Editing Equipartition theorem (section)

==General formulation of the equipartition theorem==
{{See also|Generalized coordinates|Hamiltonian mechanics|Microcanonical ensemble|Canonical ensemble}}

The most general form of the equipartition theorem states that under suitable assumptions (discussed [[#Requirement of ergodicity|below]]), for a physical system with [[Hamiltonian function|Hamiltonian]] energy function {{mvar|H}} and degrees of freedom {{math|''x<sub>n</sub>''}}, the following equipartition formula holds in thermal equilibrium for all indices {{mvar|m}} and {{mvar|n}}:<ref name="pathria_1972" /><ref name="huang_1987" /><ref name="tolman_1918">{{cite journal | last = Tolman | first = RC | author-link = Richard C. Tolman | year = 1918 | title = A General Theory of Energy Partition with Applications to Quantum Theory | journal = [[Physical Review]] | volume = 11 | issue = 4 | pages = 261–275 | doi = 10.1103/PhysRev.11.261|bibcode = 1918PhRv...11..261T | url = https://authors.library.caltech.edu/9471/1/TOLpr18.pdf }}</ref>
<math display="block">\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} k_\text{B} T.</math>
Here {{math|''δ<sub>mn</sub>''}} is the [[Kronecker delta]], which is equal to one if {{math|1=''m'' = ''n''}} and is zero otherwise. The averaging brackets <math>\left\langle \ldots \right\rangle</math> is assumed to be an [[ensemble average]] over phase space or, under an assumption of [[ergodic]]ity, a time average of a single system.

The general equipartition theorem holds in both the [[microcanonical ensemble]],<ref name="huang_1987">{{cite book | last = Huang | first = K | author-link = Kerson Huang | year = 1987 | title = Statistical Mechanics | edition = 2nd | publisher = John Wiley and Sons | pages = 136–138 | isbn = 0-471-81518-7}}</ref> when the total energy of the system is constant, and also in the [[canonical ensemble]],<ref name="pathria_1972">{{cite book | last = Pathria | first = RK | year = 1972 | title = Statistical Mechanics | publisher = Pergamon Press | pages = 43–48, 73–74 | isbn = 0-08-016747-0}}</ref><ref name="tolman_1938">{{cite book | last = Tolman | first = RC | author-link = Richard C. Tolman | year = 1938 | title = The Principles of Statistical Mechanics | publisher = Dover Publications | location = New York | pages = 93–98 | isbn = 0-486-63896-0}}</ref> when the system is coupled to a [[heat bath]] with which it can exchange energy. Derivations of the general formula are given [[#Derivations|later in the article]].

The general formula is equivalent to the following two:
# <math>\left\langle x_n \frac{\partial H}{\partial x_n} \right\rangle = k_\text{B} T \quad \text{for all } n</math>
# <math>\left\langle x_m \frac{\partial H}{\partial x_n} \right\rangle = 0 \quad \text{for all } m \neq n.</math>

If a degree of freedom ''x<sub>n</sub>'' appears only as a quadratic term ''a<sub>n</sub>x<sub>n</sub>''<sup>2</sup> in the Hamiltonian ''H'', then the first of these formulae implies that
<math display="block">k_\text{B} T = \left\langle x_n \frac{\partial H}{\partial x_n}\right\rangle = 2\left\langle a_n x_n^2 \right\rangle,</math>
which is twice the contribution that this degree of freedom makes to the average energy <math>\langle H\rangle</math>. Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by ''s'', applies to energies of the form ''a<sub>n</sub>x<sub>n</sub><sup>s</sup>''.

The degrees of freedom ''x<sub>n</sub>'' are coordinates on the [[phase space]] of the system and are therefore commonly subdivided into [[canonical coordinates|generalized position]] coordinates ''q<sub>k</sub>'' and [[generalized momentum]] coordinates ''p<sub>k</sub>'', where ''p<sub>k</sub>'' is the [[conjugate momentum]] to ''q<sub>k</sub>''. In this situation, formula 1 means that for all ''k'',
<math display="block">
\left\langle p_{k} \frac{\partial H}{\partial p_{k}} \right\rangle =  \left\langle q_k \frac{\partial H}{\partial q_k} \right\rangle = k_\text{B} T.
</math>

Using the equations of [[Hamiltonian mechanics]],<ref name="goldstein_1980">{{cite book | last = Goldstein |first = H | author-link = Herbert Goldstein | year = 1980 | title = Classical Mechanics | edition = 2nd. | publisher = Addison-Wesley | isbn = 0-201-02918-9}}</ref> these formulae may also be written
<math display="block">\left\langle p_k \frac{dq_k}{dt} \right\rangle = -\left\langle q_k \frac{dp_k}{dt} \right\rangle = k_\text{B} T.</math>

Similarly, one can show using formula 2 that
<math display="block">
\left\langle p_j \frac{\partial H}{\partial p_k} \right\rangle =
\left\langle q_j \frac{\partial H}{\partial q_k} \right\rangle = 0 \quad \text{ for all } \, j \neq k.
</math>
and
<math display="block">
\left\langle p_j \frac{\partial q_k}{\partial t} \right\rangle = 
-\left\langle q_j \frac{\partial p_k}{\partial t} \right\rangle = 0 \quad \text{ for all } \, j \neq k.
</math>

===Relation to the virial theorem===
{{See also|Virial theorem|Generalized coordinates|Hamiltonian mechanics}}

The general equipartition theorem is an extension of the [[virial theorem]] (proposed in 1870<ref>{{cite journal | last = Clausius | first = R | author-link = Rudolf Clausius | year = 1870 | title = Ueber einen auf die Wärme anwendbaren mechanischen Satz | journal = Annalen der Physik | volume = 141 | issue = 9 | pages = 124–130 | url = http://gallica.bnf.fr/ark:/12148/bpt6k152258 | doi=10.1002/andp.18702170911|language=de|bibcode = 1870AnP...217..124C }}<br />{{cite journal | last = Clausius | first = RJE | author-link = Rudolf Clausius | year = 1870 | title = On a Mechanical Theorem Applicable to Heat | journal = [[Philosophical Magazine]] |series=Series 4 | volume = 40 | pages = 122–127}}</ref>), which states that
<math display="block">
\left\langle \sum_k q_k \frac{\partial H}{\partial q_{k}} \right\rangle = 
\left\langle \sum_k p_k \frac{\partial H}{\partial p_{k}} \right\rangle = 
\left\langle \sum_k p_k \frac{dq_k}{dt} \right\rangle = -\left\langle \sum_k q_k \frac{dp_k}{dt} \right\rangle,
</math>
where ''t'' denotes [[time]].<ref name="goldstein_1980" /> Two key differences are that the virial theorem relates ''summed'' rather than ''individual'' averages to each other, and it does not connect them to the [[temperature]] ''T''. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over [[phase space]].