Virial theorem

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

$\left\langle T \right\rangle = -\frac{1}{2}\,\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle$

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.

If the force between any two particles of the system results from a potential energy V(r) = αr ⁿ that is proportional to some power n of the inter-particle 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 $\left\langle T \right\rangle$ equals n times the average total potential energy $\left\langle V_\text{TOT} \right\rangle$. Whereas V(r) represents the potential energy between two particles, V_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 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.

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 equal to 1/2 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. Karl 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. The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux 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. As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars.

Statement and derivation

The moment of inertia of a particle is dependent on its perpendicular distance from a particular axis of rotation. r is the scalar distance from the axis, not from the origin. The calculation of the moment is independent from the speed of the particle, which can be zero.

For a collection of N point particles, the scalar moment of inertia I about an arbitrary axis passing through the origin is defined by the equation

$I = \sum_{k=1}^{N} m_{k} |\mathbf{r}_{k}|^{2} = \sum_{k=1}^{N} m_{k} r_{k}^{2}$

where m_k and r_k represent the mass and position of the kth particle. r_k=|r_k| is the position vector magnitude. The scalar G is defined by the equation

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

where p_k is the momentum vector of the kth particle. Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia

$\frac{1}{2} \frac{dI}{dt} = \frac{1}{2} \frac{d}{dt} \sum_{k=1}^N m_{k} \, \mathbf{r}_k \cdot \mathbf{r}_k = \sum_{k=1}^N m_{k} \, \frac{d\mathbf{r}_k}{dt} \cdot \mathbf{r}_k = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k = G\,.$

In turn, the time derivative of G can be written

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

where m_k is the mass of the k-th particle, $\mathbf{F}_k = \frac{d\mathbf{p}_k}{dt}$ is the net force on that particle, and T is the total kinetic energy of the system

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

Connection with the potential energy between particles

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

$-\frac{1}{2}\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = -\frac{1}{2}\,\sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k.$

Since no particle acts on itself (i.e., $\mathbf{F}_{jj} = 0$ for all $j$), we have

$\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = \sum_{k=2}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{k=1}^{N-1} \sum_{j>k} \mathbf{F}_{jk} \cdot \mathbf{r}_k = \sum_{k=2}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \left( \mathbf{r}_k - \mathbf{r}_j \right).$

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

It often happens that the forces can be derived from a potential energy V 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

$\mathbf{F}_{jk} = -\nabla_{\mathbf{r}_k} V = - \frac{dV}{dr} \left( \frac{\mathbf{r}_k - \mathbf{r}_j}{r_{jk}} \right),$

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

$\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = \sum_{k=2}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \left( \mathbf{r}_k - \mathbf{r}_j \right) = -\sum_{k=2}^N \sum_{j<k} \frac{dV}{dr} \frac{| \mathbf{r}_k - \mathbf{r}_j |^2}{r_{jk}} = -\sum_{k=2}^N \sum_{j<k} \frac{dV}{dr} r_{jk}.$

Thus, we have

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

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

$V(r_{jk}) = \alpha r_{jk}^n,$

where the coefficient α and the exponent n are constants. In such cases, the virial is given by the equation

$-\frac{1}{2}\,\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = \frac{1}{2}\,\sum_{k=1}^N \sum_{j<k} \frac{dV}{dr} r_{jk} = \frac{1}{2}\,\sum_{k=1}^N \sum_{j<k} n \alpha r_{jk}^{n-1} r_{jk} = \frac{1}{2}\,\sum_{k=1}^N \sum_{j<k} n V(r_{jk}) = \frac{n}{2}\, V_\text{TOT}$

where V_TOT is the total potential energy of the system

$V_\text{TOT} = \sum_{k=1}^N \sum_{j<k} V(r_{jk}).$

Thus, we have

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

For gravitating systems the exponent n equals −1, giving Lagrange's identity

$\frac{dG}{dt} = \frac{1}{2} \frac{d^2 I}{dt^2} = 2 T + V_\text{TOT}$

which was derived by Lagrange and extended by Jacobi.

Time averaging

The average of this derivative over a time, τ, 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

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

The virial theorem states that, if $\left\langle {dG}/{dt} \right\rangle_\tau = 0$, then

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

There are many reasons why the average of the time derivative might vanish, i.e., $\left\langle {dG}/{dt} \right\rangle_{\tau} = 0$. One often-cited reason applies to stably bound systems, i.e., systems that hang together forever and whose parameters are finite. In that case, velocities and coordinates of the particles of the system have upper and lower limits so that G^bound, is bounded between two extremes, G_min and G_max, and the average goes to zero in the limit of very long times τ

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

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

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

For gravitational attraction, n equals −1 and the average kinetic energy equals half of the average negative potential energy

$\langle T \rangle_\tau = -\frac{1}{2} \langle V_\text{TOT} \rangle_\tau.$

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

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 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.

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 Fock.

Evaluate the commutator of the Hamiltonian $H=V(\{X_i\})+\sum_n P_n^2/2m$ and the product $X_n P_n$ of the position operator $X_n$ and the momentum operator $P_n=-i\hbar d/dX_n$ of particle $n$:

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

so summing over all particles one finds for $Q=\sum_n X_nP_n$ the commutator

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

with kinetic energy $T=\sum_n P_n^2/2m$. The left-hand side of this equation is just $-dQ/dt$, according to the Heisenberg equation of motion. The expectation value $\langle dQ/dt\rangle$ of that time derivative vanishes in a stationary state, hence we obtain the quantum virial theorem

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

In special relativity

For a single particle in special relativity, it is not the case that $T = \frac 12 \mathbf{p} \cdot \mathbf{v}$. Instead, it is true that $T = (\gamma - 1) mc^2\,$, where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ and $\vec{\beta} = \mathbf{v}/c$. We have,

$\begin{align} \frac 12 \mathbf{p} \cdot \mathbf{v} & = \frac 12 \vec{\beta} \gamma mc \cdot \vec{\beta} c = \frac 12 \gamma \beta^2 mc^2 = \left( \frac{\gamma \beta^2}{2(\gamma-1)}\right) T \,.\end{align}$

The last expression can be simplified to either $\left(\frac{1 + \sqrt{1-\beta^2}}{2}\right) T$ or $\left(\frac{\gamma + 1}{2 \gamma}\right) T$.

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

$\frac n2 \langle V_\mathrm{TOT} \rangle_\tau = \left\langle \sum_{k=1}^N \left(\frac{1 + \sqrt{1-\beta_k^2}}{2}\right) T_k \right\rangle_\tau = \left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau \,.$

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

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

where the more relativistic systems exhibit the larger ratios.

Generalizations

Lord Rayleigh published a generalization of the virial theorem in 1903. Henri Poincaré applied a form of the virial theorem in 1911 to the problem of determining cosmological stability. 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: the statement $2\lim\limits_{\tau\rightarrow+\infty}\langle T\rangle_\tau = \lim\limits_{\tau\rightarrow+\infty}\langle U\rangle_\tau$ is true if and only if $\lim\limits_{\tau\rightarrow+\infty}{\tau}^{-2}I(\tau)=0.$ A boundary term otherwise must be added, such as in Ref.

Inclusion of electromagnetic fields

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

$\frac{1}{2}\frac{d^2I}{dt^2} + \int_Vx_k\frac{\partial G_k}{\partial t} \, d^3r = 2(T+U) + W^E + W^M - \int x_k(p_{ik}+T_{ik}) \, dS_i,$

where I is the moment of inertia, G is the momentum density of the electromagnetic field, T is the kinetic energy of the "fluid", U is the random "thermal" energy of the particles, W^E and W^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

$p_{ik} = \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma - V_iV_k\Sigma m^\sigma n^\sigma,$

and T_ik is the electromagnetic stress tensor,

$T_{ik} = \left( \frac{\varepsilon_0E^2}{2} + \frac{B^2}{2\mu_0} \right) \delta_{ik} - \left( \varepsilon_0E_iE_k + \frac{B_iB_k}{\mu_0} \right).$

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 τ. If a total mass M is confined within a radius R, then the moment of inertia is roughly MR², and the left hand side of the virial theorem is MR²/τ². The terms on the right hand side add up to about pR³, where p is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for τ, we find

$\tau\,\sim R/c_s,$

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.

In astrophysics

The virial theorem is frequently applied in astrophysics, especially relating the gravitational potential energy of a system to its kinetic or thermal energy. Some common virial relations are,

$\frac{3}{5} \frac{GM}{R} = \frac{3}{2} \frac{k_B T}{m_p} = \frac{1}{2} v^2,$

for a mass $M$, radius $R$, velocity $v$, and temperature $T$. And 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. 3/5 or 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 radius" and "virial mass" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models—e.g. 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 Keplerian orbits. Using the virial theorem, the dispersion velocity $\sigma$ can be used in a similar way. Taking the kinetic energy (per particle) of the system as, $T = (1/2) v^2 \sim (3/2) \sigma^2$, and the potential energy (per particle) as, U ~ (3/5)(GM/R), we can write

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

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.

$\frac{GM_\text{vir}}{R_\text{vir}} \approx \sigma_\max^2.$

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, $\rho_\text{crit}=\frac{3H^2}{8\pi G}$. Where $H$ is the Hubble parameter and $G$ is the gravitational constant. A common (although mostly arbitrary) choice for the factor is 200, in which case the virial radius is approximated as $r_\text{vir} \approx r_{200}= r(\rho = 200 \cdot \rho_\text{crit})$. The virial mass is then defined relative to this radius as $M_\text{vir} \approx M_{200} = (4/3)\pi r_{200}^3 \cdot 200 \rho_\text{crit}$.

See also

• Virial coefficient
• Virial stress
• Virial mass
• Derrick's theorem (Pohozaev's identity)
• Equipartition theorem

External links

• The Virial Theorem at MathPages
• Gravitational Contraction and Star Formation, Georgia State University