Joint work: Jim Ellison and Klaus Heinemann (UNM) and Gabriele Bassi (BNL)

 

Abstract:

We consider an N-particle electron bunch moving, at nearly the speed of light, through a particle accelerator system inside a vacuum chamber. Typically, N is of an order greater than ~10^9 and the bunch is small relative to the vacuum chamber cross-section. We consider the initial electron phase space positions to be given by a set of scattered data from which an initial electron phase space density can be constructed using a density estimation procedure from mathematical statistics. The evolution of the bunch is then modeled by a random initial boundary value problem (IBVP) with random, IID initial conditions with the above density and where the electron evolution is given in terms of the Lorentz force and the associated microscopic Maxwell fields. The electron phase space density is Klimontovich, i.e., a sum of delta functions. Taking expected value of the associated Klimontovich evolution equation (with respect to the random initial conditions) and making reasonable assumptions, we obtain the Vlasov equation with a correction term (as in the BBGKY hierarchy), and the associated macroscopic Maxwell equations. We then formulate the convergence question of a coarse-grained KM to the macroscopic VM for large N as a generalized SLLN (Strong-Law-of-Large-Numbers). Proof of this SLLN and estimates of the correction term are likely difficult analysis issues. We begin the talk with the much simpler non-collective case, assuming the electrons do not radiate and thus ignoring the Maxwell self-fields, in order to set the stage for the more complex KM —> VM case above.