Landau Damping - Theoretical Physics: Perturbation Theory

Theoretical Physics: Perturbation Theory

Theoretical treatment starts with Vlasov equation in the non-relativistic zero-magnetic field limit, Vlasov-Poisson set of equations. Explicit solutions are obtained in the limit of small -field. The distribution function and field are expanded in series:, and terms of equal order are collected.

To first order the Vlasov-Poisson equations read

$(\partial_t + v\partial_x)f_1 + {e\over m}E_1 f'_0 = 0, \quad \partial_x E_1 = {e\over \epsilon_0} \int f_1 {\rm d}v$ .

Landau calculated the wave caused by an initial disturbance and found by aid of Laplace transform and contour integration a damped travelling wave of the form with wave number and damping decrement

$\gamma\approx-{\pi\omega_p^3 \over 2k^2N} f'_0(v_{ph}), \quad N = \int f_0 {\rm d}v$ .

Here is the plasma oscillation frequency and is the electron density. Later Nico van Kampen proved that the same result can be obtained with Fourier transform. He showed that the linearized Vlasov-Poisson equations have a continuous spectrum of singular normal modes, now known as van Kampen modes

in which signifies principal value, is the delta function (see generalized function) and

is the plasma permittivity. 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, the energy of Langmuir waves is known: field energy multiplied by the Brillouin factor . But damping cannot be derived in this model. 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.

In these problems are studied. Because calculations for an infinite wave are deficient in second order, a wave packet is analysed. 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. Total energy, the area under the curves, is conserved.

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