# American Institute of Mathematical Sciences

March  2010, 5(1): 31-62. doi: 10.3934/nhm.2010.5.31

## A particle system in interaction with a rapidly varying environment: Mean field limits and applications

 1 Institut de Mathématiques de Toulouse, CNRS & Université de Toulouse, 31062 Toulouse cedex 9, France 2 Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario K1N 6M8, Canada 3 Microsoft Reasearch, Roger Needham Building, 7 J J Thomson Avenue, Cambridge, CB3 0FB, United Kingdom

Received  February 2009 Revised  November 2009 Published  February 2010

We study an interacting particle system whose dynamics depends on an interacting random environment. As the number of particles grows large, the transition rate of the particles slows down (perhaps because they share a common resource of fixed capacity). The transition rate of a particle is determined by its state, by the empirical distribution of all the particles and by a rapidly varying environment. The transitions of the environment are determined by the empirical distribution of the particles. We prove the propagation of chaos on the path space of the particles and establish that the limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic differential equation involves the time averages of the environment process.
We apply the results on particle systems to understand the behavior of computer networks where users access a shared resource using a distributed random Medium Access Control (MAC) algorithm. MAC algorithms are used in all Local Area Network (LAN), and have been notoriously difficult to analyze. Our analysis allows us to provide simple and explicit expressions of the network performance under such algorithms.
Citation: Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks & Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31
 [1] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006 [2] Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 [3] Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086 [4] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 [5] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [6] Giacomo Canevari, Pierluigi Colli. Solvability and asymptotic analysis of a generalization of the Caginalp phase field system. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1959-1982. doi: 10.3934/cpaa.2012.11.1959 [7] Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4269-4292. doi: 10.3934/dcds.2015.35.4269 [8] Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361 [9] Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 [10] Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 387-396. doi: 10.3934/dcds.2004.10.387 [11] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [12] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [13] Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443 [14] Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011 [15] Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279 [16] Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 [17] Chang-Shou Lin. An expository survey on the recent development of mean field equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 387-410. doi: 10.3934/dcds.2007.19.387 [18] Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic & Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661 [19] Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465 [20] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173

2018 Impact Factor: 0.871