Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T21:28:27.460Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary states of a directed planar growth process

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

D. J. Gates
D. J. Gates*
Affiliation:
CSIRO
*
*Postal address: CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra ACT 2601, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new Markov process is introduced, describing growth or spread in two dimensions, via the aggregation of particles or the filling of cells. States of the process are configurations of part of the boundary of the growing aggregate, and transitions are captures or escapes of single particles. For suitably chosen transition rates, the process is dynamically reversible, leading to an explicit stationary distribution and a statistical description of the boundary. The growth rate is calculated and growth behaviour described. Different asymptotic relations between transition rates lead to different growth patterns or regimes. Besides the regimes familiar in polymer crystal growth, several new ones are described. The aggregate can have a porous structure resembling thin solid films deposited from vapour. Two measures of porosity, one for the boundary and one for the bulk, are calculated. The process is relevant to growing colonies of bacteria or the like, to the spread of epidemics and grass or forest fires, and to voter models.

Keywords

RICHARDSON MODELSPATIAL EPIDEMICCRYSTAL GROWTHVOTER MODELSDYNAMIC REVERSIBILITY

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 857 - 877
Copyright
Copyright © Probability Trust 1997 

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

[1] Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, CA.Google Scholar
[2] Frank, F. C. (1974) Nucleation controlled growth on a one-dimensional growth of finite length. J. Crystal Growth 22, 233–236.Google Scholar
[3] Gates, D. J. (1988) Growth and decrescence of two dimensional crystals. J. Statist. Phys. 52, 245–257.Google Scholar
[4] Gates, D. J. and Westcott, M. (1988) Kinetics of polymer crystallization, I: discrete and continuum models. Proc. R. Soc. London A 416, 443–461.Google Scholar
[5] Gates, D. J. and Westcott, M. (1988) Kinetics of polymer crystallization, II: Growth regimes. Proc. R. Soc. London A 416, 463–475.Google Scholar
[6] Gates, D. J. and Westcott, M. (1990) On the stability of crystal growth. J. Statist. Phys. 59, 73–101.Google Scholar
[7] Gates, D. J. and Westcott, M. (1993) Markov models of steady crystal growth. Ann. Appl. Prob. 3, 339–355.Google Scholar
[8] Gates, D. J. and Wescott, M. (1994) Seven basic regimes of steady crystal growth in two dimensions. J. Statist. Phys. 77, 199–215.CrossRefGoogle Scholar
[9] Gates, D. J. and Westcott, M. (1994) Stationary states of crystal growth in three dimensions. J. Statist. Phys. 81, 681–715.Google Scholar
[10] Hoffman, J. D., Davis, G. T. and Lauritzen, J. L. Jr. (1976) The rate of crystallization of linear polymers with chain folding. In Treatise on Solid-State Chemistry. Vol. 3: Crystalline and Non-Crystalline Solids. ed. Hanay, N. B. Plenum, New York. pp. 335–496.Google Scholar
[11] Hoffman, J. D. (1983) Regime III crystallization in melt-crystallized polymers: the variable cluster model of chain folding. Polymer 24, 3–26.CrossRefGoogle Scholar
[12] Leamy, H. J., Gilmer, G. H. and Dirks, A. G. (1980) Current Topics in Material Science. ed. Kaldis, E. North Holland, Amsterdam.Google Scholar
[13] Richardson, D. (1973) Random growth on a tesselation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar
[14] Sadler, D. M. (1987) On the growth of two dimensional crystals. 2. Assessment of kinetic theories of crystallization of polymers. Polymer 28, 1440–1454.Google Scholar
[15] Whittle, P. (1955) Reversibility in Markov processes. Unpublished manuscript. Google Scholar
[16] Whittle, P. (1975) Reversibility and acyclicity. Perspectives in Probability and Statistics. ed. Gani, J. Academic Press, London.Google Scholar
[17] Whittle, P. (1986) Systems in Stochastic Equilibrium. Chichester, Wiley.Google Scholar