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Four-parameter coalescing ballistic annihilation

Kimberly Affeld, Christian Dean, Matthew Junge, Hanbaek Lyu, Connor Panish, Lily Reeves

TL;DR

This work determines the critical blockade density in the four-parameter coalescing ballistic annihilation model for three velocities. It derives a recursive, fixed-point characterization of the origin-visitation probability $q(p)$ by partitioning on the first particle, and expresses the blockade-survival contributions via quantities $s$ and $r$ tied to a mass-transport framework. The main result yields an explicit formula for the critical density $p_c$ and a solvable implicit equation for $q(p)$, generalizing prior three-parameter analyses to the full four-parameter space. The approach combines detailed distance-comparison analysis, renewal arguments, and a systematic organization of collision events to reveal how reaction parameters govern blockade survival, with potential extensions to broader reaction schemes and higher-parameter families.

Abstract

In coalescing ballistic annihilation, infinitely many particles move with fixed velocities across the real line and, upon colliding, either mutually annihilate or generate a new particle. We compute the critical density in symmetric three-velocity systems with four-parameter reaction equations.

Four-parameter coalescing ballistic annihilation

TL;DR

This work determines the critical blockade density in the four-parameter coalescing ballistic annihilation model for three velocities. It derives a recursive, fixed-point characterization of the origin-visitation probability by partitioning on the first particle, and expresses the blockade-survival contributions via quantities and tied to a mass-transport framework. The main result yields an explicit formula for the critical density and a solvable implicit equation for , generalizing prior three-parameter analyses to the full four-parameter space. The approach combines detailed distance-comparison analysis, renewal arguments, and a systematic organization of collision events to reveal how reaction parameters govern blockade survival, with potential extensions to broader reaction schemes and higher-parameter families.

Abstract

In coalescing ballistic annihilation, infinitely many particles move with fixed velocities across the real line and, upon colliding, either mutually annihilate or generate a new particle. We compute the critical density in symmetric three-velocity systems with four-parameter reaction equations.
Paper Structure (9 sections, 10 theorems, 75 equations, 1 figure)

This paper contains 9 sections, 10 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof Overview
  3. Further Questions
  4. Notation
  5. Recursion and Proofs of Main Results
  6. Calculations
  7. Arrival comparisons
  8. Two useful tools
  9. Recursion Quantities

Key Result

Theorem 1

For any FCBA it holds that

Figures (1)

  • Figure 1: A simulation of FCBA with 200 particles. The four reaction parameters are set to $1/3$ and $p=0.15$. \ref{['thm:main']} tells us that $p_c = 2/13 \approx 0.1538$. Space is the horizontal axis and time is the vertical axis. Lines show particle trajectories. Blue represents blockades and red arrows. The code used to generate this figure can be found at: https://github.com/algerbrex/ballistic_annihilation.

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • proof : Proof of \ref{['thm:main']} and \ref{['thm:q']}
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 13 more