7

The virial theorem involves the time-averages of the potential and kinetic energies if the motion of the system is bounded to a finite region of space.

An ergodic theorem relates the time and space averages of a quantity, in the case of thermodynamics usually the average amount of time spent in some region of phase space with constant energy is proportional to the volume of that region of phase space.

Both principles involve time averages of energy, so I would be surprised if there was no meaningful relationship between them. Someone asked a similar question on Quora, but got no answers.

What is the relationship, if any, between the virial and ergodic theorem?

(Also I apologize for asking two questions in succession -- this and my previous question were both inspired by reading the same chapter of Landau and Lifschitz.)

valerio
  • 16,751
Chill2Macht
  • 554

1 Answers1

4

The virial theorem is valid if few simple conditions are satisfied (system particles, in the course of its evolution, remain bounded in a finite-sized region) and is thus quite general.

Ergodic theorem requires evolution to be ergodic:

https://en.wikipedia.org/wiki/Ergodicity

This is a special requirement that is not guaranteed to be satisfied for all systems. For example, integrable systems with more than 1 integral of motion are not ergodic (such as two or more noninteracting simple harmonic oscillators).

Ján Lalinský
  • 44,745