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]

Overview

Definition

Let ${\displaystyle f(v,t)}$ be a one-particle Distribution function. The equation reads:

$${\displaystyle {\frac {\partial f}{\partial t}}=B{\frac {\partial }{\partial v_{i}}}\left(\int _{\mathbb {R} ^{3}}dw{\frac {\left(u^{2}\delta _{ij}-u_{i}u_{j}\right)}{u^{3}}}\left({\frac {\partial }{\partial v_{j}}}-{\frac {\partial }{\partial w_{j}}}\right)f(v)f(w)\right)}$$

$${\displaystyle u=v-w}$$

The right-hand side of the equation is known as the Landau collision integral (in parallel to the Boltzmann collision integral).

${\displaystyle B}$ is obtained by integrating over the intermolecular potential ${\displaystyle U(r)}$:

$${\displaystyle B={\frac {1}{8\pi }}\int _{0}^{\infty }dr\,r^{3}{\hat {U}}(r)^{2}}$$

$${\displaystyle {\hat {U}}(|k|)=\int _{\mathbb {R} ^{3}}dx\,U(|x|)e^{ikx}}$$

For many intermolecular potentials (most notably power laws where ${\textstyle U(r)\propto {\frac {1}{r^{n}}}}$), the expression for ${\displaystyle B}$ diverges. Landau's solution to this problem is to introduce Cutoffs at small and large angles.

Derivations

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 ${\displaystyle f(v,t)}$, one may define a corrected potential for Coulomb interactions, ${\textstyle {\hat {U}}_{ij}=U_{ij}\exp \left(-{\frac {r_{ij}}{r_{D}}}\right)}$, where ${\displaystyle U_{ij}}$ is the Coulomb potential, ${\textstyle U_{ij}={\frac {e_{i}e_{j}}{|x_{i}-x_{j}|}}}$, and ${\displaystyle r_{D}}$ is the Debye radius. The potential ${\displaystyle {\hat {U_{ij}}}}$ 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 ${\displaystyle r_{D}\rightarrow \infty }$.

In 1946, the first formal derivation of the equation from the BBGKY hierarchy was published by Nikolay Bogolyubov.^[9]

The Fokker-Planck-Landau equation

In 1957, the equation was derived independently by Marshall Rosenbluth.^[10] Solving the Fokker–Planck equation under an inverse-square force, one may obtain:

$${\displaystyle {\frac {1}{4\pi L}}{\frac {\partial f_{i}}{\partial t}}={\frac {\partial }{\partial v_{\alpha }}}\left(-f_{i}{\frac {\partial h_{i}}{\partial v_{\alpha }}}+{\frac {1}{2}}{\frac {\partial }{\partial v_{\beta }}}\left(f_{i}{\frac {\partial ^{2}g_{i}}{\partial v_{\alpha }\partial v_{\beta }}}\right)\right)}$$

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.^[11]

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

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

where the kernel is given by ${\displaystyle \Phi ^{ij}=\mathrm {A} (p,q)S^{ij}(p,q)}$ 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. ^[3]

See also

• Boltzmann equation
• Vlasov equation

References