Landau damping
In physics, Landau damping, named after its discoverer,[1] Soviet physicist Lev Davidovich Landau (1908–68), is the effect of damping (exponential decrease as a function of time) of longitudinal space charge waves in plasma or a similar environment.
[2] 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,[3] where the gas of electrons interacting by electrostatic forces is replaced by a "gas of stars" interacting by gravitational forces.
[5] It was proved to exist experimentally by Malmberg and Wharton in 1964,[6] almost two decades after its prediction by Landau in 1946.
[7] Landau damping occurs because of the energy exchange between an electromagnetic wave with phase velocity
[9] In an ideal magnetohydrodynamic (MHD) plasma the particle velocities are often taken to be approximately a Maxwellian distribution function.
However, in the case of waves with finite amplitude, there is a simple physical interpretation[2]: §7.5 which, though not strictly correct, helps to visualize this phenomenon.
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.
A somewhat more detailed picture is obtained by considering particles' trajectories in phase space, in the wave's frame of reference.
1 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.
This is natural, since trapping involves diverging time scales for such waves (specifically
and found by aid of Laplace transform and contour integration a damped travelling wave of the form
Later Nico van Kampen proved[11] that the same result can be obtained with Fourier transform.
Decomposing the initial disturbance in these modes he obtained the Fourier spectrum of the resulting wave.
Damping is explained by phase-mixing of these Fourier modes with slightly different frequencies near
It was not clear how damping could occur in a collisionless plasma: where does the wave energy go?
In fluid theory, in which the plasma is modeled as a dispersive dielectric medium,[12] the energy of Langmuir waves is known: field energy multiplied by the Brillouin factor
To calculate energy exchange of the wave with resonant electrons, Vlasov plasma theory has to be expanded to second order and problems about suitable initial conditions and secular terms arise.
Second-order initial conditions are found that suppress secular behavior and excite a wave packet of which the energy agrees with fluid theory.
The figure shows the energy density of a wave packet traveling at the group velocity, its energy being carried away by electrons moving at the phase velocity.
The rigorous mathematical theory is based on solving the Cauchy problem for the evolution equation (here the partial differential Vlasov–Poisson equation) and proving estimates on the solution.
[14] Going beyond the linearized equation and dealing with the nonlinearity has been a longstanding problem in the mathematical theory of Landau damping.
Previously one mathematical result at the non-linear level was the existence of a class of exponentially damped solutions of the Vlasov–Poisson equation in a circle which had been proved in[15] by means of a scattering technique (this result has been recently extended in[16]).
However these existence results do not say anything about which initial data could lead to such damped solutions.
In a paper published by French mathematicians Cédric Villani and Clément Mouhot,[17] the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation.
It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time.
This shift, well known in linear theory, proves to hold in the non-linear case.
The mechanical N-body description, originally deemed impossible, enables a rigorous calculation of Landau damping using Newton’s second law of motion and Fourier series.
The calculation of the energy (more precisely momentum) exchange of the wave with electrons is done similarly.
This calculation makes intuitive the interpretation of Landau damping as the synchronization of almost resonant passing particles.
