---
tag: ZVY046R4XB
kind: concept
name: Arnold diffusion
msc: [37J40, 70H08]
sources:
  - url: https://en.wikipedia.org/wiki/Arnold_diffusion
same_as:
  wikidata: Q4795328
  wikipedia: Arnold diffusion
---

## Summary

Arnold diffusion is a slow drift of the action variables in a nearly [[integrable system|N0N7CMRCMH]] with three or more degrees of freedom.
Along such an orbit the actions change by an amount of order one, although the perturbation is arbitrarily small.
It shows that the stability given by [[KAM theory|3ZZSXKM16Z]] for most initial conditions does not hold for all of them.
Arnold gave the first example in 1964.

## Explanations

### L1 human

In a nearly integrable system most motions stay on invariant tori, which act like walls.
With two degrees of freedom these walls cut each energy level into separate rooms, so nothing can wander far.
With three or more degrees of freedom the walls are too thin to separate anything, like circles in three-dimensional space.
Then an orbit can slip through the gaps between the tori and, very slowly, travel a long way.
This slow wandering is Arnold diffusion.