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.

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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+{n)~~ju 5/2 for fj. =k/r 2 -* 0, c(r)~r -1 , and the number of clusters increases with time t as Inf. The effects of cluster drift and the presence of a sink are also considered.PACS numbers: 82.20.-w, 05.40.+J, 82.70.-y With a steady feed-in of monomers, a coagulating system can reach a steady state. 1-6 This process typifies a wide variety of physical situations, ranging from the mass distribution of stars, 2 to cluster distributions in chemical reactors. 3 Theoretical approaches to elucidate these phenomena include mean-field theories, suitably modified to account for the input, 1-6 field theoretic treatments of annihilation reactions, 7 and analytic solutions in one dimension. 8 These treatments all assume that the source is spatially homogeneous. For such an input the steady-state concentration of clusters of mass k has the form Ck-k~p, and in mean-field theory /?= f for reaction kernels with homogeneity index equal to zero. l This value of p appears to be a ubiquitous feature of steady-state coagulation in the mean-field limit.In this Letter, we investigate steady-state coagulation in the presence of a spatially localized monomer input. We envision that this describes a smoke plume, where there is a steady input of small clusters at the lower end of the plume, with coagulation occurring as the smoke clusters rise. The distance from the source is therefore a relevant parameter in describing the cluster size distribution in this process. Concomitantly, our study also describes how the steady state of pure diffusion from a point source, i.e., the solution to Laplace's equation, is modified when clusters can coalesce.Our study is based on an idealized lattice model of coagulation in which a new monomer is introduced at a single point (the origin) with probability s at each time step, and all the previously introduced clusters in the system undergo one random walk hop (Fig. 1). If two clusters happen to occupy the same lattice site, they are combined into a single point cluster whose mass is the sum of the masses of the two incident clusters. 9 We impose a constant coagulation rate by taking the merging of clusters to be independent of their masses. If the number of clusters is considered without regard to their masses, then this reduces to the two-body reaction A+ A -• A with a point source.First consider our model within the framework of a continuum reaction-diffusion equation. Within this approximation, the concentration of clusters at a distance r from the source at time /, c(r,f), obeysHere d is the spatial dimension, and the...

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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