‘‘Shattering’’ transition in fragmentation

McGrady, E. D.; Ziff, Robert M.

doi:10.1103/physrevlett.58.892

Cited by 280 publications

(195 citation statements)

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“…In other words, the entire mass is shattered into dust and there are no particles with positive mass [8,18,21]. Near the shattering transition, i.e., as τ → ∞, the mass distribution approaches…”

Section: A Random Particle Splitsmentioning

confidence: 99%

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Shattering transitions in collision-induced fragmentation

Krapivsky

Ben‐Naim

2003

Phys. Rev. E

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We investigate the kinetics of nonlinear collision-induced fragmentation. We obtain the fragment mass distribution analytically by utilizing its travelling wave behavior. The system undergoes a shattering transition in which a finite fraction of the mass is lost to infinitesimal fragments (dust). The nature of the shattering transition depends on the fragmentation process. When the larger of the two colliding fragments splits, the transition is discontinuous and the entire mass is transformed into dust at the transition point. When the smaller fragment splits, the transition is continuous with the dust gaining mass steadily on the account of the fragments. At the transition point, the fragment mass distribution diverges algebraically for small masses, c(m) ∼ m −α , with α = 1.20191 . . .

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Section: A Random Particle Splitsmentioning

confidence: 99%

“…Since n = log 2 (1/m), the mass distribution becomes log-normal, a behavior typical to fragmentation and cascade processes [2,8,18,21]. …”

Section: A Random Particle Splitsmentioning

confidence: 99%

Shattering transitions in collision-induced fragmentation

Krapivsky

Ben‐Naim

2003

Phys. Rev. E

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“…Rate equations [8][9][10][11] describe the time evolution of the mass distribution of a system of particles subject to fragmentation.…”

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confidence: 99%

Fragmentation of percolation clusters at the percolation threshold

Gyure

Edwards

1992

Phys. Rev. Lett.

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A scaling theory and simulation results are presented for fragmentation of percolation clusters by random bond dilution. At the percolation threshold, scaling forms describe the average number of fragmenting bonds and the distribution of cluster masses produced by fragmentation.A relationship between the scaling exponents and standard percolation exponents is verified in one dimension, on the Bethe lattice, and for Monte Carlo simulations on a square lattice. These results further describe the structure of percolation clusters and provide kernels relevant to rate equations for fragmentation. The structure of percolation clusters has received considerable attention in recent years because of the desire to understand transport properties through random media.For this reason, much of the recent eA'ort has focused on the structure of clusters at and near the percolation threshold p, where the infinite cluster dominates the behavior of the system. In particular, the link, node, and blob picture of the infinite cluster has helped to clarify the conduction of electricity through random networks and the flow of water through random porous materials [3,4]. Since the backbone of the infinite cluster carries current across the cluster, the structure and connectivity of the backbone has been examined carefully [5][6][7].Comparatively little effort has been spent, however, on characterizing overall cluster connectivity, an important issue for fragmentation.In this spirit, we ask about the consequences of removing a bond of mass 1 from a percolation cluster of finite mass s, given by the number of bonds, at the percolation threshold p, . Is

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“…However, in the subsequent years, this model received considerable attention as it was responsible for the shattering transition (formation of dust), a phenomenon considered by Ziff as the counterpart of gelation phenomenon (infinite cluster formation) in coagulation processes. See McGrady and Ziff (1987) and Jeon (2002), for example. In contrast with our model, size-dependence of successive fragmentations appears here in the rates, not in the splitting probability.…”

Section: On Two Fragmentation Schemes With Algebraic Splitting Probabmentioning

confidence: 99%