The Heat Equation

Temperature diffusion on curves and surfaces — the simplest geometric flow

The Heat Equation

Put a hot spot on a cold bar and the heat spreads until everything is the same temperature. The heat equation that describes this is the simplest geometric flow.

Heat Diffusion on a Bar

Click the bar to place heat sources, then watch hot spots cool and cold spots warm until the temperature evens out.

Go deeper:The Heat Equation

The heat equation ∂u/∂t = Δu is the simplest and most fundamental geometric flow. It describes how a temperature distribution evolves over time, smoothing out hot spots and cold spots until the temperature becomes uniform.

This equation is the prototype for all geometric flows: it takes something irregular and makes it smoother. Understanding the heat equation is the gateway to curve shortening flow, mean curvature flow, and ultimately Ricci flow.

Go deeper:The Laplacian Operator

The key ingredient is the Laplacian Δu, which measures how much a function's value at a point differs from its average in a small neighborhood:

Δu = ∂²u/∂x² (in 1D)
Δu = ∂²u/∂x² + ∂²u/∂y² (in 2D)

When Δu > 0, the point is cooler than its neighbors and heats up. When Δu < 0, the point is warmer and cools down. This is why the heat equation always smooths — it pulls every point toward the local average.

Go deeper:The Maximum Principle

A cornerstone result: under the heat equation, the maximum temperature can only decrease and the minimum can only increase.

  • No new extrema: Hot spots cool, cold spots warm
  • Monotone bounds: max(u) is non-increasing in time
  • Uniqueness: Solutions are determined by initial data
Go deeper:Conservation of Energy

The total heat (integral of u) is conserved. Heat is never created or destroyed — it merely redistributes.

d/dt ∫ u(x,t) dx = 0

This is the mathematical expression of the first law of thermodynamics applied to an insulated system.

The Heat Kernel

Watch a single spike of heat spread into ever-wider bell curves. Notice that the total area under the curve stays exactly 1.

Go deeper:The Gaussian Heat Kernel

The heat kernel G(x,t) is the fundamental solution — the response to a single point source of heat.

G(x,t) = (1/√(4πt)) · exp(-x²/4t)

Any initial temperature distribution u(x,0) = f(x) can be recovered by convolving with the heat kernel:

u(x,t) = ∫ G(x-y,t) · f(y) dy

This convolution formula shows that the heat equation is a smoothing machine: convolving with a Gaussian blurs sharp features. As t grows, the Gaussian widens and smoothing intensifies.

Curve Smoothing via Heat Flow

Run heat flow on a noisy closed curve: each vertex moves toward the average of its neighbors, so bumps flatten and corners round off.

Key Takeaways

  • The heat equation ∂u/∂t = Δu smooths out hot and cold spots.
  • The Laplacian compares a value with the average of its neighbors.
  • The maximum temperature can only fall and the minimum can only rise.
  • The heat kernel is a spreading Gaussian whose total area stays 1.

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