In physics, Landau damping is a mechanism by which oscillations in a charged medium (typically a plasma) are damped by non-collisional interactions with said medium. It is named after its discoverer, Soviet physicist Lev Davidovich Landau (1908–68). As the oscillation moves through the medium with phase velocity v ph {\displaystyle v_{\text{ph}}} it will accelerate slightly slower particles and decelerate slightly faster particles; if the former outnumber the latter (such as if the oscillation is travelling faster than the modal velocity of a Maxwell–Boltzmann distribution) the oscillation will lose its energy to drag and thus be damped. This phenomenon prevents an instability from developing, and creates a region of stability in the parameter space. It was later argued by Donald Lynden-Bell 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 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 with phase velocity v ph {\displaystyle v_{\text{ph}}} and particles in the plasma with velocity approximately equal to v ph {\displaystyle v_{\text{ph}}} , which can interact strongly with the wave. Those particles having velocities slightly less than v ph {\displaystyle v_{\text{ph}}} 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 v ph {\displaystyle v_{\text{ph}}} will be decelerated losing energy to the wave: particles tend to synchronize with the wave. This is proved experimentally with a traveling-wave tube.
In an ideal magnetohydrodynamic (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 § Mathematical treatment). However, in the case of waves with finite amplitude, there is a simple physical interpretation which, though not strictly correct, helps to visualize this phenomenon.
It is possible to imagine Langmuir waves 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 they 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 they move 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 faster than the waves does not exchange much energy with the wave.
A somewhat more detailed picture is obtained by considering particles' trajectories in phase space, in the wave's frame of reference. Particles near the phase velocity become trapped and are forced to move with the wavefronts, at the phase velocity. Any such particles that were initially below the phase velocity have thus been accelerated, while any particles that were initially above the phase velocity have been decelerated. Because, for a Maxwellian plasma, there are initially more particles below the phase velocity than above it, the plasma has net gained energy, and the wave has therefore lost energy. A simple mechanical description of particle dynamics provides a quantitative estimate of the synchronization of particles with the wave. 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. 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 T trap ∼ A − 1 / 2 {\displaystyle T_{\text{trap}}\sim A^{-1/2}} for a wave amplitude A {\displaystyle A} ).
Mathematical treatment
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