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 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.
History The first derivation was given in Landau's original paper. In 1946, the first formal derivation of the equation from the BBGKY hierarchy was published by Nikolay Bogolyubov. In 1949, William Allis suggested that Landau’s derivation contained an error. Landau responded the following year, demonstrating that Allis’s calculations were themselves incorrect.
Overview
Definition Let f ( v , t ) {\displaystyle f(v,t)} be a one-particle Distribution function. The equation reads:
∂ f ∂ t = B ∂ ∂ v i ( ∫ R 3 d w ( u 2 δ i j − u i u j ) u 3 ( ∂ ∂ v j − ∂ ∂ w j ) f ( v ) f ( w ) ) {\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)}
u = v − w {\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).
B {\displaystyle B} is obtained by integrating over the intermolecular potential U ( r ) {\displaystyle U(r)} :
B = 1 8 π ∫ 0 ∞ d r r 3 U ^ ( r ) 2 {\displaystyle B={\frac {1}{8\pi }}\int _{0}^{\infty }dr\,r^{3}{\hat {U}}(r)^{2}}
U ^ ( | k | ) = ∫ R 3 d x U ( | x | ) e i k x {\displaystyle {\hat {U}}(|k|)=\int _{\mathbb {R} ^{3}}dx\,U(|x|)e^{ikx}}
For many intermolecular potentials (most notably power laws where U ( r ) ∝ 1 r n {\textstyle U(r)\propto {\frac {1}{r^{n}}}} ), the expression for B {\displaystyle B} diverges. Landau's solution to this problem is to introduce Cutoffs at small and large angles.
Uses 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. It has also seen use in modeling of Active matter . The equation and its properties have been studied in depth by Alexander Bobylev.
Derivations The rough idea for the derivation: Assuming a spatially homogenous gas of point particles with unit mass described by f ( v , t ) {\displaystyle f(v,t)} , one may define a corrected potential for Coulomb interactions, U ^ i j = U i j exp ( − r i j r D ) {\textstyle {\hat {U}}_{ij}=U_{ij}\exp \left(-{\frac {r_{ij}}{r_{D}}}\right)} , where U i j {\displaystyle U_{ij}} is the Coulomb potential, U i j = e i e j | x i − x j | {\textstyle U_{ij}={\frac {e_{i}e_{j}}{|x_{i}-x_{j}|}}} , and r D {\displaystyle r_{D}} is the Debye radius. The potential U i j ^ {\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 r D → ∞ {\displaystyle r_{D}\rightarrow \infty } .
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:
1 4 π L ∂ f i ∂ t = ∂ ∂ v α ( − f i ∂ h i ∂ v α + 1 2 ∂ ∂ v β ( f i ∂ 2 g i ∂ v α ∂ v β ) ) {\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 h i , g i {\displaystyle h_{i},g_{i}} are the Rosenbluth potentials:
h i = ∑ j = 1 n K i j ∫ d w f i ( w , t ) | v − w | {\displaystyle h_{i}=\sum _{j=1}^{n}K_{ij}\int dw{\frac {f_{i}(w,t)}{|v-w|}}}
g i = ∑ j = 1 n K i j m j m i ∫ d w 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 K i j = e i 2 e j 2 m i m j , i = 1 , 2 , … , n {\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 p = ( p 1 , p 2 , p 3 ) ∈ R 3 {\displaystyle p=(p^{1},p^{2},p^{3})\in \mathbb {R} ^{3}} and energy p 0 = 1 + | p | 2 {\textstyle p^{0}={\sqrt {1+|p|^{2}}}} , the equation reads:
∂ f ∂ t = ∂ ∂ p i ∫ R 3 d q Φ i j ( p , q ) [ h ( q ) ∂ ∂ p j g ( p ) − ∂ ∂ q j h ( q ) g ( p ) ] {\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 Φ i j = A ( p , q ) S i j ( p , q ) {\displaystyle \Phi ^{ij}=\mathrm {A} (p,q)S^{ij}(p,q)} such that:
A = ( ρ − + 1 ) 2 p 0 q 0 ( ρ + ρ − ) − 3 / 2 {\displaystyle \mathrm {A} ={\frac {\left(\rho _{-}+1\right)^{2}}{p^{0}q^{0}}}\left(\rho _{+}\rho _{-}\right)^{-3/2}}
S i j = ρ + ρ − δ i j − ( p i − q i ) ( p j − q j ) + ρ − ( p i q j + p j q i ) {\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)}
ρ ± = p 0 q 0 − p q ± 1 {\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.
See also Boltzmann equation Vlasov equation
References