# Vlasov kinetic equation

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