Virial theorem

In mechanics, the virial theorem provides a general equation relating 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

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

where F_k represents the force on the kth particle, which is located at position r_k. The word "virial" derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Clausius in 1870.^[1]

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 adopts a 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 of the virial theorem

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 his "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.^[2] The theorem was later utilized, popularized, generalized and further developed by persons such as 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

Definitions of the virial and its time derivative

For a collection of N point particles, the scalar moment of inertia I about 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 virial 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, the virial 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 the virial 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 force term of the virial time derivative can be written

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

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

$\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = \sum_{k=1}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \mathbf{r}_k + \sum_{k=1}^N \sum_{j>k} \mathbf{F}_{jk} \cdot \mathbf{r}_k = \sum_{k=1}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \left( \mathbf{r}_k - \mathbf{r}_j \right).$^[3]

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, the force term of the virial time derivative is

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

Thus, we have

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

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 force term of the virial time derivative is given by the equation

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

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 and also for electrostatic 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 and the virial theorem

The average of this derivative over a time τ is defined as

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

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 $\left\langle \frac{dG}{dt} \right\rangle_\tau = 0$, 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., $\left\langle \frac{dG}{dt} \right\rangle_{\tau} = 0$. One often-cited reason applies to stable 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 the virial 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 $\left\langle \frac{dG}{dt} \right\rangle_{\tau} \approx 0$ 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.

Although derived for classical mechanics, the virial theorem also holds for quantum mechanics, which was proved by Fock^[4] (the quantum equivalent of the l.h.s. $\left\langle \frac{dG}{dt} \right\rangle_\tau$ vanishes for energy eigenstates).

The virial theorem and 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 $T = (\gamma - 1) mc^2\,$ and

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

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 of the virial theorem

Lord Rayleigh published a generalization of the virial theorem in 1903.^[5] Henri Poincaré applied a form of the virial theorem in 1911 to the problem of determining cosmological stability.^[6] A variational form of the virial theorem was developed in 1945 by Ledoux.^[7] A tensor form of the virial theorem was developed by Parker,^[8] Chandrasekhar^[9] and Fermi.^[10] The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law^{[11] [12]}: 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.^[13]

Inclusion of electromagnetic fields

The virial theorem can be extended to include electric and magnetic fields. The result is^[14]

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

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.

Virial radius

In astronomy, the term virial radius 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 $\rho_\text{crit}=\frac{3H^2}{8\pi G}$. (Here, H is the Hubble parameter and G is the gravitational constant.) A common choice for the factor is 200, in which case the virial radius is approximated as r₂₀₀.

See also

• Virial stress
• Virial mass
• Equipartition theorem

References

1. ^ Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4 40: 122–127. 
2. ^ Collins, G. W. (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press. Introduction
3. ^ Proof of this equation
4. ^ Fock, V. (1930). "Bemerkung zum Virialsatz". Zeitschrift für Physik A 63 (11): 855–858. Bibcode 1930ZPhy...63..855F. doi:10.1007/BF01339281. 
5. ^ Lord Rayleigh (1903). Unknown. 
6. ^ Poincaré, H. Lectures on Cosmological Theories. Paris: Hermann. 
7. ^ Ledoux, P. (1945). "On the Radial Pulsation of Gaseous Stars". Ap. J. 102: 143–153. Bibcode 1945ApJ...102..143L. doi:10.1086/144747. 
8. ^ Parker, E.N. (1954). "Tensor Virial Equations". Physical Review 96 (6): 1686–1689. Bibcode 1954PhRv...96.1686P. doi:10.1103/PhysRev.96.1686. 
9. ^ Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids". Ap. J. 136: 1037–1047. Bibcode 1962ApJ...136.1037C. doi:10.1086/147456. 
10. ^ Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field". Ap. J. 118: 116. Bibcode 1953ApJ...118..116C. doi:10.1086/145732. 
11. ^ Pollard, H. (1964). "A sharp form of the virial theorem". Bull. Amer. Math. Soc. LXX (5): 703–705. doi:10.1090/S0002-9904-1964-11175-7. 
12. ^ Pollard, Harry (1966). Mathematical Introduction to Celestial Mechanics. Englewood Cliffs, NJ: Prentice–Hall, Inc. 
13. ^ Kolár, M.; O'Shea, S. F. (1996). "A high-temperature approximation for the path-integral quantum Monte Carlo method". Journal of Physics A: Mathematical and General 29 (13): 3471. Bibcode 1996JPhA...29.3471K. doi:10.1088/0305-4470/29/13/018.  edit
14. ^ Schmidt, George (1979). Physics of High Temperature Plasmas (Second ed.). Academic Press. pp. 72. 

Further reading

• Goldstein, H. (1980). Classical Mechanics (2nd ed.). Addison–Wesley. ISBN 0201029189 
• Collins, G. W. (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press. http://ads.harvard.edu/books/1978vtsa.book/ 

External links

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