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Virial theorem
From Wikipedia, the free encyclopedia
In mechanics, the virial theorem provides a general equation relating the average over time of 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 kth particle, which is located at position rk. 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
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, VTOT 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.
Contents
1 History of the virial theorem
2 Statement and derivation
2.1 Definitions of the virial and its time derivative
2.2 Connection with the potential energy between particles
2.3 Special case of power-law forces
2.4 Time averaging and the virial theorem
3 The virial theorem and special relativity
4 Generalizations of the virial theorem
5 Inclusion of electromagnetic fields
6 Virial radius
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7 See also
8 References
9 Further reading
10 External links
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
where mk and rk represent the mass and position of the kth particle. rk=|rk| is the position vector magnitude.
The scalar virial G is defined by the equation
where pk 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
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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 the total
kinetic energy of the system
Connection with the potential energy between particles
The total force
on particle k is the sum of all the forces from the other particles j in the system
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 j = k), we have
[3]
where we have assumed that Newton's third law of motion holds, i.e.,
reaction).
(equal and opposite
It often happens that the forces can be derived from a potential energy V that is a function only of the
distance rjk between the point particles j and k. Since the force is the negative gradient of the potential
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energy, we have in this case
which is clearly equal and opposite to
, 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
Thus, we have
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
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
where VTOT 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
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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
from which we obtain the exact equation
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 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 Gbound 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
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
For gravitational attraction, n equals −1 and the average kinetic energy equals half of the average negative
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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, 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.
vanishes for energy eigenstates).
The virial theorem and special relativity
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 N particles with a power law potential is
In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an
interval:
where the more relativistic systems exhibit the larger ratios.
Generalizations of the virial theorem
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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
is true if and only if
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]
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, WE and WM 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 Tik is the electromagnetic stress tensor,
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
where cs 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
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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
. (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.
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
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