Editing Equipartition theorem (section)

==Basic concept and simple examples==
{{See also|Kinetic energy|Heat capacity}}

[[File:MaxwellBoltzmann-en.svg|right|thumb|upright=1.7|Figure 2. Probability density functions of the molecular speed for four [[noble gas]]es at a [[temperature]] of 298.15 [[Kelvin|K]] (25 [[Celsius|°C]]). The four gases are [[helium]] (<sup>4</sup>He), [[neon]] (<sup>20</sup>Ne), [[argon]] (<sup>40</sup>Ar) and [[xenon]] (<sup>132</sup>Xe); the superscripts indicate their [[mass number]]s. These probability density functions have [[dimensional analysis|dimensions]] of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.]]

The name "equipartition" means "equal division," as derived from the [[Latin]] ''equi'' from the antecedent, æquus ("equal or even"), and partition from the noun, ''partitio'' ("division, portion").<ref>{{cite web|url=http://www.etymonline.com/index.php?search=equi&searchmode=none|title=equi-|publisher=Online Etymology Dictionary|access-date=2008-12-20}}</ref><ref>{{cite web|url=http://www.etymonline.com/index.php?search=Partition&searchmode=none|title=partition|publisher=Online Etymology Dictionary|access-date=2008-12-20}}.</ref> The original concept of equipartition was that the total [[kinetic energy]] of a system is shared equally among all of its independent parts, ''on the average'', once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of an inert [[noble gas]], in thermal equilibrium at temperature {{mvar|T}}, has an average translational kinetic energy of {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}, where {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]]. As a consequence, since kinetic energy is equal to {{1/2}}(mass)(velocity)<sup>2</sup>, the heavier atoms of [[xenon]] have a lower average speed than do the lighter atoms of [[helium]] at the same temperature. Figure&nbsp;2 shows the [[Maxwell–Boltzmann distribution]] for the speeds of the atoms in four noble gases.

In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any [[degrees of freedom (physics and chemistry)|degree of freedom]] (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of {{math|{{frac|1|2}}''k''<sub>B</sub>''T''}} and therefore contributes {{math|{{frac|1|2}}''k''<sub>B</sub>}} to the system's [[heat capacity]]. This has many applications.

===Translational energy and ideal gases===
{{See also|Ideal gas}}

The (Newtonian) kinetic energy of a particle of mass {{mvar|m}}, velocity {{math|'''v'''}} is given by
<math display="block">H_{\text{kin}} = \tfrac 1 2 m |\mathbf{v}|^2 = \tfrac{1}{2} m\left( v_x^2 + v_y^2 + v_z^2 \right),</math>
where {{math|''v<sub>x</sub>''}}, {{math|''v<sub>y</sub>''}} and {{math|''v<sub>z</sub>''}} are the Cartesian components of the velocity {{math|'''v'''}}. Here, {{mvar|H}} is short for [[Hamiltonian (quantum mechanics)|Hamiltonian]], and used henceforth as a symbol for energy because the [[Hamiltonian mechanics|Hamiltonian formalism]] plays a central role in the most [[#General formulation of the equipartition theorem|general form]] of the equipartition theorem.

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute {{math|{{frac|1|2}}''k''<sub>B</sub>''T''}} to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}, as in the example of noble gases above.

More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the total energy of an ideal gas of {{mvar|N}} particles is {{math|{{sfrac|3|2}} ''N''&thinsp;''k''<sub>B</sub>&thinsp;''T''}}.

It follows that the [[heat capacity]] of the gas is {{math|{{sfrac|3|2}} ''N''&thinsp;''k''<sub>B</sub>}} and hence, in particular, the heat capacity of a [[mole (unit)|mole]] of such gas particles is {{math|1={{sfrac|3|2}}''N''<sub>A</sub>''k''<sub>B</sub> = {{sfrac|3|2}}''R''}}, where ''N''<sub>A</sub> is the [[Avogadro constant]] and ''R'' is the [[gas constant]]. Since ''R'' ≈ 2 [[calorie|cal]]/([[mole (unit)|mol]]·[[Kelvin|K]]), equipartition predicts that the [[molar heat capacity]] of an ideal gas is roughly 3&nbsp;cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases.<ref name="kundt_1876" />

The mean kinetic energy also allows the [[root mean square speed]] {{math|''v''<sub>rms</sub>}} of the gas particles to be calculated:
<math display="block">v_{\text{rms}} = \sqrt{\left\langle v^2 \right\rangle} = \sqrt{\frac{3 k_\text{B} T}{m}} = \sqrt{\frac{3 R T}{M}},</math>
where {{math|1=''M'' = ''N''<sub>A</sub>''m''}} is the mass of a mole of gas particles. This result is useful for many applications such as [[Graham's law]] of [[effusion]], which provides a method for [[enriched uranium|enriching]] [[uranium]].<ref>[https://www.nrc.gov/reading-rm/doc-collections/fact-sheets/enrichment.html Fact Sheet on Uranium Enrichment] U.S. Nuclear Regulatory Commission. Accessed 30 April 2007</ref>

===Rotational energy and molecular tumbling in solution===
{{See also|Angular velocity|Rotational diffusion}}

A similar example is provided by a rotating molecule with [[principal moments of inertia]] {{math|''I''<sub>1</sub>}}, {{math|''I''<sub>2</sub>}} and {{math|''I''<sub>3</sub>}}. According to classical mechanics, the [[rotational energy]] of such a molecule is given by
<math display="block">H_{\mathrm{rot}} = \tfrac{1}{2} ( I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 ),</math>
where {{math|''ω''<sub>1</sub>}}, {{math|''ω''<sub>2</sub>}}, and {{math|''ω''<sub>3</sub>}} are the principal components of the [[angular velocity]]. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.<ref name="pathria_1972" />

The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the [[relaxation (NMR)|relaxation]]s observed by [[nuclear magnetic resonance]], particularly [[protein nuclear magnetic resonance spectroscopy|protein NMR]] and [[residual dipolar coupling]]s.<ref>{{cite book |vauthors=Cavanagh J, Fairbrother WJ, Palmer AG 3rd, Skelton NJ, Rance M | year = 2006 | title = Protein NMR Spectroscopy: Principles and Practice | edition = 2nd | publisher = Academic Press | isbn = 978-0-12-164491-8}}</ref> Rotational diffusion can also be observed by other biophysical probes such as [[fluorescence anisotropy]], [[flow birefringence]] and [[dielectric spectroscopy]].<ref>{{cite book | last = Cantor | first = CR |author2=Schimmel PR | year = 1980 | title = Biophysical Chemistry. Part II. Techniques for the study of biological structure and function | publisher = W. H. Freeman | isbn = 978-0-7167-1189-6}}</ref>

===Potential energy and harmonic oscillators===
Equipartition applies to [[potential energy|potential energies]] as well as kinetic energies: important examples include [[harmonic oscillator]]s such as a [[spring (device)|spring]], which has a quadratic potential energy
<math display="block">H_{\text{pot}} = \tfrac 1 2 a q^2,\,</math>
where the constant {{mvar|a}} describes the stiffness of the spring and {{mvar|q}} is the deviation from equilibrium. If such a one-dimensional system has mass {{mvar|m}}, then its kinetic energy {{math|''H''<sub>kin</sub>}} is
<math display="block">H_{\text{kin}} = \frac{1}{2}mv^2 = \frac{p^2}{2m},</math>
where {{mvar|v}} and {{math|1=''p'' = ''mv''}} denote the velocity and momentum of the oscillator. Combining these terms yields the total energy<ref name="goldstein_1980" />
<math display="block">H = H_{\text{kin}} + H_{\text{pot}} = \frac{p^2}{2m} + \frac{1}{2} a q^2.</math>
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy
<math display="block">
\langle H \rangle = 
\langle H_{\text{kin}} \rangle + \langle H_{\text{pot}} \rangle = 
\tfrac{1}{2} k_\text{B} T + \tfrac{1}{2} k_\text{B} T = k_\text{B} T,
</math>
where the angular brackets <math>\left\langle \ldots \right\rangle</math> denote the average of the enclosed quantity,<ref name="huang_1987" />

This result is valid for any type of harmonic oscillator, such as a [[pendulum]], a vibrating molecule or a passive [[electronic oscillator]]. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy {{math|''k''<sub>B</sub>''T''}} and hence contributes {{math|''k''<sub>B</sub>}} to the system's [[heat capacity]]. This can be used to derive the formula for [[Johnson–Nyquist noise]]<ref name="mandl_1971">{{cite book | last = Mandl | first = F | year = 1971 | title = Statistical Physics | publisher = John Wiley and Sons | pages = [https://archive.org/details/statisticalphysi00fman/page/213 213–219] | isbn = 0-471-56658-6 | url = https://archive.org/details/statisticalphysi00fman/page/213 }}</ref> and the [[Dulong–Petit law]] of solid heat capacities. The latter application was particularly significant in the history of equipartition.

[[Image:Vätskefas.png|frame|left|Figure 3. Atoms in a crystal can vibrate about their equilibrium positions in the [[crystal structure|lattice]]. Such vibrations account largely for the [[heat capacity]] of crystalline [[dielectric]]s; with [[metal]]s, [[electron]]s also contribute to the heat capacity.]]

===Specific heat capacity of solids===
{{for multi|more details on the molar specific heat capacities of solids|Einstein solid|the Debye model|Debye model}}

An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of {{math|3''N''}} independent [[simple harmonic oscillator]]s, where {{mvar|N}} denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy {{math|''k''<sub>B</sub>''T''}}, the average total energy of the solid is {{math|3''N''&thinsp;''k''<sub>B</sub>''T''}}, and its heat capacity is {{math|3''N''&thinsp;''k''<sub>B</sub>}}.

By taking {{math|''N''}} to be the [[Avogadro constant]] {{math|''N''<sub>A</sub>}}, and using the relation {{math|1=''R'' = ''N''<sub>A</sub>''k''<sub>B</sub>}} between the [[gas constant]] {{math|''R''}} and the Boltzmann constant {{math|''k''<sub>B</sub>}}, this provides an explanation for the [[Dulong–Petit law]] of [[specific heat capacity|specific heat capacities]] of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its [[atomic weight]]. A modern version is that the molar heat capacity of a solid is ''3R''&nbsp;≈ 6&nbsp;cal/(mol·K).

However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived [[third law of thermodynamics]], according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero.<ref name="mandl_1971" /> A more accurate theory, incorporating quantum effects, was developed by [[Albert Einstein]] (1907) and [[Peter Debye]] (1911).<ref name="pais_1982" />

Many other physical systems can be modeled as sets of [[oscillation#Coupled oscillations|coupled oscillators]]. The motions of such oscillators can be decomposed into [[normal mode]]s, like the vibrational modes of a [[piano string]] or the [[resonance]]s of an [[organ pipe]]. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ''[[ergodicity]]'', is important for the law of equipartition to hold.

===Sedimentation of particles===
{{See also|Sedimentation|Mason–Weaver equation|Brewing}}

Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom {{math|''x''}} contributes only a multiple of {{math|''x''<sup>s</sup>}} (for a fixed real number {{math|''s''}}) to the energy, then in thermal equilibrium the average energy of that part is {{math|''k''<sub>B</sub>''T''/''s''}}.

There is a simple application of this extension to the [[sedimentation]] of particles under [[gravitation|gravity]].<ref name="tolman_1918" /> For example, the haze sometimes seen in [[beer]] can be caused by clumps of [[protein]]s that [[Rayleigh scattering|scatter]] light.<ref>{{cite journal |vauthors=Miedl M, Garcia M, Bamforth C |title=Haze formation in model beer systems |journal=J. Agric. Food Chem. |volume=53 |issue=26 |pages=10161–5 |year=2005 |pmid=16366710 |doi=10.1021/jf0506941|bibcode=2005JAFC...5310161M }}</ref> Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also [[diffusion|diffuse]] back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of [[buoyant mass]] {{math|''m''<sub>b</sub>}}. For an infinitely tall bottle of beer, the [[gravitational energy|gravitational potential energy]] is given by
<math display="block">H^{\mathrm{grav}} = m_\text{b} g z </math>
where {{mvar|z}} is the height of the protein clump in the bottle and ''[[Earth's gravity|g]]'' is the [[acceleration]] due to gravity. Since {{math|1=''s'' = 1}}, the average potential energy of a protein clump equals {{math|''k''<sub>B</sub>''T''}}. Hence, a protein clump with a buoyant mass of 10&nbsp;[[Dalton (unit)|MDa]] (roughly the size of a [[virus]]) would produce a haze with an average height of about 2&nbsp;cm at equilibrium. The process of such sedimentation to equilibrium is described by the [[Mason–Weaver equation]].<ref name="mason_1924">{{cite journal | last = Mason | first = M |author2=Weaver W | year = 1924 | title = The Settling of Small Particles in a Fluid | journal = [[Physical Review]] | volume = 23 | issue = 3 | pages = 412–426 | doi = 10.1103/PhysRev.23.412|bibcode = 1924PhRv...23..412M }}</ref>