NationMaster - Encyclopedia: Virial theorem

In mechanics, the virial theorem provides a general equation relating the average total kinetic energy $leftlangle T rightrangle$ of a system with its average total potential energy $leftlangle V_{TOT} rightrangle$ , where angle brackets represent the average of the enclosed quantity. Mathematically, the virial theorem states Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ...

$2 leftlangle T rightrangle = -sum_{k=1}^{N} leftlangle mathbf{F}_{k} cdot mathbf{r}_{k} rightrangle$

where F_k represents the force on the k^th 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]. Fritz Zwicky was the first to use the virial theorem to deduce the existence of unseen matter, what is now called dark matter. In physics, force is an influence that may cause an object to accelerate. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... Rudolf Clausius _ physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... Fritz Zwicky (February 14, 1898 â€“ February 8, 1974) was an American-based Swiss astronomer. ...

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. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Fig. ... Figure 1. ... Fig. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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 Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ...

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

Thus, twice the average total kinetic energy $leftlangle T rightrangle$ equals n times the average total potential energy $leftlangle V_{TOT} rightrangle$ . 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. 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. The Chandrasekhar limit, is the maximum mass possible for a white dwarf (one of the end stages of stars when they cool down) and is approximately 3 Ã— 1030 kg, around 1. ... This article or section does not adequately cite its references or sources. ... STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ...

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 Generalizations of the virial theorem
6 Inclusion of electromagnetic fields
7 See also
8 References
9 Additional reading

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 In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

$I = sum_{k=1}^{N} m_{k} mathbf{r}_{k}^{2} = sum_{k=1}^{N} m_{k} r_{k}^{2}$

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 In classical mechanics, momentum (pl. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...

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

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

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

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

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, $mathbf{F}_{k} = frac{dmathbf{p}_{k}}{dt}$ is the net force on that particle and $T$ is the total kinetic energy of the system The kinetic energy of an object is the extra energy which it possesses due to its motion. ...

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

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

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

where we have assumed that Newton's third law of motion holds, i.e., $mathbf{F}_{jk} = -mathbf{F}_{kj}$ (equal and opposite reaction). Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

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

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

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 Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ...

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

which was derived by Lagrange and extended by Jacobi.

Time averaging and the virial theorem

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

$leftlangle frac{dG}{dt} rightrangle_{tau} = frac{1}{tau} int_{0}^{tau} frac{dG}{dt},dt = frac{1}{tau} int_{0}^{tau} dG = frac{G(tau) - G(0)}{tau},$

from which we obtain the exact equation

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

The virial theorem states that, if $leftlangle frac{dG}{dt} rightrangle_{tau} = 0$ , then

$2 leftlangle T rightrangle_{tau} = -sum_{k=1}^{N} leftlangle mathbf{F}_{k} cdot mathbf{r}_{k} rightrangle_{tau}.$

There are many reasons why the average of the time derivative might vanish, i.e., $leftlangle frac{dG}{dt} rightrangle_{tau} = 0$ . 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 $τ$

$lim_{tau rightarrow infty} left| leftlangle frac{dG^{mathrm{bound}}}{dt} rightrangle_{tau} right| = lim_{tau rightarrow infty} left| frac{G(tau) - G(0)}{tau} right| le lim_{tau rightarrow infty} frac{G_max - G_min}{tau} = 0.$

Even if the average of the time derivative $leftlangle frac{dG}{dt} rightrangle_{tau} approx 0$ 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 â€œGravityâ€ redirects here. ...

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

This general result is useful for complex gravitating systems such as solar systems or galaxies. Major features of the Solar System (not to scale; from left to right): Pluto, Neptune, Uranus, Saturn, Jupiter, the asteroid belt, the Sun, Mercury, Venus, Earth and its Moon, and Mars. ... NGC 4414, a typical spiral galaxy in the constellation Coma Berenices, is about 17,000 parsecs in diameter and approximately 20 million parsecs distant. ...

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. In statistical mechanics, the ensemble average is defined as the weighted average of a molecular property of a system, over the set of states available to the system. ...

Although derived for classical mechanics, the virial theorem also holds for quantum mechanics.

Generalizations 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] Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...

Inclusion of electromagnetic fields

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

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

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 Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... The Poynting vector describes the energy flux (JÂ·mâˆ’2Â·sâˆ’1) of an electromagnetic field. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ...

$p_{ik} = Sigma n^sigma m^sigma langle v_iv_krangle^sigma - V_iV_kSigma m^sigma n^sigma$ ,

and T_ik is the electromagnetic stress tensor,

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

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 A plasmoid is a coherent structure of plasma and magnetic fields. ...

$tau,sim R/c_s,$

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. An ion acoustic wave is a longitudinal oscillation of the ions (and the electrons) in an unmagnetized plasma or in a magnetized plasma parallel to the magnetic field. ... An Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic (or hydromagnetic) wave. ...

References

^ Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4 40.
^ Lord Rayleigh (1903). "Unknown".
^ Poincaré, 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

John William Strutt, 3rd Baron Rayleigh (12 November 1842 â€“ 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Chandrasekhar redirects here. ... Chandrasekhar redirects here. ...

Additional reading

Goldstein, H (1980). Classical Mechanics, 2nd. ed, Addison-Wesley. ISBN 0-201-02918-9.

Collins, GW (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press.

Categories: Classical mechanics | Physics theorems | Fundamental physics concepts

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Definitions of the virial and its time derivative var ACE_AR = {Site: '756743', Size: '300250'};