Equilibrium fluctuations in an N-particle coasting beam: Schottky noise effects

We discuss the longitudinal dynamics of an unbunched beam with a collective effect due to the vacuum chamber and with the discreteness of an N-particle beam (Schottky noise) included. We start with the 2N equations of motion (in angle and energy) with random initial conditions. The 2D phase space density (Klimontovich density) for the N particles is a sum of delta functions and satisfies the Klimontovich equation and the Vlasov equation. An arbitrary function of the energy also satisfies the Vlasov equation and we linearize about a convenient equilibrium density taking the initial conditions to be independent, identically distributed random variables with the equilibrium distribution. The linearized equations can be solved using a Laplace transform in time and a Fourier series in angle. The resultant stochastic process for the phase space density is analyzed and compared with a known result. Work is in progress to study the full nonlinear problem.