General moment model of beam transport

Using the Hamiltonian structure of the Vlasov equation, we develop a general, relativistic, three-dimensional model of beam transport based on phase space moments of the beam particle distribution. Evolution equations for these moments are derived from the non-canonical Poisson bracket for the Vlasov equation. In this model, the beam centroid experiences the full non-linear forces in the system while the higher order moments are coupled to both the centroid and to various spatial derivatives of the applied fields. For example, when moments up to second order are retained, the physics content is similar to considering linearized forces. Given the large number of equations (there are 27 equations when all second order moments are kept) and their algebraic complexity, the use of symbolic computation in the derivation was critical to ensuring the correctness of the equations. This approach also allows for analytical verification of conservation laws associated with the model. The initial investigations have considered only externally applied fields, however in principle space-charge forces can also be included. We discuss the necessary extensions to the basic theory needed to model ionization cooling for the muon collider