Editing Equipartition theorem (section)

==Limitations==
[[Image:1D normal modes (280 kB).gif|frame|right|Figure 9. Energy is ''not'' shared among the various [[normal modes]] in an isolated system of ideal coupled [[harmonic oscillator|oscillators]]; the energy in each mode is constant and independent of the energy in the other modes. Hence, the equipartition theorem does ''not'' hold for such a system in the [[microcanonical ensemble]] (when isolated), although it does hold in the [[canonical ensemble]] (when coupled to a heat bath). However, by adding a sufficiently strong nonlinear coupling between the modes, energy will be shared and equipartition holds in both ensembles.]]

===Requirement of ergodicity===
{{See also|Ergodicity|Chaos theory|Kolmogorov–Arnold–Moser theorem|Solitons}}

The law of equipartition holds only for [[ergodic hypothesis|ergodic]] systems in [[thermal equilibrium]], which implies that all states with the same energy must be equally likely to be populated.<ref name="huang_1987" /> Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external [[heat bath]] in the [[canonical ensemble]]. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the [[dynamical billiards|hard-sphere system]] of [[Yakov G. Sinai|Yakov Sinai]].<ref>{{cite book | last = Arnold | first = VI | author-link = Vladimir Arnold |author2=Avez A | year = 1957 | title = Théorie ergodique des systèms dynamiques | publisher = Gauthier-Villars, Paris. (English edition: Benjamin-Cummings, Reading, Mass. 1968)|language=fr}}</ref> The requirements for isolated systems to ensure [[ergodic theory|ergodicity]]—and, thus equipartition—have been studied, and provided motivation for the modern [[chaos theory]] of [[dynamical system]]s. A chaotic [[Hamiltonian system]] need not be ergodic, although that is usually a good assumption.<ref name="reichl_1998" />

A commonly cited counter-example where energy is ''not'' shared among its various forms and where equipartition does ''not'' hold in the microcanonical ensemble is a system of coupled harmonic oscillators.<ref name="reichl_1998">{{cite book | last = Reichl | first = LE|author-link= Linda Reichl | year = 1998 | title = A Modern Course in Statistical Physics | edition = 2nd | publisher = Wiley Interscience | isbn = 978-0-471-59520-5 | pages = 326–333}}</ref> If the system is isolated from the rest of the world, the energy in each [[normal mode]] is constant; energy is not transferred from one mode to another. Hence, equipartition does not hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the [[energy]] function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the [[Kolmogorov–Arnold–Moser theorem]] states that energy will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes.

Another simple example is an ideal gas of a finite number of colliding particles in a round vessel. Due to the vessel's symmetry, the angular momentum of such a gas is conserved. Therefore, not all states with the same energy are populated. This results in the mean particle energy being dependent on the mass of this particle, and also on the masses of all the other particles.<ref>{{Cite journal |last1=Naplekov |first1=Dmitry M. |last2=Yanovsky |first2=Vladimir V. |date=2023-02-28 |title=Distribution of energy in the ideal gas that lacks equipartition |journal=Scientific Reports |language=en |volume=13 |issue=1 |pages=3427 |doi=10.1038/s41598-023-30636-6 |issn=2045-2322 |pmc=9974969 |pmid=36854979|bibcode=2023NatSR..13.3427N }}</ref>

Another way ergodicity can be broken is by the existence of nonlinear [[soliton]] symmetries. In 1953, [[Enrico Fermi|Fermi]], [[John Pasta|Pasta]], [[Stanislaw Ulam|Ulam]] and [[Mary Tsingou|Tsingou]] conducted [[Fermi–Pasta–Ulam–Tsingou problem|computer simulations]] of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the [[Korteweg–de Vries equation]] led to the development of soliton mathematics.

===Failure due to quantum effects===
{{See also|Ultraviolet catastrophe|History of quantum mechanics|Identical particles}}

The law of equipartition breaks down when the thermal energy {{math|''k''<sub>B</sub>''T''}} is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth [[continuum (theory)|continuum]], which is required in the [[#Derivations|derivations of the equipartition theorem above]].<ref name="pathria_1972" /><ref name="huang_1987" /> Historically, the failures of the classical equipartition theorem to explain [[specific heats]] and [[black-body radiation]] were critical in showing the need for a new theory of matter and radiation, namely, [[quantum mechanics]] and [[quantum field theory]].<ref name="pais_1982" />

[[Image:Et fig2.png|left|thumb|upright=1.45|Figure 10. Log–log plot of the average energy of a quantum mechanical oscillator (shown in red) as a function of temperature. For comparison, the value predicted by the equipartition theorem is shown in black. At high temperatures, the two agree nearly perfectly, but at low temperatures when {{math|''k''<sub>B</sub>''T'' ≪ ''hν''}}, the quantum mechanical value decreases much more rapidly. This resolves the problem of the [[ultraviolet catastrophe]]: for a given temperature, the energy in the high-frequency modes (where {{math|''hν'' ≫ ''k''<sub>B</sub>''T''}}) is almost zero.]]

To illustrate the breakdown of equipartition, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Neglecting the irrelevant [[zero-point energy]] term since it can be factored out of the exponential functions involved in the probability distribution, the quantum harmonic oscillator  energy levels are given by {{math|1=''E<sub>n</sub>'' = ''nhν''}}, where {{math|''h''}} is the [[Planck constant]], {{mvar|ν}} is the [[fundamental frequency]] of the oscillator, and {{mvar|n}} is an integer. The probability of a given energy level being populated in the [[canonical ensemble]] is given by its [[Boltzmann factor]]
<math display="block">P(E_n) = \frac{e^{-n\beta h\nu}}{Z},</math>
where {{math|1=''β'' = 1/''k''<sub>B</sub>''T''}} and the denominator {{math|''Z''}} is the [[partition function (statistical mechanics)|partition function]], here a [[geometric series]]
<math display="block">Z = \sum_{n=0}^{\infty} e^{-n\beta h\nu} = \frac{1}{1 - e^{-\beta h\nu}}.</math>

Its average energy is given by
<math display="block">
\langle H \rangle = \sum_{n=0}^{\infty} E_{n} P(E_{n}) =
\frac{1}{Z} \sum_{n=0}^{\infty} nh\nu \ e^{-n\beta h\nu} = 
-\frac{1}{Z} \frac{\partial Z}{\partial \beta} = 
-\frac{\partial \log Z}{\partial \beta}.
</math>

Substituting the formula for {{math|''Z''}} gives the final result<ref name="huang_1987" />
<math display="block">\langle H \rangle = h\nu \frac{e^{-\beta h\nu}}{1 - e^{-\beta h\nu}}.</math>

At high temperatures, when the thermal energy {{math|''k''<sub>B</sub>''T''}} is much greater than the spacing {{math|''hν''}} between energy levels, the exponential argument {{math|''βhν''}} is much less than one and the average energy becomes {{math|''k''<sub>B</sub>''T''}}, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when {{math|''hν'' ≫ ''k''<sub>B</sub>''T''}}, the average energy goes to zero—the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy {{math|''k''<sub>B</sub>''T''}} (roughly 0.025&nbsp;[[electronvolt|eV]]) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10&nbsp;eV).

Similar considerations apply whenever the energy level spacing is much larger than the thermal energy. This reasoning was used by [[Max Planck]] and [[Albert Einstein]], among others, to resolve the [[ultraviolet catastrophe]] of [[black-body radiation]].<ref name="Einstein1905">{{cite journal | last = Einstein | first = A | author-link = Albert Einstein | year = 1905 | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (A Heuristic Model of the Creation and Transformation of Light) | journal = [[Annalen der Physik]] | volume = 17 | issue = 6 | pages = 132–148 | url = http://gallica.bnf.fr/ark:/12148/bpt6k2094597 | doi = 10.1002/andp.19053220607|bibcode = 1905AnP...322..132E |language=de| doi-access = free }}. An [[s:A Heuristic Model of the Creation and Transformation of Light|English translation]] is available from [[Wikisource]].</ref> The paradox arises because there are an infinite number of independent modes of the [[electromagnetic field]] in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy {{math|''k''<sub>B</sub>''T''}}, there would be an infinite amount of energy in the container.<ref name="Einstein1905" /><ref>{{cite journal | last = Rayleigh | first = JWS | author-link = John Strutt, 3rd Baron Rayleigh | year = 1900 | title = Remarks upon the Law of Complete Radiation | journal = [[Philosophical Magazine]] | volume = 49 | pages = 539–540 | doi=10.1080/14786440009463878| bibcode = 1900PMag...49..539R | url = https://zenodo.org/record/1430616 }}</ref> However, by the reasoning above, the average energy in the higher-frequency modes goes to zero as ''ν'' goes to infinity; moreover, [[Planck's law]] of black-body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.<ref name="Einstein1905" />

Other, more subtle quantum effects can lead to corrections to equipartition, such as [[identical particles]] and [[symmetry|continuous symmetries]]. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the [[valence electron]]s in a metal can have a mean kinetic energy of a few [[electronvolt]]s, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that the [[Pauli exclusion principle]] invalidates the classical approach, is called a [[degenerate matter|degenerate fermion gas]]. Such gases are important for the structure of [[white dwarf]] and [[neutron star]]s.{{citation needed|date=March 2018}} At low temperatures, a [[fermionic condensate|fermionic analogue]] of the [[Bose–Einstein condensate]] (in which a large number of identical particles occupy the lowest-energy state) can form; such [[superfluid]] electrons are responsible{{dubious|date=May 2018}} for [[superconductivity]].