Dynamics and self-similarity in min-driven clustering

We study a mean-field model for a clustering process that may be described informally as follows. At each step a random integer $k$ is chosen with probability $p_k$, and the smallest cluster merges with $k$ randomly chosen clusters. We prove that the model determines a continuous dynamical system on the space of probability measures supported in $(0,\infty)$, and we establish necessary and sufficient conditions for approach to self-similar form. We also characterize eternal solutions for this model via a Levy-Khintchine formula. The analysis is based on an explicit solution formula discovered by Gallay and Mielke, extended using a careful choice of time scale

A stochastic min-driven coalescence process and its hydrodynamical limit

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalised version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models

Approach to self-similarity in Smoluchowski's coagulation equations

We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels $K(x,y)=2$, $x+y$ and $xy$. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for $K=x+y$ and $K=xy$ it is a power-law rescaling of a maximally skewed $\alpha$-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.Comment: Latex2e, 42 pages with 1 figur

Universality for one-dimensional hierarchical coalescence processes with double and triple merges

We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different "epoch" and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range. Assuming that the distribution at the beginning of the first epoch is described by a renewal simple point process, we prove limit theorems for the domain length and for the position of the leftmost point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377-1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175-180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.Comment: Published in at http://dx.doi.org/10.1214/12-AAP917 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mathematical aspects of coagulation-fragmentation equations

We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Section 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Section 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the functional spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sections 3 and 4 we are concerned with several aspects of the solutions behaviour.We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof

Coarsening dynamical systems: dynamic scaling, universality and mean-field theories

We study three distinct coarsening dynamical systems (CDS) and probe the underlying scaling laws and universal scaling functions. We employ a variety of computational methods to discover and analyse these intrinsic statistical objects. We consider mean-field type models, similar in nature to those used in the seminal work of Lifshitz, Slyozov and Wagner (LSW theory), and statistical information is then derived from these models. We first consider a simple particle model where each particle possesses a continuous positive parameter, called mass, which itself determines the particle’s velocity through a prescribed law of motion. The varying speeds of particles, caused by their differing masses, causes collisions to take place, in which the colliding particles then merge into a single particle while conserving mass. We computationally discover the presence of scaling laws of the characteristic scale (mean mass) and universal scaling functions for the distribution of particle mass for a family of power-law motion rules. We show that in the limit as the power-law exponent approaches infinity, this family of models approaches a probabilistic min-driven model. This min-driven model is then analysed through a mean-field type model, which yields a prediction of the universal scaling function. We also consider the conserved Kuramoto-Sivashinsky (CKS) equation and provide, in particular, a critique of the effective dynamics derived by Politi and ben-Avraham. We consider several different numerical methods for solving the CKS equation, both on fixed and adaptive grids, before settling on an implicit-explicit hybrid scheme. We then show, through a series of detailed numerical simulations of both the CKS equation and the proposed dynamics, that their particular reduction to a length-based CDS does not capture the effective dynamics of the CKS equation. Finally, we consider a faceted CDS derived from a one-dimensional geometric partial differential equation. Unusually, an obvious one-point mean-field theory for this CDS is not present. As a result, we consider the two-point distribution of facet lengths. We derive a mean-field evolution equation governing the two-point distribution, which serves as a two-dimensional generalisation of the LSW theory. Through consideration of the two-point theory, we subsequently derive a non-trivial one-point sub-model which we analytically solve. Our predicted one-point distribution bears a significant resemblance to the LSW distribution and stands in reasonable agreement with the underlying faceted CDS