[Paper Review] Particles approximations of Vlasov equations with singular forces : Propagation of chaos

Pierre‐Emmanuel Jabin, Maxime Hauray|arXiv (Cornell University)|Jul 19, 2011

Gas Dynamics and Kinetic TheoryMathematics17 citations

TL;DR

This paper establishes the mean field limit and propagation of chaos for particle systems with singular interaction forces of the form $1/|x|^\alpha$, where $\alpha < 1$ in dimensions $d \geq 3$. It further extends the result to forces with stronger singularities ($\alpha < d-1$) under a small, $N$-dependent cut-off, nearly covering Coulomb and gravitational interactions. The key contribution is a rigorous derivation of the Vlasov equation as the limit of $N$-particle dynamics under these singular force laws.

ABSTRACT

We obtain the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1/|x|^α$, with $α&lt;1$ in dimension $d \geq 3$. We also provide results for forces with singularity up to $α&lt; d-1$ but with large enough cut-off. This last result thus almost includes the most interesting case of Coulombian or gravitational interaction, but it is also interesting when the strength of the singularity $α$ is larger but close to one, in which case it allows for very small cut-off.

Motivation & Objective

• To establish the mean field limit for $N$-particle systems with singular forces $F(x) \sim 1/|x|^\alpha$ in $d \geq 3$ dimensions.
• To prove propagation of chaos for such systems, showing that particle marginals converge to the solution of the Vlasov equation.
• To extend the analysis to forces with stronger singularities ($\alpha < d-1$) by introducing an $N$-dependent cut-off near the origin.
• To remove the need for Hamiltonian structure or potential-based forces, allowing general singular kernels.
• To provide a rigorous foundation for particle simulations of kinetic equations in plasma and astrophysics.

Proposed method

• Use of a mean-field scaling with $1/N$ factor in the force term to keep total mass finite as $N \to \infty$.
• Definition of global solutions to the $N$-particle system via integral equations, assuming existence and continuity.
• Application of probabilistic estimates via binomial laws and exponential moments to control particle density fluctuations.
• Covering the phase space with cubes of size $L\varepsilon$ and bounding the supremum of particle counts over these cubes.
• Use of Chebyshev's inequality and optimal exponential moment bounds to derive large deviation estimates for the sup-norm of the empirical measure.
• Establishing existence and regularity of solutions to the Vlasov equation via characteristics and Lipschitz continuity of the electric field $E = F * \rho$.

Experimental results

Research questions

• RQ1 Under what conditions does the $N$-particle system with singular forces $F(x) \sim 1/|x|^\alpha$ converge to the Vlasov equation as $N \to \infty$?
• RQ2 Can propagation of chaos be established for singular forces that are not derived from a potential or do not satisfy Hamiltonian structure?
• RQ3 How can the analysis be extended to forces with stronger singularities, such as those near the Coulomb or gravitational potential?
• RQ4 What role does an $N$-dependent cut-off play in regularizing the interaction and enabling the mean field limit?
• RQ5 What is the rate of convergence in the propagation of chaos for such singular systems?

Key findings

• The mean field limit holds for forces satisfying $|F(x)| \leq C/|x|^\alpha$ with $\alpha < 1$, ensuring the potential is continuous and bounded near the origin.
• For forces with $\alpha < d-1$, the mean field limit is established under a cut-off $N^{-m}$, with $|F_N(x)| \leq N^{m\alpha}$ for $|x| \leq N^{-m}$, allowing near-Coulombic interactions.
• The propagation of chaos is proven via large deviation estimates on the empirical measure, with the probability of large deviations decaying faster than any inverse power of $N$.
• The bound on the sup-norm of the empirical measure $\|f_N^Z\|_\infty$ is controlled via exponential moments and covering arguments, yielding $\mathbb{P}(\|f_N^Z\|_\infty \geq \beta c_\phi \|f\|_\infty) \leq c_0 N^\gamma e^{-(\beta \ln \beta - \beta + 1)(2L)^n \|f\|_\infty N^{1-\gamma}}$.
• The solution to the Vlasov equation exists globally and is given by the method of characteristics, with $E$ being Lipschitz due to $\alpha < d-1$ and $\rho \in L^\infty$.
• The $L^1$ and $L^\infty$ norms of the distribution function $f$ are preserved along characteristics, ensuring stability of the solution.