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Interacting Particle Systems

Last update: 07 Jul 2025 13:33
First version: 16 February 2006, major expansion 29 September 2007

In the obvious sense, all of statistical mechanics is about "interacting particle systems". More technically, however, the name has come to refer to a class of spatio-temporal stochastic processes, in which time is continuous, space may or may not be discrete, and each spatial location can be in one of a discrete number of states --- interpreted as the number or type of particles at that point-instant. The global configuration evolves according to a Markov process. These are natural generalizations of cellular automata to continuous time, which raises some interesting mathematical issues, and adds a little more realism.

Standard CA update all cells synchronously, but changing this updating scheme can change the qualitative behavior of a rule considerably. (Fates and Morvan have a nice paper on this, with a review of the published literature on the question, which is a small slice of the unpublished folklore.) Query: When synchronous and asynchronous updating in a discrete-time CA give very different behaviors, which one matches the continuous-time interacting particle system? This sounds like a question which could be resolved through the usual Kurtz et al. machinery for proving that a sequence of Markov processes converge by manipulating their generators.

Particle filtering from state estimation goes here. The idea in that case is to represent possible hidden states of the system through a large but finite number of particles, located in the state space. In between observations, particles move independently, in accordance with the dynamics your model assumes on the state space. When observations are made, particles get re-sampled, with weights proportional to the likelihood of getting the current observation from the represented state. Particles at different locations (states) thus interact with each other through the population-averaged likelihood, rather than through the local interactions typical of physical models. Many people have noticed that this sounds like evolution, or at least a genetic algorithm....

    Recommended:
  • P. Del Moral and L. Miclo, "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear Filtering", in Jacques Azéma, Michel Émery, Michel Ledoux and Marc Yor (eds)., Séminaire de Probabilités XXXIV (Springer-Verlag, 2003), pp. 1--145 [Postscript preprint. Looks like a trial run for Del Moral's book.]
  • Rick Durrett
  • Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory [Contains a good one-chapter account of the basics of interacting particle systems, but presumes knowledge of measure-theoretic probability and stochastic processes --- such as you'd get from reading the earlier chapters!]
  • David Griffeath, Additive and Cancellative Interacting Particle Systems

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