In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. Mathematically, the theorem states
⟨
T
⟩
=
−
1
2
∑
k
=
1
N
⟨
F
k
⋅
r
k
⟩
{\displaystyle \left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }}
for the total kinetic energy ⟨T⟩ of N particles, where Fk represents the force on the kth particle, which is located at position rk, and angle brackets represent the average over time of the enclosed quantity. 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.
If the force between any two particles of the system results from a potential energy V(r) = αrn that is proportional to some power n of the interparticle distance r, the virial theorem takes the simple form
2
⟨
T
⟩
=
n
⟨
V
TOT
⟩
.
{\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .}
Thus, twice the average total kinetic energy ⟨T⟩ equals n times the average total potential energy ⟨VTOT⟩. Whereas V(r) represents the potential energy between two particles, VTOT represents the total potential energy of the system, i.e., the sum of the potential energy V(r) over all pairs of particles in the system. A common example of such a system is a star held together by its own gravity, where n equals −1.
Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.