Virial Theorem (Statistical Mechanics)

Equivalent forms

2\langle T \rangle + \langle V \rangle = 0

\langle T \rangle = -\frac{1}{2}\sum_i \langle \mathbf{F}_i \cdot \mathbf{r}_i \rangle

PV = Nk_BT - \frac{1}{3}\left\langle \sum_{i<j} r_{ij}\frac{dU}{dr_{ij}} \right\rangle

The theorem is purely mechanical — no statistics needed — yet its ensemble average gives the equation of state of real gases via the virial expansion.

Symbol	Name	SI unit	Dimension	Range
$⟨T⟩$	Time-averaged kinetic energyoutput Average kinetic energy of all particles in the system	J	ML²T⁻²	0 – 1e+30
$⟨V⟩$	Time-averaged potential energy Average potential energy; −2⟨T⟩ for gravity, −⟨T⟩ for harmonic	J	ML²T⁻²	-1e+30 – 0

Unit systems

SI:: J for energies
astrophysics:: M☉, pc, km/s; M ∼ $\sigma _v^{2} R / G$
plasma:: thermal pressure = magnetic pressure in MHD equilibrium $(\beta = 1)$

Where it holds

Bound, stationary (time-averaged) systems. Fails for unbound systems, expanding systems (cosmological), or during rapid collapse.

Discovery

Rudolf Clausius · 1870

Clausius introduced the virial in 1870 to study equations of state for real gases. Poincaré applied it to stellar dynamics; it became the cornerstone of gravitational astrophysics and plasma confinement.

Try this

Why don't clusters of galaxies fly apart?

The virial theorem relates the time-averaged kinetic energy of a bound system to the time-averaged potential energy. It is how astronomers estimate the mass of galaxy clusters — by measuring the velocity dispersion of member galaxies and invoking ⟨T⟩ = −(1/2)⟨V⟩.

Research status: stable

Common misconceptions

The virial theorem for gravity gives ⟨T⟩

= -(1/2)

⟨V⟩, but total energy

E =

⟨T⟩ + ⟨V⟩

= -

⟨T⟩ < 0. Paradoxically, adding energy (loosening binding) increases T — hence self-gravitating systems have negative heat capacity.

Derivation

Define virial

G = \Sigma _i p_i \cdot r_i

.

Then

dG/dt = 2T + \Sigma _i F_i \cdot r_i

.

For a bound system ⟨dG/dt⟩

= 0

, so 2⟨T⟩

= -

⟨

\Sigma F_i \cdot r_i

⟩

= -

⟨virial of forces⟩.

For inverse-square gravity

F_i \cdot r_i = -GM^{2}/r

summed pairwise gives 2⟨T⟩

= GM^{2}/R

.

Limiting cases

Harmonic potential

V \propto r^{2}

⟶ ⟨T⟩ = ⟨V⟩; total energy

E = 2

⟨T⟩

(not -

⟨T⟩)

Gravitational potential

V \propto -1/r

⟶ ⟨T⟩

= -E

> 0; tighter binding → faster motion (negative heat capacity!)

van der Waals gas⟶

PV = Nk_BT + B(T) \rho

+ … virial expansion where B, C, … are virial coefficients

Ideal Gas Law→Equipartition Theorem→