---
tag: KD404MF042
kind: concept
name: Heat kernel
msc: [35K08, 35K05]
sources:
  - url: https://en.wikipedia.org/wiki/Heat_kernel
same_as:
  wikidata: Q3345669
  wikipedia: Heat kernel
---

## Summary

The heat kernel is the solution of the [[heat equation|VP2HN86VAG]] that starts from a unit amount of heat concentrated at one point.
On $\R^n$ it is the Gaussian $\Phi(x, t) = (4 \pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$.
Every bounded continuous initial temperature evolves by convolution with it.
On domains and on Riemannian manifolds the heat kernel encodes the spectrum of the Laplacian.
It is a basic tool in geometric analysis and index theory.

## Explanations

### L1 human

Touch a cold infinite metal plate with a hot needle for an instant and watch the heat spread.
At each later time the temperature is a bell-shaped curve centred at the touched point, getting wider and flatter as time goes on.
The total amount of heat stays the same, so the area under the bell is always one.
Any initial temperature is a sum of many such touches, so its evolution is the matching sum of bells.