Virial theorem

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy, ${\displaystyle \left\langle T\right\rangle }$, of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy, ${\displaystyle \left\langle V_{\text{TOT}}\right\rangle }$, where angle brackets represent the average over time of the enclosed quantity. Mathematically, the theorem states

where F_k represents the force on the kth particle, which is located at position r_k. The word virial for the right-hand side of the equation derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870.

The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics; this average total kinetic energy is related to the temperature of the system by the equipartition theorem. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium. The virial theorem has been generalized in various ways, most notably to a tensor form.

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