Abstract
Binary fragmentation is the reverse process of binary aggregation: a cluster with i monomers splits into two clusters with masses j and i − j such that 1 ≤ j ≤ i − 1. This is represented schematically by the irreversible mass-conserving reaction
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- 1.
There are n i clusters with mass i and each produces i − 1 ordered fragments (fragmentation events). The total number of fragmentation events is
$$\displaystyle \begin{aligned} \sum_{i=1}^{\infty} (i-1)n_i = M-N . \end{aligned}$$ - 2.
Equation (10.5) as written is for the offspring ensemble \(\mu \mathcal C(M,N)\). In the parent ensemble \(\mu \mathcal C(M,N-1)\), N must be replaced by N − 1.
- 3.
- 4.
Notice that f(x)dx = ϕ(z)dz as an immediate consequence of Eq. (10.46).
- 5.
The thermodynamic limit requires M to be large and N fixed. This condition ensures homogeneous behavior of ϕ at any N, even small N of the order 1. If we allow N to be large, but still much smaller than M, then we can treat \(\log \Omega \) as a continuous function of M and N to write \(\log \Omega = M\beta +N\log q\).
- 6.
Here we work with the mean rather than the most probable distribution. For large M and N the two are equal to each other.
- 7.
This scaling presumes that the cluster distribution is concentrated in a relatively narrow range of cluster masses.
- 8.
Function A(M) is of no particular interest once \(\log \omega ^*({\bar z})\) has been determined.
- 9.
The physical impossibility of zero mass is a consequence of the passage to continuous space. In discrete finite systems the smallest mass in scaled coordinates is finite and equal to 1∕M.
- 10.
See McGrady and Ziff (1987), Singh and Hassan (1996), Ziff (1992), Ernst and Szamel (1993). These works study shattering by examining the kinetic equation for the mean distribution (our Eq. (10.44)) and the conditions under which it fails to preserve mass, a breakdown that identifies the presence of shattering.
References
M.H. Ernst, G. Szamel, Fragmentation kinetics. J. Phys. A Math. Gen. 26(22), 6085 (1993). URL http://stacks.iop.org/0305-4470/26/i=22/a=011
E.D. McGrady, R.M. Ziff, “Shattering” transition in fragmentation. Phys. Rev. Lett. 58(9), 892–895 (1987). URL http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.58.892
P. Singh, M.K. Hassan, Kinetics of multidimensional fragmentation. Phys. Rev. E 53, 3134–3144 (1996). https://doi.org/10.1103/PhysRevE.53.3134. URL http://link.aps.org/doi/10.1103/PhysRevE.53.3134
R.M. Ziff, An explicit solution to a discrete fragmentation model. J. Phys. A Math. Gen. 25(9), 2569 (1992). URL http://stacks.iop.org/0305-4470/25/i=9/a=027
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Appendix: Derivations
Appendix: Derivations
10.1.1 Equation (10.24)
Consider parent and offspring distributions for the fragmentation reaction
Only the cluster masses involved in this reaction have different number of clusters in the parent and offspring distributions. Then we have
where we have used the parent–offspring relationships with i ≠ j. Repeating with i = j, n i > 1:
and we notice that we no longer need to specify n i > 1 since n i = 1 makes this ratio 0. The results can be summarized into a single expression as follows:
which is Eq. (10.24) in the text.
10.1.2 Equation (10.44)
We start with Eq. (10.42)
and calculate each term on the right-hand side separately. For the first term we have
with the last result obtained by the normalization condition for \(\mathcal P^{\prime }_{i-j,j}\). The second term gives
The third and fourth terms are identical due to symmetry between all ordered pairs of fragments. The two terms combined are
Incorporating these results into Eq. (10.42) we obtain
which leads directly to Eq. (10.44) in the main text.
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Matsoukas, T. (2018). Fragmentation and Shattering. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_10
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DOI: https://doi.org/10.1007/978-3-030-04149-6_10
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