---
tag: NKNSCK1JY2
kind: concept
name: Ergodic hypothesis
msc: [37A60, 82B03]
sources:
  - url: https://en.wikipedia.org/wiki/Ergodic_hypothesis
same_as:
  wikidata: Q174196
  wikipedia: Ergodic hypothesis
---

## Summary

The ergodic hypothesis is an assumption of statistical mechanics about the long-run behaviour of a mechanical system.
It says that the system spends time in each region of its energy surface in proportion to the volume of that region.
In other words, time averages of an observable along a trajectory equal its average over the energy surface.
It justifies replacing hard time averages by phase-space averages, which are the averages that equilibrium statistical mechanics computes.
In mathematical terms it says that the [[Hamiltonian|0EDKGA2FAK]] flow restricted to an energy surface is [[ergodic|AZRPHS478B]] for the Liouville measure.

## Explanations

### L1 human

Think of a gas in a box: one cannot follow every molecule, but one can measure averages over a long time.
The ergodic hypothesis says that one long history of the system already visits all the states of the same energy, in a fair way.
So the average of a quantity over one long history is the same as its average over all possible states of that energy.
A simple model is a point moving around a circle at constant speed: the fraction of time it spends in an arc equals the length of the arc.
For real mechanical systems the hypothesis is hard to prove, and for many systems it is false.