STATISTICAL METHOD FOR NONEQUILIBRIUM SYSTEMS WITH APPLICATION TO ACCELERATOR BEAM DYNAMICS (THERMODYNAMICS, LIMIT CYCLE, INSTABILITY, BUNCH LENGTHENING, KINETICS)

In this thesis, a method is developed for calculating the limit cycle distribution of a many-particle system in weak contact with a heat bath. Both externally driven systems and unstable systems with mean-field collective interaction are considered.; The system is described by a Fokker-Planck equation, and then the single particle motion is transformed to action-angle coordinates to separate the thermal and mechanical time dependencies. The equation is then averaged over angle variables to remove the mechanical motion and produce an equation with only thermal motion in action space. The limit cycle is the time-independent solution of the averaged equation.; As an example of a driven system, the distribution of driven oscillators is calculated in the region of action space near a nonlinear resonance, and the perpetual currents known as resonance streaming are shown. As an example of collective instability, the thermodynamic stability of a system of oscillators with a long range cosine potential is considered. For the case of an attractive potential, time dependent limit cycles are found with lower free energy than equilibrium. Hence, this is a conservative many-body system which oscillates spontaneously when placed in contact with a heat bath. This prediction is verified with numerical simulations.; The phenomenon of accelerator bunch lengthening is then explained as an example of thermal instability which has been enhanced by the nonconservative nature of the wake field coupling force. The threshold of thermal instability is shown to be related to the total energy loss of the charge bunch, rather than to the collective frequency shift as predicted for the threshold of mechanical instability by the linearized Vlasov equation. Numerical calculations of bunch lengthening in the electron storage ring SPEAR are presented, and compared with simulations.