Editing Equipartition theorem (section)

{{Short description|Theorem in classical statistical mechanics}}
{{pp-move-indef|small=yes}}
{{refimprove|date=April 2015}}
[[Image:Thermally Agitated Molecule.gif|thumb|right|'''Thermal motion''' of an [[alpha helix|α-helical]] [[peptide]]. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the ''average'' [[kinetic energy]] of each atom to be computed, as well as the average potential energies of many vibrational modes. The grey, red and blue spheres represent [[atom]]s of [[carbon]], [[oxygen]] and [[nitrogen]], respectively; the smaller white spheres represent atoms of [[hydrogen]].]]

In [[classical physics|classical]] [[statistical mechanics]], the '''equipartition theorem''' relates the [[temperature]] of a system to its average [[energy|energies]]. The equipartition theorem is also known as the '''law of equipartition''', '''equipartition of energy''', or simply '''equipartition'''. The original idea of equipartition was that, in [[thermal equilibrium]], energy is shared equally among all of its various forms; for example, the average [[kinetic energy]] per [[Degrees of freedom (physics and chemistry)|degree of freedom]] in [[translation (physics)|translational motion]] of a molecule should equal that in [[rotational motion]].

The equipartition theorem makes quantitative predictions. Like the [[virial theorem]], it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's [[heat capacity]] can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a single [[spring (device)|spring]]. For example, it predicts that every atom in a [[monatomic]] [[ideal gas]] has an average kinetic energy of {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}} in thermal equilibrium, where {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]] and ''T'' is the [[Thermodynamic temperature|(thermodynamic) temperature]]. More generally, equipartition can be applied to any [[classical physics|classical system]] in [[thermal equilibrium]], no matter how complicated. It can be used to derive the [[ideal gas law]], and the [[Dulong–Petit law]] for the [[specific heat capacity|specific heat capacities]] of solids.<ref>Stone, A. Douglas, “Einstein and the Quantum,” Chapter 13, “Frozen Vibrations,” 2013. {{ISBN|978-0691139685}}</ref> The equipartition theorem can also be used to predict the properties of [[star]]s, even [[white dwarf]]s and [[neutron star]]s, since it holds even when [[special relativity|relativistic]] effects are considered.

Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when [[quantum physics|quantum effects]] are significant, such as at low temperatures.  When the [[thermal energy]] {{math|''k''<sub>B</sub>''T''}} is smaller than the quantum energy spacing in a particular [[degrees of freedom (physics and chemistry)|degree of freedom]], the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the heat capacity of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model [[black-body radiation]]—also known as the [[ultraviolet catastrophe]]—led [[Max Planck]] to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of [[quantum mechanics]] and [[quantum field theory]].