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Virial theorem 

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Contents

[hide] 

  1 History 

  2 Statement and derivation 

o  2.1 Definitions of the virial and its time derivative 

o  2.2 Connection with the potential energy between particles 

o  2.3 Special case of power-law forces 

o  2.4 Time averaging   3 In special relativity 

  4 Generalizations 

  5 Inclusion of electromagnetic fields 

  6 In astrophysics 

o  6.1 Galaxies and cosmology (virial mass and radius) 

  7 See also 

  8 References 

  9 Further reading 

  10 External links 

In mechanics, the virial theorem provides a general equation that relates the average over timeof the total kinetic energy,  , of a stable system consisting of N  particles, bound by potential

forces, with that of the total potential energy, , where angle brackets represent the

average over time of the enclosed quantity. Mathematically, the theorem states

where Fk  represents the force on the k th particle, which is located at position rk . The word

"virial" derives from vis, the Latin word for "force" or "energy", and was given its technicaldefinition by Rudolf 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

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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

Thus, twice the average total kinetic energy equals n times the average total potential energy

. 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[edit]

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 tothe 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 ofthermodynamics and classical dynamics.

[2] 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 calleddark 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[edit]

Definitions of the virial and its time derivative[edit] 

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

the equation

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where mk  and rk  represent the mass and position of the k th particle. r k =|rk | is the position vector

magnitude. The scalar virial G is defined by the equation

where pk  is the momentum vector  of the k th particle. Assuming that the masses are constant, the

virial G is one-half the time derivative of this moment of inertia

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

where mk  is the mass of the k -th particle, is the net force on that particle, and T  is thetotal kinetic energy of the system

Connection with the potential energy between particles[edit]

The total force on particle k  is the sum of all the forces from the other particles j in the system

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where is the force applied by particle j on particle k . Hence, the force term of the virial time

derivative can be written

Since no particle acts on itself (i.e., whenever ), we have

[3] 

where we have assumed that  Newton's third law of motion holds, i.e., (equal and

opposite reaction).

It often happens that the forces can be derived from a potential energy V  that is a function only ofthe 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

which is clearly equal and opposite to , the force applied by particle on particle j, as may be confirmed by explicit calculation. Hence, the force term of the virial timederivative is

Thus, we have

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Special case of power-law forces[edit] 

In a common special case, the potential energy V  between two particles is proportional to a

 power n of their distance r  

where the coefficient α and the exponent n are constants. In such cases, the force term of thevirial time derivative is given by the equation

where V TOT is the total potential energy of the system

Thus, we have

For gravitating systems and also for  electrostatic systems, the exponent n equals −1, giving

Lagrange's identity 

which was derived by Lagrange and extended by Jacobi.

Time averaging[edit] 

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

from which we obtain the exact equation

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The virial theorem states that, if   , then

There are many reasons why the average of the time derivative might vanish, i.e.,

. One often-cited reason applies to stably bound systems, i.e., systems that hangtogether 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, Gmin and Gmax, and the average goes to zero in the limit of very long times τ  

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

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

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

negative potential energy

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, andthe 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.

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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. vanishes for energy

eigenstates).

In special relativity[edit]

For a single particle in special relativity, it is not the case that . Instead, it is true

that and

The last expression can be simplified to either or .

Thus, under the conditions described in earlier sections (including  Newton's third law of motion, 

, despite relativity), the time average for particles with a power law potential is

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

where the more relativistic systems exhibit the larger ratios.

Generalizations[edit]

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 thevirial 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

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inverse square law:[11][12]

 the statement is true if and only if

A boundary term otherwise must be added, such as in Ref .[13]

 

Inclusion of electromagnetic fields[edit]The virial theorem can be extended to include electric and magnetic fields. The result is

[14] 

where I  is the moment of inertia, G is the momentum density of the electromagnetic field, T  isthe 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, pik  is the fluid- pressure tensor expressed in the local moving coordinate system

and T ik  is the electromagnetic stress tensor,

A  plasmoid is a finite configuration of magnetic fields and plasma. With the virial theorem it iseasy 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  isconfined within a radius R, then the moment of inertia is roughly MR

2, and the left hand side of

the virial theorem is MR2/τ

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 forτ, we find 

where c s is the speed of the ion acoustic wave (or the Alfvén wave, if the magnetic pressure ishigher 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[edit]

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,

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,

for a mass , radius , velocity , and temperature . And the constants are  Newton's

constant  , the Boltzmann constant  , and proton mass . Note that these relations are onlyapproximate, and often the leading numerical factors (e.g. 3/5 or 1/2) are neglected entirely.

Galaxies and cosmology (virial mass and radius)[edit] 

In astronomy, the mass and size of a galaxy (or general overdensity) is often defined in terms ofthe "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  can be used in a similar way. Taking the kinetic energy (per particle) of the system as,

T = (1/2) v2 ~ (3/2)

2, and the potential energy (per particle) as, U ~ (3/5)(GM/R), we can write

.

Here is the radius at which the velocity dispersion is being measured, and is the masswithin that radius. The virial mass and radius are generally defined for the radius at which the

velocity dispersion is a maximum, i.e.

.

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 usedto 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,  . Where is the Hubble parameter  and is the

gravitational constant. A common (although mostly arbitrary) choice for the factor is 200, in

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which case the virial radius is approximated as . The virial

mass is then defined relative to this radius as .

Virial stress 

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Virial stress is a measure of  mechanical stress on an atomic scale. It is given by

where

  and are atoms in the domain,

  is the volume of the domain,

  is the mass of atom k ,

  is the i th

 component of the velocity of atom k ,  is the  j 

th component of the average velocity of atoms in the volume,

  is the i th

 component of the position of atom k , and

  is the i th

 component of the force applied on atom by atom .

At zero kelvin, all velocities are zero so we have

.

This can be thought of as follows. The τ11 component of stress is the force in the x1-directiondivided by the area of a plane perpendicular to that direction. Consider two adjacent volumes

separated by such a plane. The 11-component of stress on that interface is the sum of all pairwise

forces between atoms on the two sides