Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-17T01:29:42.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Stably coalescent stochastic froths

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

J. M. C. Clark  and
V. Katsouros
J. M. C. Clark*
Affiliation:
Imperial College
V. Katsouros*
Affiliation:
Imperial College
*
∗ Postal address: Centre for Process Systems Engineering, Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BT, UK.
∗ Postal address: Centre for Process Systems Engineering, Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BT, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

A model of a stochastic froth is introduced in which the rate of random coalescence of a pair of bubbles depends on an inverse power law of their sizes. The main question of interest is whether froths with a large number of bubbles can grow in a stable fashion; that is, whether under some time-varying change of scale the distributions of rescaled bubble sizes become approximately stationary. It is shown by way of a law of large numbers for the froths that the question can be re-interpreted in terms of a measure flow solving a nonlinear Boltzmann equation that represents an idealized deterministic froth. Froths turn out to be stable in the sense that there are scalings in which the rescaled measure flow is tight and, for a particular case, stable in the stronger sense that the rescaled flow converges to an equilibrium measure. Precise estimates are also given for the degree of tightness of the rescaled measure flows.

Keywords

CoalescencecoagulationfrothbubblesBoltzmann equationSmoluchowski's equationscaling lawsflotation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K40: Other physical applications of random processes 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 199 - 219
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.Google Scholar
van Dongen, P. G. J. and Ernst, M. H. (1988). Scaling solutions of Smoluchowski's coagulation equation. J. Statist. Phys., 50, 295–329.Google Scholar
Dubovskii, P. B. and Stewart, I. W. (1996). Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Methods Appl. Sci., 19, 571–591.Google Scholar
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks Cole, Andover, UK.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Ferland, R. (1994). Laws of large numbers for pairwise interacting particle systems. Math. Models Methods. Appl. Sci., 4, 1–15.Google Scholar
Flyvbjerg, H. (1993). Model of coarsening froths and foams Physical Rev. E, 47, 4037–4054.Google Scholar
Wagner, W. (1994). Stochastic systems of particles with weights and approximation of the Boltzmann equation. The Markov process in the spatially homogeneous case. Stochast. Anal. Appl., 12, 639–659.Google Scholar