Virial theorem

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 a conservative force, with that of the total potential energy of the system. Mathematically, the theorem states that

⟨

T

⟩

=

−

1

2

∑

k

=

1

N

⟨

F

k

⋅

r

k

⟩

,

{\displaystyle \langle T\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,}

where

T

{\displaystyle T}

is the total kinetic energy of the

N

{\displaystyle N}

particles,

F

k

{\displaystyle F_{k}}

represents the force on the

k

{\displaystyle k}

th 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

)

=

α

r

n

{\displaystyle V(r)=\alpha r^{n}}

that is proportional to some power

n

{\displaystyle n}

of the interparticle distance

r

{\displaystyle 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

⟩

{\displaystyle \langle T\rangle }

equals

n

{\displaystyle n}

times the average total potential energy

⟨

V

TOT

⟩

{\displaystyle \langle V_{\text{TOT}}\rangle }

. Whereas

V

(

r

)

{\displaystyle V(r)}

represents the potential energy between two particles separated by distance

r

{\displaystyle r}

,

V

TOT

{\displaystyle V_{\text{TOT}}}

represents the total potential energy of the system, i.e., the sum of the potential energy

V

(

r

)

{\displaystyle 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

=

−

1

{\displaystyle n=-1}

.