Correlationless Plasma with Time-Dependent Background State

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Abstract
A Volterra type (second kind) integral equation is derived for the electric charge density starting from the Vlasov equation linearized with respect to a time‐dependent spatially uniform background state. The kernel of this integral equation vanishes identically in the case when the background state is time‐independent (i.e., in the usual Landau problem), and the inhomogeneous term reduces to the exact solution of the initial value problem in the same case. It is shown that the inhomogeneous term of this integral equation corresponds to the adiabatic approximation; hence, a Neumann series type solution can be constructed for the charge density by iteration of this inhomogeneous term for the case when the background state is varying slowly in time. Approximate solutions of this integral equation are obtained for the small time and asymptotic limits. It appears that for some particular cases it is possible to have nondecaying solutions for the continuum mode. This is in contrast with one of the basic assumptions of the quasi‐linear theory of plasmas.

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