Persistence in cluster-cluster aggregation

Abstract

Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For gamma>0 the universal persistences decay algebraically with the exponent 2/(2-gamma). For the empty site case the exponent remains the same for gamma<0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.

Asymptotic Models for Studying Kinetics of Formation of Compact Objects with Strong Internal Bonds

“…Every power exponent Z with t obtained in this work are positive which is indicative of suppression of an inverse damage wave and formation of resultant direct wave of object growth even in case of cluster interaction between themselves. This result is in agreement with the concept of "persistence" in cluster-cluster aggregation [8]. By analogy with the theory of dispersive waves [71] one can try to introduce the concept of "phase" and "group" velocities in dimensionless space These considerations bring to a conclusion that the developed asymptotic method for investigation into kinetics of formation of compact objects having strong internal bonds has a certain degree of generality to use it in solving problems in physics of high density energy and high-intensive processes.…”

Persistence in systems with conserved order parameter

“…that the order parameter should remain constant throughout coarsening. The robustness of this behaviour has been confirmed in a variety of aggregation systems [8,13]. By contrast, in the critical µ = 1/2 limit considered, changes in the relative mobility of the two phases lead to different persistence characteristics.…”

“…It should be noted that the data in Fig.7 consistently fall slightly above the predicted value θ = 1. Since the exponent is known to be 1 in the limiting cases R 0 = 1 and R 0 = 0 [8] it may be that the numerics have not yet reached the true asymptotic value. Nevertheless, even considering this small discrepancy along all values of R 0 , γ = 0 remains the only case for which the values of all four exponents match in the limits R 0 = 0 and R 0 = 1.…”

“…For negative γ the exponents governing the algebraic decay of the cluster density and empty persistence probability have been calculated [8] to be φ cc = 1/(2 − γ) and θ E = 2φ…”

Flux control in networks of diffusion paths

“…An example of particular interest is when the diffusion is restricted to narrow one-dimensional paths. Networks of such domains are frequently used to model porous media [1][2][3][4][5] and fiber networks in brain white matter [6]. Also, a discrete model of diffusion path network, in which particles exhibit random-walk steps between nodes of some graph [13], is used to study computer and social networks [8][9][10][11][12][13], as well as city traffic [14].…”