Jamming and Tiling in Fragmentation of Rectangles

TL;DR

This paper studies a stochastic fragmentation process of rectangles with discrete dimensions, revealing a jammed state with sticks, a size-independent growth of sticks, and phase transitions in asymmetric cases.

Contribution

It introduces a model of rectangle fragmentation with discrete sizes, analyzes the jammed state, and uncovers phase transitions based on asymmetry in breakage probabilities.

Findings

Average number of sticks grows as $A/\sqrt{2\pi\ln A}$ in large areas.

Stick length distribution has a power-law tail with nonlinear scaling exponents.

Phase transition occurs between size-independent and size-dependent distributions in asymmetric breakage.

Abstract

We investigate a stochastic process where a rectangle breaks into smaller rectangles through a series of horizontal and vertical fragmentation events. We focus on the case where both the vertical size and the horizontal size of a rectangle are discrete variables. Because of this constraint, the system reaches a jammed state where all rectangles are sticks, that is, rectangles with minimal width. Sticks are frozen as they can not break any further. The average number of sticks in the jammed state, $S$, grows as $S\simeq A/\sqrt{2\pi\ln A}$ with rectangle area $A$ in the large-area limit, and remarkably, this behavior is independent of the aspect ratio. The distribution of stick length has a power-law tail, and further, its moments are characterized by a nonlinear spectrum of scaling exponents. We also study an asymmetric breakage process where vertical and horizontal fragmentation events are realized with different probabilities. In this case, there is a phase transition between a weakly asymmetric phase where the length distribution is independent of system size, and a strongly asymmetric phase where this distribution depends on system size.

Tables

Table 1. Table 1: The number of jammed states for small squares.

$n$	1	2	3	4	5	6	7	8
$T(n,n)$	1	1	4	33	436	9,524	354,224	23,097,969

Equations

$$S\simeq \frac{A}{\sqrt{2\pi \ln A}}.$$

$$(m,n)\to \{\begin{array}{ll}(i,n)+(m-i,n)& \mathrm{with}prob.1/2,\\ (m,j)+(m,n-j)& \mathrm{with}prob.1/2.\end{array}$$

$$S(m,n)=\frac{1}{m-1}\sum _{i=1}^{m-1}S(i,n)+\frac{1}{n-1}\sum _{j=1}^{n-1}S(m,j).$$

$$S(m,1)=S(1,n)=1$$

$$\begin{array}{rl}& S(2,2)=2,S(2,3)=\frac{5}{2},S(2,4)=\frac{17}{6},\\ & S(3,3)=\frac{7}{2},S(3,4)=\frac{17}{4}S(4,4)=\frac{97}{18}.\end{array}$$

$$S(2,n)=1+H_{n-1},$$

$$S(m,n)\simeq \frac{(\ln n{)}^{m-1}}{(m-1){!}^2},$$

$$S(m,n)=\frac{1}{m}{\int }_1^{m}diS(i,n)+\frac{1}{n}{\int }_1^{n}djS(m,j),$$

$${\partial }_m{\partial }_n\left[mnS(m,n)\right]={\partial }_m\left[mS(m,n)\right]+{\partial }_n\left[nS(m,n)\right].$$

$$\mu =\ln m,\nu =\ln n.$$

$${\partial }_{\mu }{\partial }_{\nu }S(\mu ,\nu )=S(\mu ,\nu ),$$

$$\hat{S}(p,q)={\int }_0^{\infty }d\mu e^{-p\mu }{\int }_0^{\infty }d\nu e^{-q\nu }S(\mu ,\nu ).$$

$$\hat{S}(p,q)=\frac{1}{pq-1}.$$

$$S(\mu ,\nu )$$

$$I={\int }_{-\mathrm{i}\infty }^{\mathrm{i}\infty }\frac{dq}{2\pi \mathrm{i}}F(q)e^{\nu f(q)}.$$

$$f(q)\simeq f(q_{\ast })+\frac{1}{2}(q-q_{\ast }{)}^2{f}^{\prime \prime }(q_{\ast }).$$

$$I\simeq \frac{F(q_{\ast })e^{\nu f(q_{\ast })}}{\sqrt{2\pi \nu f^{\prime \prime }(q_{\ast })}}.$$

$$f(q)=q+q^{-1}$$

$$S(\nu ,\nu )\simeq \frac{e^{2\nu }}{\sqrt{4\pi \nu }}.$$

$$S(\mu ,\nu )\simeq \frac{e^{2\sqrt{\mu \nu }}}{\sqrt{4\pi \sqrt{\mu \nu }}}.$$

$$S(m,n)=\sum _{k\ge 2}{S}_k(m,n),A=\sum _{k\ge 2}k{S}_k(m,n).$$

$$S_k(m,1)={\delta }_{m,k}{S}_k(1,n)={\delta }_{n,k}.$$

$$\begin{array}{rl}{S}_k(\mu ,0)& =e^{-\mu }\delta (\mu -\ln k),\\ S_k(0,\nu )& =e^{-\nu }\delta (\nu -\ln k).\end{array}$$

$${\hat{S}}_k(p,q)=\frac{p{k}^{-1-p}+q{k}^{-1-q}}{pq-1}.$$

$$f(q)=q+q^{-1}-(1+q)x,x=\frac{\ln k}{\ln n}.$$

$$S_k(\nu ,\nu )\simeq \frac{\exp \left[\nu \left(2\sqrt{1-x}-x\right)\right]}{\sqrt{\pi \nu (1-x{)}^{3/2}}}.$$

$$\frac{\ln \left(\frac{1}{2}{P}_k\right)}{\ln n}\simeq \Phi \left(\frac{\ln k}{\ln n}\right)$$

$$\Phi (x)=2\left(\sqrt{1-x}-1\right)-x.$$

$$P_k\simeq 2k^{-2}\exp \left[-\frac{(\ln k{)}^2}{4\ln n}\right].$$

$$P_k=\frac{2}{k(k+1)},$$

Full text

I Introduction

Fragmentation processes where large objects break into smaller ones underlie an ever growing number of physical and natural phenomena bdh ; bkb ; zxg ; kbg ; fl ; tfg ; khp . In particular, fragmentation occurs in soft matter systems such as polymers mrt , active matter mo ; gtd , granular media hp ; sas , and brittle materials kh ; alt ; ghb .

Experimental and theoretical studies of fragmentation generally focus on the distribution of fragment size. Typically, this distribution is self-similar throughout the breakage process, and it is characterized by a single quantity, for example, the average fragment size af ; zm ; cr . Self-similarity extends to discrete fragments and continuous ones, one-dimensional fragments and multi-dimensional ones krb . However, while the fragment size is a fluctuating quantity throughout the breakage process, the fragment size becomes deterministic in the final state as all fragments have the same size. In this sense, the final state can be trivial.

Recently, non-trivial final states have been reported in a multi-dimensional fragmentation process cb ; qpb ; bch ; ch which models martensitic phase transformations rss ; bk ; tivp . The system reaches a jammed state where the two-dimensional fragments are characterized two sizes: one size is a deterministic quantity, but the second size is a stochastic quantity. Here, we study this planar fragmentation process analytically, and we present a comprehensive statistical analysis of the jammed state.

We study fragmentation of rectangles with discrete horizontal and vertical sizes (Fig. 1). A rectangle can break vertically or horizontally into two smaller rectangles. Due to discreteness, rectangles with minimal vertical or horizontal size can not break, and hence, are frozen. We refer to these frozen rectangles as “sticks.” Through a sequence of random fragmentation events, the system which initially consists of a single rectangle, reaches a jammed state where all rectangles are sticks (Fig. 2).

We find that, up to a logarithmic correction, the average number of frozen sticks in the jammed state, $S$, grows linearly with the area $A$,

[TABLE]

Interestingly, this asymptotic behavior is universal as it applies regardless of aspect ratio. We also study the distribution of stick length and find that this distribution has a power-law tail. Further, this length distribution exhibits multi-scaling asymptotic behavior as its moments are characterized by a nonlinear spectrum of exponents.

We also investigate an asymmetric process where horizontal and vertical cuts are realized with different probabilities. We find a phase transition at a critical value of the asymmetry parameter. In the weakly asymmetric phase, the length distribution does not depend on system size, while in the strongly asymmetric phase, this distribution does depend on system size.

The rest of this paper is organized as follows. In Sec. II, we introduce the fragmentation process and develop the theoretical techniques used throughout this investigation to obtain the leading asymptotic behavior in the large system-size limit. We first analyze the average number of sticks and then consider the length distribution of sticks. In Sec. III, we generalize the results to the case where horizontal and vertical fragmentation occur at different rates. Next, in section IV, we analyze a closely related process of fragmentation into four, rather than two, rectangles. In this case, the outcome of a fragmentation event is deterministic, and we can also address the total number of jammed configurations. We conclude with a discussion in Sec. V. Details of several technical derivations are presented in the Appendix.

II Fragmentation of Rectangles

Initially, the system consists of a single rectangle with horizontal size $m$, vertical size $n$, and hence, area $A=mn$. Both the horizontal size and the vertical size are integer. It is convenient to envision a square grid with $(m-1)(n-1)$ internal grid points embedded within the rectangle (Fig. 1). In each fragmentation event, an internal grid point is selected, and then a cut is made along the horizontal or the vertical direction. As a result, the rectangle breaks into two smaller ones,

[TABLE]

The grid point with $1\leq i\leq m-1$ and $1\leq j\leq n-1$ is chosen at random, as is the fragmentation direction. Of course, the total area is conserved.

Fragmentation requires an internal grid point. Therefore, rectangles with $m>1$ and $n>1$ are active, and otherwise, rectangles with $m=1$ or $n=1$, referred to as sticks, are frozen. The fragmentation process (2) is repeated for every active rectangle until the system reaches a jammed state with sticks only (Fig. 2). In this study, we focus on the jammed state.

Let $S(m,n)$ be the average number of sticks in the jammed state when the initial rectangle has dimensions $m\times n$. This average is taken over all realizations of the random breakage process. Since the fragmentation process (2) is symmetric with respect to the horizontal and the vertical direction, we expect $S(m,n)=S(n,m)$. The average number of frozen sticks obeys the recursion tivp

[TABLE]

The first term on the right-hand side accounts for the $m-1$ possible cuts in the vertical direction, and similarly, the second term accounts for the $n-1$ possible cuts in the horizontal direction. The recursion equation is linear as each fragmentation event involves a single rectangle, and it is subject to the boundary conditions

[TABLE]

for all $m\geq 1$ and $n\geq 1$.

Equations (3) and (4) yield the average number of frozen sticks for small rectangles,

[TABLE]

Moreover, for ladders ($m=2$), equation (3) simplifies to $S(2,n)-S(2,n-1)=\tfrac{1}{n-1}$ and therefore,

[TABLE]

where $H_{N}=\sum_{1\leq i\leq N}i^{-1}$ is the harmonic number. For long ladders, $S(2,n)\simeq\ln n+1+\gamma$, where $\gamma=0.57721$ is the Euler constant. It is also possible to show that as long as $m$ is finite, the leading asymptotic behavior remains logarithmic,

[TABLE]

in the limit $n\to\infty$.

Our main interest is the behavior for large rectangles, and specifically, the leading asymptotics when $m\to\infty$ and $n\to\infty$. Hence, we treat $m$ and $n$ as continuous variables, and replace the sums in (3) with integrals. The average number of sticks satisfies the integral equation

[TABLE]

within this continuous framework. Next, we multiply this integral equation by the area $mn$ and then differentiate the resulting equation with respect to $m$ and $n$. That shows the quantity $S(m,n)$ satisfies the partial differential equation

[TABLE]

Hereinafter, we use $\partial_{m}=\frac{\partial}{\partial m}$ and $\partial_{n}=\frac{\partial}{\partial n}$ to denote partial derivatives. Further simplification can be achieved by introducing the logarithmic variables

[TABLE]

With this transformation, Eq. (9) reduces to a partial differential equation with constant coefficients,

[TABLE]

that should be solved subject to the boundary conditions $S(\mu,0)=1$ and $S(0,\nu)=1$.

The central quantity throughout our analysis is the double Laplace transform

[TABLE]

It is obtained by multiplying both sides of the governing equation (11) by $e^{-p\mu-q\nu}$ and then integrating over the logarithmic variables $\mu$ and $\nu$. By using the boundary conditions $S(\mu,0)=1$ and $S(\nu,0)=1$, we find that double Laplace transform is remarkably compact,

[TABLE]

Therefore, the average number of sticks in the jammed state, $S(\mu,\nu)$, equals the inverse Laplace transform

[TABLE]

We perform the inversion first with respect to the conjugate variable $p$ and then with respect to the conjugate variable $q$. The inversion with respect to $p$ is immediate as the integrand in the first line has a pole at $p=q^{-1}$.

The integral over the variable $q$ in (14) has the form

[TABLE]

The exponential dominates the integrand in the limit $\nu\to\infty$. Further, the function $f(q)$ is maximal at the saddle point $q_{*}$ which is determined from $f^{\prime}(q_{*})=0$, and in the vicinity of this saddle point we have

[TABLE]

The integration contour in (15) can be along any line that parallels the imaginary axis in the complex plane as long as $\text{Re}(q)$ is greater than the real part of any singularity the integrand may have. We conveniently choose a line parallel to the imaginary axis that passes through the saddle point $q_{*}$. With the transformation of variables $q=q_{*}+{\rm i}y/\sqrt{f^{\prime\prime}(q_{*})}$, the integral (15) reduces to a Gaussian integral. As long as $\text{Re}(q_{*})$ exceeds the real part of the singularities of $F(q)$, we have

[TABLE]

Here, we used $\int_{-\infty}^{\infty}\exp(-y^{2}/2)dy=\sqrt{2\pi}$.

First, we discuss squares, $\mu=\nu$, for which $S(\nu,\nu)=I$ with $I$ given in (15). The quantities $F(q)=q^{-1}$ and

[TABLE]

specify the integral (15). The saddle point is $q_{*}=1$, and furthermore, $f(q_{*})=f^{\prime\prime}(q_{*})=2$ and $F(q_{*})=1$. By substituting these values into the general expression (17), we obtain the leading asymptotic behavior

[TABLE]

Equation (1) expresses this behavior in terms of the area $A=e^{2\nu}$. Results of numerical evaluation of the recursion equation (3) are in excellent agreement with the theoretical prediction, and we conclude that the continuum framework yields exact results for the leading asymptotic behavior (Fig. 3).

We now consider squares of arbitrary size for which $S(\mu,\nu)=I$ with $f(q)=q+(\mu/\nu)q^{-1}$ and $F(q)=q^{-1}$. It is straightforward to repeat the steps leading to (19), and obtain the general behavior

[TABLE]

As expected, the average number of sticks is symmetric, $S(\mu,\nu)=S(\nu,\mu)$, and moreover, the quantity $\nu$ in (19) is now replaced with the geometric average $\sqrt{\mu\nu}$. In the limit $m\to\infty$ and $n\to\infty$ with the aspect ratio $r=\tfrac{m}{n}$ kept fixed, we have $\sqrt{\mu\nu}\simeq(\mu+\nu)/2$. In this limit, the leading asymptotic behavior (20) is identical to (19). Interestingly, the average number of jammed sticks is universal in the large-area limit—all rectangles with the same area behave similarly. The only requirement is that the aspect ratio is finite. In view of this universality, we henceforth quote results for squares without loss of generality.

We can immediately deduce the average length of a stick in the jammed state, $\langle k\rangle$. Since the rectangle is covered entirely by sticks, we have $A=S\langle k\rangle$. Therefore, the average stick length grows logarithmically with area, $\langle k\rangle\simeq\sqrt{2\pi\ln A}$, and this behavior is independent of the aspect ratio in the large-area limit.

In the jammed state, the original rectangle is covered with rectangles of size $1\times k$ or $k\times 1$ with $k\geq 2$, and we now analyze the distribution of stick length $k$. Let $S_{k}(m,n)$ be the average number sticks of length $k$ in the jammed state, when the initial rectangle has dimensions $m\times n$. This quantity satisfies two sum rules,

[TABLE]

For all $k$, the quantity $S_{k}(m,n)$ satisfies the recursion equation (3), although the boundary condition does depend on length

[TABLE]

With the boundary condition $S_{n}(1,n)=1$, the recursion equation (3) implies $S_{n}(m,n)=1$ for all $m<n$, and it is also possible to show that $S_{n}(n,n)=2$.

For large rectangles, we utilize the continuum approach once again. As a function of the logarithmic variables defined in (10), the average number of frozen sticks with a given length $S_{k}(\mu,\nu)$ satisfies the partial differential equation (11), subject to the boundary conditions

[TABLE]

To obtain these boundary conditions, we first rewrite (22) as $S_{k}(m,1)=\delta(m-k)$ and $S_{k}(1,n)=\delta(n-k)$ and then perform the transformation of variables (10) by using $\delta[{\cal F}(x)]=\delta(x-x_{0})/|{\cal F}^{\prime}(x_{0})|$.

Next, we repeat the steps leading to (13), using the partial differential equation (11) which also governs $S_{k}(\mu,\nu)$ and the boundary condition (23), and arrive at

[TABLE]

We note that this double Laplace transform is symmetric, $\widehat{S}_{k}(p,q)=\widehat{S}_{k}(q,p)$ as the two terms in the numerator are equivalent. For squares, $\mu=\nu$, it suffices to invert only one of these terms. We thus perform the inverse Laplace transform of the term $qk^{-1-q}/(pq-1)$ first with respect to $p$ and then with respect to $q$, thereby leading to $S_{k}(\nu,\nu)=I$ with $I$ given by (17). The integrand is specified by $F(q)=2$ and

[TABLE]

The saddle point is $q_{*}=1/\sqrt{1-x}$, and by substituting $f(q_{*})=2\sqrt{1-x}-x$ and $f^{\prime\prime}(q_{*})=2(1-x)^{3/2}$ into the general formula (17), we obtain

[TABLE]

By definition, $P_{k}(\nu)=S_{k}(\nu,\nu)/S(\nu,\nu)$ is the fraction of frozen sticks with length $k$. This distribution is normalized, $\sum_{k\geq 2}P_{k}=1$, and its first moment equals the average length, $\langle k\rangle=\sum_{k\geq 2}kP_{k}$. By using equations (19) and (26) we find that the length distribution adheres to the scaling form

[TABLE]

with the scaling function

[TABLE]

Equation (27) constitutes an unusual scaling form as the scaled logarithm of the length distribution $P_{k}$ is a universal function of the scaled logarithm of the length $k$. As a result, the convergence toward the ultimate asymptotic behavior is extremely slow as it involves logarithm of system size (Fig. 4).

The scaling behavior (27)-(28) describes the distribution at large length scales, that is, $\ln k={\cal O}(\ln n)$. Still, the small-$x$ behavior $\Phi(x)\simeq-2x-\tfrac{1}{4}x^{2}$ yields the behavior at smaller length scales,

[TABLE]

Therefore, the length distribution decays as a power-law, $P_{k}\simeq 2k^{-2}$ tivp , at sufficiently small length scales, $\ln k\ll\sqrt{\ln n}$. Beyond this length scale, the power-law tail is suppressed by a log-normal term. We also note that log-normal distributions naturally arise in multiplicative random processes lsa ; mm , and that the fragmentation process (2) can be formulated as such.

In a finite system, the power-law tail holds over a limited range (Fig. 5). Further, the log-normal term is relevant at length scales $k_{2}$ determined by $\ln k_{2}\sim\sqrt{\ln n}$. Similarly there is a series of length scales $k_{b}$ that are specified by $\ln k_{b}\sim(\ln n)^{(b-1)/b}$. For $k\gg k_{b}$, the $b^{\text{th}}$ term in the Taylor expansion of $\Phi(x)$ affects the length distribution. Ultimately, at sufficiently large scales, the entire scaling function (28) characterizes the length distribution (Fig. 4).

In appendix A, we derive the exact length distribution

[TABLE]

which is realized in the limit $n\to\infty$. This distribution is properly normalized, $\sum_{k\geq 2}P_{k}=1$, and its power-law tail $P_{k}\simeq 2k^{-2}$ agrees with the asymptotic behavior (29). Furthermore, results of numerical evaluation of the recursion equation (3) with the boundary condition (22) are in excellent agreement with this theoretical prediction (Fig. 5). The exact length distribution (30) can be expressed a ratio of Gamma functions, the discrete counterpart of a power-law. It can be derived by treating the variables $m$ and $n$ as discrete, in contrast with the continuum analysis leading to (29).

Finally, we investigate the moments of the length distribution, defined by $\langle k^{h}\rangle=\sum_{k\geq 2}k^{h}P_{k}$. It is convenient to normalize these moments by the average length,

[TABLE]

with $h>1$. By using the definition $P_{k}=S_{k}/S$ and the second sum rule in (21), we can express the normalized moments through $S_{k}(n,n)$, the average number of frozen sticks with length $k$ in a square of of size $n\times n$,

[TABLE]

We now substitute (26) into this expression and convert the sum over the discrete variable $k$ into an integral over the continuous variable $x$ by using $k=e^{\nu x}$. With this transformation of variables, the moments are given by

[TABLE]

with $\phi(x)=\Phi(x)+(h+1)x$. The exponential term dominates the integral in the limit $\nu\to\infty$. The function $\phi(x)$ is maximal at the saddle point, $x_{*}=1-h^{-2}$, and from the quadratic behavior $\phi(x)=\phi(x_{*})+\tfrac{1}{2}\phi^{\prime\prime}(x_{*})(x-x_{*})^{2}$, we deduce the leading asymptotic behavior (Fig. 6)

[TABLE]

Results of numerical evaluation of low-order moments are in excellent agreement with this theoretical prediction.

The scaling exponent $\mu$ is a nonlinear function of the index $h$, and therefore, the scaling behavior of the first moment does not characterize high-order moments. Hence, the moments exhibit multi-scaling asymptotic behavior. Qualitatively similar behavior is found for the continuous version of the fragmentation process (2). To compare the two cases, we note that the aspect ratio of a frozen rectangle $r$ equals its length, $r=k$. For the continuous version, the normalized moments of the aspect ratio, $M_{h}=\langle r^{h}\rangle/\langle r\rangle$, also exhibit multi-scaling asymptotic behavior $M_{h}\sim(A/\langle A\rangle)^{\mu_{\text{cont}}}$, where $\langle A\rangle$ is the average rectangle area and

[TABLE]

is the nonlinear scaling exponent kb ; bk . Surprisingly, the two spectrums of exponents are different, $\mu\neq\mu_{\text{cont}}$ although both become linear at high orders, $\mu\simeq\mu_{\text{cont}}\simeq h$ as $h\to\infty$. Figure 7 shows that multi-scaling is more pronounced in the discrete case.

III Asymmetric Fragmentation

We now generalize the fragmentation process (2) and consider the case where the probabilities of horizontal and vertical cuts may differ tivp . The asymmetric fragmentation process can be represented schematically as

[TABLE]

The parameter $\alpha$ controls the degree of asymmetry, and without loss of generality, we assume $0\leq\alpha\leq 1$. The fragmentation process (36) reduces to (2) when there is no asymmetry, $\alpha=0$, and it becomes one-dimensional, when the asymmetry parameter is maximal, $\alpha=1$.

In the completely asymmetric case, the jammed state contains $n$ identical sticks of length $n$. Hence, the number of sticks is not proportional to the area, and also, there is no logarithmic dependence on system size, unlike (1). Below, we show that the logarithmic dependence on system size disappears when the asymmetry parameter exceeds the critical value

[TABLE]

The average number of frozen sticks obeys

[TABLE]

subject to the boundary condition (4). This recurrence reduces to (3) when the asymmetry parameter vanishes.

Once again, we employ the continuum approach. In terms of the logarithmic variables (10), the quantity $S\equiv S(\mu,\nu)$ satisfies the partial differential equation

[TABLE]

We now repeat the steps leading to (13) and find that the double Laplace transform, defined in (12), is given by

[TABLE]

Since the governing equation (39) is no longer symmetric in the variables $\mu$ and $\nu$, the Laplace transform (40) is not symmetric when $\alpha\neq 0$. We thus reiterate that results are quoted only for squares.

To invert the Laplace transform (40), we split the numerator $1+\alpha(p^{-1}-q^{-1})$ into $q-$dependent and $p-$dependent terms: $1-\alpha\,q^{-1}$ and $\alpha\,p^{-1}$. Due to asymmetry, these two are no longer equivalent, and the average number of sticks in the jammed state $S\equiv S(\nu,\nu)$ is given by

[TABLE]

The quantity $I\equiv I(\nu,\nu)$ is obtained by inverting \big{(}1-\alpha\,q^{-1}\big{)}/[pq+\alpha(q-p)-1] first with respect to the conjugate variable $p$ and then with respect to the conjugate variable $q$. Similarly, the quantity $J\equiv J(\nu,\nu)$ is obtained by inverting $\alpha\,p^{-1}/[pq+\alpha(q-p)-1]$ first with respect to $q$ and then, with respect to $p$.

To compute the first term in (41), we follow the calculations in the symmetric case, and find that $I$ is an integral of the form (15), specified by $F(q)=q^{-1}$ and

[TABLE]

The saddle point is simply $q_{*}=\beta+\alpha$, and from the general formula (17) we obtain

[TABLE]

This asymptotic behavior resembles (1) in that the leading exponential behavior is suppressed by a logarithmic term.

The second quantity in (41) is analogous in form to (15): it is given by a integral over $p$, rather than $q$,

[TABLE]

with the functions

[TABLE]

However, the saddle point of the function $g(p)$ is different, $p_{*}=\beta-\alpha$. To evaluate the integral $J$, we simply replace $F(q_{*})$, $f(q_{*})$, and $f^{\prime\prime}(q_{*})$ in (17) with $G(p_{*})$, $g(p_{*})$, and $g^{\prime\prime}(p_{*})$ respectively, to find

[TABLE]

Therefore, the two terms in the sum (41) are proportional to each other, and by adding (43) and (46), we arrive at

[TABLE]

with the critical point $\alpha_{c}$ given in (37). The constant $C$ diverges as $\alpha\uparrow\alpha_{c}$, thereby indicating that the result (47) is valid only when the asymmetry is sufficiently weak, $\alpha<\alpha_{c}$. Indeed, the integrand in (44) has two simple poles: one at $p=-\alpha$ and another at $p=0$. The first pole is located to the left of the saddle point $p_{*}$, irrespective of $\alpha$. However, the second pole is located to the left of the saddle point only when $\alpha<\alpha_{c}$, and consequently, Eq. (46) holds only in this regime.

To evaluate the integral $J$ when $\alpha>\alpha_{c}$, we deform the integration contour so that it consists of a line parallel to the imaginary axis which passes through the saddle point $p_{*}$ and a small circle enclosing the origin, $p=0$. The residue at the origin gives the dominant contribution, $J\simeq e^{\nu/\alpha}$, which is valid when the asymmetry is sufficiently strong, $\alpha>\alpha_{c}$. In this regime, the quantity $I$ in (43) is negligible, and consequently, $S(\nu,\nu)\simeq e^{\nu/\alpha}$.

In summary, the number of sticks in the jammed state state grows algebraically with area $A$

[TABLE]

As long as there is some asymmetry, the growth is sub-linear: $\gamma<1$ when $\alpha>0$. The exponent $\gamma$ is continuous at the critical point, but its first derivative is discontinuous at that point. Furthermore, the exponent $\gamma$ is concave when $\alpha<\alpha_{c}$ but it is convex when $\alpha>\alpha_{c}$. The critical value is $\gamma_{c}=\alpha_{c}$, and the convergence toward the leading asymptotic behavior is slower near the critical point (Fig. 8).

The prefactor $U\equiv U(A)$ in (48) depends logarithmically on area in the weakly asymmetric phase, and it is given by

[TABLE]

The constant $C$ is quoted in (47), and the behavior at the critical point is derived in Appendix B. Hence, there is logarithmic correction in the weakly asymmetric phase, but the logarithmic correction disappears in the strongly asymmetric phase. The critical prefactor $U_{c}=\sqrt{e/(4\pi)}$ obeys $0<U_{c}<1$, and we note the “double discontinuity”: the critical value $U_{c}$ does not match either of the limiting behaviors $U\to 0$ as $\alpha\uparrow\alpha_{c}$ or $U\to 1$ as $\alpha\downarrow\alpha_{c}$.

The length distribution decays algebraically

[TABLE]

for $k\gg 1$. This behavior is derived in Appendix C. The power-law tail (50) generally holds for infinitely large systems. However, for finite systems, this behavior holds in the range $\ln k\ll\sqrt{\ln n}$, as discussed above. Generally, the exponent $\sigma$ decreases monotonically as $\alpha$ increases, and it vanishes in the completely asymmetric case (Fig. 9). Therefore, as the asymmetry parameter becomes smaller, the tail of the length distribution decays more sharply (see also Ref. tivp ).

Interestingly, the exponent $\sigma$ which characterizes the tail of the length distribution has the same form (50) in the weakly asymmetric phase, $\alpha\leq\alpha_{c}$ and the strongly asymmetric phase $\alpha\geq\alpha_{c}$. Yet, the prefactor $V\equiv V(A)$ in (50) depends algebraically on area in the strongly asymmetric phase, and it is given by

[TABLE]

Therefore, the length distribution depends on system size in the strongly asymmetric phase, but it is independent of system size in the weakly asymmetric phase.

From the length distribution $P_{k}$, it is also possible to evaluate the moments $M_{h}$ defined in (31). We find that the moments grow algebraically with the area as in (34), $M_{h}\sim A^{\mu}$. For asymmetric fragmentation, the spectrum of scaling exponents is given by

[TABLE]

This spectrum reduces to (34) when fragmentation is symmetric, but in contrast, the scaling exponents are linear, $\mu=\tfrac{h-1}{2}$, when fragmentation is completely asymmetric. Figure 10 demonstrates how multi-scaling becomes less pronounced as the asymmetric nature of the fragmentation process becomes stronger.

IV Deterministic Fragmentation

The fragmentation process (2) incorporates two stochastic elements as both the fragmentation point and the fragmentation direction are selected at random. The latter element can be eliminated by generating four rectangles, rather than two, in each fragmentation event (Fig. 11). A continuous version of this planar fragmentation process was introduced in tv and analyzed in a number of subsequent studies kb ; rh ; btv ; btv97 ; bk97 .

We now address this natural counterpart of the fragmentation process (2) where first an internal grid point is selected at random, and then, two simultaneous cuts are made, one in the horizontal direction and one in the vertical direction. As a result, each fragmentation event generates four rectangles (Fig. 11)

[TABLE]

with randomly selected $1\leq i\leq n-1$ and $1\leq j\leq m-1$. In contrast with (2), once the grid point is selected, the outcome is deterministic. Again, rectangles with $m>1$ and $n>1$ are active and otherwise, rectangles with $m=1$ or $n=1$ are frozen. Starting with a single $m\times n$ rectangle, the system eventually reaches a jammed state where all rectangles are sticks, including minimal $1\times 1$ rectangles (Fig. 12).

The average number of frozen sticks in the jammed state, $S(m,n)$, satisfies the recursion equation

[TABLE]

subject to the boundary conditions (4). For small rectangles, the recursion equation gives $S(2,2)=4$, $S(3,3)=7$, $S(4,4)=\frac{32}{3}$, and so on. In contrast with the fragmentation process (2) where the average number of sticks for narrow but long rectangles diverge logarithmically, these quantities are now finite, and for example, $S(2,n)=4$ and $S(3,n)=10-\tfrac{6}{n-1}$. In general, we find the limiting values

[TABLE]

from the recursion equation (54).

For large rectangles, the recursion equation (54) turns into the partial differential equation

[TABLE]

Using the logarithmic variables (10), we transform this equation into a partial differential equation with constant coefficients, $\partial_{\mu}\partial_{\nu}S+\partial_{\mu}S+\partial_{\nu}S=3S$. By repeating the steps leading to (13), we obtain the Laplace transform

[TABLE]

Next, we rewrite this expression as a sum of two terms: $(2^{-1}+q^{-1})/(pq+p+q+3)$ and $(2^{-1}+p^{-1})/(pq+p+q+3)$. For squares, these two terms are equivalent and it suffices to perform the inverse Laplace transform of the first term with respect to $p$ and then, with respect to $q$. We thus obtain $S(\nu,\nu)=I$ where $I$ is given by the general integral (17). The integrand is specified by the functions

[TABLE]

From the condition $f^{\prime}(q_{*})=0$ we notice that the saddle point remains the same, $q_{*}=1$, and by using (17), we find the leading asymptotic behavior of the number of frozen sticks in the jammed state (Fig. 13)

[TABLE]

In comparison with (1), the average number of frozen sticks is now $3/\sqrt{2}\approx 2.12132$ times larger. Results of numerical evaluation of the recursion equation (54) are in excellent agreement with the theoretical prediction (59). As was the case for stochastic fragmentation, the average number of jammed rectangles (59) extends to all rectangles with a finite aspect ratio in the large-area limit.

For completeness, we quote the exact distribution of stick length

[TABLE]

for $k>1$ and $P_{1}=\frac{1}{3}$. This form, which is realized in the limit $n\to\infty$, can be obtained using the method outlined in Appendix A. Sticks with $k\geq 2$ are doubly degenerate compared with minimal $1\times 1$ rectangles and hence, the quantity $P_{1}$ is suppressed by a factor 2. The length distribution has a power-law tail, $P_{k}\simeq\tfrac{4}{3}k^{-2}$, which can be established using continuum analysis used to obtain (29). For a finite system, the power-law tail holds when $1\ll k\ll\sqrt{\ln n}$, while at larger length scales the distribution is strongly suppressed by a log-normal term.

The planar fragmentation processes considered in this investigation generate special tilings of two-dimensional domains. Indeed, in the jammed configuration, sticks of unit width and variable length cover the original rectangle (Figures 2 & 12). The jammed configurations differ from those in the heavily studied dimer tiling pwk ; ft ; mef ; ehl ; rjb ; cep ; ckp in two respects. First, the lengths of the sticks do vary dramatically hl ; mj ; fw ; gjl . Second, whereas in equilibrium problems jammed configurations are given equal weights, fragmentation is a dynamical process, and the different tiling configurations are generally realized with different probabilities. In this sense, the jammed configurations considered here can be viewed as nonequilibrium tilings.

The central quantity in tiling problem is the total number of jammed configurations which typically grows exponentially with area. For the deterministic process (53), it is straightforward to show that $T(m,n)$, the total number of jammed configurations for a rectangle of size $m\times n$, satisfies the recursion equation

[TABLE]

This recursion applies for all $m\geq 2$ and $n\geq 2$, and it is subject to the boundary conditions $T(m,1)=1$ for all $m\geq 1$ and $T(1,n)=1$ for all $n\geq 1$. As illustrated in Fig. 14, for any jammed configuration, the first fragmentation event can be uniquely identified. This first fragmentation event divides the original rectangle to four smaller and independent rectangles, thereby leading to the recursion (61). The same does not hold true for the fragmentation process (2) and for this reason, it is not possible to write closed recursion equations for the corresponding of number of jammed states.

A single iteration of the recursion (61) yields the number of jammed configurations for ladders, $T(2,n)=n-1$, and a second iteration yields $T(3,n)=\tfrac{1}{3}(n-2)(n^{2}-4n+15)$. The exact expression for $T(4,n)$ is a seventh-order polynomial, and $T(m,n)$ quickly become unwieldy when $m$ increases.

Table I lists the number of jammed configurations for squares with $n\leq 7$. Numerical iteration of the recursion (61) shows that the number of jammed configurations grows exponentially with area (see Fig. 15)

[TABLE]

For squares, we obtain $\lambda=0.2805$ by fitting the quantity $T(n,n)$ to an exponential. We also mention another numerical observation: the numeric prefactor in (62) has finite upper and lower bounds.

V Discussion

In summary, we studied planar fragmentation, which can be viewed as dual to planar aggregation bbk . We obtained analytically several properties of the jammed state including the average number of rectangles, and the length distribution of rectangles. In general, statistical properties become independent of the aspect ratio in the large-area limit. Moreover, the length distribution of rectangles in the jammed state has a power-law tail, and the moments of this distribution exhibit multiscaling.

We also found a phase transition when the fragmentation process is asymmetric. Generally, the average number of jammed rectangles grows sub-linearly with system size, and the exponent characterizing this growth varies continuously with the asymmetry parameter. This exponent is concave in the weakly asymmetric phase and convex in the strongly asymmetric phase. In addition, The length distribution is independent of system size in the weakly asymmetric phase, but it does depend on system size in the strongly asymmetric phase.

Our theoretical analysis relies on recursion equations that describe the final state of the system. Since each fragmentation event involves a single rectangle, the recursion equations are linear. For large systems, we employed the continuum approach and then applied the Laplace transform to obtain exact results for the leading asymptotic behavior. Numerical evaluation of the recursion equations support the theoretical predictions.

The recursion equations bypass the evolution toward the jammed state and hence, directly yield statistics of the final configuration. The fragmentation rate may be an arbitrary function of the area, yet, as long as the fragmentation point is selected at random, the recursion equation (3) holds. For specific fragmentation rates, it is natural to study the evolution towards the jammed state, including in particular the average jamming time, and the distribution of jamming times.

Our analysis yields statistics of single fragments in the jammed states such as the average length and the length distribution. Missing from our analysis, however, are statistics of multiple fragments such as correlations between the orientations of neighboring sticks. Both sets of statistics are relevant for characterizing the geometrical structure of planar fragmentation patterns found in martensitic transformations tivp , breakage of brittle objects vv ; dp , cracking of soils bpc , and drying of suspensions nm .

The behavior in higher dimensions can be studied as well. In the three-dimensional generalization of (2), the jammed state consists of rectangular plates, that is, boxes with unit width. In this case, we find that the average number $F$ of frozen boxes grows as

[TABLE]

with $V$ the volume of the original box. The area distribution of jammed plates represents an interesting challenge.

The deterministic process (53) can be also generalized to $d$ dimensions. Here, each fragmentation event generates $2^{d}$ boxes. The jammed state consists of frozen boxes, each of which has at least one minimal side. The number of frozen boxes $F$ grows as

[TABLE]

This result, which generalizes (59), is derived in Appendix D. A frozen box is characterized by $d-1$ nontrivial lengths and it is an interesting challenge to characterize the distribution of these lengths.

We thank Jean-Marc Luck for making helpful comments. This work was supported in part by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

Appendix A Length Distribution

To obtain the length distribution for an infinite system, we treat $m$ and $n$ as discrete variables. To this end, we introduce the generating function

[TABLE]

The generating function $\mathcal{S}\equiv\mathcal{S}(x,y)$ satisfies the partial differential equation

[TABLE]

To obtain this equation, we multiply the recursion equation (3), that governs the averages $S(m,n)$, by $(m-1)(n-1)x^{m-2}y^{n-2}$ and sum over $m\geq 2$ and $n\geq 2$. Furthermore, equation (66) is subject to the boundary conditions $\mathcal{S}(x,0)=x/(1-x)$ and $\mathcal{S}(0,y)=y/(1-y)$.

We now introduce the variables

[TABLE]

This transformation turns (66) into a partial differential equation with constant coefficients

[TABLE]

while the boundary conditions become $\mathcal{S}(\xi,0)=e^{\xi}-1$ and $\mathcal{S}(0,\eta)=e^{\eta}-1$. Next, we introduce the double Laplace transform

[TABLE]

Using the governing equation (68), we obtain

[TABLE]

We now invert the Laplace transform with respect to one of the conjugate variables, to obtain the sum

[TABLE]

Thus far, our analysis is exact and in particular, it applies to all $m$ and $n$. We now restrict our attention to squares, $\xi=\eta$ and further, we focus on the leading asymptotic behavior for large systems which is captured by the leading behavior when $\eta\to\infty$. By performing the inverse Laplace transform over the second conjugate variable, we obtain

[TABLE]

Next, we analyze the length distribution $S_{k}(m,n)$. Its corresponding generating function satisfies Eq.(68), subject to the boundary conditions $\mathcal{S}_{k}(\xi,0)=(1-e^{-\xi})^{k-1}$ $S_{k}(0,\eta)=(1-e^{-\eta})^{k-1}$. These two boundary conditions follow from $\mathcal{S}_{k}(x,0)=x^{k-1}$ and $\mathcal{S}_{k}(0,y)=y^{k-1}$. By repeating the steps leading to (70), we obtain

[TABLE]

We can verify that $\mathcal{S}(p,q)=\sum_{k\geq 2}\mathcal{S}_{k}(p,q)$, whereas the corresponding quantity in (24), which is obtained by treating the variables $m$ and $n$ as continuous, violates this normalization. The leading asymptotic behavior in the limit $\eta\to\infty$ is given by

[TABLE]

The average number of sticks does not depend on aspect ratio and thus, we assume $S_{k}(m,n)\simeq P_{k}S(m,n)$ at large sizes. Then, according to the definition (65) and the leading asymptotic behavior (71) we have $\mathcal{S}_{k}(\eta,\eta)\simeq P_{k}e^{4\eta}/\sqrt{4\pi\eta}$. By comparing this expression with (72), we deduce the length distribution $P_{k}$ in (30).

Appendix B Critical Behavior

Here, we derive the critical behavior for the asymmetric fragmentation process (36). The average number of jammed sticks is generally given by the sum (41). First, we compute the integral $J$ defined in (44). At the critical point, $\alpha=\alpha_{c}$, the factor $p^{-1}$ in the integrand (45) becomes large near the saddle point, and thus, we incorporate this term into the exponential

[TABLE]

The saddle point $p_{*}$ vanishes algebraically with $\nu$,

[TABLE]

in the $\nu\to\infty$ limit. To perform the integration, we again chose an integration contour parallel to the imaginary axis which passes through $p_{*}$ and compute

[TABLE]

Next we evaluate Eq. (43) at the critical point. The value $I\simeq 2^{-3/4}(4\pi\nu)^{-1/2}e^{\sqrt{2}\,\nu}$ is negligible compared with $J$, and thus, $S\simeq J$ with $J$ given by (75).

For infinite systems, the critical behavior applies strictly at the critical point. For finite systems, the critical behavior characterizes a small, yet finite, region near the critical point. The size of this region, which often referred to as the “scaling window,” shrinks as the system size increases, as follows from the saddle point (74),

[TABLE]

The scaling window decays very slowly with system size, and hence, it is relevant even for reasonably large system.

Appendix C Asymmetric fragmentation

The average number of sticks with a given length satisfies the partial differential equation (3) subject to the boundary conditions (23). The double Laplace transform, defined by (12), is given by

[TABLE]

To perform the double inverse Laplace transform, we rewrite the numerator as a sum of the $q$-dependent quantity $(q-\alpha)k^{-1-q}$ and the $p$-dependent quantity $(p+\alpha)k^{-1-p}$. The quantity $S_{k}(p,q)$ is therefore a sum of two terms as in (41).

The first term in the sum (41) is the integral $I$ defined in (15) with $F(q)=1$ and

[TABLE]

Here, we again used the notations $\beta=\sqrt{1-\alpha^{2}}$ and $x=\ln k/\ln n$. The saddle point of the function $f(q)$ is $q_{*}=\beta(1-x)^{-1/2}+\alpha$ and with (17), we arrive at

[TABLE]

By evaluating the small-$x$ behavior of the general expression in the first line, we obtained the large-$k$ behavior in the second line.

The second term in the sum (41) is the integral $J$ defined in (44) with $G(p)=1$ and

[TABLE]

The saddle point of the function $g(p)$ is $p_{*}=\beta(1-x)^{-1/2}-\alpha$. By using the analog of the general expression (17), we obtain

[TABLE]

The large-$k$ behavior in the second line follows from the small-$x$ behavior in the first line. By comparing the tails (78) and (80), we conclude that $I$ is negligible compared with $J$, and therefore $S_{k}(\nu,\nu)\simeq J$ for sufficiently large $k$. The power-law tail (50) follows from $P_{k}=S_{k}(\nu,\nu)/S(\nu,\nu)$.

Appendix D Arbitrary Dimensions

The deterministic process (53) can be generalized to $d$ dimensions where in each fragmentation event a box breaks into $2^{d}$ boxes. This elementary event is repeated until a jammed state is reached. The recursion equation for the number of frozen boxes is a straightforward generalization of Eq. (54) and it includes $d$ sums.

The multivariate Laplace Transform is given by a straightforward generalization of (57)

[TABLE]

The inverse Laplace transform is a $d-$fold integral, and we first invert this multivariate transform with respect to the conjugate variable $q_{d}$. Further, we restrict our attention hyper-cubes, $\ln n_{\ell}=\nu$ for all $\ell$, and then

[TABLE]

Here, we introduced the shorthand notation ${\bf q}=(q_{1},\ldots,q_{d-1})$ and

[TABLE]

The saddle point is ${\bf q}_{*}=(1,\ldots,1)$, and at this point, it is easy to show that $\phi({\bf q}_{*})=d$ and $\Phi({\bf q}_{*})=(2^{d}-1)/2^{d-1}$. We tacitly do not display the function $\Phi({\bf q})$ because only its value at saddle point is needed.

To evaluate the integral (82), expand $\phi({\bf q})$ near the saddle point using $q_{\ell}=1+{\rm i}u_{\ell}/\sqrt{\nu}$, and then,

[TABLE]

The Gaussian integral can be computed in arbitrary dimension,

[TABLE]

The computation of the Gaussian integral relies on the fact that the $(d-1)\times(d-1)$ matrix associated with the quadratic form $U({\bf u})$ has eigenvalues $\frac{1}{2}(d,1,\dots,1)$. The integral (85) completes the derivation of (64).