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. [2][3]
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 ), the expression for diverges. Landau's solution to this problem is to introduce Cutoffs at small and large angles.
The equation and its properties have been studied in depth by Alexander Bobylev.[8]
Derivationsedit
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, , where is the Coulomb potential, , and is the Debye radius. The potential 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 .
^ abLandau, L.D. (1936). "Kinetic equation for the case of coulomb interaction". Phys. Z. Sowjetunion. 10: 154–164.
^Bobylev, Alexander (2015). "On some properties of the landau kinetic equation". Journal of Statistical Physics. 161 (6): 1327. Bibcode:2015JSP...161.1327B. doi:10.1007/s10955-015-1311-0. S2CID 39781.
^ abRobert M. Strain, Maja Tasković (2019). "Entropy dissipation estimates for the relativistic Landau equation, and applications". Journal of Functional Analysis. 277 (4): 1139–1201. arXiv:1806.08720. doi:10.1016/j.jfa.2019.04.007. S2CID 119323748.
^Landau kinetic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Landau_kinetic_equation&oldid=47573