🪔Understanding Diffusion from a Point Source

Even though the magnitude of the current gets weaker with distance, a remarkable thing happens: the total flux—the total amount of substance passing through an imaginary sphere drawn around the source—is always the same, no matter how big or small you make the sphere. Here is why this is the case: The surface area of a sphere grows in proportion to the square of its radius. The magnitude of the current decreases in proportion to the inverse square of the radius. These two effects—the growing area and the weakening current—perfectly cancel each other out.

Imagine a tiny, magical bead placed in the center of a large, still tank of water. This bead isn't just sitting there; it's constantly releasing a colored dye at a perfectly steady rate. The dye immediately begins to spread out—or diffuse—in all directions. At first, the color is intense near the bead, but it becomes fainter as it moves further away.

This document aims to explain the physics of this process. We will explore how a substance produced at a single point spreads out and describe what the "current," or flow, of that substance looks like at any distance from its source once the system has reached a stable state. To build this model, we first need to define a few key concepts.

🎬Programmatic demo

the stationary solution to Poisson's equation driven by a Dirac delta function source | Animated Resultsviadean on Notion