fbpx
Wikipedia

Virial theorem

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. Mathematically, the theorem states

T = 1 2 k = 1 N F k r k {\displaystyle \left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }}

for the total kinetic energyT ofN particles, whereFk represents the force on thekth particle, which is located at positionrk, and angle brackets represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870.

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 energyV(r) = αrn that is proportional to some powern of the interparticle distancer, the virial theorem takes the simple form

2 T = n V TOT . {\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .}

Thus, twice the average total kinetic energyT equalsn times the average total potential energyVTOT. WhereasV(r) represents the potential energy between two particles,VTOT represents the total potential energy of the system, i.e., the sum of the potential energyV(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, wheren 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

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 to1/2 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 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. The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux, Richard Bader 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. Richard Bader showed the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. 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.

ConsiderN = 2 particles with equal massm, acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radiusr. The velocities arev1(t) andv2(t) = −v1(t), which are normal to forcesF1(t) andF2(t) = −F1(t). The respective magnitudes are fixed atv andF. The average kinetic energy of the system is

T = k = 1 N 1 2 m k | v k | 2 = 1 2 m | v 1 | 2 + 1 2 m | v 2 | 2 = m v 2 . {\displaystyle \langle T\rangle =\sum _{k=1}^{N}{\frac {1}{2}}m_{k}\left|\mathbf {v} _{k}\right|^{2}={\frac {1}{2}}m|\mathbf {v} _{1}|^{2}+{\frac {1}{2}}m|\mathbf {v} _{2}|^{2}=mv^{2}.}

Taking center of mass as the origin, the particles have positionsr1(t) andr2(t) = −r1(t) with fixed magnituder. The attractive forces act in opposite directions as positions, soF1(t) {\displaystyle \cdot } r1(t) = F2(t) {\displaystyle \cdot } r2(t) = −Fr. Applying the centripetal force formulaF = mv2/r results in:

1 2 k = 1 N F k r k = 1 2 ( F r F r ) = F r = m v 2 r r = m v 2 = T , {\displaystyle -{\frac {1}{2}}\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }=-{\frac {1}{2}}(-Fr-Fr)=Fr={\frac {mv^{2}}{r}}\cdot r=mv^{2}=\langle T\rangle ,}

as required. Note: If the origin is displaced then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forcesF1(t),F2(t) results in net cancellation.

For a collection ofN point particles, the scalar moment of inertiaI about the origin is defined by the equation

I = k = 1 N m k | r k | 2 = k = 1 N m k r k 2 {\displaystyle I=\sum _{k=1}^{N}m_{k}\left|\mathbf {r} _{k}\right|^{2}=\sum _{k=1}^{N}m_{k}r_{k}^{2}}

wheremk andrk represent the mass and position of thekth particle.rk = |rk| is the position vector magnitude. The scalarG is defined by the equation

G = k = 1 N p k r k {\displaystyle G=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k}}

wherepk is the momentum vector of thekth particle. Assuming that the masses are constant,G is one-half the time derivative of this moment of inertia

1 2 d I d t = 1 2 d d t k = 1 N m k r k r k = k = 1 N m k d r k d t r k = k = 1 N p k r k = G . {\displaystyle {\begin{aligned}{\frac {1}{2}}{\frac {dI}{dt}}&={\frac {1}{2}}{\frac {d}{dt}}\sum _{k=1}^{N}m_{k}\mathbf {r} _{k}\cdot \mathbf {r} _{k}\\&=\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}=G\,.\end{aligned}}}

In turn, the time derivative ofG can be written

d G d t = k = 1 N p k d r k d t + k = 1 N d p k d t r k = k = 1 N m k d r k d t d r k d t + k = 1 N F k r k = 2 T + k = 1 N F k r k {\displaystyle {\begin{aligned}{\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}\\&=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}\,\end{aligned}}}

wheremk is the mass of thekth particle,Fk = dpk/dt is the net force on that particle, andT is the total kinetic energy of the system according to thevk = drk/dt velocity of each particle

T = 1 2 k = 1 N m k v k 2 = 1 2 k = 1 N m k d r k d t d r k d t . {\displaystyle 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}}.}

Connection with the potential energy between particles

The total forceFk on particlek is the sum of all the forces from the other particlesj in the system

F k = j = 1 N F j k {\displaystyle \mathbf {F} _{k}=\sum _{j=1}^{N}\mathbf {F} _{jk}}

whereFjk is the force applied by particlej on particlek. Hence, the virial can be written

1 2 k = 1 N F k r k = 1 2 k = 1 N j = 1 N F j k r k . {\displaystyle -{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=-{\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j=1}^{N}\mathbf {F} _{jk}\cdot \mathbf {r} _{k}\,.}

Since no particle acts on itself (i.e.,Fjj = 0 for1 ≤ jN), we split the sum in terms below and above this diagonal and we add them together in pairs:

k = 1 N F k r k = k = 1 N j = 1 N F j k r k = k = 2 N j = 1 k 1 ( F j k r k + F k j r j ) = k = 2 N j = 1 k 1 ( F j k r k F j k r j ) = k = 2 N j = 1 k 1 F j k ( r k r j ) {\displaystyle {\begin{aligned}\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}=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\left(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}+\mathbf {F} _{kj}\cdot \mathbf {r} _{j}\right)\\&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\left(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}-\mathbf {F} _{jk}\cdot \mathbf {r} _{j}\right)=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot \left(\mathbf {r} _{k}-\mathbf {r} _{j}\right)\end{aligned}}}

where we have assumed that Newton's third law of motion holds, i.e.,Fjk = −Fkj (equal and opposite reaction).

It often happens that the forces can be derived from a potential energyVjk that is a function only of the distancerjk between the point particlesj andk. Since the force is the negative gradient of the potential energy, we have in this case

F j k = r k V j k = d V j k d r j k ( r k r j r j k ) , {\displaystyle \mathbf {F} _{jk}=-\nabla _{\mathbf {r} _{k}}V_{jk}=-{\frac {dV_{jk}}{dr_{jk}}}\left({\frac {\mathbf {r} _{k}-\mathbf {r} _{j}}{r_{jk}}}\right),}

which is equal and opposite toFkj = −∇rjVkj = −∇rjVjk, the force applied by particlek on particlej, as may be confirmed by explicit calculation. Hence,

k = 1 N F k r k = k = 2 N j = 1 k 1 F j k ( r k r j ) = k = 2 N j = 1 k 1 d V j k d r j k | r k r j | 2 r j k = k = 2 N j = 1 k 1 d V j k d r j k r j k . {\displaystyle {\begin{aligned}\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot \left(\mathbf {r} _{k}-\mathbf {r} _{j}\right)\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}{\frac {|\mathbf {r} _{k}-\mathbf {r} _{j}|^{2}}{r_{jk}}}\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.\end{aligned}}}

Thus, we have

d G d t = 2 T + k = 1 N F k r k = 2 T k = 2 N j = 1 k 1 d V j k d r j k r j k . {\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.}

Special case of power-law forces

In a common special case, the potential energyV between two particles is proportional to a powern of their distancerij

V j k = α r j k n , {\displaystyle V_{jk}=\alpha r_{jk}^{n},}

where the coefficientα and the exponentn are constants. In such cases, the virial is given by the equation

1 2 k = 1 N F k r k = 1 2 k = 1 N j < k d V j k d r j k r j k = 1 2 k = 1 N j < k n α r j k n 1 r j k = 1 2 k = 1 N j < k n V j k = n 2 V TOT {\displaystyle {\begin{aligned}-{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}n\alpha r_{jk}^{n-1}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}nV_{jk}={\frac {n}{2}}\,V_{\text{TOT}}\end{aligned}}}

whereVTOT is the total potential energy of the system

V TOT = k = 1 N j < k V j k . {\displaystyle V_{\text{TOT}}=\sum _{k=1}^{N}\sum _{j<k}V_{jk}\,.}

Thus, we have

d G d t = 2 T + k = 1 N F k r k = 2 T n V TOT . {\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-nV_{\text{TOT}}\,.}

For gravitating systems the exponentn equals −1, giving Lagrange's identity

d G d t = 1 2 d 2 I d t 2 = 2 T + V TOT {\displaystyle {\frac {dG}{dt}}={\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}=2T+V_{\text{TOT}}}

which was derived by Joseph-Louis Lagrange and extended by Carl Jacobi.

Time averaging

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

d G d t τ = 1 τ 0 τ d G d t d t = 1 τ G ( 0 ) G ( τ ) d G = G ( τ ) G ( 0 ) τ , {\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle _{\tau }={\frac {1}{\tau }}\int _{0}^{\tau }{\frac {dG}{dt}}\,dt={\frac {1}{\tau }}\int _{G(0)}^{G(\tau )}\,dG={\frac {G(\tau )-G(0)}{\tau }},}

from which we obtain the exact equation

d G d t τ = 2 T τ + k = 1 N F k r k τ . {\displaystyle \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 ifdG/dtτ = 0, then

2 T τ = k = 1 N F k r k τ . {\displaystyle 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,dG/dtτ = 0. One often-cited reason applies to stably-bound systems, that is to say 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 thatGbound, is bounded between two extremes,Gmin andGmax, and the average goes to zero in the limit of very long timesτ:

lim τ | d G b o u n d d t τ | = lim τ | G ( τ ) G ( 0 ) τ | lim τ G max G min τ = 0. {\displaystyle \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|\leq \lim _{\tau \rightarrow \infty }{\frac {G_{\max }-G_{\min }}{\tau }}=0.}

Even if the average of the time derivative ofG is only approximately zero, the virial theorem holds to the same degree of approximation.

For power-law forces with an exponentn, the general equation holds:

T τ = 1 2 k = 1 N F k r k τ = n 2 V TOT τ . {\displaystyle {\begin{aligned}\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_{\text{TOT}}\rangle _{\tau }.\end{aligned}}}

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

T τ = 1 2 V TOT τ . {\displaystyle \langle T\rangle _{\tau }=-{\frac {1}{2}}\langle V_{\text{TOT}}\rangle _{\tau }.}

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 effect 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.

If the ergodic hypothesis holds for the system under consideration, the averaging need not be taken over time; an ensemble average can also be taken, with equivalent results.

Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Fock using the Ehrenfest theorem.

Evaluate the commutator of the Hamiltonian

H = V ( { X i } ) + n P n 2 2 m {\displaystyle H=V{\bigl (}\{X_{i}\}{\bigr )}+\sum _{n}{\frac {P_{n}^{2}}{2m}}}

with the position operatorXn and the momentum operator

P n = i d d X n {\displaystyle P_{n}=-i\hbar {\frac {d}{dX_{n}}}}

of particlen,

[ H , X n P n ] = X n [ H , P n ] + [ H , X n ] P n = i X n d V d X n i P n 2 m . {\displaystyle [H,X_{n}P_{n}]=X_{n}[H,P_{n}]+[H,X_{n}]P_{n}=i\hbar X_{n}{\frac {dV}{dX_{n}}}-i\hbar {\frac {P_{n}^{2}}{m}}~.}

Summing over all particles, one finds for

Q = n X n P n {\displaystyle Q=\sum _{n}X_{n}P_{n}}

the commutator amounts to

i [ H , Q ] = 2 T n X n d V d X n {\displaystyle {\frac {i}{\hbar }}[H,Q]=2T-\sum _{n}X_{n}{\frac {dV}{dX_{n}}}}

where T = n P n 2 2 m {\displaystyle T=\sum _{n}{\frac {P_{n}^{2}}{2m}}} is the kinetic energy. The left-hand side of this equation is justdQ/dt, according to the Heisenberg equation of motion. The expectation valuedQ/dt of this time derivative vanishes in a stationary state, leading to the quantum virial theorem,

2 T = n X n d V d X n . {\displaystyle 2\langle T\rangle =\sum _{n}\left\langle X_{n}{\frac {dV}{dX_{n}}}\right\rangle ~.}

Pokhozhaev's identity

This section does not cite any sources. Please help improve this section by . Unsourced material may be challenged and removed.(April 2020) ()

In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary nonlinear Schrödinger equation or Klein–Gordon equation, is Pokhozhaev's identity, also known as Derrick's theorem.

Let g ( s ) {\displaystyle g(s)} be continuous and real-valued, with g ( 0 ) = 0 {\displaystyle g(0)=0} .

Denote G ( s ) = 0 s g ( t ) d t {\displaystyle G(s)=\int _{0}^{s}g(t)\,dt} . Let

u L l o c ( R n ) , u L 2 ( R n ) , G ( u ( ) ) L 1 ( R n ) , n N , {\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}

be a solution to the equation

2 u = g ( u ) , {\displaystyle -\nabla ^{2}u=g(u),} in the sense of distributions.

Then u {\displaystyle u} satisfies the relation

( n 2 ) R n | u ( x ) | 2 d x = n R n G ( u ( x ) ) d x . {\displaystyle (n-2)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx.}
This section does not cite any sources. Please help improve this section by . Unsourced material may be challenged and removed.(April 2020) ()

For a single particle in special relativity, it is not the case thatT = 1/2p · v. Instead, it is true thatT = (γ − 1) mc2, whereγ is the Lorentz factor

γ = 1 1 v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}

andβ = v/c. We have,

1 2 p v = 1 2 β γ m c β c = 1 2 γ β 2 m c 2 = ( γ β 2 2 ( γ 1 ) ) T . {\displaystyle {\begin{aligned}{\frac {1}{2}}\mathbf {p} \cdot \mathbf {v} &={\frac {1}{2}}{\boldsymbol {\beta }}\gamma mc\cdot {\boldsymbol {\beta }}c\\[5pt]&={\frac {1}{2}}\gamma \beta ^{2}mc^{2}\\[5pt]&=\left({\frac {\gamma \beta ^{2}}{2(\gamma -1)}}\right)T\,.\end{aligned}}}

The last expression can be simplified to

( 1 + 1 β 2 2 ) T or ( γ + 1 2 γ ) T {\displaystyle \left({\frac {1+{\sqrt {1-\beta ^{2}}}}{2}}\right)T\qquad {\text{or}}\qquad \left({\frac {\gamma +1}{2\gamma }}\right)T} .

Thus, under the conditions described in earlier sections (including Newton's third law of motion,Fjk = −Fkj, despite relativity), the time average forN particles with a power law potential is

n 2 V T O T τ = k = 1 N ( 1 + 1 β k 2 2 ) T k τ = k = 1 N ( γ k + 1 2 γ k ) T k τ . {\displaystyle {\frac {n}{2}}\left\langle V_{\mathrm {TOT} }\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\frac {1+{\sqrt {1-\beta _{k}^{2}}}}{2}}\right)T_{k}\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\frac {\gamma _{k}+1}{2\gamma _{k}}}\right)T_{k}\right\rangle _{\tau }\,.}

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

2 T T O T n V T O T [ 1 , 2 ] , {\displaystyle {\frac {2\langle T_{\mathrm {TOT} }\rangle }{n\langle V_{\mathrm {TOT} }\rangle }}\in \left[1,2\right]\,,}

where the more relativistic systems exhibit the larger ratios.

Lord Rayleigh published a generalization of the virial theorem in 1903. Henri Poincaré proved and applied a form of the virial theorem in 1911 to the problem of formation of the solar system from a proto-stellar cloud (then known as cosmogony). A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of the virial theorem was developed by Parker, Chandrasekhar and Fermi. The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:

2 lim τ + T τ = lim τ + U τ if and only if lim τ + τ 2 I ( τ ) = 0 . {\displaystyle 2\lim \limits _{\tau \rightarrow +\infty }\langle T\rangle _{\tau }=\lim \limits _{\tau \rightarrow +\infty }\langle U\rangle _{\tau }\qquad {\text{if and only if}}\quad \lim \limits _{\tau \rightarrow +\infty }{\tau }^{-2}I(\tau )=0\,.}

A boundary term otherwise must be added.

The virial theorem can be extended to include electric and magnetic fields. The result is

1 2 d 2 I d t 2 + V x k G k t d 3 r = 2 ( T + U ) + W E + W M x k ( p i k + T i k ) d S i , {\displaystyle {\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}+\int _{V}x_{k}{\frac {\partial G_{k}}{\partial t}}\,d^{3}r=2(T+U)+W^{\mathrm {E} }+W^{\mathrm {M} }-\int x_{k}(p_{ik}+T_{ik})\,dS_{i},}

whereI 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 andWM 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

p i k = Σ n σ m σ v i v k σ V i V k Σ m σ n σ , {\displaystyle p_{ik}=\Sigma n^{\sigma }m^{\sigma }\langle v_{i}v_{k}\rangle ^{\sigma }-V_{i}V_{k}\Sigma m^{\sigma }n^{\sigma },}

andTik is the electromagnetic stress tensor,

T i k = ( ε 0 E 2 2 + B 2 2 μ 0 ) δ i k ( ε 0 E i E k + B i B k μ 0 ) . {\displaystyle T_{ik}=\left({\frac {\varepsilon _{0}E^{2}}{2}}+{\frac {B^{2}}{2\mu _{0}}}\right)\delta _{ik}-\left(\varepsilon _{0}E_{i}E_{k}+{\frac {B_{i}B_{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 timeτ. If a total massM is confined within a radiusR, then the moment of inertia is roughlyMR2, and the left hand side of the virial theorem isMR2/τ2. The terms on the right hand side add up to aboutpR3, wherep is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving forτ, we find

τ R c s , {\displaystyle \tau \,\sim {\frac {R}{c_{\mathrm {s} }}},}

wherecs 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.

In case when in the physical system the pressure field, the electromagnetic and gravitational fields are taken into account, as well as the field of particles’ acceleration, the virial theorem is written in the relativistic form as follows:

W k 0.6 k = 1 N F k r k , {\displaystyle \left\langle W_{k}\right\rangle \approx -0.6\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,}

where the valueWkγcT exceeds the kinetic energy of the particlesT by a factor equal to the Lorentz factorγc of the particles at the center of the system. Under normal conditions we can assume thatγc ≈ 1, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 1/2, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalarG is not equal to zero and should be considered as the material derivative.

An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature:

v r m s = c 1 4 π η ρ 0 r 2 c 2 γ c 2 sin 2 ( r c 4 π η ρ 0 ) , {\displaystyle v_{\mathrm {rms} }=c{\sqrt {1-{\frac {4\pi \eta \rho _{0}r^{2}}{c^{2}\gamma _{c}^{2}\sin ^{2}\left({\frac {r}{c}}{\sqrt {4\pi \eta \rho _{0}}}\right)}}}},}

where c {\displaystyle ~c} is the speed of light, η {\displaystyle ~\eta } is the acceleration field constant, ρ 0 {\displaystyle ~\rho _{0}} is the mass density of particles, r {\displaystyle ~r} is the current radius.

Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:

E k f + 2 W f = 0 , {\displaystyle ~E_{kf}+2W_{f}=0,}

where the energy E k f = A α j α g d x 1 d x 2 d x 3 {\displaystyle ~E_{kf}=\int A_{\alpha }j^{\alpha }{\sqrt {-g}}\,dx^{1}\,dx^{2}\,dx^{3}} considered as the kinetic field energy associated with four-current j α {\displaystyle ~j^{\alpha }} , and

W f = 1 4 μ 0 F α β F α β g d x 1 d x 2 d x 3 {\displaystyle ~W_{f}={\frac {1}{4\mu _{0}}}\int F_{\alpha \beta }F^{\alpha \beta }{\sqrt {-g}}\,dx^{1}\,dx^{2}\,dx^{3}}

sets the potential field energy found through the components of the electromagnetic tensor.

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[citation needed]

3 5 G M R = 3 2 k B T m p = 1 2 v 2 {\displaystyle {\frac {3}{5}}{\frac {GM}{R}}={\frac {3}{2}}{\frac {k_{\mathrm {B} }T}{m_{\mathrm {p} }}}={\frac {1}{2}}v^{2}}

for a massM, radiusR, velocityv, and temperatureT. The constants are Newton's constantG, the Boltzmann constantkB, and proton massmp. Note that these relations are only approximate, and often the leading numerical factors (e.g. 3/5 or 1/2) are neglected entirely.

Galaxies and cosmology (virial mass and radius)

Main article: Virial mass

In astronomy, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "virial mass" and "virial radius" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as 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 velocityσ can be used in a similar way. Taking the kinetic energy (per particle) of the system asT = 1/2v2 ~ 3/2σ2, and the potential energy (per particle) asU ~ 3/5 GM/R we can write

G M R σ 2 . {\displaystyle {\frac {GM}{R}}\approx \sigma ^{2}.}

Here R {\displaystyle R} is the radius at which the velocity dispersion is being measured, andM is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e.

G M vir R vir σ max 2 . {\displaystyle {\frac {GM_{\text{vir}}}{R_{\text{vir}}}}\approx \sigma _{\max }^{2}.}

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 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

ρ crit = 3 H 2 8 π G {\displaystyle \rho _{\text{crit}}={\frac {3H^{2}}{8\pi G}}}

whereH is the Hubble parameter andG is the gravitational constant. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see Virial mass), in which case the virial radius is approximated as

r vir r 200 = r , ρ = 200 ρ crit . {\displaystyle r_{\text{vir}}\approx r_{200}=r,\qquad \rho =200\cdot \rho _{\text{crit}}.}

The virial mass is then defined relative to this radius as

M vir M 200 = 4 3 π r 200 3 200 ρ crit . {\displaystyle M_{\text{vir}}\approx M_{200}={\frac {4}{3}}\pi r_{200}^{3}\cdot 200\rho _{\text{crit}}.}

In stars

The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the main sequence convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to supports its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative specific heat. This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation withn equals −1 no longer holds.

  1. Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine. Series 4. 40 (265): 122–127. doi:10.1080/14786447008640370.
  2. Collins, G. W. (1978). "Introduction". The Virial Theorem in Stellar Astrophysics. Pachart Press. Bibcode:1978vtsa.book.....C. ISBN 978-0-912918-13-6.
  3. Bader, R. F. W.; Beddall, P. M. (1972). "Virial Field Relationship for Molecular Charge Distributions and the Spatial Partitioning of Molecular Properties". The Journal of Chemical Physics. 56 (7).
  4. Goldstein, Herbert, 1922-2005. (1980). Classical mechanics (2d ed.). Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-201-02918-9. OCLC 5675073.CS1 maint: multiple names: authors list (link)
  5. Fock, V. (1930). "Bemerkung zum Virialsatz". Zeitschrift für Physik A. 63 (11): 855–858. Bibcode:1930ZPhy...63..855F. doi:10.1007/BF01339281. S2CID 122502103.
  6. Lord Rayleigh (1903). "Unknown".Cite journal requires |journal= ();Cite uses generic title ()
  7. Poincaré, Henri (1911). Leçons sur les hypothèses cosmogoniques [Lectures on Theories of Cosmogony]. Paris: Hermann. pp. 90-91 et seq.
  8. Ledoux, P. (1945). "On the Radial Pulsation of Gaseous Stars". The Astrophysical Journal. 102: 143–153. Bibcode:1945ApJ...102..143L. doi:10.1086/144747.
  9. Parker, E.N. (1954). "Tensor Virial Equations". Physical Review. 96 (6): 1686–1689. Bibcode:1954PhRv...96.1686P. doi:10.1103/PhysRev.96.1686.
  10. Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids". Astrophys. J. 136: 1037–1047. Bibcode:1962ApJ...136.1037C. doi:10.1086/147456.
  11. Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field". Astrophys. J. 118: 116. Bibcode:1953ApJ...118..116C. doi:10.1086/145732.
  12. 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.
  13. Pollard, Harry (1966). Mathematical Introduction to Celestial Mechanics. Englewood Cliffs, NJ: Prentice–Hall, Inc. ISBN 978-0-13-561068-8.
  14. Kolár, M.; O'Shea, S. F. (July 1996). "A high-temperature approximation for the path-integral quantum Monte Carlo method". Journal of Physics A: Mathematical and General. 29 (13): 3471–3494. Bibcode:1996JPhA...29.3471K. doi:10.1088/0305-4470/29/13/018.
  15. Schmidt, George (1979). Physics of High Temperature Plasmas (Second ed.). Academic Press. p. 72.
  16. Fedosin, S. G. (2016). "The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept". Continuum Mechanics and Thermodynamics. 29 (2): 361–371. arXiv:1801.06453. Bibcode:2017CMT....29..361F. doi:10.1007/s00161-016-0536-8. S2CID 53692146.
  17. Fedosin, Sergey G. (2018-09-24). "The integral theorem of generalized virial in the relativistic uniform model". Continuum Mechanics and Thermodynamics. 31 (3): 627–638. arXiv:1912.08683. Bibcode:2018CMT...tmp..140F. doi:10.1007/s00161-018-0715-x. ISSN 1432-0959. S2CID 125180719 – via Springer Nature SharedIt.
  18. Fedosin S.G. The Integral Theorem of the Field Energy. Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). doi:10.5281/zenodo.3252783.
  19. BAIDYANATH BASU; TANUKA CHATTOPADHYAY; SUDHINDRA NATH BISWAS (1 January 2010). AN INTRODUCTION TO ASTROPHYSICS. PHI Learning Pvt. Ltd. pp. 365–. ISBN 978-81-203-4071-8.
  20. William K. Rose (16 April 1998). Advanced Stellar Astrophysics. Cambridge University Press. pp. 242–. ISBN 978-0-521-58833-1.

Virial theorem
Virial theorem Language Watch Edit In mechanics the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles bound by potential forces with that of the total potential energy of the system Mathematically the theorem states T 1 2 k 1 N F k r k displaystyle left langle T right rangle frac 1 2 sum k 1 N bigl langle mathbf F k cdot mathbf r k bigr rangle for the total kinetic energy T of N particles where Fk represents the force on the k th particle which is located at position rk and angle brackets represent the average over time of the enclosed quantity The word virial for the right hand side of the equation derives from vis the Latin word for force or energy and was given its technical definition 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 arn that is proportional to some power n of the interparticle distance r the virial theorem takes the simple form 2 T n V TOT displaystyle 2 langle T rangle n langle V text TOT rangle Thus twice the average total kinetic energy T equals n times the average total potential energy VTOT 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 2 Illustrative Special Case 3 Statement and derivation 3 1 Connection with the potential energy between particles 3 2 Special case of power law forces 3 3 Time averaging 4 In quantum mechanics 4 1 Pokhozhaev s identity 5 In special relativity 6 Generalizations 7 Inclusion of electromagnetic fields 8 Relativistic uniform system 9 In astrophysics 9 1 Galaxies and cosmology virial mass and radius 9 2 In stars 10 See also 11 References 12 Further reading 13 External linksHistory EditIn 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 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 James Clerk Maxwell Lord Rayleigh Henri Poincare Subrahmanyan Chandrasekhar Enrico Fermi Paul Ledoux Richard Bader 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 Richard Bader showed the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem 3 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 Illustrative Special Case EditConsider N 2 particles with equal mass m acted upon by mutually attractive forces Suppose the particles are at diametrically opposite points of a circular orbit with radius r The velocities are v1 t and v2 t v1 t which are normal to forces F1 t and F2 t F1 t The respective magnitudes are fixed at v and F The average kinetic energy of the system is T k 1 N 1 2 m k v k 2 1 2 m v 1 2 1 2 m v 2 2 m v 2 displaystyle langle T rangle sum k 1 N frac 1 2 m k left mathbf v k right 2 frac 1 2 m mathbf v 1 2 frac 1 2 m mathbf v 2 2 mv 2 Taking center of mass as the origin the particles have positions r1 t and r2 t r1 t with fixed magnitude r The attractive forces act in opposite directions as positions so F1 t displaystyle cdot r1 t F2 t displaystyle cdot r2 t Fr Applying the centripetal force formula F mv2 r results in 1 2 k 1 N F k r k 1 2 F r F r F r m v 2 r r m v 2 T displaystyle frac 1 2 sum k 1 N bigl langle mathbf F k cdot mathbf r k bigr rangle frac 1 2 Fr Fr Fr frac mv 2 r cdot r mv 2 langle T rangle as required Note If the origin is displaced then we d obtain the same result This is because the dot product of the displacement with equal and opposite forces F1 t F2 t results in net cancellation Statement and derivation EditFor a collection of N point particles the scalar moment of inertia I about the origin is defined by the equation I k 1 N m k r k 2 k 1 N m k r k 2 displaystyle I sum k 1 N m k left mathbf r k right 2 sum k 1 N m k r k 2 where mk and rk represent the mass and position of the k th particle rk rk is the position vector magnitude The scalar G is defined by the equation G k 1 N p k r k displaystyle G sum k 1 N mathbf p k cdot mathbf r k where pk is the momentum vector of the k th particle 4 Assuming that the masses are constant G is one half the time derivative of this moment of inertia 1 2 d I d t 1 2 d d t k 1 N m k r k r k k 1 N m k d r k d t r k k 1 N p k r k G displaystyle begin aligned frac 1 2 frac dI dt amp frac 1 2 frac d dt sum k 1 N m k mathbf r k cdot mathbf r k amp sum k 1 N m k frac d mathbf r k dt cdot mathbf r k amp sum k 1 N mathbf p k cdot mathbf r k G end aligned In turn the time derivative of G can be written d G d t k 1 N p k d r k d t k 1 N d p k d t r k k 1 N m k d r k d t d r k d t k 1 N F k r k 2 T k 1 N F k r k displaystyle begin aligned frac dG dt amp 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 amp 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 amp 2T sum k 1 N mathbf F k cdot mathbf r k end aligned where mk is the mass of the k th particle Fk dpk dt is the net force on that particle and T is the total kinetic energy of the system according to the vk drk dt velocity of each particle T 1 2 k 1 N m k v k 2 1 2 k 1 N m k d r k d t d r k d t displaystyle 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 Connection with the potential energy between particles Edit The total force Fk on particle k is the sum of all the forces from the other particles j in the system F k j 1 N F j k displaystyle mathbf F k sum j 1 N mathbf F jk where Fjk is the force applied by particle j on particle k Hence the virial can be written 1 2 k 1 N F k r k 1 2 k 1 N j 1 N F j k r k displaystyle frac 1 2 sum k 1 N mathbf F k cdot mathbf r k frac 1 2 sum k 1 N sum j 1 N mathbf F jk cdot mathbf r k Since no particle acts on itself i e Fjj 0 for 1 j N we split the sum in terms below and above this diagonal and we add them together in pairs k 1 N F k r k k 1 N j 1 N F j k r k k 2 N j 1 k 1 F j k r k F k j r j k 2 N j 1 k 1 F j k r k F j k r j k 2 N j 1 k 1 F j k r k r j displaystyle begin aligned sum k 1 N mathbf F k cdot mathbf r k amp sum k 1 N sum j 1 N mathbf F jk cdot mathbf r k sum k 2 N sum j 1 k 1 left mathbf F jk cdot mathbf r k mathbf F kj cdot mathbf r j right amp sum k 2 N sum j 1 k 1 left mathbf F jk cdot mathbf r k mathbf F jk cdot mathbf r j right sum k 2 N sum j 1 k 1 mathbf F jk cdot left mathbf r k mathbf r j right end aligned where we have assumed that Newton s third law of motion holds i e Fjk Fkj equal and opposite reaction It often happens that the forces can be derived from a potential energy Vjk 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 energy we have in this case F j k r k V j k d V j k d r j k r k r j r j k displaystyle mathbf F jk nabla mathbf r k V jk frac dV jk dr jk left frac mathbf r k mathbf r j r jk right which is equal and opposite to Fkj rjVkj rjVjk the force applied by particle k on particle j as may be confirmed by explicit calculation Hence k 1 N F k r k k 2 N j 1 k 1 F j k r k r j k 2 N j 1 k 1 d V j k d r j k r k r j 2 r j k k 2 N j 1 k 1 d V j k d r j k r j k displaystyle begin aligned sum k 1 N mathbf F k cdot mathbf r k amp sum k 2 N sum j 1 k 1 mathbf F jk cdot left mathbf r k mathbf r j right amp sum k 2 N sum j 1 k 1 frac dV jk dr jk frac mathbf r k mathbf r j 2 r jk amp sum k 2 N sum j 1 k 1 frac dV jk dr jk r jk end aligned Thus we have d G d t 2 T k 1 N F k r k 2 T k 2 N j 1 k 1 d V j k d r j k r j k displaystyle frac dG dt 2T sum k 1 N mathbf F k cdot mathbf r k 2T sum k 2 N sum j 1 k 1 frac dV jk dr jk r jk 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 rij V j k a r j k n displaystyle V jk alpha r jk n where the coefficient a and the exponent n are constants In such cases the virial is given by the equation 1 2 k 1 N F k r k 1 2 k 1 N j lt k d V j k d r j k r j k 1 2 k 1 N j lt k n a r j k n 1 r j k 1 2 k 1 N j lt k n V j k n 2 V TOT displaystyle begin aligned frac 1 2 sum k 1 N mathbf F k cdot mathbf r k amp frac 1 2 sum k 1 N sum j lt k frac dV jk dr jk r jk amp frac 1 2 sum k 1 N sum j lt k n alpha r jk n 1 r jk amp frac 1 2 sum k 1 N sum j lt k nV jk frac n 2 V text TOT end aligned where VTOT is the total potential energy of the system V TOT k 1 N j lt k V j k displaystyle V text TOT sum k 1 N sum j lt k V jk Thus we have d G d t 2 T k 1 N F k r k 2 T n V TOT displaystyle frac dG dt 2T sum k 1 N mathbf F k cdot mathbf r k 2T nV text TOT For gravitating systems the exponent n equals 1 giving Lagrange s identity d G d t 1 2 d 2 I d t 2 2 T V TOT displaystyle frac dG dt frac 1 2 frac d 2 I dt 2 2T V text TOT which was derived by Joseph Louis Lagrange and extended by Carl Jacobi Time averaging Edit The average of this derivative over a time t is defined as d G d t t 1 t 0 t d G d t d t 1 t G 0 G t d G G t G 0 t displaystyle left langle frac dG dt right rangle tau frac 1 tau int 0 tau frac dG dt dt frac 1 tau int G 0 G tau dG frac G tau G 0 tau from which we obtain the exact equation d G d t t 2 T t k 1 N F k r k t displaystyle 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 dG dt t 0 then 2 T t k 1 N F k r k t displaystyle 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 dG dt t 0 One often cited reason applies to stably bound systems that is to say 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 Gbound is bounded between two extremes Gmin and Gmax and the average goes to zero in the limit of very long times t lim t d G b o u n d d t t lim t G t G 0 t lim t G max G min t 0 displaystyle 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 leq lim tau rightarrow infty frac G max G min tau 0 Even if the average of the time derivative of G 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 T t 1 2 k 1 N F k r k t n 2 V TOT t displaystyle begin aligned langle T rangle tau amp frac 1 2 sum k 1 N langle mathbf F k cdot mathbf r k rangle tau amp frac n 2 langle V text TOT rangle tau end aligned For gravitational attraction n equals 1 and the average kinetic energy equals half of the average negative potential energy T t 1 2 V TOT t displaystyle langle T rangle tau frac 1 2 langle V text TOT rangle tau 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 effect 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 If the ergodic hypothesis holds for the system under consideration the averaging need not be taken over time an ensemble average can also be taken with equivalent results In quantum mechanics EditAlthough originally derived for classical mechanics the virial theorem also holds for quantum mechanics as first shown by Fock 5 using the Ehrenfest theorem Evaluate the commutator of the Hamiltonian H V X i n P n 2 2 m displaystyle H V bigl X i bigr sum n frac P n 2 2m with the position operator Xn and the momentum operator P n i ℏ d d X n displaystyle P n i hbar frac d dX n of particle n H X n P n X n H P n H X n P n i ℏ X n d V d X n i ℏ P n 2 m displaystyle H X n P n X n H P n H X n P n i hbar X n frac dV dX n i hbar frac P n 2 m Summing over all particles one finds for Q n X n P n displaystyle Q sum n X n P n the commutator amounts to i ℏ H Q 2 T n X n d V d X n displaystyle frac i hbar H Q 2T sum n X n frac dV dX n where T n P n 2 2 m displaystyle T sum n frac P n 2 2m is the kinetic energy The left hand side of this equation is just dQ dt according to the Heisenberg equation of motion The expectation value dQ dt of this time derivative vanishes in a stationary state leading to the quantum virial theorem 2 T n X n d V d X n displaystyle 2 langle T rangle sum n left langle X n frac dV dX n right rangle Pokhozhaev s identity Edit This section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed April 2020 Learn how and when to remove this template message In the field of quantum mechanics there exists another form of the virial theorem applicable to localized solutions to the stationary nonlinear Schrodinger equation or Klein Gordon equation is Pokhozhaev s identity also known as Derrick s theorem Let g s displaystyle g s be continuous and real valued with g 0 0 displaystyle g 0 0 Denote G s 0 s g t d t displaystyle G s int 0 s g t dt Let u L l o c R n u L 2 R n G u L 1 R n n N displaystyle u in L mathrm loc infty mathbb R n qquad nabla u in L 2 mathbb R n qquad G u cdot in L 1 mathbb R n qquad n in mathbb N be a solution to the equation 2 u g u displaystyle nabla 2 u g u in the sense of distributions Then u displaystyle u satisfies the relation n 2 R n u x 2 d x n R n G u x d x displaystyle n 2 int mathbb R n nabla u x 2 dx n int mathbb R n G u x dx In special relativity EditThis section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed April 2020 Learn how and when to remove this template message For a single particle in special relativity it is not the case that T 1 2 p v Instead it is true that T g 1 mc2 where g is the Lorentz factorg 1 1 v 2 c 2 displaystyle gamma frac 1 sqrt 1 frac v 2 c 2 and b v c We have 1 2 p v 1 2 b g m c b c 1 2 g b 2 m c 2 g b 2 2 g 1 T displaystyle begin aligned frac 1 2 mathbf p cdot mathbf v amp frac 1 2 boldsymbol beta gamma mc cdot boldsymbol beta c 5pt amp frac 1 2 gamma beta 2 mc 2 5pt amp left frac gamma beta 2 2 gamma 1 right T end aligned The last expression can be simplified to 1 1 b 2 2 T or g 1 2 g T displaystyle left frac 1 sqrt 1 beta 2 2 right T qquad text or qquad left frac gamma 1 2 gamma right T Thus under the conditions described in earlier sections including Newton s third law of motion Fjk Fkj despite relativity the time average for N particles with a power law potential is n 2 V T O T t k 1 N 1 1 b k 2 2 T k t k 1 N g k 1 2 g k T k t displaystyle frac n 2 left langle V mathrm TOT right rangle tau left langle sum k 1 N left frac 1 sqrt 1 beta k 2 2 right T k right rangle tau left langle sum k 1 N left frac gamma k 1 2 gamma k right T k right rangle tau In particular the ratio of kinetic energy to potential energy is no longer fixed but necessarily falls into an interval 2 T T O T n V T O T 1 2 displaystyle frac 2 langle T mathrm TOT rangle n langle V mathrm TOT rangle in left 1 2 right where the more relativistic systems exhibit the larger ratios Generalizations EditLord Rayleigh published a generalization of the virial theorem in 1903 6 Henri Poincare proved and applied a form of the virial theorem in 1911 to the problem of formation of the solar system from a proto stellar cloud then known as cosmogony 7 A variational form of the virial theorem was developed in 1945 by Ledoux 8 A tensor form of the virial theorem was developed by Parker 9 Chandrasekhar 10 and Fermi 11 The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law 12 13 2 lim t T t lim t U t if and only if lim t t 2 I t 0 displaystyle 2 lim limits tau rightarrow infty langle T rangle tau lim limits tau rightarrow infty langle U rangle tau qquad text if and only if quad lim limits tau rightarrow infty tau 2 I tau 0 A boundary term otherwise must be added 14 Inclusion of electromagnetic fields EditThe virial theorem can be extended to include electric and magnetic fields The result is 15 1 2 d 2 I d t 2 V x k G k t d 3 r 2 T U W E W M x k p i k T i k d S i displaystyle frac 1 2 frac d 2 I dt 2 int V x k frac partial G k partial t d 3 r 2 T U W mathrm E W mathrm 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 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 p i k S n s m s v i v k s V i V k S m s n s displaystyle p ik Sigma n sigma m sigma langle v i v k rangle sigma V i V k Sigma m sigma n sigma and Tik is the electromagnetic stress tensor T i k e 0 E 2 2 B 2 2 m 0 d i k e 0 E i E k B i B k m 0 displaystyle T ik left frac varepsilon 0 E 2 2 frac B 2 2 mu 0 right delta ik left varepsilon 0 E i E k frac B i B 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 t 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 t2 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 t we find t R c s displaystyle tau sim frac R c mathrm s where cs 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 Relativistic uniform system EditIn case when in the physical system the pressure field the electromagnetic and gravitational fields are taken into account as well as the field of particles acceleration the virial theorem is written in the relativistic form as follows 16 W k 0 6 k 1 N F k r k displaystyle left langle W k right rangle approx 0 6 sum k 1 N langle mathbf F k cdot mathbf r k rangle where the value Wk gcT exceeds the kinetic energy of the particles T by a factor equal to the Lorentz factor gc of the particles at the center of the system Under normal conditions we can assume that gc 1 then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 1 2 but rather by the coefficient close to 0 6 The difference from the classical case arises due to considering the pressure field and the field of particles acceleration inside the system while the derivative of the scalar G is not equal to zero and should be considered as the material derivative An analysis of the integral theorem of generalized virial makes it possible to find on the basis of field theory a formula for the root mean square speed of typical particles of a system without using the notion of temperature 17 v r m s c 1 4 p h r 0 r 2 c 2 g c 2 sin 2 r c 4 p h r 0 displaystyle v mathrm rms c sqrt 1 frac 4 pi eta rho 0 r 2 c 2 gamma c 2 sin 2 left frac r c sqrt 4 pi eta rho 0 right where c displaystyle c is the speed of light h displaystyle eta is the acceleration field constant r 0 displaystyle rho 0 is the mass density of particles r displaystyle r is the current radius Unlike the virial theorem for particles for the electromagnetic field the virial theorem is written as follows 18 E k f 2 W f 0 displaystyle E kf 2W f 0 where the energy E k f A a j a g d x 1 d x 2 d x 3 displaystyle E kf int A alpha j alpha sqrt g dx 1 dx 2 dx 3 considered as the kinetic field energy associated with four current j a displaystyle j alpha and W f 1 4 m 0 F a b F a b g d x 1 d x 2 d x 3 displaystyle W f frac 1 4 mu 0 int F alpha beta F alpha beta sqrt g dx 1 dx 2 dx 3 sets the potential field energy found through the components of the electromagnetic tensor In astrophysics EditThe 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 citation needed 3 5 G M R 3 2 k B T m p 1 2 v 2 displaystyle frac 3 5 frac GM R frac 3 2 frac k mathrm B T m mathrm p frac 1 2 v 2 for a mass M radius R velocity v and temperature T The constants are Newton s constant G the Boltzmann constant kB and proton mass mp Note that these relations are only approximate and often the leading numerical factors e g 3 5 or 1 2 are neglected entirely Galaxies and cosmology virial mass and radius Edit Main article Virial mass In astronomy the mass and size of a galaxy or general overdensity is often defined in terms of the virial mass and virial radius respectively Because galaxies and overdensities in continuous fluids can be highly extended even to infinity in some models such as 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 s can be used in a similar way Taking the kinetic energy per particle of the system as T 1 2 v2 3 2 s2 and the potential energy per particle as U 3 5 GM R we can write G M R s 2 displaystyle frac GM R approx sigma 2 Here R displaystyle R is the radius at which the velocity dispersion is being measured and M is the mass within that radius The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum i e G M vir R vir s max 2 displaystyle frac GM text vir R text vir approx sigma max 2 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 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 r crit 3 H 2 8 p G displaystyle rho text crit frac 3H 2 8 pi G where H is the Hubble parameter and G is the gravitational constant A common choice for the factor is 200 which corresponds roughly to the typical over density in spherical top hat collapse see Virial mass in which case the virial radius is approximated as r vir r 200 r r 200 r crit displaystyle r text vir approx r 200 r qquad rho 200 cdot rho text crit The virial mass is then defined relative to this radius as M vir M 200 4 3 p r 200 3 200 r crit displaystyle M text vir approx M 200 frac 4 3 pi r 200 3 cdot 200 rho text crit In stars Edit The virial theorem is applicable to the cores of stars by establishing a relation between gravitational potential energy and thermal kinetic energy i e temperature As stars on the main sequence convert hydrogen into helium in their cores the mean molecular weight of the core increases and it must contract to maintain enough pressure to supports its own weight This contraction decreases its potential energy and the virial theorem states increases its thermal energy The core temperature increases even as energy is lost effectively a negative specific heat 19 This continues beyond the main sequence unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with n equals 1 no longer holds 20 See also EditVirial coefficient Virial stress Virial mass Chandrasekhar tensor Chandrasekhar virial equations Derrick s theorem Equipartition theorem Ehrenfest s theorem Pokhozhaev s identityReferences Edit Clausius RJE 1870 On a Mechanical Theorem Applicable to Heat Philosophical Magazine Series 4 40 265 122 127 doi 10 1080 14786447008640370 Collins G W 1978 Introduction The Virial Theorem in Stellar Astrophysics Pachart Press Bibcode 1978vtsa book C ISBN 978 0 912918 13 6 Bader R F W Beddall P M 1972 Virial Field Relationship for Molecular Charge Distributions and the Spatial Partitioning of Molecular Properties The Journal of Chemical Physics 56 7 Goldstein Herbert 1922 2005 1980 Classical mechanics 2d ed Reading Mass Addison Wesley Pub Co ISBN 0 201 02918 9 OCLC 5675073 CS1 maint multiple names authors list link Fock V 1930 Bemerkung zum Virialsatz Zeitschrift fur Physik A 63 11 855 858 Bibcode 1930ZPhy 63 855F doi 10 1007 BF01339281 S2CID 122502103 Lord Rayleigh 1903 Unknown Cite journal requires journal help Cite uses generic title help Poincare Henri 1911 Lecons sur les hypotheses cosmogoniques Lectures on Theories of Cosmogony Paris Hermann pp 90 91 et seq Ledoux P 1945 On the Radial Pulsation of Gaseous Stars The Astrophysical Journal 102 143 153 Bibcode 1945ApJ 102 143L doi 10 1086 144747 Parker E N 1954 Tensor Virial Equations Physical Review 96 6 1686 1689 Bibcode 1954PhRv 96 1686P doi 10 1103 PhysRev 96 1686 Chandrasekhar S Lebovitz NR 1962 The Potentials and the Superpotentials of Homogeneous Ellipsoids Astrophys J 136 1037 1047 Bibcode 1962ApJ 136 1037C doi 10 1086 147456 Chandrasekhar S Fermi E 1953 Problems of Gravitational Stability in the Presence of a Magnetic Field Astrophys J 118 116 Bibcode 1953ApJ 118 116C doi 10 1086 145732 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 Pollard Harry 1966 Mathematical Introduction to Celestial Mechanics Englewood Cliffs NJ Prentice Hall Inc ISBN 978 0 13 561068 8 Kolar M O Shea S F July 1996 A high temperature approximation for the path integral quantum Monte Carlo method Journal of Physics A Mathematical and General 29 13 3471 3494 Bibcode 1996JPhA 29 3471K doi 10 1088 0305 4470 29 13 018 Schmidt George 1979 Physics of High Temperature Plasmas Second ed Academic Press p 72 Fedosin S G 2016 The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept Continuum Mechanics and Thermodynamics 29 2 361 371 arXiv 1801 06453 Bibcode 2017CMT 29 361F doi 10 1007 s00161 016 0536 8 S2CID 53692146 Fedosin Sergey G 2018 09 24 The integral theorem of generalized virial in the relativistic uniform model Continuum Mechanics and Thermodynamics 31 3 627 638 arXiv 1912 08683 Bibcode 2018CMT tmp 140F doi 10 1007 s00161 018 0715 x ISSN 1432 0959 S2CID 125180719 via Springer Nature SharedIt Fedosin S G The Integral Theorem of the Field Energy Gazi University Journal of Science Vol 32 No 2 pp 686 703 2019 doi 10 5281 zenodo 3252783 BAIDYANATH BASU TANUKA CHATTOPADHYAY SUDHINDRA NATH BISWAS 1 January 2010 AN INTRODUCTION TO ASTROPHYSICS PHI Learning Pvt Ltd pp 365 ISBN 978 81 203 4071 8 William K Rose 16 April 1998 Advanced Stellar Astrophysics Cambridge University Press pp 242 ISBN 978 0 521 58833 1 Further reading EditGoldstein H 1980 Classical Mechanics 2nd ed Addison Wesley ISBN 978 0 201 02918 5 Collins G W 1978 The Virial Theorem in Stellar Astrophysics Pachart Press Bibcode 1978vtsa book C ISBN 978 0 912918 13 6 External links EditThe Virial Theorem at MathPages Gravitational Contraction and Star Formation Georgia State University Retrieved from https en wikipedia org w index php title Virial theorem amp oldid 1037185727, wikipedia, wiki, book,

books

, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.