Virial theorem

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In mechanics, the virial theorem provides a general equation relating the average total kinetic energy $\left\langle T \right\rangle$ of a system with its average total potential energy $\left\langle V_{TOT} \right\rangle$ , where angular brackets such as $\left\langle \ldots \right\rangle$ symbolize taking the average of the enclosed quantity. Mathematically, the virial 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 k^th particle, which is located at position r_k. 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. As one example of many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars. The word "virial" derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Clausius in 1870.^[1]

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

$2 \langle T \rangle = n \langle V_{TOT} \rangle$

Thus, twice the average total kinetic energy $\left\langle T \right\rangle$ equals n times the average total potential energy $\left\langle V_{TOT} \right\rangle$ . 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; please be patient!

1 Definitions of the virial and its time derivative
2 Connection with the potential energy between particles
3 Special case of power-law forces
4 Time averaging and the virial theorem
5 Extensions of the virial theorem
6 Inclusion of electromagnetic fields
7 References
8 Additional reading

[edit] 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 k^th particle. 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 k^th particle. Assuming that the masses are constant, the virial G is the time derivative of this moment of inertia

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

In turn, the time derivative of the virial G can be written

$\frac{dG}{dt} = \sum_{k=1}^{N} \mathbf{p}_{k} \cdot \frac{d\mathbf{r}_{k}}{dt} + \sum_{k=1}^{N} \frac{d\mathbf{p}_{k}}{dt} \cdot \mathbf{r}_{k}$

$= \sum_{k=1}^{N} m_{k} \frac{d\mathbf{r}_{k}}{dt} \cdot \frac{d\mathbf{r}_{k}}{dt} + \sum_{k=1}^{N} \mathbf{F}_{k} \cdot \mathbf{r}_{k}$

or, more simply,

$\frac{dG}{dt} = 2 T + \sum_{k=1}^{N} \mathbf{F}_{k} \cdot \mathbf{r}_{k}.$

Here $m k$ 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

$T = \frac{1}{2} \sum_{k=1}^{N} m_{k} v_{k}^{2} = \frac{1}{2} \sum_{k=1}^{N} m_{k} \frac{d\mathbf{r}_{k}}{dt} \cdot \frac{d\mathbf{r}_{k}}{dt}.$

[edit] 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

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

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 $r j k$ between the point particles $j$ and $k$ . Since the force is the 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

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

[edit] 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

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

where $V T O T$ is the total potential energy of the system

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

Thus, we have

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

For gravitating systems and also for electrostatic systems, the exponent n equals -1, giving the Lagrange's identity

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

which was derived by Lagrange and extended by Jacobi.

[edit] 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 bound systems, i.e., systems that hang together forever. In that case, the virial $G bound$ is usually 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

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

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

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

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

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.

[edit] Extensions of the virial theorem

Lord Rayleigh published a generalization of the virial theorem in 1903.^[2] Henri Poincaré applied a form of the virial theorem in 1911 to the problem of determining cosmological stability.^[3] A variational form of the virial theorem was developed in 1945 by Ledoux.^[4] A tensor form of the virial theorem was developed by Parker,^[5] Chandrasekhar^[6] and Fermi.^[7]

[edit] Inclusion of electromagnetic fields

The virial theorem can be extended to include electric and magnetic fields. The result is^[8]

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

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,

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

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 Alfven 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 Alfven) transit time.

[edit] References

^ Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4 40.
^ Lord Rayleigh (1903). "Unknown".
^ Poincare, H. Lectures on Cosmological Theories. Paris: Hermann.
^ Ledoux, P. (1945). "On the Radial Pulsation of Gaseous Stars". Ap. J. 102: 143–153.
^ Parker, E.N. (1954). "Tensor Virial Equations". Physical Review 96 (6): 1686–1689.
^ Chandrasekhar, S; Lebovitz NR (1962). "Unknown". Ap. J. 136: 1037–1047.
^ Chandrasekhar, S; Fermi E (1953). "Unknown". Ap. J. 118: 116.
^ George Schmidt, Physics of High Temperature Plasmas (Second edition), Academic Press (1979), p.72