A Particle-in-Cell simulation of temporal plasma echo in the presence of Coulomb collisions

Particle-in-Cell simulation is developed to study temporal plasma echo of electron plasma wave. By imposing two external pulse electric fields to the plasma (pulse-like in time) \footnote{R.D.Sydora, Advanced Methods for Space Simulations, pp.47-60 (2007).} the echo signal is observed. Coulomb collisional effect manifests itself as a shift of the echo peak and the damping of the peak amplitude,\footnote{C.H.Su and C.Oberman, Phys. Rev. Lett. 20, 427 (1968).} which can be seen by adding (rather phenomenological) frictional force to the electron equation of motion. A first principle based binary collision model \footnote{T.Takizuka and H.Abe, J. Comput. Phys. 25, 205 (1978).} is incorporated into the numerical simulation.