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Global existence and mean-field limit for a stochastic interacting particle system of signed Coulomb charges

Patrick van Meurs, Mark A. Peletier, Thomas Slangen

Abstract

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.

Global existence and mean-field limit for a stochastic interacting particle system of signed Coulomb charges

Abstract

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.

Paper Structure

This paper contains 18 sections, 15 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Related literature
  4. Main results
  5. Outline of the proof
  6. Discussion
  7. Outlook
  8. Mean-field limit for $\frac{1}{N} \ll \gamma \ll 1$.
  9. Beyond $\gamma < \frac{1}{2} \sigma^2$.
  10. Other collision rules.
  11. Preliminaries
  12. Scaling invariance of the particle system
  13. Squared Bessel processes
  14. The case of two particles
  15. Collisions happen with positive probability
  16. ...and 3 more sections

Key Result

Proposition 2.2

If $\mathbf X$ is a weak solution to SDE:full:param, then:

Figures (3)

  • Figure 1: Difference between the fully attractive case and the signed case. Close pairs are stronger attractors than single particles in the fully attractive case, but have hardly any effect in the signed case.
  • Figure 2: Areas in the parameter space $(\sigma,\gamma)$ where weak solutions exist (left) and where sticky collisions occur (right).
  • Figure 3: By the definitions of $D_0$ and $D^{\operatorname{opp}}_0$, any particle $x_0^j$ other than $x_0^1$ or $x_0^2$ is outside of the gray regions.

Theorems & Definitions (29)

  • Definition 2.1: Weak solution
  • Proposition 2.2: Quick observations
  • Lemma 2.3: Squared Bessel processes in positive and negative dimension; RevuzYor99
  • Theorem 2.4: Zero-hitting properties
  • Theorem 3.1: Collisions happen with positive probability
  • proof : Proof of Theorem \ref{['t:coll']}
  • Remark 3.2
  • Theorem 4.1: Existence and properties of weak solutions
  • Remark 4.2
  • Lemma 4.3: Uniform bounds on $\mathbf X^{\varepsilon, \ell}$
  • ...and 19 more