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Discrete-to-continuum limits of interacting particle systems in one dimension with collisions

Patrick van Meurs

Abstract

We study a class of interacting particle systems in which $n$ signed particles move on the real line. At close range particles with the same sign repel and particles with opposite sign attract each other. The repulsion and attraction are described by the same singular interaction force $f$. Particles of opposite sign may collide in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Požár and the author, one particular particle system of this type was studied; that in which $f(x) = \frac1x$ is the Coulomb force. Global well-posedness of this particle system was shown and a discrete-to-continuum limit (i.e. $n \to \infty$) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. In the present paper we extend these results to a large class of particle systems in order to cover many new applications. Motivated by these applications, we consider the presence of an external force $g$, consider interaction forces $f$ with a large range of singularities and allow $f$ to scale with $n$. To handle this class of $f$ we develop several new proof techniques in addition to those used for the Coulomb force.

Discrete-to-continuum limits of interacting particle systems in one dimension with collisions

Abstract

We study a class of interacting particle systems in which signed particles move on the real line. At close range particles with the same sign repel and particles with opposite sign attract each other. The repulsion and attraction are described by the same singular interaction force . Particles of opposite sign may collide in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Požár and the author, one particular particle system of this type was studied; that in which is the Coulomb force. Global well-posedness of this particle system was shown and a discrete-to-continuum limit (i.e. ) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. In the present paper we extend these results to a large class of particle systems in order to cover many new applications. Motivated by these applications, we consider the presence of an external force , consider interaction forces with a large range of singularities and allow to scale with . To handle this class of we develop several new proof techniques in addition to those used for the Coulomb force.
Paper Structure (25 sections, 18 theorems, 256 equations, 7 figures)

This paper contains 25 sections, 18 theorems, 256 equations, 7 figures.

Table of Contents

  1. Introduction
  2. VanMeursPeletierPozar22: charged particles and the limit $n \to \infty$
  3. Setting and main results of the present paper
  4. Applications and literature
  5. Dislocation structures.
  6. Riesz potentials.
  7. The external potential $U$.
  8. The parameter $\alpha_n$.
  9. Organization of the paper
  10. Well-posedness of the ODE system \ref{['Pn']}
  11. Definitions of \ref{['Pn']} and ($P^m$) in terms of viscosity solutions
  12. Notation
  13. Asumptions on $U$ and $V$
  14. The integrated PDEs formally
  15. Viscosity solutions of \ref{['HJe']} with a comparison principle
  16. ...and 10 more sections

Key Result

Lemma 2.4

Let $f$ satisfy Assumption a:fg. For all $\gamma > 0$, the function $x \mapsto f(x+\gamma) - f(x)$ is nondecreasing on $(0,\infty)$.

Figures (7)

  • Figure 1: A sketch of solution trajectories to \ref{['Pn']}. Trajectories of particles with positive charge are colored red; those with negative charge blue. The black dots indicate the collision points.
  • Figure 2: Typical examples of $V$ and $U$.
  • Figure 3: Left: the graph sketches the force field (given by $-\frac{1}{n} f(x)$) that a negative particle at $x=0$ exerts on $\mathbb R$. The effect of this force field is further illustrated by force vectors acting on two positive particles in the vicinity. Right: similar as the left figure, but now for a dipole (with force field $\frac{1}{n} f(x+\gamma) - \frac{1}{n} f(x)$). Both figures: the force field yields a positive contribution to $\frac{d}{dt} d$.
  • Figure 4: Sketch of possible trajectories of the limiting particles $x_i$ around $\tau$. Each particle $x_i$ enters at $t = \tau - r$ the trapezoid around $x_i(\tau)$ and leaves the trapezoid at $t = \tau + h$. Green trajectories correspond to particles which become neutral after collision.
  • Figure 5: Plot of the staircase approximation $E_\varepsilon$ of the identity.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Example 2.1: Derivative of the Riesz potential
  • Lemma 2.4
  • proof
  • Definition 2.5: Solution to $(P_n)$
  • Remark 2.6: Uniqueness modulo relabeling
  • Theorem 2.7: Properties of $(P_n)$
  • Corollary 2.8: Multiple-particle collisions
  • Remark 2.9: Weakened assumptions on $f$
  • proof : Proof of Theorem \ref{['t:Pn']}
  • Lemma 3.2
  • ...and 32 more