Editing Equipartition theorem (section)

===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.