Virial theoremFrom Wikipedia, the free encyclopedia Jump to:navigation,searchContents [hide]1 History2 Statement and derivationo2.1 Definitions of the virial and its time derivativeo2.2 Connection with the potential energy between particles o2.3 Special case of power-law forces o2.4 Time averaging3 In special relativity4 Generalizations5 Inclusion of electromagnetic fields6 In astrophysicso6.1 Galaxies and cosmology (virial mass and radius)7 See also8 References 9 Further reading10 External linksInmechanics,the virial theoremprovides a general equation that relates the average over time of the totalkinetic energy,, of a stable system consisting of Nparticles, 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 theoremstates where F krepresents theforceon the kth particle, which is located at position r k. The word "virial" derives from vis, theLatinword for "force" or "energy", and was given its technical definition byRudolf Clausiusin 1870. [1] The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated s ystems that defy an exact solution, such as those considered instatistical mechanics;this average total kinetic energy is rela ted to the temperatureof the system by theequipartition theorem.However, the virial theorem does not depend on the
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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
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
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
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
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,
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
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