Mesoscopic description of dynamical systems: Hierarchy of recursive equations for the moments of the density distribution with an application to charged particle beam dynamics

The quantumlike formalism in the form of Madelung fluid has been applied to describe the collective dynamics of a mesoscopic aggregate of particles in a potential field. The scheme has been specialized to the one-degree-of-freedom dynamics of a charged particle beam in an accelerator in the presence of high-order multipole nonlinearities. A hierarchy of recursive equations satisfied by the moments of the density distribution has been obtained. It has been shown that the recursion relations can be in principle solved if a finite number of the first several moments is known. Beam dynamics in the presence of octupole and decapole nonlinearities has been studied as an example, demonstrating the above scheme. It has been shown that an appropriate mechanism can be introduced to control stability and coherence of states when high-order nonlinearities have to be taken into account.