The Landau kinetic equation is a transport equation of weakly coupled charged particles performing Coulomb collisions in a plasma.
The equation was derived by Lev Landau in 1936[1] as an alternative to the Boltzmann equation in the case of Coulomb interaction.
When used with the Vlasov equation, the equation yields the time evolution for collisional plasma, hence it is considered a staple kinetic model in the theory of collisional plasma.
be a one-particle Distribution function.
The equation reads:
{\displaystyle {\frac {\partial f}{\partial t}}=B{\frac {\partial }{\partial v_{i}}}\left(\int _{\mathbb {R} ^{3}}dw{\frac {\left(u^{2}\delta _{ij}-u_{i}u_{j}\right)}{u^{3}}}\left({\frac {\partial }{\partial v_{j}}}-{\frac {\partial }{\partial w_{j}}}\right)f(v)f(w)\right)}
The right-hand side of the equation is known as the Landau collision integral (in parallel to the Boltzmann collision integral).
is obtained by integrating over the intermolecular potential
For many intermolecular potentials (most notably power laws where
Landau's solution to this problem is to introduce Cutoffs at small and large angles.
The equation is used primarily in Statistical mechanics and Particle physics to model plasma.
As such, it has been used to model and study Plasma in thermonuclear reactors.
[4][5][6] It has also seen use in modeling of Active matter .
[7] The equation and its properties have been studied in depth by Alexander Bobylev.
[8] The first derivation was given in Landau's original paper.
[1] The rough idea for the derivation: Assuming a spatially homogenous gas of point particles with unit mass described by
, one may define a corrected potential for Coulomb interactions,
is the Coulomb potential,
is the Debye radius.
is then plugged it into the Boltzmann collision integral (the collision term of the Boltzmann equation) and solved for the main asymptotic term in the limit
In 1946, the first formal derivation of the equation from the BBGKY hierarchy was published by Nikolay Bogolyubov.
[9] In 1957, the equation was derived independently by Marshall Rosenbluth.
[10] Solving the Fokker–Planck equation under an inverse-square force, one may obtain:
are the Rosenbluth potentials:
{\displaystyle h_{i}=\sum _{j=1}^{n}K_{ij}\int dw{\frac {f_{i}(w,t)}{|v-w|}}}
The Fokker-Planck representation of the equation is primarily used for its convenience in numerical calculations.
A relativistic version of the equation was published in 1956 by Gersh Budker and Spartak Belyaev.
[11] Considering relativistic particles with momentum
, the equation reads:
A relativistic correction to the equation is relevant seeing as particle in hot plasma often reach relativistic speeds.