# Kinetic equation

An equation in non-equilibrium statistical physics that is used in gas theory, aerodynamics, plasma physics, the theory of the passage of particles through matter, and the theory of radiation transfer. The solution of the kinetic equation determines the distribution function of the dynamical states of a single particle, which usually depends on time, coordinates and velocity.

In 1872 L. Boltzmann formulated the fundamental kinetic equation of gas theory. It is a non-linear integro-differential equation (see Boltzmann equation), which describes the motion of the molecules as a certain random process determined by collisions between pairs of molecules. The coefficients (effective cross sections) entering into the equation are calculated from the equations of classical mechanics. By studying the properties of the solutions of the kinetic equation, Boltzmann gave a molecular-kinetic interpretation of the second law of thermodynamics and established a statistical meaning of the notion of entropy (see Boltzmann $H$-theorem). In quantum statistical physics the Boltzmann equation is described in the simplest case by analogy with the classical case, but using quantum effective cross sections and taking symmetry requirements into account. For a relativistic gas the Boltzmann equation is stated in covariant form. A method has been developed for obtaining the kinetic equation of gas theory taking into account the correlation between the dynamical states of the molecules (see Bogolyubov chain of equations). Starting from the Liouville equation one can use this method to obtain in the lowest approximation the Boltzmann equation if one uses a power series expansion in terms of the density of the gas.

The power series expansion in the strength of the interaction energy leads to the Vlasov kinetic equation with a self-adapting field, and in the subsequent approximation in the spatially-homogeneous case to the Landau kinetic equation, describing the so-called "diffusion in velocity space" .

Few exact solutions of the non-linear kinetic equation are known. It is difficult to solve it numerically, even when using a computer. More thoroughly investigated is the linearized kinetic equation, which describes small deviations from the equilibrium solution of the non-linear equation. It is the same in form as the linear transport equations occurring in the theory of radiative and neutron transfer (see Radiative transfer theory), and also in the theory of the passage of particles through matter.

The theory of radiative transfer is close, with regard to its problems and methods of their solution, to that of neutron transfer. Calculations regarding nuclear reactors and protection against nuclear radiation have required the creation of effective methods for solving the kinetic equation of the transfer of neutrons and gamma-quanta, and have also promoted the creation of a mathematical theory of the linear kinetic equation. Existence and uniqueness theorems and the asymptotic behaviour of the non-linear Boltzmann equation have been widely studied [1]. In the case of the linear equation these theorems are obtained in the most general formulation of the mathematical theory of nuclear reactors [2], [3], [4]. A number of problems have been solved by analytic methods, and in the general case many approximate methods, convenient for programming on a computer, have been developed for its solution (see Transport equations, numerical methods).

The problem of the passage of charged particles through matter often reduces to the solution of the linear transport equation in the presence of anisotropic scattering or to the solution of the linearized Landau equation (for example, in the determination of the angular distribution of the particles scattered from the surface of a body, or in the determination of the energy spectrum of charged particles in matter).

#### References

[1] | E.W. Montroll (ed.) J.L. Lebowitz (ed.) , Nonequilibrium phenomena , 1. The Boltzmann equation , North-Holland (1983) MR0724618 Zbl 0583.76004 |

[2] | V.S. Vladimirov, "Mathematical problems in the uniform-velocity theory of transport" Trudy Mat. Inst. Steklov. , 61 (1961) (In Russian) |

[3] | S.B. Shikhov, "Questions in the mathematical theory of reactors. Linear analysis" , Moscow (1973) (In Russian) |

[4] | K.M. Case, P.F. Zweifel, "Linear transport theory" , Addison-Wesley (1967) MR0225547 Zbl 0162.58903 |

[5] | M.N. Kogan, "Rarified gas dynamics" , Plenum (1969) (Translated from Russian) |

[6] | V.P. Silin, "Introduction to the kinetic theory of gases" , Moscow (1971) (In Russian) |

[7] | V.V. Sobolev, "Transfer of radiation energy in the atmospheres of stars and planets" , Moscow (1956) (In Russian) |

#### Comments

See also the references to Boltzmann equation.

#### References

[a1] | C. Cercignani, "Mathematical methods in kinetic theory" , Plenum (1969) MR0255199 Zbl 0191.25103 |

[a2] | C. Cercignani, "The Boltzmann equation and its applications" , Springer (1988) MR1313028 Zbl 0646.76001 |

[a3] | T. Carleman, "Problèmes mathématiques dans la théorie kinétique des gaz" , Mittag-Leffler Inst. (1957) |

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Kinetic equation.

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