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Is the equipartition theorem really a theorem and derivable from more basic assumptions or is it just a hypothesis. Some of the ways energy is partition is not to squared quantum numbers (e.g. vibration) although translation and rotation are

user46221
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  • Yes, it can be derived from the canonical ensemble. – knzhou Apr 24 '16 at 01:38
  • Why vibration is not squared? – velut luna Apr 24 '16 at 02:18
  • And btw it holds only in classical SM. – velut luna Apr 24 '16 at 02:20
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    Possible duplicate of For which systems is the equipartition theorem valid? – ACuriousMind Apr 24 '16 at 10:10
  • Knzhou -- could you point me to a reference where the equipartition theorem is derived from the canonical ensemble? Thanks. – user46221 Apr 24 '16 at 17:45
  • The best explanation of equipartition that I've found is in "Modern Physics" Jeremy Bernstein et. al. Prentice Hall (2000) pp. 339 -342. Each quadratic degree of freedom gives 1/2 kT to the energy. The only trick is getting vibration which has a linear dependence on vibrational quantum number into a quadratic form -- they do this by splitting vibration into kinetic and potential forms - both of which are quadratic. Then it's just a matter of integrating X^2 * exp (- X^2/kT) and dividing by the partition function. – user46221 Apr 24 '16 at 18:30
  • This question is not a duplicate of the linked question. Note one asks for the conditions for the theorem to hold (and no derivation of the theorem or the conditions) the other asks for a derivation from basic principles. The closest thing to an answer to this question on stackexchange can actually be found here: https://physics.stackexchange.com/questions/20944/the-equipartition-theorem-in-momentum-space?rq=1. – Marten Aug 31 '18 at 17:39

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