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\title{Comment on "Kinematic Scaling and Crossover to Scale 
Invariance in Martensite Growth"}
\author{E.~Ben-Naim$^1$ and P.~L.~Krapivsky$^2$}
\address{$^1$The James Franck Institute, The University of Chicago, 
Chicago, IL 60637}
\address{$^2$Courant Institute of Mathematical Sciences, 
New York University, New York, 10012}
\maketitle
{P.A.C.S. numbers: 64.60.Ht, 64.60.Ak, 81.30.Kf}
\begin{multicols}{2}
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In a recent Letter [1], Rao {\it et. al.} proposed a simple model
describing martensites formation.  In 2D, the model is defined as
follows: (i) Segments grow from seeds that are nucleated in a bounded
region; (ii) The tips of these segments move with a constant velocity
$V$ until they hit either another segment or the boundary; (iii) Any
tip grows in one of two possible orthogonal directions.  There are two
important limiting cases, uniform nucleation and simultaneous
nucleation. Uniform nucleation with $V=\infty$ can be viewed as a
multifragmentation process [2,3], and both result in similar patterns
(compare Figs.~2 of [1] and [3]).  In the following Comment, we obtain
exact asymptotic properties of the uniform model using the method of
Ref.~[3]. The process is characterized by an infinite amount of scales
and an infinite set of conserved quantities, and thus exhibits
multiscaling.  These findings disagree with the ordinary scaling
behavior of the segment length distribution reported in [1].

The case of uniform nucleation is equivalent to a stochastic
process where seeds appear uniformly in space with unit rate, and grow
with infinite velocity in the $x$ or the $y$ directions with equal
probabilities.  For simplicity, we choose the unit square as the
transformed region.  The distribution function $P(x_1,x_2,t)$,
describing rectangles of size $x_1 \times x_2$ arising in this kinetic
process, satisfies
\begin{eqnarray}
{\partial P(x_1,x_2,t)\over \partial t}=-x_1x_2 P(x_1,x_2,t)\label{ke}\\
+x_2\int_{x_1}^1 dx'_1 P(x'_1,x_2,t)
+x_1\int_{x_2}^1 dx'_2 P(x_1,x'_2,t).\nonumber
\end{eqnarray}
Performing the double Mellin transform, $M(s_1,s_2,t)=\int dx_1
 dx_2 x_1^{s_1-1} x_2^{s_2-1}P(x_1,x_2,t)$, Eq.~(\ref{ke}) reduces to
\begin{equation}
{\partial M(s_1,s_2)\over \partial t}= 
\left(s_1^{-1}+s_2^{-1}-1\right)M(s_1+1,s_2+1).
\label{me}
\end{equation}

Indeed, the total area is $M(2,2,t)=1$ and the number of rectangles is
$N=M(1,1,t)=1+t$, in agreement with (\ref{me}).  A remarkable feature of
Eq.~(\ref{me}) 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 $s_2$
on the hyperbola $s_1^{-1}+s_2^{-1}=1$, are time-independent. 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 simple scaling solutions.  Indeed, a
solution of the form $P(x_1,x_2,t)=t^wQ(t^zx_1,t^zx_2)$ to
Eq.~(\ref{ke}) would imply an infinite set of relations,
$w=z(s_1+s_2)$ when $s_1^{-1}+s_2^{-1}=1$, which cannot be satisfied
by just two scaling exponents, $w$ and $z$.

The moments of Eq.~(\ref{me}) are exactly solvable in terms of
generalized hypergeometric functions. However, we present an asymptotic
analysis since for sufficiently large $t\equiv N$, the moments depend
algebraically on time, $M(s_1,s_2,t)\sim t^{-\alpha(s_1,s_2)}$.
Substituting this form into Eq.~(\ref{me}) yields
$\alpha(s_1,s_2)+1=\alpha(s_1+1,s_2+1)$.  This difference equation is
solved subject to the boundary conditions, $\alpha(s_1,s_2)=0$ on the
hyperbola $s_1^{-1}+s_2^{-1}=1$, and one finds
\begin{equation}
\alpha(s_1,s_2)={s_1+s_2\over 2}-1-\sqrt{\left({s_1-s_2\over
2}\right)^2+1}.
\label{alpha}
\end{equation}

The nontrivial nature of the asymptotic behavior is demonstrated by
evaluating moments such as $\langle x_1^{n_1}x_2^{n_2}\rangle
\equiv \int dx_1 dx_2 x_1^{n_1} x_2^{n_2}P(x_1,x_2,t)/\int dx_1 dx_2
P(x_1,x_2,t)$.  For example, the ratio $\langle (x_1 x_2)^n\rangle/
\langle x_1^n\rangle\langle x_2^n\rangle\sim t^{-(\sqrt{n^2+4}-2)}$,
{\it depends} asymptotically on time, while for a scaling distribution
$P(x_1,x_2,t)$ such a ratio would approach a non-zero constant.  The
length distribution $P(l,t)$ can be easily obtained from the size
distribution using the evolution equation
$\partial P(l,t)/\partial t=
\int dx_1 dx_2 x_1x_2P(x_1,x_2,t)[\delta(l-x_1)+\delta(l-x_2)]/2$.
Hence, the corresponding Mellin transform $M(s,t)=\int dl l^{s-1}P(l,t)$ 
satisfies $\partial M(s,t)/\partial t=M(s+1,2,t)$.  This is solved to
find  $M(s,t)\sim t^{1-\alpha(s+1,1)}$. Substituting Eq.~(\ref{alpha}) gives 
\begin{equation}
\langle l^n\rangle\sim {\langle x_1^n\rangle}\sim
t^{-\left(n+2-\sqrt{n^2+4}\right)/2}, 
\label{non}
\end{equation} 
rather than simple scaling $\langle l^n\rangle\sim t^{-n/2}$ [1]. 
Thus, the length distribution is characterized by {\it nontrivial}
exponents. For example, one finds ${\langle l\rangle}\sim
t^{-(3-\sqrt{5})/2}\sim t^{-.382}$ and $\langle l^2\rangle\sim
t^{2-\sqrt{2}}\sim t^{-.586}$, as confirmed by simulations.  Since
all the moments still show a power-law behavior, we conclude that the
model exhibits a {\it multiscaling} asymptotic behavior.  In contrast
with the general behavior, moments of the area $A=x_1x_2$ follow
ordinary scaling, $\langle A^n\rangle\sim \langle A\rangle^n\sim
t^{-n}$, and the area distribution function approaches a simple
scaling form $P(A,t) \simeq t^2e^{-At}$.

\begin{thebibliography}{99}
\bibitem{1} M.~Rao, S.~Sengupta, and H.~K.~Sahu, {\sl Phys. Rev. Lett.}
            {\bf 75}, 2164 (1995).
\bibitem{2} G.~Tarjus, and P.~Viot, {\sl Phys. Rev. Lett.} {\bf 67}, 
            1875 (1991).
\bibitem{3} P.~L.~Krapivsky and E.~Ben-Naim, {\sl Phys. Rev.} {\bf E 50},
            3502 (1994).
\end{thebibliography} 

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