On Mean Field Limits for Dynamical Systems

Abstract

We present a purely probabilistic proof of propagation of molecular chaos for $N$-particle systems in dimension $3$ with interaction forces scaling like $1/|q{|}^{\lambda }$ with $\lambda <2$ and cut-off at $q=N^{-1/3}$. The proof yields a Gronwall estimate for the maximal distance between exact microscopic and approximate mean-field dynamics. This can be used to show propagation of molecular chaos, i.e. weak convergence of the marginals to the corresponding products of solutions of the respective mean-field equation without cut-off in a quantitative way. Our results thus lead to a derivation of the Vlasov equation from the microscopic $N$-particle dynamics with force term arbitrarily close to the physically relevant Coulomb- and gravitational forces.

17 Pages

References in corpus (3)

• Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields
• Particles approximations of Vlasov equations with singular forces : Propagation of chaos
• The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

Cited by in corpus (18)

• Mean Field Limit for Coulomb-Type Flows
• Propagation of chaos: a review of models, methods and applications. II. Applications
• Learning Mean-Field Equations from Particle Data Using WSINDy
• The Vlasov-Poisson dynamics as the mean field limit of extended charges
• Propagation of chaos for the Keller-Segel equation over bounded domains
• A new approach to the mean-field limit of Vlasov-Fokker-Planck equations
• The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation
• The discreteness-driven relaxation of collisionless gravitating systems: entropy evolution in external potentials, N-dependence and the role of chaos
• Formation and relaxation of quasi-stationary states in particle systems with power law interactions
• From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials
• Mean Field Limit and Propagation of Chaos for a Pedestrian Flow Model
• Combined Mean Field Limit and Non-relativistic Limit of Vlasov-Maxwell Particle System to Vlasov-Poisson System
• Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces
• On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system
• A mean field limit for the Hamiltonian Vlasov system
• The Mean Field Kinetic Equation for Interacting Particle Systems with Non-Lipschitz Force
• A particle approximation for the relativistic Vlasov-Maxwell dynamics
• Uniform-in-time propagation of chaos for the Cucker--Smale model