Some fundamental aspects of fluctuations and coherence in charged‐particle beams in storage rings

A conceptual survey and exposition is presented of some fundamendal aspects of fluctuations and coherence, as well as the interplay between the two, in coasting charged‐particle beams—both continuous and bunched—in storage rings. A detailed study is given of the spectral properties of the incoherent phase‐space Schottky fluctuations, their propagation as waves in the beam, and the analytic complex coherent beam electromagnetic response or transfer function. The modification or distortion of these by collective interactions is examined in terms of simple regeneration mechanisms. Collective or coherent forces in the beam–storage‐ring system are described by defining suitable impedance functions or propagators, and a brief discussion of the coherent collective modes and their stability is provided, including a general and rigorous description of the Nyquist stability criterion. The nature of the critical fluctuations near an instability threshold is explored. The concept of Landau damping and its connection with phase‐mixing within the beam is outlined. The important connection between the incoherent fluctuations and the beam response, namely the Fluctuation–Dissipation relation, is revealed. A brief discussion is given of the information degrees of freedom, and effective temperature of the fluctuation signals. Appendices provide a short résumé of some general aspects of various interactions in a charged‐particle between‐environment system in a storage ring and a general introduction to kinetic theory as applied to particle beams.