Analytical treatment of constraints in fragmentation

In fragmentation of a finite object such as an atomic nucleus or a percolation lattice, the multidimensional probability distribution of the numbers of fragments of each mass (charge, size) may be approximated as a product of Poisson distributions. In this work we present an analytical treatment of the influence of constraints (multiplicity, mass, etc.) that may be imposed on the basic product distribution.

Figure 1

Constrained probabilities $Q(N,M)$ [Eq. (3.6)] compared with results of Monte Carlo simulations (filled circles) with $X_m=e^{-0.1m}+35e^{-m}$ for $1⩽m⩽80$ $(X_m=0,m>80)$. The predictions are represented as continuous lines.

Figure 2

Probability distributions [Eq. (3.7)] of the second moment ${\kappa }_2$ for fixed values of the multiplicity $N\left(30\right)$ and total mass $M\left(130\right)$. Other specifications as in Fig. 1. Note that if $N$ is even (odd) ${\kappa }_2$ is even (odd) so that the (continuous) approximation must be multiplied by a factor of 2 before comparing with data.

Figure 3

Probability distributions corresponding to the numbers of fragments of mass $i$ $(i=l,2,\text{and}10)$. In each case the quantity displayed is $p\left(n_i\right)=W\left(n_i\right)∕Q(N,M)$ with $W\left(n_i\right)$ given by Eq. (4.1). The Monte Carlo calculations are represented as filled circles and the predictions by open circles joined by straight lines (set $\left\{X_m\right\}$ as in Fig. 1, $N=30$, $M=130$).

Figure 4

Predicted mean values $⟨n_m⟩$ [Eq. (4.2)] compared with Monte Carlo simulation results $(M=130,N=20,30,40)$. The Monte Carlo “data” are represented by crosses and the predictions by straight lines (set $\left\{X_m\right\}$ as in Fig. 1). Values for $N=20$ and $N=40$ have been displaced by $+2$ and $-2$, respectively.

Figure 5

Monte Carlo simulation results obtained by variation of $X_m$ parameters. For each value of $m$ the variation was made starting with the set $\mathbf{X}$ specified in the caption to Fig. 1 [see Eq. (4.8) and preceding text]. The straight lines represent linear least squares fits to the Monte Carlo “data.”

Figure 6

Results of search procedure: $⟨n_m⟩$ values $(M=130,N=20,30,40)$ obtained from the Monte Carlo simulations are represented by crosses and predictions (the final values obtained from the search routine) as open circles. The $X_m$ values obtained from the search routine are represented as continuous lines. Values for $N=20$ and $N=40$ have been displaced by $+2$ and $-2$, respectively.

Figure 7

Probability [defined within the subset with $N=30$ and $M=130$, Eq. (5.1)] to simultaneously observe $n_i$ fragments of mass $i$ and $n_j$ fragments of mass $j$ with $i=1$ and $j=2$. The Monte Carlo results are represented by crosses and the predictions, by open circles joined by straight lines (set $\left\{X_m\right\}$ as in Fig. 1).

Figure 8

As Fig. 7 but for $i=10$ and $j=12$.

Figure 9

Correlation ratios. The results of the Monte Carlo simulations (${10}^8$ events, set $\left\{X_m\right\}$ as in Fig. 1, $N=30$, $M=130$) are represented as filled circles, open circles, and crosses linked by thin straight lines. The predictions [Eq. (6.5)] are represented as thick dashed lines. Diagonal elements (which would be off scale in this figure) have been suppressed.