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Equipartition theorem
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===Extreme relativistic ideal gases=== {{See also|Special relativity|White dwarf|Neutron star}} Equipartition was used above to derive the classical [[ideal gas law]] from [[Newtonian mechanics]]. However, [[special relativity|relativistic effects]] become dominant in some systems, such as [[white dwarf]]s and [[neutron star]]s,<ref name="huang_1987" /> and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic [[ideal gas]].<ref name="pathria_1972" /> In such cases, the kinetic energy of a [[relativistic particle|single particle]] is given by the formula <math display="block">H_{\mathrm{kin}} \approx cp = c \sqrt{p_x^2 + p_y^2 + p_z^2}.</math> Taking the derivative of {{mvar|H}} with respect to the {{math|''p<sub>x</sub>''}} momentum component gives the formula <math display="block">p_x \frac{\partial H_{\mathrm{kin}}}{\partial p_x} = c \frac{p_x^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}}</math> and similarly for the {{math|''p<sub>y</sub>''}} and {{math|''p<sub>z</sub>''}} components. Adding the three components together gives <math display="block">\begin{align} \langle H_{\mathrm{kin}} \rangle &= \left\langle c \frac{p_x^2 + p_y^2 + p_z^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}} \right\rangle\\ &= \left\langle p_x \frac{\partial H^{\mathrm{kin}}}{\partial p_x} \right\rangle + \left\langle p_y \frac{\partial H^{\mathrm{kin}}}{\partial p_y} \right\rangle + \left\langle p_z \frac{\partial H^{\mathrm{kin}}}{\partial p_z} \right\rangle \\ &= 3 k_\text{B} T \end{align}</math> where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for {{mvar|N}} particles, it is {{math|3 ''Nk''<sub>B</sub>''T''}}.
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