# For which systems is the equipartition theorem valid?

Under what conditions does a system with many degrees of freedom satisfy the equipartition theorem?

• There are many comments on the conditions that must be satisfied in the wikipedia article: en.wikipedia.org/wiki/Equipartition_theorem – Rafael Jan 24 '11 at 13:25
• Thanks Rafael. The section on ET for systems described in terms of generalized coordinates makes all very clear. – Johannes Jan 25 '11 at 3:40

But once a system reaches the thermal equilibrium, it is guaranteed that each subsystem and/or each average degree of freedom of a certain kind carries the energy that is attributed to it by the equipartition theorem - e.g. $kT/2$ for a typical classical degree of freedom (translations or rotations). This is a statistical statement, so you need to look at many atoms or degrees of freedom of the same kind ($N$ of them) and take their average energy. When you do it right, the average energy will be given by the figure implied by the equipartition theorem plus minus an error - the relative error goes to zero as $1/\sqrt{N}$ for many atoms or degrees of freedom.
Of course, one must be careful what the actual prediction of the equipartition theorem for a "degree of freedom" is. The figure $kT/2$ only holds if the degree of freedom is de facto classical and more or less decoupled from others - at the given frequency. For very strongly coupled systems, the "number of degrees of freedom" is less clearly defined, and so is the energy predicted by the equipartition theorem for one of them.