In all derivations of the equipartition theorem I can find a thermodynamic equilibrium distribution is used to show it's validity.
To further stress it:
"More generally, it can be applied to any classical system in thermal equilibrium"
"Two physical systems are in thermal equilibrium if no heat flows between them when they are connected by a path permeable to heat."
"Systems in thermodynamic equilibrium are always in thermal equilibrium, but the converse is not always true."
Thus, how can the equipartition theorem be shown without assuming thermodynamic equilibrium?
Maybe an explanation of why this is important:
In deriving the Stokes Einstein relation: $$\gamma D = k_B T $$ ($\gamma=$ friction coeff., $D=$ diffusion const.) In some derivations the thermodynamic equilibrium is used and in some just the equipartition theorem.
If equipartition is not a thermodynamic equilibrium phenomenon then the relation holds in a much wider variety of cases. Basically it could not be used to determine if thermodynamic equilibrium is at hand by measuring $D$, $\gamma$ and $T$ which I was told is an important applicaiton of the relation.