- Print publication year: 2013
- Online publication date: June 2014

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/CBO9781139519939.004
- pp 125-209

Abstract

Interacting particle systems (IPS) are probabilistic mathematical models of complex phenomena involving a large number of interrelated components. There are numerous examples within all areas of natural and social sciences, such as traffic flow on motorways or communication networks, opinion dynamics, spread of epidemics or fires, genetic evolution, reaction diffusion systems, crystal surface growth, financial markets, etc. The central question is to understand and predict emergent behaviour on macroscopic scales, as a result of the microscopic dynamics and interactions of individual components. Qualitative changes in this behaviour depending on the system parameters are known as collective phenomena or phase transitions and are of particular interest.

In IPS the components are modelled as particles confined to a lattice or some discrete geometry. But applications are not limited to systems endowed with such a geometry, since continuous degrees of freedom can often be discretized without changing the main features. So depending on the specific case, the particles can represent cars on a motorway, molecules in ionic channels, or prices of asset orders in financial markets (see Chapter 6), to name just a few examples. In principle, such systems often evolve according to well-known laws, but in many cases microscopic details of motion are not fully accessible. Due to the large system size these influences on the dynamics can be approximated as effective random noise with a certain postulated distribution. The actual origin of the noise, which may be related to chaotic motion (see Chapter 2) or thermal interactions (see Chapter 4), is usually ignored.

Powered by UNSILO

[1] Interaction of Markov processes. Adv. Math. 5, 246–290 (1970). :

[2] Mechanisms for spatio-temporal pattern formation in highway traffic models. Philos. Transact A Math. Phys. Eng. Sci. 366(1872), 2017–2032 (2008). :

[3] Reproduction time statistics and segregation patterns in growing populations. Phys. Rev. E 85(2), 021923 (2012). , , :

[4] Genetic drift at expanding frontiers promotes gene segregation. PNAS 104(50), 19926–19930.(2007). et al.:

[5] Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001). :

[6] Probability and Random Processes, 3rd edition, Oxford (2001). , :

[7] Markov Chains, Cambridge (1997). :

[8] Foundations of Modern Probability, 2nd edition, Springer (2002). :

[9] The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. J. Phys. A: Math. Gen. 39, 12679–12705 (2006). , :

[10] Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329(4), 199–32 (2000). , and :

[11] Metapopulation dynamics as a contact process on a graph. Ecological Complexity 1, 49–63 (2004). :

[12] Nonequilibrium Phase Transitions in Lattice Models. Cambridge (2005). , :

[13] Real and Complex Analysis, McGraw-Hill (1987). :

[14] Functional Analysis, 6th edition, Springer (1980). :

[15] Interacting Particle Systems, Springer (1985). :

[16] Scaling limits of Interacting Particle Systems, Springer (1999). , :

[17] Functional Analysis, 2nd edition, McGraw-Hill (1991). :

[18] Convergence of Probability Measures, 2nd edition, Wiley (1999). :

[19] Gibbs Measures and Phase Transitions, de Gruyter (1988). :

[20] The Green-Kubo formula and power spectrum of reversible Markov processes; J. Math. Phys. 44(10), 4681–4689 (2003). , :

[21] Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976). :

[22] Non-equilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 41–53 (1981). :

[23] Hydrodynamic limit for attractive particle systems on ℤd. Commun. Math. Phys. 140, 417–448 (1991). :

[24] Hyperbolic Conservation Laws in Continuum Physics. Springer (2000). :

[25] Stochastic Interacting Systems, Springer (1999). :

[26] Limit Distributions for Sums of Indepenent Random Variables. Addison Wesley (1954). , :

[27] I-Divergence geometry of probability distributions and minimization problems. I; Ann. Prob. 3, 146–158 (1975). :

[28] Nonequilibrium statistical mechanics of the zerorange process and related models; J. Phys. A: Math. Theor. 38, R195–R240 (2005). , :

[29] Phase transitions in one-dimensional nonequilibrium systems; Braz. J. Phys. 30(1), 42–57 (2000). :

[30] Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28(3), 1162–1194 (2000). , , :

[31] Invarient measures for the zero range process. Ann. Probab. 10(3), 525–547 (1982). :