---
tag: 464NN2K4FZ
kind: concept
name: Equilibrium measure
msc: [31A15, 31B15, 30C85]
sources:
  - url: https://en.wikipedia.org/wiki/Capacity_of_a_set
---

## Summary

The equilibrium measure of a compact set $K$ is the way a unit charge distributes itself on $K$ when it minimizes its energy.
In the plane the energy is the logarithmic energy, and in $\R^n$ with $n \ge 3$ it is the Newtonian energy.
When $K$ has positive [[capacity|DBQXRTC0VN]], the minimizing measure exists and is unique.
Its potential is constant on most of $K$, and this constant determines the [[capacity of the set|4YZVFGJVSZ]].

## Explanations

### L1 human

Put one unit of electric charge on a metal body $K$ and let it move freely.
Like charges repel, so the charge spreads out until the electrostatic energy is as small as possible.
The final distribution is the equilibrium measure, and the body is then at one constant voltage.
On the unit circle the equilibrium measure is the uniform measure, with total mass $1$ spread evenly along the circle.
On the closed unit disc the charge also sits on the boundary circle and leaves the interior empty.