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A kinetic equation for electrically-charged particles in which the interaction between the particles is described through a self-consistent electro-magnetic field. The equation has the form [[#References|[1]]], [[#References|[2]]]
 
A kinetic equation for electrically-charged particles in which the interaction between the particles is described through a self-consistent electro-magnetic field. The equation has the form [[#References|[1]]], [[#References|[2]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
 
 +
\frac{\partial  f _  \alpha  }{\partial  t }
 +
+
 +
\mathbf v  \mathop{\rm grad} _ {r}  f _  \alpha  +
 +
 
 +
\frac{e _  \alpha  }{m _  \alpha  }
 +
 
 +
\left ( \mathbf E + [ \mathbf v \times \mathbf B ] \right )
 +
\mathop{\rm grad} _ {\mathbf v }  f _  \alpha  = 0,
 +
$$
 +
 
 +
where  $  f _  \alpha  ( t, r, \mathbf v ) $
 +
is the particle distribution function, while the index  $  \alpha $
 +
is indicative for the kind of particle. The self-consistent electro-magnetic field  $  \mathbf E , \mathbf B $
 +
follows from the [[Maxwell equations|Maxwell equations]]
 +
 
 +
$$ \tag{2 }
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\left .
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\begin{array}{c}
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 +
{ \mathop{\rm rot}  \mathbf B  = \epsilon _ {0}
 +
\frac{\partial  \mathbf E }{\partial  t }
 +
+ \mathbf j ,\ \
 +
\mathop{\rm div}  \mathbf E  =
 +
\frac \rho {\epsilon _ {0} }
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, }
 +
\\
 +
 
 +
{ \mathop{\rm rot}  \mathbf E  = \
 +
 
 +
\frac{\partial  \mathbf B }{\partial  t }
 +
,\ \
 +
\mathop{\rm div}  \mathbf B  = 0, }
 +
\end{array}
 +
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968302.png" /> is the particle distribution function, while the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968303.png" /> is indicative for the kind of particle. The self-consistent electro-magnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968304.png" /> follows from the [[Maxwell equations|Maxwell equations]]
+
in which the volume density of electric charge  $  \rho $
 +
and the volume density of electric current  $  \mathbf j $
 +
are related to the particle distribution function via
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{3 }
 +
\left .
 +
\begin{array}{c}
  
in which the volume density of electric charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968306.png" /> and the volume density of electric current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968307.png" /> are related to the particle distribution function via
+
{\rho  ( t, \mathbf r )  = \sum _  \alpha  e _  \alpha  \int\limits f _  \alpha  ( t, \mathbf r , \mathbf v )  d  ^ {3} \mathbf v , }
 +
\\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968308.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
{\mathbf j ( t, \mathbf r )  = \sum _  \alpha  e _  \alpha  \int\limits f _  \alpha  ( t, \mathbf r , \mathbf v ) \mathbf v  d  ^ {3} \mathbf v . }
 +
\end{array}
  
Vlasov's kinetic equation may be obtained from the [[Liouville-equation(2)|Liouville equation]] for a distribution function of all particles of a given kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096830/v0968309.png" /> if either the particle interactions are neglected or it is assumed that the multi-particle distribution function is the product of single-particle distribution functions [[#References|[3]]], [[#References|[4]]].
+
\right \}
 +
$$
 +
 
 +
Vlasov's kinetic equation may be obtained from the [[Liouville-equation(2)|Liouville equation]] for a distribution function of all particles of a given kind $  \alpha $
 +
if either the particle interactions are neglected or it is assumed that the multi-particle distribution function is the product of single-particle distribution functions [[#References|[3]]], [[#References|[4]]].
  
 
The system of equations (1), (2), (3), proposed by A.A. Vlasov, is extensively employed in plasma physics. The linear theory, based on linearization of equations (1), (2), (3), is the most fully developed. It is used in the study of small oscillations and the stability of a plasma [[#References|[5]]]. The quasi-linear theory, which makes it possible to study non-linear effects, is in full development.
 
The system of equations (1), (2), (3), proposed by A.A. Vlasov, is extensively employed in plasma physics. The linear theory, based on linearization of equations (1), (2), (3), is the most fully developed. It is used in the study of small oscillations and the stability of a plasma [[#References|[5]]]. The quasi-linear theory, which makes it possible to study non-linear effects, is in full development.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Vlasov,  "On oscillation properties of ionized gases"  ''Zh. Eksper. Teoret. Fiz.'' , '''8''' :  3  (1938)  pp. 291–318  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Vlasov,  "Many-particle theory and its applocation to plasmas" , Gordon &amp; Breach  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  "Problems of a dynamic theory in statistical physics" , North-Holland  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Silin,  "Introduction to the kinetic theory of gases" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.P. Silin,  A.A. Rukhadze,  "Electromagnetic properies of plasma and plasma-like media" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Vlasov,  "On oscillation properties of ionized gases"  ''Zh. Eksper. Teoret. Fiz.'' , '''8''' :  3  (1938)  pp. 291–318  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Vlasov,  "Many-particle theory and its applocation to plasmas" , Gordon &amp; Breach  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  "Problems of a dynamic theory in statistical physics" , North-Holland  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.P. Silin,  "Introduction to the kinetic theory of gases" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.P. Silin,  A.A. Rukhadze,  "Electromagnetic properies of plasma and plasma-like media" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Ecker,  "Theory of fully ionized plasmas" , Acad. Press  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Ecker,  "Theory of fully ionized plasmas" , Acad. Press  (1972)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


A kinetic equation for electrically-charged particles in which the interaction between the particles is described through a self-consistent electro-magnetic field. The equation has the form [1], [2]

$$ \tag{1 } \frac{\partial f _ \alpha }{\partial t } + \mathbf v \mathop{\rm grad} _ {r} f _ \alpha + \frac{e _ \alpha }{m _ \alpha } \left ( \mathbf E + [ \mathbf v \times \mathbf B ] \right ) \mathop{\rm grad} _ {\mathbf v } f _ \alpha = 0, $$

where $ f _ \alpha ( t, r, \mathbf v ) $ is the particle distribution function, while the index $ \alpha $ is indicative for the kind of particle. The self-consistent electro-magnetic field $ \mathbf E , \mathbf B $ follows from the Maxwell equations

$$ \tag{2 } \left . \begin{array}{c} { \mathop{\rm rot} \mathbf B = \epsilon _ {0} \frac{\partial \mathbf E }{\partial t } + \mathbf j ,\ \ \mathop{\rm div} \mathbf E = \frac \rho {\epsilon _ {0} } , } \\ { \mathop{\rm rot} \mathbf E = \ \frac{\partial \mathbf B }{\partial t } ,\ \ \mathop{\rm div} \mathbf B = 0, } \end{array} \right \} $$

in which the volume density of electric charge $ \rho $ and the volume density of electric current $ \mathbf j $ are related to the particle distribution function via

$$ \tag{3 } \left . \begin{array}{c} {\rho ( t, \mathbf r ) = \sum _ \alpha e _ \alpha \int\limits f _ \alpha ( t, \mathbf r , \mathbf v ) d ^ {3} \mathbf v , } \\ {\mathbf j ( t, \mathbf r ) = \sum _ \alpha e _ \alpha \int\limits f _ \alpha ( t, \mathbf r , \mathbf v ) \mathbf v d ^ {3} \mathbf v . } \end{array} \right \} $$

Vlasov's kinetic equation may be obtained from the Liouville equation for a distribution function of all particles of a given kind $ \alpha $ if either the particle interactions are neglected or it is assumed that the multi-particle distribution function is the product of single-particle distribution functions [3], [4].

The system of equations (1), (2), (3), proposed by A.A. Vlasov, is extensively employed in plasma physics. The linear theory, based on linearization of equations (1), (2), (3), is the most fully developed. It is used in the study of small oscillations and the stability of a plasma [5]. The quasi-linear theory, which makes it possible to study non-linear effects, is in full development.

References

[1] A.A. Vlasov, "On oscillation properties of ionized gases" Zh. Eksper. Teoret. Fiz. , 8 : 3 (1938) pp. 291–318 (In Russian)
[2] A.A. Vlasov, "Many-particle theory and its applocation to plasmas" , Gordon & Breach (1961) (Translated from Russian)
[3] N.N. Bogolyubov, "Problems of a dynamic theory in statistical physics" , North-Holland (1962) (Translated from Russian)
[4] V.P. Silin, "Introduction to the kinetic theory of gases" , Moscow (1971) (In Russian)
[5] V.P. Silin, A.A. Rukhadze, "Electromagnetic properies of plasma and plasma-like media" , Moscow (1961) (In Russian)

Comments

References

[a1] G. Ecker, "Theory of fully ionized plasmas" , Acad. Press (1972)
How to Cite This Entry:
Vlasov kinetic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vlasov_kinetic_equation&oldid=49157
This article was adapted from an original article by D.P. Kostomarov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article