Landau kinetic equation

Derivations

The first derivation was given in Landau's original paper. The rough idea for the derivation:

Assuming a spatially homogenous gas of point particles with unit mass described by ${\displaystyle f(v,t)}$, one may define a corrected potential for Coulomb interactions, ${\hat{U}}_{ij}=U_{ij}\exp \left(-\frac{r_{ij}}{r_D}\right)$, where ${\displaystyle U_{ij}}$ is the Coulomb potential, $U_{ij}=\frac{e_i{e}_j}{|x_i-x_j|}$, and ${\displaystyle r_D}$ is the Debye radius. The potential ${\displaystyle \widehat{U_{ij}}}$ 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 ${\displaystyle r_D\to \mathrm{\infty }}$.

In 1946, the first formal derivation of the equation from the BBGKY hierarchy was published by Nikolay Bogolyubov.

The Fokker-Planck-Landau equation

In 1957, the equation was derived independently by Marshall Rosenbluth. Solving the Fokker–Planck equation under an inverse-square force, one may obtain:

$${\displaystyle \frac{1}{4\pi L}\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}=\frac{\mathrm{\partial }}{\mathrm{\partial }{v}_{\alpha }}\left(-f_i\frac{\mathrm{\partial }{h}_i}{\mathrm{\partial }{v}_{\alpha }}+\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }{v}_{\beta }}\left(f_i\frac{{\mathrm{\partial }}^2{g}_i}{\mathrm{\partial }{v}_{\alpha }\mathrm{\partial }{v}_{\beta }}\right)\right)}$$

where ${\displaystyle h_i,g_i}$ are the Rosenbluth potentials:

$${\displaystyle h_i=\sum _{j=1}^n{K}_{ij}\int dw\frac{f_i(w,t)}{|v-w|}}$$ $${\displaystyle g_i=\sum _{j=1}^n{K}_{ij}\frac{m_j}{m_i}\int dw\frac{f_i(w,t)}{|v-w|}}$$

for ${\displaystyle K_{ij}=\frac{e_i^2{e}_j^2}{m_i{m}_j},i=1,2,\dots ,n}$

The Fokker-Planck representation of the equation is primarily used for its convenience in numerical calculations.

The relativistic Landau kinetic equation

A relativistic version of the equation was published in 1956 by Gersh Budker and Spartak Belyaev.

Considering relativistic particles with momentum ${\displaystyle p=(p^1,p^2,p^3)\in {\mathbb{R}}^3}$ and energy $p^0=\sqrt{1+|p{|}^2}$, the equation reads:

$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}=\frac{\mathrm{\partial }}{\mathrm{\partial }{p}_i}{\int }_{{\mathbb{R}}^3}dq{\mathrm{\Phi }}^{ij}(p,q)\left[h(q)\frac{\mathrm{\partial }}{\mathrm{\partial }{p}_j}g(p)-\frac{\mathrm{\partial }}{\mathrm{\partial }{q}_j}h(q)g(p)\right]}$$

where the kernel is given by ${\displaystyle {\mathrm{\Phi }}^{ij}=\mathrm{A}(p,q)S^{ij}(p,q)}$ such that:

$${\displaystyle \mathrm{A}=\frac{{\left({\rho }_{-}+1\right)}^2}{p^0{q}^0}{\left({\rho }_{+}{\rho }_{-}\right)}^{-3/2}}$$ $${\displaystyle S^{ij}={\rho }_{+}{\rho }_{-}{\delta }_{ij}-\left(p_i-q_i\right)\left(p_j-q_j\right)+{\rho }_{-}\left(p_i{q}_j+p_j{q}_i\right)}$$ $${\displaystyle {\rho }_{\pm }=p^0{q}^0-pq\pm 1}$$

A relativistic correction to the equation is relevant seeing as particle in hot plasma often reach relativistic speeds.