PROBABILISTIC MODEL OF BEAM–PLASMA INTERACTION IN RANDOMLY INHOMOGENEOUS PLASMA
Résumé
We propose a new model that describes beam–plasma interaction in the presence of random density fluctuations
with a known probability distribution. We use the property that, for the given frequency, the probability
distribution of the density fluctuations uniquely determines the probability distribution of the phase velocity of
waves. We present the system as discrete and consisting of small, equal spatial intervals with a linear density
profile. This approach allows one to estimate variations in wave energy density and particle velocity, depending on
the density gradient on any small spatial interval. Because the characteristic time for the evolution of the electron
distribution function and the wave energy is much longer than the time required for a single wave–particle resonant
interaction over a small interval, we determine the description for the relaxation process in terms of averaged
quantities. We derive a system of equations, similar to the quasi-linear approximation, with the conventional
velocity diffusion coefficient D and the wave growth rate γ replaced by the average in phase space, by making use
of the probability distribution for phase velocities and by assuming that the interaction in each interval is
independent of previous interactions. Functions D and γ are completely determined by the distribution function for
the amplitudes of the fluctuations. For the Gaussian distribution of the density fluctuations, we show that the
relaxation process is determined by the ratio of beam velocity to plasma thermal velocity, the dispersion of the
fluctuations, and the width of the beam in the velocity space.
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