# Why is the Virial Theorem not a Special Case of the Ergodic Theorem? What is their Relationship?

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.)

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).

• Wait so ergodic transformations aren't considered general in physics? What about the ergodic hypothesis? Commented May 4, 2016 at 21:19
• I know it is a dumb question, but I thought that the evolution of state was in general an ergodic transformation. Since that is not the case, that is why the virial theorem is not a special case of the ergodic theorem? In the case that the ergodic transformation hypothesis holds, do the two theorems have any relationship then? Thank you again so much for your time and help -- I really appreciate it! Commented May 5, 2016 at 1:00
• @William, few systems are known to be ergodic, the status of most is unknown. There are important systems that are known to be not ergodic. Ergodic theory is only one possible way to try and justify algorithms of statistical physics. Commented May 5, 2016 at 15:35
• Also, the time scale of ergodicity may be very unphysical, so it is not clear whether ergodicity is of practical importance for a given physical system. Commented Jan 23, 2019 at 1:56
• @MaxLein indeed. In description of nonequilibrium processes, things including probabilities depend on time, so infinite time averages are probably not very interesting as a tool to estimate experimental values. Commented Jan 23, 2019 at 2:40