Virial theorem

 ( Physics Theorems ) 

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 (where the work done is independent of path), with that of the total potential energy of the system. Mathematically, the theorem states that

$$$$

\langle T \rangle = -\frac12\,\sum_{k=1}^N \langle\mathbf{F}_k \cdot \mathbf{r}_k\rangle,
     

where $T$ is the total kinetic energy of the $N$ particles, $F\_k$ represents the force on the $k$th particle, which is located at position , 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 $V(r)=\backslash alpha\; r^n$ that is proportional to some power $n$ of the interparticle distance $r$, the virial theorem takes the simple form

$$$$

2 \langle T \rangle = n \langle V_\text{TOT} \rangle.
     

Thus, twice the average total kinetic energy $\backslash langle\; T\backslash rangle$ equals $n$ times the average total potential energy $\backslash langle\; V\_\backslash text\{TOT\}\backslash rangle$. Whereas $V(r)$ represents the potential energy between two particles of distance $r$, $V\_\backslash text\{TOT\}$ 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 $n=-1$.

---

History In 1870, Rudolf Clausius delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean vis viva of the system is equal to its virial, or that the average kinetic energy is one half of the average potential energy. The virial theorem can be obtained directly from Lagrange's identity as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772. Carl Jacobi's generalization of the identity to $N$ bodies and to the present form of Laplace's identity closely resembles the classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics.

G. W. Collins (1978). 9780912918136, Pachart Press. . ISBN 9780912918136

The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux, Richard Bader and Eugene Parker. Fritz Zwicky was the first to use the virial theorem to deduce the existence of unseen matter, which is now called dark matter. Richard Bader showed that the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf .

---

Illustrative special case Consider $N=2$ particles with equal mass $m$, acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radius $r$. The velocities are $\backslash mathbf\{v\}\_1(t)$ and $\backslash mathbf\{v\}\_2(t)=-\backslash mathbf\{v\}\_1(t)$, which are normal to forces $\backslash mathbf\{F\}\_1(t)$ and $\backslash mathbf\{F\}\_2(t)=-\backslash mathbf\{F\}\_1(t)$. The respective magnitudes are fixed at $v$ and $F$. The average kinetic energy of the system in an interval of time from $t\_1$ to $t\_2$ is

$$$$

\langle T \rangle =
\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} \sum_{k=1}^N \frac12 m_k |\mathbf{v}_k(t)|^2 \,dt =
\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} \left( \frac12 m|\mathbf{v}_1(t)|^2 + \frac12 m|\mathbf{v}_2(t)|^2 \right) \,dt = mv^2.
     

Taking center of mass as the origin, the particles have positions $\backslash mathbf\{r\}\_1(t)$ and $\backslash mathbf\{r\}\_2(t)=-\backslash mathbf\{r\}\_1(t)$ with fixed magnitude $r$. The attractive forces act in opposite directions as positions, so $\backslash mathbf\; F\_1(t)\; \backslash cdot\; \backslash mathbf\; r\_1(t)\; =\; \backslash mathbf\; F\_2(t)\; \backslash mathbf\; r\_2(t)\; =\; -Fr$. Applying the centripetal force formula $F=mv^2/r$ results in

$$$$

-\frac12 \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle =
-\frac12(-Fr - Fr) = Fr = \frac{mv^2}{r} \cdot r = mv^2 = \langle T \rangle,
     

as required. Note: If the origin is displaced, then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces $\backslash mathbf\{F\}\_1(t)$, $\backslash mathbf\{F\}\_2(t)$ results in net cancellation.

---

Statement and derivation Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

For a collection of $N$ point particles, the scalar moment of inertia $I$ about the origin is

$$$$

I = \sum_{k=1}^N m_k |\mathbf{r}_k|^2 = \sum_{k=1}^N m_k r_k^2,
     

where $m\_k$ and $\backslash mathbf\{r\}\_k$ represent the mass and position of the $k$th particle and $r\_k=|\backslash mathbf\{r\}\_k|$ is the position vector magnitude. Consider the scalar

$$$$

G = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k,
     

where $\backslash mathbf\{p\}\_k$ is the momentum vector of the $k$th particle.

Assuming that the masses are constant, $G$ is one-half the time derivative of this moment of inertia:

$$\backslash begin\{align\}$$

\frac12 \frac{dI}{dt} &= \frac12 \frac{d}{dt} \sum_{k=1}^N m_k \mathbf{r}_k \cdot \mathbf{r}_k \\
                      &= \sum_{k=1}^N m_k \, \frac{d\mathbf{r}_k}{dt} \cdot \mathbf{r}_k \\
                      &= \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k = G.
     

\end{align}

In turn, the time derivative of $G$ is

$$\backslash begin\{align\}$$

\frac{dG}{dt} &= \sum_{k=1}^N \mathbf{p}_k \cdot \frac{d\mathbf{r}_k}{dt} +
                 \sum_{k=1}^N \frac{d\mathbf{p}_k}{dt} \cdot \mathbf{r}_k \\
              &= \sum_{k=1}^N m_k \frac{d\mathbf{r}_k}{dt} \cdot \frac{d\mathbf{r}_k}{dt} +
                 \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k \\
              &= 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k,
     

\end{align}

where $m\_k$ is the mass of the $k$th particle, $\backslash mathbf\{F\}\_k=\backslash frac\{d\backslash mathbf\{p\}\_k\}\{dt\}$ is the net force on that particle, and $T$ is the total kinetic energy of the system according to the $\backslash mathbf\{v\}\_k=\backslash frac\{d\backslash mathbf\{r\}\_k\}\{dt\}$ velocity of each particle,

$$$$

T = \frac12 \sum_{k=1}^N m_k v_k^2 =
\frac12 \sum_{k=1}^N m_k \frac{d\mathbf{r}_k}{dt} \cdot \frac{d\mathbf{r}_k}{dt}.
     

---

Connection with the potential energy between particles The total force $\backslash mathbf\{F\}\_k$ on particle $k$ is the sum of all the forces from the other particles $j$ in the system:

$$$$

\mathbf{F}_k = \sum_{j=1}^N \mathbf{F}_{jk},
     

where $\backslash mathbf\{F\}\_\{jk\}$ is the force applied by particle $j$ on particle $k$. Hence, the virial can be written as

$$$$

-\frac12\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
-\frac12\,\sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k.
     

Since no particle acts on itself (i.e., $\backslash mathbf\{F\}\_\{jj\}=0$ for $1\backslash leq\; j\backslash leq\; N$), we split the sum in terms below and above this diagonal and add them together in pairs:

$$\backslash begin\{align\}$$

\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
 &= \sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k =
    \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{k=1}^{N-1} \sum_{j=k+1}^{N} \mathbf{F}_{jk} \cdot \mathbf{r}_k \\
 &= \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{j=2}^N \sum_{k=1}^{j-1} \mathbf{F}_{jk} \cdot \mathbf{r}_k =
    \sum_{k=2}^N \sum_{j=1}^{k-1} (\mathbf{F}_{jk} \cdot \mathbf{r}_k + \mathbf{F}_{kj} \cdot \mathbf{r}_j) \\
 &= \sum_{k=2}^N \sum_{j=1}^{k-1} (\mathbf{F}_{jk} \cdot \mathbf{r}_k - \mathbf{F}_{jk} \cdot \mathbf{r}_j) =
    \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot (\mathbf{r}_k - \mathbf{r}_j),
     

\end{align}

where we have used Newton's third law of motion, i.e., $\backslash mathbf\{F\}\_\{jk\}=-\backslash mathbf\{F\}\_\{kj\}$ (equal and opposite reaction).

It often happens that the forces can be derived from a potential energy $V\_\{jk\}$ that is a function only of the distance $r\_\{jk\}$ between the point particles $j$ and $k$. Since the force is the negative gradient of the potential energy, we have in this case

$$$$

\mathbf{F}_{jk} = -\nabla_{\mathbf{r}_k} V_{jk} =
-\frac{dV_{jk}}{dr_{jk}} \left(\frac{\mathbf{r}_k - \mathbf{r}_j}{r_{jk}}\right),
     

which is equal and opposite to $\backslash mathbf\{F\}\_\{kj\}=-\backslash nabla\_\{\backslash mathbf\{r\}\_j\}V\_\{kj\}=-\backslash nabla\_\{\backslash mathbf\{r\}\_j\}V\_\{jk\}$, the force applied by particle $k$ on particle $j$, as may be confirmed by explicit calculation. Hence,

$$\backslash begin\{align\}$$

\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
 &= \sum_{k=2}^N \sum_{j=1}^{k-1} \mathbf{F}_{jk} \cdot (\mathbf{r}_k - \mathbf{r}_j) \\
 &= -\sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} \frac{|\mathbf{r}_k - \mathbf{r}_j|^2}{r_{jk}} \\
 & =-\sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} r_{jk}.
     

\end{align}

Thus

$$$$

\frac{dG}{dt} = 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
2 T - \sum_{k=2}^N \sum_{j=1}^{k-1} \frac{dV_{jk}}{dr_{jk}} r_{jk}.
     

---

Special case of power-law forces In a common special case, the potential energy $V$ between two particles is proportional to a power $n$ of their distance $r\_\{ij\}$:

$$$$

V_{jk} = \alpha r_{jk}^n,
     

where the coefficient $\backslash alpha$ and the exponent $n$ are constants. In such cases, the virial is

$$\backslash begin\{align\}$$

-\frac12\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k
 &= \frac12\,\sum_{k=1}^N \sum_{j
\end{align}
     

where
     

     $$$$
V_\text{TOT} = \sum_{k=1}^N \sum_{j
     
     

is the total potential energy of the system.
     

Thus
     

     $$$$
\frac{dG}{dt} = 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = 2 T - n V_\text{TOT}.
     
     
     

For gravitating systems the exponent $n=-1$, giving  Lagrange's identity
     

     $$$$
\frac{dG}{dt} = \frac12 \frac{d^2 I}{dt^2} = 2 T + V_\text{TOT},
     
     
     

which was derived by Joseph-Louis Lagrange and extended by Carl Jacobi.
     

 

 
 
  

  Time averaging
 
 

 
 
The average of this derivative over a duration $\backslash tau$ is defined as
     

     $$$$
\left\langle \frac{dG}{dt} \right\rangle_\tau = \frac{1}{\tau} \int_0^\tau \frac{dG}{dt} \,dt = \frac{1}{\tau} \int_{G(0)}^{G(\tau)} \,dG = \frac{G(\tau) - G(0)}{\tau},
     
     
     

from which we obtain the exact equation
     

     $$$$
\left\langle \frac{dG}{dt} \right\rangle_\tau =
2 \langle T \rangle_\tau + \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
     
     
     

The  virial theorem states that if $\backslash langle\; dG/dt\backslash rangle\_\backslash tau=0$, then
     

     $$$$
2 \langle T \rangle_\tau = -\sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau.
     
     
     

There are many reasons why the average of the time derivative might vanish. One often-cited reason applies to stably bound systems, that is, to systems that hang together forever and whose parameters are finite. In this case, velocities and coordinates of the particles of the system have upper and lower limits, so that $G^\{\backslash text\{bound\}\}$ is bounded between two extremes, $G\_\{\backslash text\{min\}\}$ and $G\_\{\backslash text\{max\}\}$, and the average goes to zero in the limit of infinite $\backslash tau$:
     

     $$$$
\lim_{\tau \to \infty} \left| \left\langle \frac{dG^{\text{bound}}}{dt} \right\rangle_\tau \right| =
\lim_{\tau \to \infty} \left| \frac{G(\tau) - G(0)}{\tau} \right| \le
\lim_{\tau \to \infty} \frac{G_\max - G_\min}{\tau} = 0.
     
     
     

Even if the average of the time derivative of $G$ is only approximately zero, the virial theorem holds to the same degree of approximation.
     

For power-law forces with an exponent $n$, the general equation holds:
     

     $$$$
\langle T \rangle_\tau = -\frac12 \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau
= \frac{n}{2} \langle V_\text{TOT} \rangle_\tau.
     
     
     

For  attraction, $n=-1$, and the average kinetic energy equals half of the average negative potential energy:
     

     $$$$
\langle T \rangle_\tau = -\frac12 \langle V_\text{TOT} \rangle_\tau.
     
     
     

This general result is useful for complex gravitating systems such as  or galaxy.
     

A simple application of the virial theorem concerns galaxy clusters. If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. Doppler effect measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter.
     

If the Ergodicity holds for the system under consideration, the averaging need not be taken over time; an ensemble average can also be taken, with equivalent results.
     

 

 
 
  

  In quantum mechanics
 
 

 
 
Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Vladimir Fock using the Ehrenfest theorem.
     

Evaluate the commutator of the Hamiltonian
     

     $$$$
H = V\bigl(\{X_i\}\bigr) + \sum_n \frac{P_n^2}{2m_n}
     
     
     

with the position operator $X\_n$ and the momentum operator
     

     $$$$
P_n = -i\hbar \frac{d}{dX_n}
     
     
     

of particle $n$,
     

     $$$$
[H, X_n P_n] = X_n [H, P_n] + [H, X_n] P_n = i\hbar X_n \frac{dV}{dX_n} - i\hbar\frac{P_n^2}{m_n}.
     
     
     

Summing over all particles, one finds that for
     

     $$$$
Q = \sum_n X_n P_n
     
     
     

the commutator is
     

     $$$$
\frac{i}{\hbar} [H, Q] = 2 T - \sum_n X_n \frac{dV}{dX_n},
     
     
     

where $T\; =\; \backslash sum\_n\; P\_n^2/2m\_n$ is the kinetic energy. The left-hand side of this equation is just
     $dQ/dt$, according to the Heisenberg equation of motion. The expectation value $\backslash langle\; dQ/dt\backslash rangle$ of this time derivative vanishes in a stationary state, leading to the  quantum virial theorem:
     

     $$$$
2\langle T\rangle = \sum_n \left\langle X_n \frac{dV}{dX_n}\right\rangle.
     
     
     

 

 
 
  

  Pokhozhaev's identity
 
 

 
 
In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary nonlinear Schrödinger equation or Klein–Gordon equation, is Pokhozhaev's identity, also known as Derrick's theorem.
Let $g(s)$ be continuous and real-valued, with $g(0)\; =\; 0$.
     

Denote $G(s)\; =\; \backslash int\_0^s\; g(t)\backslash ,dt$. Let
     

     $$$$
u \in L^\infty_{\text{loc}}(\R^n), \quad
\nabla u \in L^2(\R^n), \quad
G(u(\cdot)) \in L^1(\R^n), \quad
n \in \N
     
     
     

be a solution to the equation
     

     $$$$
-\nabla^2 u = g(u),
     
     
     

in the sense of distributions.
Then $u$ satisfies the relation
     

     $$$$
\left(\frac{n - 2}{2}\right) \int_{\R^n} |\nabla u(x)|^2 \,dx = n \int_{\R^n} G\big(u(x)\big) \,dx.
     
     
     

 

 
 
  

  In special relativity
 
 

 
 
For a single particle in special relativity, it is not the case that
     $T=\backslash frac\{1\}\{2\}\backslash mathbf\{p\}\backslash cdot\; \backslash mathbf\{v\}$. Instead, it is true that $T=(\backslash gamma-1)mc^2$, where $\backslash gamma$ is the Lorentz factor
     

     $$$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}},
     
     
     

and $\backslash mathbf\backslash beta\; =\; \backslash frac\{\backslash mathbf\{v\}\}\{c\}$. We have
     

     $$\backslash begin\{align\}$$
\frac 12 \mathbf{p} \cdot \mathbf{v}
 &= \frac 12 \boldsymbol{\beta} \gamma mc \cdot \boldsymbol{\beta} c \\
 &= \frac 12 \gamma \beta^2 mc^2 \\[5pt]
 &= \left(\frac{\gamma \beta^2}{2(\gamma - 1)}\right) T.
     
\end{align}
     

The last expression can be simplified to
     

     $$$$
\left(\frac{1 + \sqrt{1 - \beta^2}}{2}\right) T = \left(\frac{\gamma + 1}{2 \gamma}\right) T.
     
     
     

Thus, under the conditions described in earlier sections (including Newton's third law of motion,
     $\backslash mathbf\{F\}\_\{jk\}\; =\; -\backslash mathbf\{F\}\_\{kj\}$, despite relativity), the time average for $N$ particles with a power law potential is
     

     $$$$
\frac{n}{2} \left\langle V_\text{TOT} \right\rangle_\tau =
\left\langle \sum_{k=1}^N \left(\tfrac{1 + \sqrt{1 - \beta_k^2}}{2}\right) T_k \right\rangle_\tau =
\left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau.
     
     
     

In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
     

     $$$$
\frac{2 \langle T_\text{TOT} \rangle}{n \langle V_\text{TOT} \rangle} \in [1, 2],
     
     
     

where the more relativistic systems exhibit the larger ratios.
     

 

 
 
  

  Examples
 
 

 
The virial theorem has a particularly simple form for periodic motion. It can be used to perform perturbative calculation for nonlinear oscillators.
     

It can also be used to study motion in a central potential. If the central potential is of the form $U\; \backslash propto\; r^n$, the virial theorem simplifies to $\backslash langle\; T\; \backslash rangle=\; \backslash frac\{n\}\{2\}\; \backslash langle\; U\; \backslash rangle$. In particular, for gravitational or electrostatic (Coulomb law) attraction, $\backslash langle\; T\; \backslash rangle=\; -\backslash frac\{1\}2\; \backslash langle\; U\; \backslash rangle$.
     

 

 
 
  

  Driven damped harmonic oscillator
 
 

 
 
Analysis based on Sivardiere, 1986. For a one-dimensional oscillator with mass $m$, position $x$, driving force $F\backslash cos(\backslash omega\; t)$, spring constant $k$, and damping coefficient $\backslash gamma$, the equation of motion is
     

     $$$$
m \underbrace{\frac{d^2x}{dt^2}}_{\text{acceleration}} = \underbrace{-kx \vphantom{\frac dd}}_\text{spring}\ \underbrace{-\ \gamma \frac{dx}{dt}}_\text{friction}\ \underbrace{+\ F\cos(\omega t) \vphantom{\frac dd}}_\text{external driving}.
     
     
     

When the oscillator has reached a steady state, it performs a stable oscillation $x\; =\; X\backslash cos(\backslash omega\; t\; +\; \backslash varphi)$, where $X$ is the amplitude, and $\backslash varphi$ is the phase angle.
     

Applying the virial theorem, we have $m\; \backslash langle\; \backslash dot\; x\; \backslash dot\; x\; \backslash rangle\; =\; k\backslash langle\; xx\; \backslash rangle\; +\; \backslash gamma\; \backslash langle\; x\backslash dot\; x\; \backslash rangle\; -\; F\; \backslash langle\; \backslash cos(\backslash omega\; t)\; x\; \backslash rangle$, which simplifies to $F\backslash cos(\backslash varphi)\; =\; m(\backslash omega\_0^2\; -\; \backslash omega^2)X$, where $\backslash omega\_0\; =\; \backslash sqrt\{k/m\}$ is the natural frequency of the oscillator.
     

To solve the two unknowns, we need another equation. In steady state, the power lost per cycle is equal to the power gained per cycle:
     

     $$$$
\underbrace{\langle \dot x \, \gamma \dot x\rangle}_\text{power dissipated} =
\underbrace{\langle \dot x \, F \cos \omega t \rangle}_\text{power input},
     
     
     

which simplifies to $\backslash sin\; \backslash varphi\; =\; -\backslash frac\{\backslash gamma\; X\; \backslash omega\}\{F\}$.
     

Now we have two equations that yield the solution
     

     $$\backslash begin\{cases\}$$
X = \sqrt{\dfrac{F^2}{\gamma^2 \omega^2 + m^2 (\omega_0^2 - \omega^2)^2}}, \\
\tan\varphi = -\dfrac{\gamma \omega}{m(\omega_0^2 - \omega^2)}.
     
\end{cases}
     

 

 
 
  

  Ideal-gas law
 
 

 
Consider a container filled with an ideal gas consisting of point masses. The only forces applied to the point masses are due to the container walls. In this case, the expression in the virial theorem equals
     

     $$$$
\Big \langle \sum_i \mathbf{F}_i \cdot \mathbf{r}_i \Big \rangle =
     
- P \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA,
     
     

since, by definition, the pressure  P is the average force per area exerted by the gas upon the walls, which is normal to the wall. There is a minus sign because $\backslash hat\{\backslash mathbf\{n\}\}$ is the unit normal vector pointing outwards, and the force to be used is the one upon the particles by the wall.
     

 
Then the virial theorem states that
     

     $$$$
\langle T \rangle =
\frac{P}{2} \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA.
     
     
     

By the divergence theorem, $\backslash oint\; \backslash hat\{\backslash mathbf\{n\}\}\; \backslash cdot\; \backslash mathbf\{r\}\; \backslash ,dA\; =\; \backslash int\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{r\}\; \backslash ,dV\; =\; 3\; \backslash int\; dV\; =\; 3V$.
     

From equipartition, the average total kinetic energy $\backslash langle\; T\; \backslash rangle\; =\; N\; \backslash big\backslash langle\; \backslash frac\; 12\; mv^2\; \backslash big\backslash rangle\; =\; N\; \backslash cdot\; \backslash frac\; 32\; kT$. Hence, $PV\; =\; NkT$, the ideal gas law.
     

 

 
 
  

  Dark matter
 
 

 
In 1933, Fritz Zwicky applied the virial theorem to estimate the mass of Coma Cluster, and discovered a discrepancy of mass of about 450, which he explained as due to "dark matter". He refined the analysis in 1937, finding a discrepancy of about 500.
     

 

 
 
  

  Theoretical analysis
 
 

 
He approximated the Coma cluster as a spherical "gas" of $N$ stars of roughly equal mass $m$, which gives $\backslash langle\; T\; \backslash rangle=\; \backslash frac\; 12\; Nm\; \backslash langle\; v^2\; \backslash rangle$. The total gravitational potential energy of the cluster is , giving . Assuming the motion of the stars are all the same over a long enough time (ergodicity), $\backslash langle\; U\backslash rangle\; =\; -\backslash frac\{1\}2\; N^2\; Gm^2\backslash langle\; \{1\}/\{r\}\backslash rangle$.
     

Zwicky estimated $\backslash langle\; U\backslash rangle$ as the gravitational potential of a uniform ball of constant density, giving $\backslash langle\; U\backslash rangle\; =\; -\backslash frac\; 35\; \backslash frac\{GN^2m^2\}\{R\}$.
     

So by the virial theorem, the total mass of the cluster is
     

     $$Nm\; =\; \backslash frac\{5\backslash langle\; v^2\backslash rangle\}\{3G\backslash langle\; \backslash frac\{1\}\{r\}\backslash rangle\}$$
     

 

 
 
  

  Data
 
 

 
Zwicky$\_\{1933\}$ estimated that there are $N\; =\; 800$ galaxies in the cluster, each having observed stellar mass $m\; =\; 10^9\; M\_\{\backslash odot\}$ (suggested by Hubble), and the cluster has radius $R\; =\; 10^6\; \backslash text\{ly\}$. He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be $\backslash langle\; v\_r^2\backslash rangle\; =\; (1000\; \backslash text\{km/s\})^2$. Assuming equipartition of kinetic energy, $\backslash langle\; v^2\backslash rangle\; =\; 3\; \backslash langle\; v\_r^2\backslash rangle$.
     

By the virial theorem, the total mass of the cluster should be $\backslash frac\{5R\; \backslash langle\; v\_r^2\backslash rangle\}\{G\}\; \backslash approx\; 3.6\backslash times\; 10^\{14\}\; M\_\backslash odot$. However, the observed mass is $Nm\; =\; 8\; \backslash times\; 10^\{11\}\; M\_\backslash odot$, meaning the total mass is 450 times that of observed mass.
     

 

 
 
  

  Generalizations
 
 

 
 
Lord Rayleigh published a generalization of the virial theorem in 1900, which was partially reprinted in 1903. Henri Poincaré proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as cosmogony). A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of the virial theorem was developed by Parker, Chandrasekhar and Fermi. The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:
Harry Pollard (1966). 9780135610688, Prentice–Hall, Inc.. ISBN 9780135610688
     $$$$
2\lim_{\tau\to+\infty} \langle T\rangle_\tau =
\lim_{\tau\to+\infty} \langle U\rangle_\tau \quad \text{if and only if} \quad \lim_{\tau\to+\infty}{\tau}^{-2}I(\tau) = 0.
     
     
A  boundary term otherwise must be added.
     

 

 
 
  

  Inclusion of electromagnetic fields
 
 

 
 
The virial theorem can be extended to include electric and magnetic fields. The result is
     

     $$\backslash frac12\backslash frac\{d^2I\}\{dt^2\}\; +\; \backslash int\_Vx\_k\backslash frac\{\backslash partial\; G\_k\}\{\backslash partial\; t\}\; \backslash ,\; d^3r\; =\; 2(T+U)\; +\; W^\backslash mathrm\{E\}\; +\; W^\backslash mathrm\{M\}\; -\; \backslash int\; x\_k(p\_\{ik\}+T\_\{ik\})\; \backslash ,\; dS\_i,$$
     

where $I$ is the moment of inertia, $G$ is the Poynting vector, $T$ is the kinetic energy of the "fluid", $U$ is the random "thermal" energy of the particles,
     $W^\{\backslash text\{E\}\}$ and $W^\{\backslash text\{M\}\}$ are the electric and magnetic energy content of the volume considered. Finally,
     $p\_\{ik\}$ is the fluid-pressure tensor expressed in the local moving coordinate system
     

     $$p\_\{ik\}\; =\; \backslash Sigma\; n^\backslash sigma\; m^\backslash sigma\; \backslash langle\; v\_iv\_k\backslash rangle^\backslash sigma\; -\; V\_iV\_k\backslash Sigma\; m^\backslash sigma\; n^\backslash sigma,$$
     

and $T\_\{ik\}$ is the electromagnetic stress tensor,
     

     $$T\_\{ik\}\; =\; \backslash left(\; \backslash frac\{\backslash varepsilon\_0E^2\}\{2\}\; +\; \backslash frac\{B^2\}\{2\backslash mu\_0\}\; \backslash right)\; \backslash delta\_\{ik\}\; -\; \backslash left(\; \backslash varepsilon\_0E\_iE\_k\; +\; \backslash frac\{B\_iB\_k\}\{\backslash mu\_0\}\; \backslash right).$$
     

A plasmoid is a finite configuration of magnetic fields and plasma. With the virial theorem it is easy to see that any such configuration will expand if not contained by external forces. In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish. Since all the other terms on the right hand side are positive, the acceleration of the moment of inertia will also be positive. It is also easy to estimate the expansion time $\backslash tau$. If a total mass $M$ is confined within a radius $R$, then the moment of inertia is roughly $MR^2$, and the left hand side of the virial theorem is $\backslash frac\{MR^2\}\{\backslash tau^2\}$. The terms on the right hand side add up to about $pR^3$, where $p$ is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for $\backslash tau$, we find
     

     $$\backslash tau\backslash ,\backslash sim\; \backslash frac\{R\}\{c\_\backslash mathrm\{s\}\},$$
     

where $c\_s$ is the speed of the ion acoustic wave (or the Alfvén wave, if the magnetic pressure is higher than the plasma pressure). Thus the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time.
     

 

 
 
  

  Relativistic uniform system
 
 

 
In case when in the physical system the pressure field, the electromagnetic and gravitational fields are taken into account, as well as the field of particles’ acceleration, the virial theorem is written in the relativistic form as follows:
     

     $$\backslash left\backslash langle\; W\_k\; \backslash right\backslash rangle\; \backslash approx\; -\; 0.6\; \backslash sum\_\{k=1\}^N\backslash langle\backslash mathbf\{F\}\_k\backslash cdot\backslash mathbf\{r\}\_k\backslash rangle\; ,$$
     

where the value $W\_k=\backslash gamma\_c\; T$ exceeds the kinetic energy of the particles $T$ by a factor equal to the Lorentz factor
     $\backslash gamma\_c$ of the particles at the center of the system. Under normal conditions we can assume that
     $\backslash gamma\_c\backslash approx\; 1$, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient $\backslash frac\{1\}\{2\}$, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar $G$ is not equal to zero and should be considered as the material derivative.
     

An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature:
     

     $$v\_\backslash mathrm\{rms\}\; =\; c\; \backslash sqrt\{1-\; \backslash frac\; \{4\; \backslash pi\; \backslash eta\; \backslash rho\_0\; r^2\}\{c^2\; \backslash gamma^2\_c\; \backslash sin^2\; \backslash left(\; \backslash frac\; \{r\}\{c\}\; \backslash sqrt\; \{4\; \backslash pi\; \backslash eta\; \backslash rho\_0\}\; \backslash right)\; \}\; \}\; ,$$
     

where $~\; c$ is the speed of light, $~\; \backslash eta$ is the acceleration field constant, $~\; \backslash rho\_0$ is the mass density of particles, $~\; r$ is the current radius.
     

Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:
     

     $$~\; E\_\{kf\}\; +\; 2\; W\_f\; =0\; ,$$
     

where the energy $~\; E\_\{kf\}\; =\; \backslash int\; A\_\backslash alpha\; j^\backslash alpha\; \backslash sqrt\; \{-g\}\; \backslash ,dx^1\; \backslash ,dx^2\; \backslash ,dx^3$ considered as the kinetic field energy associated with four-current $j^\backslash alpha$, and
     

     $$~\; W\_f\; =\; \backslash frac\; \{1\}\{4\; \backslash mu\_0\; \}\; \backslash int\; F\_\{\backslash alpha\; \backslash beta\}\; F^\{\backslash alpha\; \backslash beta\}\; \backslash sqrt\; \{-g\}\; \backslash ,dx^1\; \backslash ,dx^2\; \backslash ,dx^3$$
     

sets the potential field energy found through the components of the electromagnetic tensor.
     

 

 
 
  

  In astrophysics
 
 

 
The virial theorem is frequently applied in astrophysics, especially relating the gravitational potential energy of a system to its kinetic energy or thermal energy. Some common virial relations are
     $$\backslash frac35\; \backslash frac\{GM\}\{R\}\; =\; \backslash frac32\; \backslash frac\{k\_\backslash mathrm\{B\}\; T\}\{m\_\backslash mathrm\{p\}\}\; =\; \backslash frac12\; v^2$$
for a mass $M$, radius $R$, velocity $v$, and temperature $T$. The constants are Newton's constant $G$, the Boltzmann constant $k\_B$, and proton mass $m\_p$. Note that these relations are only approximate, and often the leading numerical factors (e.g. $\backslash frac\{3\}\{5\}$ or $\backslash frac\{1\}\{2\}$) are neglected entirely.
     

 

 
 
  

  Galaxies and cosmology (virial mass and radius)
 
 

 
 
In astronomy, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "virial mass" and "virial radius" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an isothermal sphere), it can be hard to define specific, finite measures of their mass and size. The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties.
     

In galaxy dynamics, the mass of a galaxy is often inferred by measuring the rotation velocity of its gas and stars, assuming circular orbit. Using the virial theorem, the velocity dispersion $\backslash sigma$ can be used in a similar way. Taking the kinetic energy (per particle) of the system as $T=\backslash frac\{1\}\{2\}v^2\backslash sim\; \backslash frac\{3\}\{2\}\backslash sigma^2$, and the potential energy (per particle) as $U\backslash sim\; \backslash frac\{3\}\{5\}\backslash frac\{GM\}\{R\}$ we can write
     

     $$\backslash frac\{GM\}\{R\}\; \backslash approx\; \backslash sigma^2.$$
     

Here $R$ is the radius at which the velocity dispersion is being measured, and $M$ is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e.
     

     $$\backslash frac\{GM\_\backslash text\{vir\}\}\{R\_\backslash text\{vir\}\}\; \backslash approx\; \backslash sigma\_\backslash max^2.$$
     

As numerous approximations have been made, in addition to the approximate nature of these definitions, order-unity proportionality constants are often omitted (as in the above equations). These relations are thus only accurate in an order of magnitude sense, or when used self-consistently.
     

An alternate definition of the virial mass and radius is often used in cosmology where it is used to refer to the radius of a sphere, centered on a galaxy or a galaxy cluster, within which virial equilibrium holds. Since this radius is difficult to determine observationally, it is often approximated as the radius within which the average density is greater, by a specified factor, than the critical density
     $$\backslash rho\_\backslash text\{crit\}=\backslash frac\{3H^2\}\{8\backslash pi\; G\}$$
where $H$ is the Hubble parameter and $G$ is the gravitational constant. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see Virial mass), in which case the virial radius is approximated as
     

     $$r\_\backslash text\{vir\}\; \backslash approx\; r\_\{200\}=\; r,\; \backslash qquad\; \backslash rho\; =\; 200\; \backslash cdot\; \backslash rho\_\backslash text\{crit\}.$$
     

The virial mass is then defined relative to this radius as
     

     $$M\_\backslash text\{vir\}\; \backslash approx\; M\_\{200\}\; =\; \backslash frac43\backslash pi\; r\_\{200\}^3\; \backslash cdot\; 200\; \backslash rho\_\backslash text\{crit\}\; .$$
     

 

 
 
  

  Stars
 
 

 
The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the main sequence convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to support its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative specific heat.
 (2010). 9788120340718, PHI Learning Pvt. Ltd.. . ISBN 9788120340718 This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with $n=-1$ no longer holds.
William K. Rose (1998). 9780521588331, Cambridge University Press. . ISBN 9780521588331
     

 

 
 
  

  See also
 
 

 
 

  

Virial coefficient
  
  

     Virial stress
  
  

     Virial mass
  
  

Chandrasekhar tensor
  
  

Chandrasekhar virial equations
  
  

Derrick's theorem
  
  

Equipartition theorem
  
  

Ehrenfest theorem
  
  

Pokhozhaev's identity
  
  

Statistical mechanics
  
 

 
 
 

 
 
  

  Further reading
 
 

 
 

  

   
H. Goldstein (1980). 9780201029185, Addison–Wesley. ISBN 9780201029185
  
  

   
G. W. Collins (1978). 9780912918136, Pachart Press. . ISBN 9780912918136
  
 

 

 
 
  

  External links
 
 

 
 

  

      The Virial Theorem at MathPages
  
  

        Gravitational Contraction and Star Formation, Georgia State University
  
 

 
Categories: Physics Theorems, Dynamics, Solid Mechanics, Concepts In Physics, Equations Of Astronomy

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