\langle
x_1x_2\rangle} \sim t^{-1/2}$.  We observe that the distribution
function $P(x_1,x_2;t)$ 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.






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. Indeed, the $n^{\rm th}$ moment of the aspect
ratio, 
\begin{equation}
{\langle (x_1/x_2)^n \rangle} \sim t^{\sqrt{n^2+4}-2} \qquad |n|<1, 
\end{equation} 
is a growing function of time. In other words, rectangles tend to
break into thin and long rectangles. 

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


Let us consider the {\it area} distribution function, $P(A,t)$, with
$P(A,t) = \int_0^{\infty}\int_0^{\infty} dx_1 dx_2
\delta(x_1x_2-A)P(x_1,x_2;t)$. The corresponding ``diagonal'' moments
$M(s;t)\equiv M(s,s;t)$ can be evaluated from the general solution
$M(s,t)\sim t^{2-s}$. Consequently, $\langle A^n\rangle\sim \langle
A\rangle^n\sim t^{-n}$,  indicating that the area distribution
function reaches a scaling form asymptotically, 
\begin{equation}
P(A,t) \simeq t^2\Phi_2(At). 
\end{equation}
The exact scaling function is given by $\Phi_2(z) = 6\int_0^1 d\zeta
\left({1\over \zeta}-1\right)e^{-z/\zeta}$, and has the following limiting 
behaviors
\begin{equation}
\Phi_2(z) \to  \cases {6 z^{-2}e^{-z},  &if $z \gg 1$,\cr 
                 6\ln(1/z),       &if $z \ll 1$.\cr}
\end{equation}
Similar to the random scission model, large areas are exponentially
suppressed.  However, in contrast with the one-dimensional case, there
is a weak logarithmic cusp in the limit of small areas. 

The above results can be easily generalized to $d$ dimensions.  In the
generalized version of the random scission model, each fragmentation
event results in $2^d$ fragments. We denote the moments by $M({\bf
s};t)$ with ${\bf s}\equiv (s_1,\ldots,s_n)$ and assume a power-law
dependence, $M({\bf s};t)\sim t^{-\alpha(\bf s)}$. The exponents
satisfy the recursion relation $\alpha({\bf s})+1=\alpha({\bf s}+{\bf
1})$, with ${\bf 1}=(1,\ldots,1)$. Meanwhile, the exponents should
also reflect the hidden conserved integrals, {\it i.e.}, $\alpha({\bf
s}^*)=0$ on the hypersurface ${\bf s}^*$, where $\Pi_j s^*_j=2^d$.
The solution to the recursion relations satisfying the boundary
conditions is given by the formal expression  
\begin{equation}
\alpha({\bf  s})=\alpha({\bf  s}^*+k{\bf 1})=k.  
\end{equation} 
Hence, the problem is reduced to finding roots of the algebraic
equation $\Pi_j [s_j-\alpha({\bf s})]=2^d$.  Since this equation is of
degree $d$, an analytic solution is feasible only for $d\le4$.

All features found for the two-dimensional case such as multiscaling
occur for higher dimensions as well. For example, nonuniversal
behavior is found for the various moments of the length
distribution. In the limit $d\gg 1$, we find that the 
moments decay asymptotically according to
\begin{equation}
\langle l^n\rangle\sim t^{-2\ln(1+n/2)/d}.
\end{equation}
The $n$ dependence is logarithmic in contrast to the linear dependence
expected for scaling distributions.  The diagonal exponents
$\alpha(s,\ldots,s)=2-s$ indicate that the volume distribution
exhibits scaling $P(V,t) \simeq t^2\Phi_d(Vt)$. The limiting behavior
of the scaling function can be evaluated as well 
\begin{equation}
\Phi_d(z) \sim \cases { z^{-2}e^{-z},& $z \gg 1$,\cr 
                  \ln^{d-1}(1/z),    & $z \ll 1$.\cr}
\end{equation} 
While the power law behavior is general, the logarithmic cusp occurs
for $d>1$. We conclude that the volume distribution generally obeys
scaling, while for $d>1$ other geometric characteristics such as the
length, the surface area, {\it etc.} obey multiscaling. 

One can also consider shape-dependent fragmentation rates and study
homogeneous kernels $K={\bf x}^{\bf m}\equiv x_1^{m_1}\cdots
x_d^{m_d}$ (the case ${\bf m}={\bf 1}$ corresponds to the random
scission model). When $m_i>0$, this generalization also results in
multiscaling of the fragment distribution. The moments $M({\bf s}^*)$
with $\Pi_j (s^*_j+m_j-1)=2^d$ are conserved. Asymptotically, moments
depend algebraically upon time, $M({\bf s};t)\sim t^{-\alpha({\bf
s})}$, and the exponents can be found by solving the equation $\Pi_j
[s^*_j+m_j-1-\alpha({\bf s})]=2^d$. The kernel $K$ plays the role of
the volume $V$ as it exhibits regular scaling, $P(K,t)=t^2\Phi(Kt)$, or
equivalently, $\langle K^n \rangle \sim \langle K\rangle^n\sim
t^{-n}$. Of course, other geometric properties show multiscaling. 

The special situation where the kernel is a homogeneous function of
the volume $K=V^{\lambda}$ is of particular interest. The total number
of fragments $N(t)\equiv M({\bf 1};t)$ grows according to $N(t)\sim
t^{1/\lambda}$.  Hence, the case $\lambda=0$ is critical, 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 [11,12] and has been examined in the context of
two-dimensional fragmentation as well [19].

The above techniques are useful in a number of problems.
Surprisingly, one can map random sequential adsorption of needles in
2D to a multifragmentation problem.  Using a rate equation similar to
equation (3), the needle density is found $n(t)\sim t^{\sqrt{2}-1}$ [16]. A
recently suggested model for martensitic transformations can also be
reduced to a two-dimensional fragmentation process where only two
fragments are created at each event. Although the rate equations are
different, the moment technique is still applicable and the resulting
behavior is similar: the length multiscales, while the area
scales [4,20]. Stochastic counterparts of canonical fractals such as the
Cantor set and the Serpinsky gasket can also be considered using
multifragmentation techniques, and one can obtain expressions for the
fractal dimension and other multifractal properties [21,22].

In conclusion, fragmentation processes in spatial dimensions larger
than one exhibit multiscaling 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 volume distribution function also exhibits a weak
logarithmic singularity near the origin. 


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\end{multicols}
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