Moderator: Fraydoun Rezakhanlou (Berkeley)
Jean Bertoin (Paris): Some aspects of random fragmentations
This talk will survey different probabilistic aspects of a model which is used to describe the evolution of an object which falls apart randomly as time passes. Each point of view yields useful techniques to establish properties of such random fragmentation processes.
Luis L. Bonilla (Madrid): An asymptotic and numerical study of time-dependent homogeneous nucleation modelled by the Becker-Doering equations
Among many other physical, chemical or biological processes, the precipitation of lipids in aqueous solutions to form micelles, the formation of crystals in undercooled liquids or crystallization of glasses may be examples of homogeneous nucleation. In the limit of large critical sizes, we carry out an asymptotic and numerical study of the Becker-Doering kinetic equations which model transient homogeneous nucleation. These equations are a particular case of coagulation-fragmentation processes in which clusters can grow or decrease only by adding or losing a single monomer. The two kinetic constants of the model are related through the assumption of detailed balance with the usual free energy containing cluster volume and surface terms. This leaves only one free constant, a discrete diffusivity, which depends on the physical phenomena to be described. Starting from pure monomers, three eras of transient nucleation are characterized in the Becker-Doering equations with two different models of discrete diffusivity: the classic Turnbull-Fisher formula and an expression describing thermally driven growth of the nucleus. The latter diffusivity yields time lags for nucleation which are much closer to values measured in experiments with disilicate glasses. After an initial stage in which the number of monomers decreases, many clusters of small size are produced and a continuous size distribution is created. During the second era, nucleii are increasing steadily in size in such a way that their distribution appears as a wave front advancing towards the critical size for steady nucleation. The nucleation rate at critical size is negligible during this era. After the wave front reaches critical size, it ignites the creation of supercritical clusters at a rate that increases monotonically until its steady value is reached. Analytical formulas for the transient nucleation rate and the time lag are obtained that improve classical ones and compare very well with direct numerical solutions.
Fernando P. Costa (Lisboa): Convergence to self-similar behaviour in a Becker-Doring model with input of monomers
For a special case of the Becker-Doring equations without fragmentation
and with time-independent monomer input, we use methods from the qualitative
theory of ODEs (invariant regions, Poincaré compactification, centre manifold
theory), and a generation function approach, to study the detailed long-time
behaviour of non-negative solutions. Then, by an convenient change of time
scale and careful estimates on an integral expression for $c_{j}(\tau)$ we
prove the convergence to similarity profiles. These results constitute a first
step towards the rigorous analysis of the plethora of scaling regimes holding
in these type of equations, whose formal study has been recently done by
Wattis.
This is a loint work with HJv Roessel (Edmonton) and JAD Wattis
(Nottingham).
Nicolas Fournier (Nancy): On self-similar solutions to Smoluchowski'scoagulation equations
The existence of self-similar solutions to Smoluchowski's coagulation equation is conjectured since several years by physicists and numerical simulations have confirmed the validity of this conjecture. Still, there was no existence result up to now, except for the constant and additive kernels for which explicit formulae are available. We prove the existence of self-similar solutions decaying rapidly at infinity for a wide class of homogeneous coagulation kernels. In the case of a sum kernel, we also study the smoothness and the small and large mass behaviour of the scaling profile. (with P. Laurençot).
Stéphane Mischler (Paris): Qualitative behavior of the Smoluchowski coagulation equation
We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space $\dot L^1_1$ for a class of homogeneous coagulation rates of degree lambda in [0,2). For any initial datum fin in $\dot L^1_1$ we build a weak solution which conserves the mass when lambda <= 1 and loses mass in finite time (gelation phenomena) when lambda > 1. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we prove existence of self-similar solutions to the Smoluchowski's coagulation equation when lambda < 1 and we investigate their qualitative properties. We prove regularity results and sharp uniform small and large size behavior for the self-similar profiles.
James Norris (Cambridge): Brownian coagulation
The talk will discuss the derivation of a spatial version of Smoluchowski's coagulation equation with Brownian kernel, starting from a stochastic model of coagulating Brownian spheres.
Benoit Perthame (Paris): The cell-division equation and generalized entropy method
Fraydoun Rezakhanlou (Berkeley): A stochastic model for coagulation and Smoluchowski equation
We consider a natural model for the coagulation process. In this model we have $N$ Brownian particles that are travelling in the $d$--dimensional Euclidean space with a diffusion coefficient that is a function of the size of the particle. We regard the location of each particle as the center of a cluster and when two particles are sufficiently close, they coagulate to a larger cluster. In a joint work with Alan Hammond, we establish a scaling limit for the above model. If the dimension is 3 or more, we show that if the range of the interaction is scaled like $N^{\frac 1{2-d}}$, and as the number of particles $N$ goes to infinity, the microscopic particle densities converge to solutions of the Smoluchoski's equation. A similar result holds in dimension 2 but now the range of interaction is of order $\exp (-N)$.