Linear Response Theory of Longitudinal Plasma Excitations

Abstract

AbstractThe plasma convolution equation equivalent to the initial problem of linearized BOLTZMANN‐VLASOV and POISSON'S equations of an isotropic, one‐dimensional and nonrelativistic plasma is derived. The integral equation obtained involves both the time t and space coordinate x.The solution of this equation is exhibited in terms of a forcing function and a resolvent kernel. The forcing function is an exciting electric field caused by the initial disturbance of plasma equilibrium. The resolvent kernel obeys an integral kernel equation which also involves the kernel of the convolution plasma equation and the resolvent is interpreted as a plasma response to the unit impulse disturbance. The plasma kernel is directly expressed by the equilibrium velocity distribution. The plasma response and the equilibrium distribution are related in the same way as the response of a transmission‐line is related to its transfer function and the integral‐kernel equation plays a key role in the formulation of problems of plasma analysis and synthesis. To analyse plasma, is to determine its electric response for a given equilibrium velocity distribution of plasma components and to synthesize plasma is to design an equilibrium distribution of plasma components for a required plasma response.In this paper the plasma analysis is carried out and the example of Lorentzian and Maxwellian plasma equilibrium distributions are considered. The plasma convolution equation equivalent to the one‐point boundary problem, (x = 0), of linearized BOLTZMANN‐VLASOV and POISSON'S equations is presented. The absence of discrete spectrum solutions is evident from the plasma convolution equations but the oscillation properties of plasma are preserved for the examples considered. The complete analytical results are obtained for a Lorentzian plasma. The electric response of the plasma is shown to be a temporal oscillation travelling with the velocity U = x/t and the amplitude of the travelling oscillation depending on U and decaying as t−1. For a Maxwellian plasma the asymptotic expansion of the electric response is obtained for short and long times and a fixed x. The result for the short‐time limit is the same as in the Lorentzian plasma case and for the long time approximation the plasma response is shown to be an oscillation decaying as t−1.