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Landau kinetic equation

From Wikipedia, the free encyclopedia

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

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Definition

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Let be a one-particle Distribution function. The equation reads:

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.

Uses

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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

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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 .

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

The Fokker-Planck-Landau equation

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

where are the Rosenbluth potentials:

for

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

The relativistic Landau kinetic equation

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A relativistic version of the equation was published in 1956 by Gersh Budker and Spartak Belyaev.[11]

Considering relativistic particles with momentum and energy , the equation reads:

where the kernel is given by such that:

A relativistic correction to the equation is relevant seeing as particle in hot plasma often reach relativistic speeds. [3]

See also

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References

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  1. ^ a b 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. Bibcode:2015JSP...161.1327B. doi:10.1007/s10955-015-1311-0. S2CID 39781.
  3. ^ a b Robert 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.
  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. arXiv:2010.12213. Bibcode:2021JSMTE2021c3210P. doi:10.1088/1742-5468/abe410. S2CID 225062056.
  8. ^ Alexander Bobylev. ResearchGate. URL: https://www.researchgate.net/profile/Alexander-Bobylev
  9. ^ Bogolyubov, N.N. (1946). Problems of a Dynamical Theory in Statistical Physics. USSR: State Technical Press.
  10. ^ Rosenbluth, M.N. (1957). "Fokker-Planck equation for an inverse-square force". Phys. Rev. 107 (1): 1–6. Bibcode:1957PhRv..107....1R. doi:10.1103/PhysRev.107.1.
  11. ^ S. T. Belyaev and G. I. Budker. Relativistic kinetic equation. Dokl. Akad. Nauk SSSR (N.S.), 107:807–810, 1956.