Propagation of chaos for topological interactions

We consider a $N$-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit $N \to \infty$, as following from the previous analysis in [A. Blanchet, P. Degond, Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), pp. 41-60.] can be rigorously derived. This means that the statistical independence (propagation of chaos) is indeed recovered in the limit, provided it is assumed at time zero.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[3]  J. Gillis,et al.  Probability and Related Topics in Physical Sciences , 1960 .

[4]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[5]  O. Lanford.  Time evolution of large classical systems , 1975 .

[6]  W. Braun,et al.  The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles , 1977 .

[7]  H. Spohn.  Kinetic equations from Hamiltonian dynamics: Markovian limits , 1980 .

[8]  C. Cercignani.  The grad limit for a system of soft spheres , 1983 .

[9]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[10]  Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum , 1986 .

[11]  R. Illner,et al.  Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum , 1986 .

[12]  M. Pulvirenti,et al.  A stochastic system of particles modelling the Euler equations , 1990 .

[13]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[14]  W. Wagner,et al.  Convergence of particle schemes for the Boltzmann equation , 1994 .

[15]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[16]  C. Graham,et al.  Stochastic particle approximations for generalized Boltzmann models and convergence estimates , 1997 .

[17]  M. Pulvirenti,et al.  Stationary particle systems approximating stationary solutions to the Boltzmann equation , 1998 .

[18]  S. Ukai.  The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem , 2001 .

[19]  G. Parisi,et al.  Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study , 2007, Proceedings of the National Academy of Sciences.

[20]  Francois Bolley Jos.  Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming , 2010 .

[21]  Daniel W Franks,et al.  Limited interactions in flocks: relating model simulations to empirical data , 2011, Journal of The Royal Society Interface.

[22]  G. Parisi,et al.  FROM EMPIRICAL DATA TO INTER-INDIVIDUAL INTERACTIONS: UNVEILING THE RULES OF COLLECTIVE ANIMAL BEHAVIOR , 2010 .

[23]  马剑,et al.  Multi-grid simulation of pedestrian counter flow with topological interaction , 2010 .

[24]  H. Chaté,et al.  Relevance of metric-free interactions in flocking phenomena. , 2010, Physical review letters.

[25]  Pierre-Emmanuel Jabin,et al.  Particles approximations of Vlasov equations with singular forces : Propagation of chaos , 2011, 1107.3821.

[26]  Charlotte K. Hemelrijk,et al.  Some Causes of the Variable Shape of Flocks of Birds , 2011, PloS one.

[27]  Giorgio Parisi,et al.  Spatially balanced topological interaction grants optimal cohesion in flocking models , 2012, Interface Focus.

[28]  Jan Haskovec,et al.  Flocking dynamics and mean-field limit in the Cucker–Smale-type model with topological interactions , 2013, 1301.0925.

[29]  M. Pulvirenti,et al.  On the validity of the Boltzmann equation for short range potentials , 2013, 1301.2514.

[30]  Yilun Shang,et al.  Consensus reaching in swarms ruled by a hybrid metric-topological distance , 2014, The European Physical Journal B.

[31]  Isabelle Gallagher,et al.  From Newton to Boltzmann: Hard Spheres and Short-range Potentials , 2012, 1208.5753.

[32]  Yilun Shang,et al.  Influence of the number of topologically interacting neighbors on swarm dynamics , 2014, Scientific Reports.

[33]  Samuel Martin,et al.  Multi-agent flocking under topological interactions , 2013, Syst. Control. Lett..

[34]  Yukio-Pegio Gunji,et al.  Emergence of the scale-invariant proportion in a flock from the metric-topological interaction , 2014, Biosyst..

[35]  Pierre Degond,et al.  Topological Interactions in a Boltzmann-Type Framework , 2015, Journal of Statistical Physics.

[36]  S. Mischler,et al.  A new approach to quantitative propagation of chaos for drift, diffusion and jump processes , 2011, 1101.4727.

[37]  Pierre-Emmanuel Jabin,et al.  Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces , 2015, 1511.03769.

[38]  L. Saint-Raymond,et al.  One-sided convergence in the Boltzmann-Grad limit , 2016, 1612.03722.

[39]  Matthias Abend.  Stochastic Numerics For The Boltzmann Equation , 2016 .

[40]  Guanrong Chen,et al.  Synchronization of multi-agent systems with metric-topological interactions. , 2016, Chaos.

[41]  P. Degond,et al.  Kinetic Models for Topological Nearest-Neighbor Interactions , 2017, Journal of Statistical Physics.

[42]  Tsuyoshi Murata,et al.  {m , 1934, ACML.