[Paper Review] Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces

Pierre‐Emmanuel Jabin, Zhenfu Wang|arXiv (Cornell University)|Nov 12, 2015

Gas Dynamics and Kinetic Theory29 references6 citations

TL;DR

This paper establishes the mean field limit and propagation of chaos for Vlasov systems with bounded interaction kernels using relative entropy estimates in the Liouville framework. It proves strong convergence of all particle marginals to the solution of the Vlasov equation, even for rough, non-smooth forces, by combining combinatorial analysis with weak-strong uniqueness arguments based on entropy control.

ABSTRACT

We consider large systems of particles interacting through rough but bounded interaction kernels. We are able to control the relative entropy between the $N$-particle distribution and the expected limit which solves the corresponding Vlasov system. This implies the Mean Field limit to the Vlasov system together with Propagation of Chaos through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.

Motivation & Objective

• To establish the mean field limit for large systems of particles interacting via bounded, rough interaction kernels.
• To prove propagation of chaos through strong convergence of all particle marginals to the Vlasov solution.
• To extend the validity of the mean field limit beyond classical regularity assumptions, including for oscillatory or discontinuous kernels.
• To develop a robust analytical framework based on relative entropy and weak solutions of the Liouville equation.
• To unify the treatment of deterministic, fixed-noise, and vanishing-noise stochastic systems in the mean field limit.

Proposed method

• Uses the Liouville equation to describe the evolution of the N-particle joint distribution, allowing weak solutions even when the force kernel is only bounded.
• Applies relative entropy estimates between the N-particle distribution and the limiting Vlasov solution to quantify convergence.
• Employs combinatorial techniques to control the number of particle interactions in the entropy expansion, especially for marginal distributions.
• Establishes weak-strong uniqueness for the Vlasov equation via a control of the exponential moment of the gradient of the logarithmic density.
• Derives a differential inequality for the entropy functional involving the gradient of the logarithmic density, which is then controlled via characteristic tracing.
• Handles both deterministic and stochastic systems (with vanishing or fixed noise) uniformly by treating the diffusion parameter εN as a general scaling.

Experimental results

Research questions

• RQ1 Can the mean field limit be rigorously established for Vlasov systems with only bounded interaction kernels, without requiring Lipschitz or smoothness conditions?
• RQ2 Does propagation of chaos hold in the sense of strong convergence of all k-particle marginals for such systems?
• RQ3 Can relative entropy be used as a quantitative tool to control the convergence to the Vlasov equation in the presence of rough forces?
• RQ4 Is the weak-strong uniqueness principle for the Vlasov equation sufficient to imply convergence when combined with entropy estimates?
• RQ5 How does the method extend to stochastic systems with non-vanishing noise in the N→∞ limit?

Key findings

• The relative entropy between the N-particle distribution and the limiting Vlasov solution decays to zero as N→∞, implying the mean field limit.
• All k-particle marginals converge strongly to the corresponding marginals of the Vlasov solution, establishing propagation of chaos.
• The method applies uniformly to deterministic systems (εN=0), systems with fixed noise (εN→ε>0), and systems with vanishing noise (εN→0), ensuring robustness.
• The key technical innovation is the use of combinatorial estimates in the Liouville framework to control the entropy expansion for general bounded kernels.
• The weak-strong uniqueness argument is validated via a priori control of the exponential moment of |∇v log f|, which remains bounded in time under the Vlasov dynamics.
• The result holds even for highly oscillatory or discontinuous interaction kernels, significantly broadening the class of admissible forces beyond the classical Coulomb or gravitational cases.