Interacting Particle Systems

14 Jul 2008 09:20

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have a nice paper on this, with a review of the published literature on the question, which is a small slice of the unpublished folklore.)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 Trotter/Kurtz/etc. 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....

P. Del Moral and L. Miclo, "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear Filtering", in J. Azema, M. Emery, M. Ledoux and M. Yor (eds)., Semainaire de Probabilites XXXIV (Springer-Verlag, 2000), pp. 1--145 [Postscript preprint. Looks like a trial run for Del Moral's book.]
Richard Durrett, Lectures Notes on Particle Systems and Percolation
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

E. Andjel, G. Maillard, T.S. Mountford, "A note on 'signed voter models'", arxiv:0709.3468
Alexei Andreanov, Giulio Biroli, Jean-Philippe Bouchaud, and Alexandre Lefevre, "Field theories and exact stochastic equations for interacting particle systems", cond-mat/0602307
Chalee Asavathiratham, The influence model: a tractable representation for the dynamics of networked Markov chains [Ph.D. thesis, MIT, 2001; online]
Anne-Severine Boudou, Pietro Caputo, Paolo Dai Pra and Gustavo Posta, "Spectral gap estimates for interacting particle systems via a Bakry & Emery-type approach", math.PR/0505533 ["We develop a general technique, based on the Bakry-Emery approach, to estimate spectral gaps of a class of Markov operators. We apply this technique to various interacting particle systems."]
Xavier Bressaud and Nicolas Fournier, "On the invariant distribution of a one-dimensional avalanche process", math.PR/0703750
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Pierre Del Moral
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- Feynman-Kac Formulae
Paul Doukhan, Gabriel Lang, Sana Louhichi, Bernard Ycart, "A functional central limit theorem for interacting particle systems on transitive graphs", math-ph/0509041
Rick Durrett, Stochastic Spatial Models: A Hyper-Tutorial
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Malte Henkel, "Ageing, dynamical scaling and its extensions in many-particle systems without detailed balance", cond-mat/0609672
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Julio Largo, Piero Tartaglia, Francesco Sciortino, "Effective non-additive pair potential for lock-and-key interacting particles: the role of the limited valence", cond-mat/0703383
Alexandre Lefevre, Giulio Biroli, "Dynamics of interacting particle systems: stochastic process and field theory", arxiv:0709.1325
Thomas M. Liggett
- Interacting Particle Systems
- Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes
E. Locherbach, "Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps", Statistical Inference for Stochastic Processes 5 (2002): 153--177
A. V. Skorohod, Stochastic Equations for Complex Systems [chapter 2 being "Randomly Interacting Systems of Particles"]
Anja Sturm and Jan Swart, "Voter models with heterozygosity selection", math.PR/0701555
Biao Wu, "Interacting Agent Feedback Finance Model", math.PR/0703827