particle-field.qmd

---
title: "Particle–Field Interactions"
subtitle: "Particles modify local concentrations"
---

## Overview

Particles are not passive tracers—they actively modify the chemical fields through **consumption** and **production**. This creates bidirectional coupling and enables emergent feedback dynamics.

## Equations

Particles locally consume field 1 and produce (or consume) field 2:

$$
\frac{\partial C_1}{\partial t}\bigg|_{\text{particle}} = -\gamma_c \sum_{j=1}^{N_p} w(\|\mathbf{x} - \mathbf{x}_j\|)
$$

$$
\frac{\partial C_2}{\partial t}\bigg|_{\text{particle}} = +\gamma_p \sum_{j=1}^{N_p} w(\|\mathbf{x} - \mathbf{x}_j\|)
$$

where $w(r) = \exp(-r^2 / 2r_{\text{inf}}^2)$ is a Gaussian influence kernel centered on each particle.

## Parameters

| Symbol | Description | Typical Value |
|--------|-------------|---------------|
| $\gamma_c$ | Consumption rate | 180 |
| $\gamma_p$ | Production rate | -180 |
| $r_{\text{inf}}$ | Influence radius | 0.05 |

## Feedback Loop

The particle–field coupling creates a feedback loop:

1. Fields develop gradients (Turing patterns)
2. Particles migrate along gradients (diffusiophoresis)
3. Particles modify local concentrations (consumption/production)
4. Modified fields change gradient landscape
5. Repeat...

::: {.callout-warning}
## Coupling Challenge
For Gray-Scott, particle consumption can overwhelm the slow $UV^2$ production, causing field instability. Brusselator's stronger reaction rates are more robust to particle feedback.
:::