The Landau kinetic equation is a transport equation of weakly coupled charged particles performing
Coulomb collisions in a
plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
.
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.
Overview
Definition
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
The equation is used primarily in
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
Particle physics to model plasma. As such, it has been used to model and study
Plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
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 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 ''
'', one may define a corrected potential for
Coulomb interactions,
, where
is the
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
,
, 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.
The Fokker-Planck-Landau equation
In 1957, the equation was derived independently by
Marshall Rosenbluth
Marshall Nicholas Rosenbluth (5 February 1927 – 28 September 2003) was an American plasma physicist and member of the National Academy of Sciences, and member of the American Philosophical Society. In 1997 he was awarded the National Medal of ...
. Solving the
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
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
A
relativistic version of the equation was published in 1956 by
Gersh Budker
Gersh Itskovich Budker (Герш Ицкович Будкер), also named Andrey Mikhailovich Budker (1 May 1918 – 4 July 1977), was a Soviet physicist, specialized in nuclear physics and accelerator physics.
Biography
He was elected a Correspo ...
and
Spartak Belyaev
Spartak Timofeyevich Belyaev (October 27, 1923 – January 5, 2017) was a Soviet and Russian theoretical physicist who was awarded a Lomonosov Gold Medal.
Biography
Belyaev was born on October 27, 1923 in Moscow, Russia. When World War II began, h ...
.
[S. T. Belyaev and G. I. Budker. Relativistic kinetic equation. Dokl. Akad. Nauk SSSR (N.S.), 107:807–810, 1956.]
Considering relativistic particles with momentum
and energy
, the equation reads: