CERN Accelerating science

Plasma acceleration with electron or proton driver beams is a challenging opportunity for high energy physics. An energy doubling experiment with electron drivers was successfully performed at SLAC and a key experiment AWAKE with proton drivers is on schedule at CERN. Simulations play an important role in choosing the best experimental conditions and in interpreting the results. The Vlasov equation is the theoretical tool to describe the interaction of a driver particle beam or a driver laser pulse with a plasma. Collective effects, such as tune shift and mismatch instabilities, appear in high intensity standard accelerators and are described by the Poisson-Vlasov equation. In the paper we review the Vlasov equation in electrostatic and fully electromagnetic case. The general framework of variational principles is used to derive the equation, the local form of the balance equations and related conservation laws. In the electrostatic case we remind the analytic Kapchinskij-Vladimirskij (K-V) model and we propose an extension of of the adiabatic theory for Hamiltonian systems, which ensures stability for perturbation of size ǫ on times of order 1/ǫ . The variational framework is used to derive the Maxwell-Vlasov equations and related conservation laws and to briefly sketch the particle-in-cell (PIC) approximation schemes. Finally the proton-driven acceleration is examined in the linear and quasi-linear regime. A PIC simulation with the code ALaDyn developed at the Bologna University is presented to illustrate the longitudinal and transverse fields evolution which allow a witness electron bunch to be accelerated with a gradient of a few GeV/m. We also present some remarks on future perspectives.