In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Landau damping, named after its discoverer,
[Landau, L. "On the vibration of the electronic plasma". ''JETP'' 16 (1946), 574. English translation in ''J. Phys. (USSR)'' 10 (1946), 25. Reproduced in Collected papers of L.D. Landau, edited and with an introduction by D. ter Haar, ]Pergamon Press
Pergamon Press was an Oxford-based publishing house, founded by Paul Rosbaud and Robert Maxwell, that published scientific and medical books and journals. Originally called Butterworth-Springer, it is now an imprint of Elsevier.
History
The cor ...
, 1965, pp. 445–460; and in Men of Physics: L.D. Landau, Vol. 2, Pergamon Press, D. ter Haar, ed. (1965). Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
physicist
Lev Davidovich Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet- Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His a ...
(1908–68), is the effect of
damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
(
exponential decrease as a function of time) of
longitudinal space charge waves in
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 ...
or a similar environment.
[Chen, Francis F. ''Introduction to Plasma Physics and Controlled Fusion''. Second Ed., 1984 Plenum Press, New York.] This phenomenon prevents an instability from developing, and creates a region of stability in the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the ...
. It was later argued by
Donald Lynden-Bell
Donald Lynden-Bell CBE FRS (5 April 1935 – 6 February 2018) was a British theoretical astrophysicist. He was the first to determine that galaxies contain supermassive black holes at their centres, and that such black holes power quasars. ...
that a similar phenomenon was occurring in galactic dynamics, where the gas of electrons interacting by electrostatic forces is replaced by a "gas of stars" interacting by gravitational forces. Landau damping can be manipulated exactly in numerical simulations such as
particle-in-cell
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in a Lagrangian frame are tracked in continuous ph ...
simulation. It was proved to exist experimentally by Malmberg and Wharton in 1964, almost two decades after its prediction by Landau in 1946.
Wave–particle interactions
Landau damping occurs because of the energy exchange between an electromagnetic
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
with phase velocity
and particles in the plasma with velocity approximately equal to
, which can interact strongly with the wave. Those particles having velocities slightly less than
will be accelerated by the electric field of the wave to move with the wave phase velocity, while those particles with velocities slightly greater than
will be decelerated losing energy to the wave: particles tend to synchronize with the wave. This is proved experimentally with a
traveling-wave tube
A traveling-wave tube (TWT, pronounced "twit") or traveling-wave tube amplifier (TWTA, pronounced "tweeta") is a specialized vacuum tube that is used in electronics to amplify radio frequency (RF) signals in the microwave range. The TWT belongs t ...
.
![Maxwell dist ress partic landau](https://upload.wikimedia.org/wikipedia/commons/f/fa/Maxwell_dist_ress_partic_landau.svg)
In an ideal
magnetohydrodynamic
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
(MHD) plasma the particle velocities are often taken to be approximately a
Maxwellian distribution function. If the slope of the function is negative, the number of particles with velocities slightly less than the wave phase velocity is greater than the number of particles with velocities slightly greater. Hence, there are more particles gaining energy from the wave than losing to the wave, which leads to wave damping.
If, however, the slope of the function is positive, the number of particles with velocities slightly less than the wave phase velocity is smaller than the number of particles with velocities slightly greater. Hence, there are more particles losing energy to the wave than gaining from the wave, which leads to a resultant increase in the wave energy. Then Landau damping is substituted with Landau growth.
Physical interpretation
The mathematical theory of Landau damping is somewhat involved—see the section below. However, there is a simple physical interpretation
_with_a_caveat.html" ;"title="ntroduced in section 7.5 of
with a caveat">ntroduced in section 7.5 of
with a caveat which, though not strictly correct, helps to visualize this phenomenon.
![Phys interp landau damp](https://upload.wikimedia.org/wikipedia/commons/2/2c/Phys_interp_landau_damp.svg)
It is possible to imagine
Langmuir waves Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
as waves in the sea, and the particles as surfers trying to catch the wave, all moving in the same direction. If the surfer is moving on the water surface at a velocity slightly less than the waves he will eventually be caught and pushed along the wave (gaining energy), while a surfer moving slightly faster than a wave will be pushing on the wave as he moves uphill (losing energy to the wave).
It is worth noting that only the surfers are playing an important role in this energy interactions with the waves; a beachball floating on the water (zero velocity) will go up and down as the wave goes by, not gaining energy at all. Also, a boat that moves very fast (faster than the waves) does not exchange much energy with the wave.
A simple mechanical description of particle dynamics provides a quantitative estimate of the synchronization of particles with the wave
.html" ;"title="quation (1) of
">quation (1) of
A more rigorous approach shows the strongest synchronization occurs for particles with a velocity in the wave frame proportional to the damping rate and independent of the wave amplitude [section 3.2 of
[ Escande, D. F., Bénisti, D., Elskens, Y., Zarzoso, D., & Doveil, F. (2018). Basic microscopic plasma physics from N-body mechanics, A tribute to Pierre-Simon de Laplace, Reviews of Modern Plasma Physics, 2, 1-68]]. Since Landau damping occurs for waves with arbitrarily small amplitudes, this shows the most active particles in this damping are far from being trapped. This is natural, since trapping involves diverging time scales for such waves (specifically
for a wave amplitude
).
Theoretical physics: perturbation theory in a Vlasovian frame
Theoretical treatment starts with the
Vlasov equation The Vlasov equation is a differential equation describing time evolution of the Distribution function (physics), distribution function of plasma (physics), plasma consisting of charged particles with long-range interaction, e.g. Coulomb's law, Coulo ...
in the non-relativistic zero-magnetic field limit, the Vlasov–Poisson set of equations. Explicit solutions are obtained in the limit of a small
-field. The distribution function
and field
are expanded in a series:
,
and terms of equal order are collected.
To first order the Vlasov–Poisson equations read
:
.
Landau calculated
the wave caused by an initial disturbance
and found by aid of
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
and
contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
a damped travelling wave of the form