where E is a linear operator, S is a source term due to the particles, and Φ Φ is the Fourier-Laplace transform of the electrostatic potential. When substituting an integral over a smooth distribution function for the discrete sum over the particles in E , one gets ϵ ϵ ( k , ω ω ) Φ Φ ( k , ω ω ) = S ( k , ω ω ) , where ϵ ϵ ( k , ω ω ) is the plasma permittivity, or dielectric function, classically obtained by a linearized Vlasov-Poisson equation, k is the wave vector, ω ω is the frequency, and S ( k , ω ω ) is the sum of N source terms due to the particles.