Dielectric conductivity of a bounded plasma and its rate of convergence towards its infinite-geometry value

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known and independent of time. Then, a first-order Taylor expansion in the velocity variable is performed, thanks to an approximation of the exact characteristics by the unperturbed ones. The resulting first-order correction to the distribution function is finally integrated over velocities to yield the dielectric function. The special case of a plane wave for the electric field is examined and the results are compared with the more usual unbounded case: the integral does not present any singularity in the vicinity of resonant particles and the dielectric function depends on the distance to the boundary and tends to the usual infinite-geometry value when this distance tends to infinity, with a rate of convergence proportional to its inverse square root. Numerical examples are provided for illustration.