Landau kinetic equation explained

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

f(v, t) be a one-particle Distribution function. The equation reads:

$$\backslash frac\; =\; B\; \backslash frac\backslash left(\backslash int\_dw\; \backslash frac\backslash left(\backslash frac\; -\; \backslash frac\backslash right)f(v)f(w)\backslash right)$$$$u\; =\; v\; -\; w$$

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

B is obtained by integrating over the intermolecular potential U(r) :

$$B\; =\; \backslash frac\backslash int\_0^\backslash infty\; dr\; \backslash ,\; r^3\; \backslash hat(r)^2$$$$\backslash hat(|k|)\; =\; \backslash int\_\; dx\; \backslash ,\; U(|x|)\; e^$$

For many intermolecular potentials (most notably power laws where $U(r)\; \backslash propto\; \backslash frac$), the expression for

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.^{[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]

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

f(v, t) , one may define a corrected potential for Coulomb interactions, $\backslash hat\_\; =\; U\_\; \backslash exp\backslash left(-\backslash frac\backslash right)$, where U _{ij} is the Coulomb potential, $U\_\; =\; \backslash frac$, and r _{D} is the Debye radius. The potential \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} → inf in .

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:

$$\backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash left(-f\_i\; \backslash frac+\backslash frac\; \backslash frac\; \backslash left(f\_i\; \backslash frac\backslash right)\backslash right)$$

where

h _{i,} g _{i} are the Rosenbluth potentials:

$$h\_i\; =\; \backslash sum^n\_\; K\_\; \backslash int\; dw\; \backslash frac$$

$$g\_i\; =\; \backslash sum^n\_\; K\_\; \backslash frac\; \backslash int\; dw\; \backslash frac$$

for

K _{ij} = (e ^{2} _{i} e ^{2} _{j})/(m _{i} m _{j}) , i = 1 , 2 , ... , 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

p = (p ^{1,} p ^{2,} p ^{3)} \in R ^{3} and energy $p^0\; =\; \backslash sqrt$, the equation reads:

$$\backslash frac\; =\; \backslash frac\backslash int\_\; dq\; \backslash ,\; \backslash Phi^(p,\backslash \; q)\; \backslash left[h(q)\backslash frac\{\backslash partial\}\{\backslash partial\; p\_j\}g(p)-\backslash frac\{\backslash partial\}\{\backslash partial\; q\_j\}h(q)g(p)\backslash right]$$

where the kernel is given by

\Phi ^{ij} = \Alpha(p, q)S ^{ij} (p, q) such that:

$$\backslash Alpha\; =\; \backslash frac\; \backslash left(\backslash rho\_+\; \backslash rho\_-\backslash right)^$$$$S^\; =\; \backslash rho\_+\; \backslash rho\_-\; \backslash delta\_\; -\; \backslash left(p\_i-q\_i\backslash right)\backslash left(p\_j-q\_j\backslash right)+\backslash rho\_-\backslash left(p\_i\; q\_j\; +\; p\_j\; q\_i\backslash right)$$$$\backslash rho\_\; =\; p^0\; q^0\; -\; pq\; \backslash pm\; 1$$

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

See also

• Boltzmann equation
• Vlasov equation

Notes and References

1. Landau. L.D.. 1936. Kinetic equation for the case of coulomb interaction. Phys. Z. Sowjetunion. 10. 154–164.
2. Bobylev. Alexander. 2015. On some properties of the landau kinetic equation . Journal of Statistical Physics. 161. 6. 1327. 10.1007/s10955-015-1311-0 . 2015JSP...161.1327B . 39781.
3. Robert M. Strain, Maja Tasković. 2019. Entropy dissipation estimates for the relativistic Landau equation, and applications. Journal of Functional Analysis. 277. 4 . 1139–1201. 10.1016/j.jfa.2019.04.007. 1806.08720. 119323748.
4. Landau kinetic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Landau_kinetic_equation&oldid=47573
5. J. Killeen, K.D. Marx, "Methods in computational physics", 9, Acad. Press (1970)
6. J. Killeen, A.A. Mirin, M.E. Rensink, "Methods in computational physics", 16, Acad. Press (1976)
7. Patelli. Aurelio. 2021. Landau kinetic equation for dry aligning active models . J. Stat. Mech. . 2021 . 3. 033210 . 10.1088/1742-5468/abe410. 2010.12213. 2021JSMTE2021c3210P. 225062056.
8. Alexander Bobylev. ResearchGate. URL: https://www.researchgate.net/profile/Alexander-Bobylev
9. Book: Bogolyubov, N.N.. Problems of a Dynamical Theory in Statistical Physics. State Technical Press. 1946. USSR.
10. Rosenbluth. M.N.. 1957. Fokker-Planck equation for an inverse-square force. Phys. Rev.. 107. 1. 1–6. 10.1103/PhysRev.107.1. 1957PhRv..107....1R.
11. S. T. Belyaev and G. I. Budker. Relativistic kinetic equation. Dokl. Akad. Nauk SSSR (N.S.), 107:807–810, 1956.