Landau kinetic equation

A kinetic equation for a weakly interacting gas, in particular, the transfer equation of charged particles in a plasma with Coulomb collisions taken into account. It was obtained by L.D. Landau (see , [2]). For systems with Coulomb interaction, in the derivation of the Landau kinetic equation the coefficients of the equation contain a divergent integral (the "Coulomb logarithmCoulomb logarithm" : $ \mathop{\rm ln} \Lambda _ {ab} $, the logarithm of the ratio of the maximum and minimum impact parameter in the collision of two charged particles $ a $ and $ b $). To obtain an approximate non-divergent result one "cuts off" the integral: for the upper limit of integration one takes the length of the Debye electrostatic screening, and for the lower limit one takes the distance of neighbouring interaction (or the quantum-mechanical wave length). The ad hoc "cutting off" of the integral, which does not follow from the derivation of the Landau kinetic equation itself, remains an open question concerning the construction of an adequate kinetic equation for systems with Coulomb interaction. Different forms of such equations have been proposed (see [3], for example) (they also are not free from divergences). In these equations one takes account of dynamic screening, which depends on the velocities of the particles.

For a rarefied gas of test particles interacting with an equilibrium background, the Landau kinetic equation becomes the linear Fokker–Planck equation. For an inhomogeneous plasma the Landau collision integral must be added to the right-hand side of the Vlasov kinetic equation. The resulting equation is called the Vlasov–Landau equation (see [3]).

For a mixture of particles of several types the system of Landau kinetic equations can be written in the form

$$ \tag{1 } \frac{d f _ {a} }{d t } = \ I _ {a} + S _ {a} , $$

where $ f _ {a} \equiv f _ {a} ( \mathbf r , \mathbf v , t ) $ is the distribution function for particles of type $ a $ in the $ 6 $-dimensional phase space of the coordinates $ \mathbf r $ and velocities $ \mathbf v $ ($ t $ is the time). The function $ S _ {a} ( \mathbf r , \mathbf v , t ) \equiv S $ describes the sources of the particles, $ I _ {a} $ is the collision integral, which can be reduced to the form

$$ \tag{2 } I _ {a} = \Gamma _ {a} \left [ - \frac \partial {\partial v _ {i} } \left ( f _ {a} \frac{\partial h _ {a} }{\partial v _ {i} } \right ) + \frac{1}{2} \frac{\partial ^ {2} }{\partial v _ {i} \partial v _ {j} } \left ( f _ {a} \frac{\partial ^ {2} g _ {a} }{\partial v _ {i} \partial v _ {j} } \right ) \right ] , $$

where the summation is assumed to be over identical indices $ i , j $ from 1 to $ 3 $, $ \Gamma _ {a} = 4 \pi Z _ {a} ^ {4} e ^ {4} / m _ {a} ^ {2} $; $ m _ {a} $ and $ Z _ {a} e $ are the mass and charge of particles of type $ a $, $ e $ is the charge of an electron,

$$ \tag{3 } \left . \begin{array}{c} g _ {a} = \sum _ { b } \left ( \frac{Z _ {b} }{Z _ {a} } \right ) ^ {2} \mathop{\rm ln} \Lambda _ {ab} \int\limits f _ {b} ( \mathbf r , \mathbf v ^ \prime , t ) | \mathbf v - \mathbf v ^ \prime | d \mathbf v ^ \prime , \\ h _ {a} = \sum _ { b } \left ( \frac{m _ {a} + m _ {b} }{m _ {b} } \right ) \left ( \frac{Z _ b}{Z _ a} \right ) ^ {2} \mathop{\rm ln} \Lambda _ {ab} \times \\ \times \int\limits f _ {b} ( \mathbf r , \mathbf v ^ \prime , t ) | \mathbf v - \mathbf v ^ \prime | ^ {- 1} d \mathbf v ^ \prime \end{array} \right \} $$

are the potentials introduced in [2], and $ \mathop{\rm ln} \Lambda _ {ab} $ are the Coulomb logarithms, which depend on the mean energies of the particles. The collision integral (2) contains an elliptic differential operator with respect to the velocity whose coefficients are expressed in terms of integral operators of potential type of $ f _ {a} ( \mathbf r , \mathbf v , t ) $. For an inhomogeneous plasma,

$$ \frac{d f _ {a} }{dt} \equiv \ \frac{\partial f _ {a} }{\partial t } + \mathbf v \frac{\partial f _ {a} }{\partial \mathbf r } + \frac{Z _ {a} e }{m _ {a} } \mathbf F \frac{\partial f _ {a} }{\partial \mathbf v } , $$

where $ Z _ {a} e \mathbf F $ is the force acting on particles of type $ a $.

Landau's kinetic equations make it possible to obtain hydrodynamic conservation equations for the densities of mass, momentum and internal energy, and also the Boltzmann $ H $-theorem.

The existence of a generalized solution of the Landau kinetic equation has been proved locally (see [4]).

A numerical solution of the Landau kinetic equation has been carried out on a computer to calculate the loss of particles from open magnetic traps (see [5]), to determine the coefficient of multiplication of energy in toroidal thermonuclear reactors (see [6]), and to evaluate additional methods of heating a plasma in tokamaks. In conditions of good confinement of a plasma in magnetic traps it is necessary to use completely-conservative difference schemes (see [7]), which exactly conserve the total number of particles and their energy in solving of the Landau kinetic equation.

References

[1a]	L.D. Landau, "Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung" Phys. Z. Sowjetunion , 10 : 2 (1936) pp. 154–164
[1b]	L.D. Landau, "The kinetic equation in the case of Coulomb interaction" Zh. Eksper. i Teoret. Fiz. , 7 : 2 (1937) pp. 203–209 (Translated from German)
[2]	B.A. Trubnikov, "Problems of plasma theory" M.A. Leontovich (ed.) , Reviews in plasma physics , New York (1965) pp. 105–204
[3]	R. Balescu, "Equilibrium and non-equilibrium statistical mechanics" , 1–2 , Wiley (1975)
[4]	A.A. Arsen'ev, N.V. Peskov, "On the existence of a generalized solution of Landau's equation" USSR Comp. Math. Math. Phys. , 17 : 4 (1977) pp. 241–245 Zh. Vychisl. Mat. i Mat. Fiz. , 17 (1977) pp. 1063–1068
[5]	J. Killeen, K.D. Marx, "Methods in computational physics" , 9 , Acad. Press (1970)
[6]	J. Killeen, A.A. Mirin, M.E. Rensink, "Methods in computational physics" , 16 , Acad. Press (1976)
[7]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)