Zipf's law in nuclear multifragmentation and percolation theory

We investigate the average sizes of the $n$ largest fragments in nuclear multifragmentation events near the critical point of the nuclear matter phase diagram. We perform analytic calculations employing Poisson statistics as well as Monte Carlo simulations of the percolation type. We find that previous claims of manifestations of Zipf's Law in the rank-ordered fragment size distributions are not borne out in our result, in neither finite nor infinite systems. Instead, we find that Zipf-Mandelbrot distributions are needed to describe the results, and we show how one can derive them in the infinite size limit. However, we agree with previous authors that the investigation of rank-ordered fragment size distributions is an alternative way of looking for the critical point in the nuclear matter diagram.

Figure 1

Symbols indicate $\langle A_n\rangle$ for $\tau =2$ and different system sizes calculated with Eq. (8), the solid line indicates $\langle A_n^{\infty }\rangle$ from Eq. (24), the dashed line indicates Zipf's law [Eq. (2)].

Figure 2

Top: λ as a function of τ as fitted to Zipf's law in Eq. (3) for different system sizes. Bottom: Error of the fit.

Figure 3

λ as a function of τ as fitted to the Zipf-Mandelbrot distribution (25) for different system sizes. Middle: Extracted parameter $k$. Bottom: Error of the fit.

Figure 4

Average sizes of the $n$th largest cluster as a function of $n$ for the lattice with 216 sites for different bond probabilities $p_{\mathrm{bond}}$. The dashed lines are fits for Zipf's law (3), the solid lines are fits for the Zipf-Mandelbrot distribution (25).

Figure 5

Fitted values for λ using the different fits described in the text. Solid symbols are for fits including the first fragment, open symbols are for fits omitting $\langle A_n\rangle$. Upper figure is for the small lattice (216 sites), lower figure is for the large lattice (${10}^6$ sites). Approximate values for $p_{\mathrm{c}}$ are also indicated by a dashed line for both cases.

Figure 6

Ratio of the average size of the ($n+1$)th largest fragment to average size of the $n$th largest fragment as a function of $p_{\mathrm{bond}}$ for a large system with ${10}^6$ sites. The result for $n=1$ is omitted.

Figure 7

$\langle A_{n+1}\rangle /\langle A_n\rangle$ for different finite system sizes $V$ (symbols) using Eq. (8) and for the infinite system (line) using Eq. (23) for $\tau =2.18$.

Figure 8

Same as Fig. 7, but for $\tau =2$.

Figure 9

Ratio of the average size of the ($n+1$)th largest fragment to average size of the $n$th largest fragment as a function of $p_{\mathrm{bond}}$ normalized by the factor in Eq. (28) for a large system (${10}^6$ sites). The result for $n=1$ is omitted.

Figure 10

Same as Fig. 9, but for the small system (216 sites).