Editing Equipartition theorem (section)

===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.