## 1. Introduction

## 2. Random Fragmentation

#### 2.1. One-Component Random Fragmentation

#### 2.1.1. Probability of Random Fragment Distribution

**Proposition**

**1.**

**Proof.**

#### 2.1.2. Mean Fragment Distribution

**Proposition**

**2.**

**Proof.**

#### 2.2. Two-Component Random Fragmentation

#### 2.2.1. Representations of Bicomponent Populations

#### 2.2.2. The Ensemble of Random Fragment Distributions

**Proposition**

**3.**

**Proof.**

#### 2.2.3. Mean Fragment Distribution

**Proposition**

**4.**

**Proof.**

**Alternative**

**Proof.**

#### 2.3. Any Number of Components

## 3. Nonrandom Bicomponent Fragmentation

#### 3.1. Linear Ensemble

**Proposition**

**5.**

**Proof.**

#### 3.2. Composition-Independent Bias

## 4. Simulation of Biased Fragmentation

#### 4.1. Monte Carlo Sampling by Exchange Reaction

#### 4.2. Two Examples

- Case I (positive deviations)$${w}_{a,b}={(a+1)}^{\alpha}+{(b+1)}^{\alpha}$$
- Case II (negative deviations)$${w}_{a,b}={(a+1)}^{\alpha}{(b+1)}^{\alpha}$$

## 5. Concluding Remarks

## Funding

## Conflicts of Interest

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**Figure 1.**Random fragmentation of integer mass M into N pieces is equivalent to breaking a string with M beads at $N-1$ random points. With $M=10$, $N=3$ the number of possible partitions is $\left(\genfrac{}{}{0pt}{}{9}{2}\right)=36$. If the mass is made up of two colors, every permutation of the beads is equally probable; with ${M}_{A}=6$, ${M}_{B}=4$ the number of partitions increases by the factor $\left(\genfrac{}{}{0pt}{}{6+4}{4}\right)=210$ and the total number of permutations is 7560.

**Figure 2.**The compositional distribution $\overline{{c}_{a|k}}$ in particles of mass $k=2,3$, and 4. The parent particle contains ${M}_{A}=4$ units of A, ${M}_{B}=3$ units of B, and breaks into $N=4$ pieces. Lines are from Equation (37) and points are from MC simulation after 20,000 fragmentation events. The excellent agreement between MC and theory demonstrates the exact nature of Equation (37) and validates the MC method. The binomial distribution is only asymptotically valid and in this case is a poor approximation because the size of the fragments is small.

**Figure 3.**Size and compositional distributions at fragment sizes $k=2,4$, and 8 for two bias functionals: (

**a**–

**d**): ${w}_{a,b}={(1+a)}^{4}{(1+b)}^{4}$; (

**e**–

**h**): ${w}_{a,b}={(1+a)}^{4}+{(1+b)}^{4}$. In both cases, the particle contains ${M}_{A}=20$ units of A, ${M}_{B}=20$ units of B, and breaks into $N=4$ fragments.

${\mathit{w}}_{\mathit{a},\mathit{k}-\mathit{a}}$ | ${\mathit{\Omega}}_{\mathit{M};\mathit{N}}^{\left(1\right)}$ | |
---|---|---|

Case 1 | 1 | $\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ |

Case 2 | $\frac{{k}^{k-1}}{k!}$ | ${m}^{M-N}\frac{N!}{M!}\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ |

Case 3 ^{‡} | $2\frac{{\left(2k\right)}^{k-2}}{k!}$ | ${\left({m}^{M-N}\frac{N!}{M!}\right)}^{2}\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ |

^{‡}asymptotically for $M,N\gg 1$, $M/N<2$.

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