Forward-scattering vs Vlasov methods for computing the linear and nonlinear response functions: Electrostatic approximation in unmagnetized plasmas

The present paper examines the issue of whether the forward-scattering method and the traditional Vlasov method to compute the linear and nonlinear susceptibilities of plasmas are equivalent. For the sake of simplicity, the present paper employs the simplest situation, namely, the electrostatic plasma susceptibilities in unmagnetized plasma. It is shown that both approaches lead to the same expression for linear and second-order nonlinear susceptibilities, but as far as the third-order susceptibility is concerned, it is found that the forward-scattering method fails to reproduce the known expression rigorously derived under the Vlasov approach. The present finding thus indicates the limitation of the forward-scattering method as it relates to the computation of higher-order nonlinear response functions.