Diffusion Models Intro: Concepts, Formulas and Examples

Diffusion is the spreading of something from where it is concentrated to where it is scarce — a drop of ink in water, heat through a metal rod, or a pollutant in the air. Diffusion models describe this spreading with partial differential equations, most famously the heat equation.

Diffusion is one of nature's great smoothers: it explains how smells cross a room, how heat evens out in an engine block, and how drugs spread from a pill into the bloodstream.

Key Concepts

• Concentration u(x, t) — amount per unit length (or volume) at position x and time t.
• Diffusion coefficient D — how fast spreading happens (units: length²/time, e.g. m²/s). Ink in water has a small D; heat in copper has a large one.
• Flux — the flow of material per unit time, driven by concentration differences (Fick's law: flux = −D × gradient).
• Heat equation — the master diffusion equation, relating the rate of change in time to the curvature in space.
• Steady state — when spreading balances out and the profile stops changing in time.

Formulas

• ∂u/∂t = D ∂²u/∂x² — the heat (diffusion) equation in one dimension. The second spatial derivative measures curvature: where the profile bulges upward, material flows away. Assumes D constant and no sources.
• Fundamental solution: u(x,t) = (M/√(4πDt)) e^(−x²/(4Dt)) — a point release of mass M spreads as a widening bell curve (Gaussian).
• Spreading width ∼ √(Dt) — the characteristic distance reached grows with the square root of time: doubling the distance takes four times as long.
• Steady state with fixed ends: d²u/dx² = 0 — the profile becomes a straight line between the boundary values.

Worked Example

A drop of ink (mass M = 1 unit) is released at x = 0 in still water with D = 10⁻⁹ m²/s (a realistic value for small molecules). How wide has it spread after 1 hour?

1. Step 1: Time t = 3,600 s. Spreading width ∼ √(Dt) = √(10⁻⁹ × 3,600) = √(3.6 × 10⁻⁶) ≈ 0.0019 m ≈ 1.9 mm.
2. Step 2: So after a full hour, molecular diffusion alone has spread the ink only about 2 mm — diffusion is slow at human scales.
3. Step 3: Compare with heat in a copper rod: D ≈ 10⁻⁴ m²/s, so in 1 hour the width ∼ √(10⁻⁴ × 3,600) = √0.36 = 0.6 m. Heat spreads vastly faster than ink.
4. Step 4: Lesson — the √(Dt) scaling and the value of D together decide whether diffusion matters on your timescale.

Common Misconceptions

• "Diffusion moves things at a constant speed." It does not. The front advances as √t — fast at first, ever slower. There is no fixed "speed of diffusion".
• "Stirring is a kind of diffusion." Stirring is advection (bulk flow carrying material), which is far faster. True diffusion is random molecular motion; models often include both terms.
• "The heat equation only applies to heat." It applies to anything that spreads down a gradient: chemicals, pollutants, genes in a population, even option prices in finance (the Black-Scholes equation transforms into it).

Real-World Applications

• Drug delivery — diffusion models predict how a medicine spreads from an injection site or a slow-release implant.
• Environmental science — pollutant plumes in groundwater and the atmosphere are modelled with diffusion (plus advection).
• Materials engineering — doping silicon chips relies on precisely controlled diffusion of impurities into the crystal.

Diffusion models turn "stuff spreads out" into the heat equation ∂u/∂t = D∂²u/∂x², whose bell-curve solutions and √t spreading law appear across science. Whenever a sharp difference smooths itself away — ink, heat, pollution, prices — diffusion mathematics is at work.