Abstract. A general discussion of the kinetics of continuous, irreversible fragmentation processes is presented. For a linear process, where particle breakup is driven by an external force, we develop a scaling theory to describe the evolution of the cluster size distribution. We treat the general case where the breakup rate of a cluster of mass x varies as xi. When I > 0, corresponding to larger clusters more likely to break up, the scaled cluster size distribution, $(x), decays with the scaled mass, x, as x-* exp(-ax"), as x -+ 00. For small mass, $(x) has the log-normal form, exp(-a In2 x), if the breakup kernel has a small-size cutoff, while $(x) has a power-law tail in the absence of a cutoff. We also show that a conventional scaling picture applies only for the case I > 0. For I < 0, we develop an alternative formulation for the cluster size distribution, in which the typical mass scale is determined by the initial condition. In this regime, we also investigate the nature of a 'shattering' transition, where mass is lost to a 'dust' phase of zero-mass particles. We also study the kinetics of a nonlinear, collision-induced fragmentation process. We analyse the asymptotic behaviours of a simple-minded class of models in which a two-particle collision results in either: (1) both particles splitting into two equal pieces, (2) only the larger particle splitting in two, or (3) only the smaller particle splitting. We map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations. Scaling is found to hold for different ranges of homogeneity index for the three models.
We investigate the phenomenon of coagulation with constant feed-in of monomers at a single point. For spatial dimension d > 4, the steady-state cluster concentration, c(r), obeys Laplace's equation, while for d < 4, the steady-state concentration of clusters of mass k a distance r from the source scales as c k (r)~-k~T 2, and z-2, T-1+/2 for d < 2. For the linear chain, we outline an exact solution for which c*(r)~-A: _3/2 0(/i), with
We investigate the fluctuations in N a (R), the number of lattice points n £ Z 2 inside a circle of radius R centered at a fixed point α e [0, I) 2 . Assuming that R is smoothly (e.g., uniformly) distributed on a segment 0 ^ R ^ T 9 we prove that the random variable ---7= -has a limit distribution as T -> oo (independent of the distribution of R\ which is absolutely continuous with respect to Lebesgue measure. The density p u (x) is an entire function of x which decays, for real x, faster than exρ ( -|x| 4~ε ). We also obtain a lower bound on the distribution function P α (x) = J^ p Λ (y)dy which shows that P α (x) and 1 -P Λ (x) decay when x -> oo not faster than exp ( -x 4+ε ). Numerical studies show that the profile of the density p α (x) can be very different for different α. For instance, it can be both unimodal and bimodal. We show that J^ xp α (x)dx = 0, and the variance D α = J^ x 2 p (X (x)dx depends continuously on α. However, the partial derivatives of D a are infinite at every rational point α e Q 2 , so D α is analytic nowhere. ContentsI. Introduction 434 II. Ergodic Theorem 443 III. Almost Periodicity of the Error Function 445 IV. Upper Bound on the Error Term Distribution Density 450 V. Lower Bound on the Error Term Distribution Function 454 Appendix A. Proof of Theorem 4.1 456 Appendix B. Proof of Theorem 4.3 460 References 468
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.