Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Equipartition theorem
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
===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 [[electronvolt|eV]]) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 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]].
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)