Self-organized criticality in fragmenting

Shattering transitions in collision-induced fragmentation

We investigate the kinetics of nonlinear collision-induced fragmentation. We obtain the fragment mass distribution analytically by utilizing its traveling 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) approximately m(-alpha), with alpha=1.201 91 em leader.

Segmentation of genomic DNA through entropic divergence: power laws and scaling

Genomic DNA is fragmented into segments using the Jensen-Shannon divergence. Use of this criterion results in the fragments being entropically homogeneous to within a predefined level of statistical significance. Application of this procedure is made to complete genomes of organisms from archaebacteria, eubacteria, and eukaryotes. The distribution of fragment lengths in bacterial and primitive eukaryotic DNAs shows two distinct regimes of power-law scaling. The characteristic length separating these two regimes appears to be an intrinsic property of the sequence rather than a finite-size artifact, and is independent of the significance level used in segmenting a given genome. Fragment length distributions obtained in the segmentation of the genomes of more highly evolved eukaryotes do not have such distinct regimes of power-law behavior.

Crack propagation in thin glass plates caused by high velocity impact

Crack propagation within thin glass plates under high shock loading is directly observed using a high speed camera. The fractal dimension of cracks and the power-law exponents of the fragment area distributions are investigated as a function of time. Two models of the fragmentation process are proposed: in one case the cracks are net-like, while in the other the cracks are tree-like, and the relations between fractal dimension and power-law exponent are estimated and compared with the experimental results. It appears that at early stages of the fragmentation process the relation is described by the latter case, while at later stages it approaches that of the former case.

Scaling behavior in explosive fragmentation

We investigate the explosive fragmentation process in two dimensions using molecular-dynamics simulations. We show that the mass distribution of fragments follows a power law, with a scaling exponent that is strongly dependent on the macroscopic characteristics of the system prior to the explosion process. In particular, for thermalized initial configurations at low temperatures, we observe that the exponent is close to -1. We suggest that this result can be interpreted in terms of a multiplicative fracture process.

Quantum tunneling fragmentation model

A nonthermal quantum mechanical statistical fragmentation model based on tunneling of particles through potential barriers is studied in compact two- and three-dimensional systems. It is shown that this fragmentation dynamics gives origin to several static and dynamic scaling relations. The critical exponents are found and compared with those obtained in classical statistical models of fragmentation of general interest, in particular with thermal fragmentation involving classical processes over potential barriers. Besides its general theoretical interest, the fragmentation dynamics discussed here is complementary to classical fragmentation dynamics of interest in chemical kinetics and can be useful in the study of a number of other dynamic processes such as nuclear fragmentation.

Universality in fragmentation

Fragmentation of a two-dimensional brittle solid by impact and "explosion," and a fluid by "explosion" are all shown to become critical. The critical points appear at a nonzero impact velocity, and at infinite explosion duration, respectively. Within the critical regimes, the fragment-size distributions satisfy a scaling form qualitatively similar to that of the cluster-size distribution of percolation, but they belong to another universality class. Energy balance arguments give a correlation length exponent that is exactly one-half of its percolation value. A single crack dominates fragmentation in the slow-fracture limit, as expected.

Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0<p<1 leads to a purely algebraic size distribution, P(x) approximately x(-2p), in one dimension. In d dimensions, the volume distribution diverges algebraically in the small fragment limit, P(V) approximately V-gamma, with gamma=2p(1/d). Hence, the entire range of exponents allowed by mass conservation is realized. We demonstrate that this fragmentation process is non-self-averaging as the moments Y(alpha)= summation operator(i)x(alpha)(i) exhibit significant sample to sample fluctuations.