First‐ and Second‐Order Perturbations of Plasmas with Cauchy Equilibrium Distributions

The space—time behavior of initial density perturbations (acoustic mode) in hot plasmas is analyzed, including second‐order effects, using a one‐dimensional Cauchy velocity distribution of the form (v² + a²)⁻² for the equilibrium distribution. The initial density perturbation is assumed to be sinusoidal in space and to have the same Cauchy velocity distribution. The first‐order solution has an exact analytic form which gives damped oscillations at a thermally shifted plasma frequency. The nonlinear interference between the density wave and its electric field produces second harmonics in both space and time which appear in the analytic second‐order solution. The harmonic structure suggests a ``spectral decay'' of the initial perturbation energy. In general, the oscillation frequency and damping decrement increase with temperature so that at sufficiently high temperatures, all forms of ordered motion are destroyed by the random thermal motion.