Virial

Virial for the set $N$ ${\ displaystyle N}$ $N$ Point particles in mechanics are defined as a scalar function:

\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k},

{\ displaystyle \ sum _ {k = 1} ^ {N} \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k},}

{\ displaystyle \ sum _ {k = 1} ^ {N} \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k},}

Where $\mathbf {r} _{k}$ ${\ displaystyle \ mathbf {r} _ {k}}$ ${\ mathbf {r}} _ {k}$ and $\mathbf {F} _{k}$ ${\ displaystyle \ mathbf {F} _ {k}}$ ${\ mathbf {F}} _ {k}$ - spatial vectors of coordinates and forces for $k$ ${\ displaystyle k}$ $k$ th particles.

The expression "virial" comes from the Latin words "vis" , "viris" - "strength" or "energy". It was introduced by Clausius in 1870 .

Content

Virial theorem

For a stable system connected by potential forces, the viriale theorem ^{[1] is} valid:

2\langle T\rangle =-\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,

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

Where $\langle T\rangle$ ${\ displaystyle \ langle T \ rangle}$ represents the average total kinetic energy and $\mathbf {F} _{k}$ ${\ displaystyle \ mathbf {F} _ {k}}$ - force acting on $k$ ${\ displaystyle k}$ particle

In the particular case when the potential force corresponding to the force $V(r)$ ${\ displaystyle V (r)}$ proportional to $n$ ${\ displaystyle n}$ -th degree of distance between particles $r$ ${\ displaystyle r}$ , virial theorem takes a simple form

2\langle T\rangle =n\langle U\rangle .

{\ displaystyle 2 \ langle T \ rangle = n \ langle U \ rangle.}

In other words, twice the average total kinetic energy $T$ ${\ displaystyle T}$ equals $n$ ${\ displaystyle n}$ -fold average total potential energy $U$ ${\ displaystyle U}$ .

The significance of the virial theorem is that it makes it possible to calculate the average total kinetic energy even for very complex systems that are inaccessible for an exact solution, which are considered, for example, by statistical mechanics . For example, the virial theorem can be used to derive an equipartial theorem (a theorem on the uniformity of energy distribution over degrees of freedom) or to calculate the Chandrasekhar limit for the stability of a white dwarf .

Time derivative and averaging

Another scalar function is closely related to virial:

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

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

Where $\mathbf {p} _{k}$ ${\ displaystyle \ mathbf {p} _ {k}}$ there is momentum $k$ ${\ displaystyle k}$ particles

Time derivative of the function $G$ ${\ displaystyle G}$ can be written like this:

{\frac {dG}{dt}}=\sum _{k=1}^{N}{\frac {d\mathbf {p} _{k}}{dt}}\cdot \mathbf {r} _{k}+\sum _{k=1}^{N}\mathbf {p} _{k}\cdot {\frac {d\mathbf {r} _{k}}{dt}}=

{\ displaystyle {\ frac {dG} {dt}} = \ sum _ {k = 1} ^ {N} {\ frac {d \ mathbf {p} _ {k}} {dt}} \ cdot \ mathbf { r} _ {k} + \ sum _ {k = 1} ^ {N} \ mathbf {p} _ {k} \ cdot {\ frac {d \ mathbf {r} _ {k}} {dt}} = }

=\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}+\sum _{k=1}^{N}m_{k}{\frac {d\mathbf {r} _{k}}{dt}}\cdot {\frac {d\mathbf {r} _{k}}{dt}}

{\ displaystyle = \ sum _ {k = 1} ^ {N} \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k} + \ sum _ {k = 1} ^ {N} m_ {k} {\ frac {d \ mathbf {r} _ {k}} {dt}} \ cdot {\ frac {d \ mathbf {r} _ {k}} {dt}}}

or in a simpler form

{\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}.

{\ displaystyle {\ frac {dG} {dt}} = 2T + \ sum _ {k = 1} ^ {N} \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k}.}

Here $m_{k}$ ${\ displaystyle m_ {k}}$ weight $k$ ${\ displaystyle k}$ particles $\mathbf {F} _{k}={\frac {d\mathbf {p} _{k}}{dt}}$ ${\ displaystyle \ mathbf {F} _ {k} = {\ frac {d \ mathbf {p} _ {k}} {dt}}}$ - the total force acting on the particle, and $T$ ${\ displaystyle T}$ - total kinetic energy of the system

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}}.

{\ 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}}.}

Averaging this derivative over time $\tau$ ${\ displaystyle \ tau}$ is defined as follows:

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

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

where do we get the exact solution

\left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=2\langle T\rangle _{\tau }+\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.

{\ displaystyle \ left \ langle {\ frac {dG} {dt}} \ right \ rangle _ {\ tau} = 2 \ langle T \ rangle _ {\ tau} + \ sum _ {k = 1} ^ {N } \ langle \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k} \ rangle _ {\ tau}.}

Virial theorem

The virial theorem states:

If a $\left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=0$ ${\ displaystyle \ left \ langle {\ frac {dG} {dt}} \ right \ rangle _ {\ tau} = 0}$ then
$2\langle T\rangle _{\tau }=-\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.$ ${\ displaystyle 2 \ langle T \ rangle _ {\ tau} = - \ sum _ {k = 1} ^ {N} \ langle \ mathbf {F} _ {k} \ cdot \ mathbf {r} _ {k} \ rangle _ {\ tau}.}$

There are several reasons why time averaging disappears, that is, $\left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=0$ ${\ displaystyle \ left \ langle {\ frac {dG} {dt}} \ right \ rangle _ {\ tau} = 0}$ . One often cited reason appeals to related systems , that is, systems that remain limited in space. In this case, the function $G^{\mathrm {bound} }$ ${\ displaystyle G ^ {\ mathrm {bound}}}$ usually limited to two limits $G_{\min }$ ${\ displaystyle G _ {\ min}}$ and $G_{\max }$ ${\ displaystyle G _ {\ max}}$ and the average tends to zero in the limit of very long times $\tau$ ${\ displaystyle \ tau}$ :

\lim _{\tau \to \infty }\left|\left\langle {\frac {dG^{\mathrm {bound} }}{dt}}\right\rangle _{\tau }\right|=\lim _{\tau \to \infty }\left|{\frac {G(\tau )-G(0)}{\tau }}\right|\leqslant \lim _{\tau \to \infty }{\frac {G_{\max }-G_{\min }}{\tau }}=0.

{\ displaystyle \ lim _ {\ tau \ to \ infty} \ left | \ left \ langle {\ frac {dG ^ {\ mathrm {bound}}} {dt}} \ right \ rangle _ {\ tau} \ right | = \ lim _ {\ tau \ to \ infty} \ left | {\ frac {G (\ tau) -G (0)} {\ tau}} \ right | \ leqslant \ lim _ {\ tau \ to \ infty} {\ frac {G _ {\ max} -G _ {\ min}} {\ tau}} = 0.}

This conclusion is valid only for those systems in which the function $G$ ${\ displaystyle G}$ depends only on time and does not significantly depend on the coordinates. If the average value of the time derivative $\left\langle {\frac {dG}{dt}}\right\rangle _{\tau }\approx 0$ ${\ displaystyle \ left \ langle {\ frac {dG} {dt}} \ right \ rangle _ {\ tau} \ approx 0}$ , virial theorem has the same degree of approximation.

Ratio to potential energy

Full power $\mathbf {F} _{k}$ ${\ displaystyle \ mathbf {F} _ {k}}$ acting on the particle $k$ ${\ displaystyle k}$ , is the sum of all the forces acting by other particles $j$ ${\ displaystyle j}$ in system

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

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

Where $\mathbf {F} _{jk}$ ${\ displaystyle \ mathbf {F} _ {jk}}$ - force acting on the particle $j$ ${\ displaystyle j}$ from the particle side $k$ ${\ displaystyle k}$ . Hence, the term in the time derivative of the function $G$ ${\ displaystyle G}$ containing strength can be rewritten as:

\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}.

{\ displaystyle \ 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 there is no self-action (i.e. $\mathbf {F} _{jk}=0$ ${\ displaystyle \ mathbf {F} _ {jk} = 0}$ where $j=k$ ${\ displaystyle j = k}$ ), we'll get:

\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 (\mathbf {r} _{k}-\mathbf {r} _{j}),

{\ displaystyle \ 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 ( \ mathbf {r} _ {k} - \ mathbf {r} _ {j}),}

^[2]

where we assume that Newton's third law is fulfilled, that is, $\mathbf {F} _{jk}=-\mathbf {F} _{kj}$ ${\ displaystyle \ mathbf {F} _ {jk} = - \ mathbf {F} _ {kj}}$ (equal in magnitude and opposite in direction).

It often happens that forces can be obtained from potential energy $V$ ${\ displaystyle V}$ which is a function of distance only $r_{jk}$ ${\ displaystyle r_ {jk}}$ between point particles $j$ ${\ displaystyle j}$ and $k$ ${\ displaystyle k}$ . Since force is a gradient of potential energy with the opposite sign, we have in this case

\mathbf {F} _{jk}=-\nabla _{\mathbf {r} _{k}}V=-{\frac {dV}{dr}}{\frac {\mathbf {r} _{k}-\mathbf {r} _{j}}{r_{jk}}},

{\ displaystyle \ mathbf {F} _ {jk} = - \ nabla _ {\ mathbf {r} _ {k}} V = - {\ frac {dV} {dr}} {\ frac {\ mathbf {r} _ {k} - \ mathbf {r} _ {j}} {r_ {jk}}},}

which is equal in magnitude and opposite in direction to the vector $\mathbf {F} _{kj}=-\nabla _{\mathbf {r} _{j}}V$ ${\ displaystyle \ mathbf {F} _ {kj} = - \ nabla _ {\ mathbf {r} _ {j}} V}$ - force acting on the part of the particle $k$ ${\ displaystyle k}$ on a particle $j$ ${\ displaystyle j}$ as can be shown by simple calculations. Hence the force term in the derivative of the function $G$ ${\ displaystyle G}$ time equals

\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}-\mathbf {r} _{j})=-\sum _{k=1}^{N}\sum _{j<k}{\frac {dV}{dr}}{\frac {(\mathbf {r} _{k}-\mathbf {r} _{j})^{2}}{r_{jk}}}=-\sum _{k=1}^{N}\sum _{j<k}{\frac {dV}{dr}}r_{jk}.

{\ displaystyle \ 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} - \ mathbf {r} _ {j}) = - \ sum _ {k = 1} ^ { N} \ sum _ {j <k} {\ frac {dV} {dr}} {\ frac {(\ mathbf {r} _ {k} - \ mathbf {r} _ {j}) ^ {2}} {r_ {jk}}} = - \ sum _ {k = 1} ^ {N} \ sum _ {j <k} {\ frac {dV} {dr}} r_ {jk}.}

Application to force depending on distance in a power-law manner

It often turns out that potential energy $V$ ${\ displaystyle V}$ has the form of a power function

V(r_{jk})=\alpha r_{jk}^{n},

{\ displaystyle V (r_ {jk}) = \ alpha r_ {jk} ^ {n},}

where is the coefficient $\alpha$ ${\ displaystyle \ alpha}$ and indicator $n$ ${\ displaystyle n}$ - constants. In this case, the power term in the derivative of the function $G$ ${\ displaystyle G}$ time is given by the following equations

-\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}nV(r_{jk})=nU,

{\ displaystyle - \ 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} nV (r_ {jk}) = nU,}

Where $U$ ${\ displaystyle U}$ - total potential energy of the system:

U=\sum _{k=1}^{N}\sum _{j<k}V(r_{jk}).

{\ displaystyle U = \ sum _ {k = 1} ^ {N} \ sum _ {j <k} V (r_ {jk}).}

In cases where the average of the time derivative $\left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=0$ ${\ displaystyle \ left \ langle {\ frac {dG} {dt}} \ right \ rangle _ {\ tau} = 0}$ , equation is executed

\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 U\rangle _{\tau }.

{\ displaystyle \ 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 U \ rangle _ {\ tau}.}

The most commonly cited example is gravitational attraction , for which $n=-1$ ${\ displaystyle n = -1}$ . In that case, the average kinetic energy is half the average negative potential energy.

\langle T\rangle _{\tau }=-{\frac {1}{2}}\langle U\rangle _{\tau }.

{\ displaystyle \ langle T \ rangle _ {\ tau} = - {\ frac {1} {2}} \ langle U \ rangle _ {\ tau}.}

This result is remarkably useful for complex gravitational systems, such as a solar system or a galaxy , and is also performed for an electrostatic system , for which $n=-1$ ${\ displaystyle n = -1}$ also.

Although this expression was obtained for classical mechanics, the virial theorem is also true for quantum mechanics .

Electromagnetic field accounting

The virial theorem can be generalized to the case of electric and magnetic fields. Result: ^[3]

{\frac {1}{2}}{\frac {d^{2}}{dt^{2}}}I+\int \limits _{V}x_{k}{\frac {\partial P_{k}}{\partial t}}\,d^{3}r=2(T+U)+W^{E}+W^{M}-\int x_{k}(p_{ik}+T_{ik})\,dS_{i},

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

Where $I$ ${\ displaystyle I}$ - moment of inertia $P$ ${\ displaystyle P}$ - Poynting vector , $T$ ${\ displaystyle T}$ - the kinetic energy of the "liquid" $U$ ${\ displaystyle U}$ - random thermal energy of particles, $W^{E}$ ${\ displaystyle W ^ {E}}$ and $W^{M}$ ${\ displaystyle W ^ {M}}$ - the energy of the electric and magnetic fields in the considered volume of the system, $p_{ik}$ ${\ displaystyle p_ {ik}}$ - fluid pressure tensor, expressed in the local moving coordinate system, the accompanying fluid:

p_{ik}=\Sigma n^{\sigma }m^{\sigma }\langle v_{i}v_{k}\rangle ^{\sigma }-V_{i}V_{k}\Sigma m^{\sigma }n^{\sigma }

{\ 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 $T_{ik}$ ${\ displaystyle T_ {ik}}$ - the energy-momentum tensor of the electromagnetic field:

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).

{\ 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 limited configuration of magnetic fields and plasma. Using the virial theorem, it is easy to show that any such configuration expands, if not constrained by external forces. In the final configuration, the surface integral will disappear without applying walls or magnetic coils. Since all other terms on the right are positive, the acceleration of the moment of inertia will also be positive. Easy to estimate expansion time $\tau$ ${\ displaystyle \ tau}$ . If the total mass $M$ ${\ displaystyle M}$ limited within radius $R$ ${\ displaystyle R}$ then the moment of inertia is approximately $MR^{2}$ ${\ displaystyle MR ^ {2}}$ , and the left side in the virial theorem - $MR^{2}/\tau ^{2}$ ${\ displaystyle MR ^ {2} / \ tau ^ {2}}$ . The terms on the right add up to about $pR^{3}$ ${\ displaystyle pR ^ {3}}$ where $p$ ${\ displaystyle p}$ - greater of plasma pressure or magnetic pressure. Equating these two terms and considering that $M=m_{i}nV\sim m_{i}nR^{3}$ ${\ displaystyle M = m_ {i} nV \ sim m_ {i} nR ^ {3}}$ , $p\sim nkT$ ${\ displaystyle p \ sim nkT}$ , $c_{s}^{2}\sim {\frac {kT}{m_{i}}}$ ${\ displaystyle c_ {s} ^ {2} \ sim {\ frac {kT} {m_ {i}}}}$ where $m_{i}$ ${\ displaystyle m_ {i}}$ there is a mass of ion $n$ ${\ displaystyle n}$ - ion concentration $V\sim R^{3}$ ${\ displaystyle V \ sim R ^ {3}}$ - the volume of the plasmoid, $k$ ${\ displaystyle k}$ - Boltzmann constant, $T$ ${\ displaystyle T}$ - temperature, for $\tau$ ${\ displaystyle \ tau}$ we find:

\tau \sim R/c_{s},

{\ displaystyle \ tau \ sim R / c_ {s},}

Where $c_{s}$ ${\ displaystyle c_ {s}}$ is the speed of the ion acoustic wave (or the Alphen wave , if the magnetic pressure is higher than the plasma pressure). Thus, the plasmoid lifetime is expected to be equal in order of magnitude to the acoustic (Alfen) transit time.

Relativistic homogeneous system

In the case when the pressure, electromagnetic and gravitational fields, as well as the particle acceleration fields are taken into account in the physical system, the virial theorem in relativistic form is written as: ^[4]

~\langle W_{k}\rangle \approx -0,6\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,

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

and the magnitude $~W_{k}\approx \gamma _{c}T$ ${\ displaystyle ~ W_ {k} \ approx \ gamma _ {c} T}$ exceeds the kinetic energy of the particles $T$ ${\ displaystyle ~ T}$ by a factor equal to the Lorenz factor $~\gamma _{c}$ ${\ displaystyle ~ \ gamma _ {c}}$ particles in the center of the system. Under normal conditions, we can assume that $~\gamma _{c}\approx 1$ ${\ displaystyle ~ \ gamma _ {c} \ approx 1}$ and then it can be seen that in the virial theorem the kinetic energy is related to the potential energy not by a factor of 0.5, but rather by a factor close to 0.6. The difference from the classical case arises due to taking into account the pressure field and the particle acceleration field inside the system, while the derivative of the scalar function $G$ ${\ displaystyle ~ G}$ is not equal to zero and should be considered as a derivative of Lagrange .

An analysis of the integral theorem of a generalized virial allows us to find, on the basis of field theory, a formula for the root-mean-square velocity of typical particles of a system, without using the concept of temperature:

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)}}}}},

{\ 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 $~\eta$ ${\ displaystyle ~ \ eta}$ - constant field of acceleration, $~\rho _{0}$ ${\ displaystyle ~ \ rho _ {0}}$ - the mass density of particles, $r$ ${\ displaystyle ~ r}$ - current radius.

Notes

↑ Sivukhin D.V. General Physics Course. Mechanics. - M .: Science, 1979. - Circulation 50 000 copies. - with. 141.
↑ Proof of this equality
↑ Schmidt G. Physics of High Temperature Plasmas. - Second edition. - Academic Press, 1979. - p. 72.
↑ Fedosin, SG, viV.R. - 2016. - Vol. 29 , no. 2 - P. 361-371 . - DOI : 10.1007 / s00161-016-0536-8 . - . - arXiv : 1801.06453 .
↑ Fedosin, Sergey G. The integral theorem of generalized virial in the relativistic uniform model (Eng.) // Continuum Mechanics and Thermodynamics: journal. - 2018. - September 24 ( vol. 31 , no. 3 ). - P. 627-638 . - ISSN 1432-0959 . - DOI : 10.1007 / s00161-018-0715-x . - .

Literature

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

[1] Sivukhin D.V. General Physics Course. Mechanics. - M .: Science, 1979. - Circulation 50 000 copies. - with. 141.

[2] Proof of this equality

[3] Schmidt G. Physics of High Temperature Plasmas. - Second edition. - Academic Press, 1979. - p. 72.

[4] Fedosin, SG, viV.R. - 2016. - Vol. 29 , no. 2 - P. 361-371 . - DOI : 10.1007 / s00161-016-0536-8 . - . - arXiv : 1801.06453 .

[5] Fedosin, Sergey G. The integral theorem of generalized virial in the relativistic uniform model (Eng.) // Continuum Mechanics and Thermodynamics: journal. - 2018. - September 24 ( vol. 31 , no. 3 ). - P. 627-638 . - ISSN 1432-0959 . - DOI : 10.1007 / s00161-018-0715-x . - .

Virial

Content

Virial theorem

Time derivative and averaging

Virial theorem

Ratio to potential energy

Application to force depending on distance in a power-law manner

Electromagnetic field accounting

Relativistic homogeneous system

See also

Notes

Literature

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