Exact solutions to planar emittance growth problems

This paper is the first in a series which develops the theory of emittance dynamics based on simple statistical reasoning. Emittance is a central quantity used to characterize the quality of electron microscopes, photon sources and particle beams. Emittance growth in high intensity charged particle beams is a particularly challenging non-equilibrium statistical physics problem in which effects such as disordered-induced heating and charge reorganization can lead to very rapid degradation of emittance and beam quality. The concepts of free energy and entropy have been utilized to improve conceptual understanding of emittance dynamics. Here we develop a theory based on the second order cumulant of particle distributions and use this formulation to exactly solve several one dimensional problems. These solutions are important extensions of the existing results for the free expansion dynamics of pancake bunches used in ultrafast electron microscopy, which at short times are known to expand quadratically with time[1-3]. Here we show that the squared emittance of a strictly planar expanding bunch increases as a quadratic polynomial of time. We compare theories based on individual particle trajectories with theories based on distributions and expand the foundations of theories based on individual particle trajectories, which we call the "sample picture". Our later work uses this formulation to derive generalized envelope equations which capture emittance growth effects in two and three dimensional systems. 1. B. J. Siwick, J. R. Dwyer, R. E. Jordan, and R. J. Dwayne Miller, Journal of Applied Physics 92, 1643 (2002). 2. B. W. Reed, Journal of Applied Physics 100, 034916 (2006). 3. B. Zerbe, X. Xiang, C.-Y. Ruan, S. Lund, and P. Duxbury, Physical Review Accelerators and Beams 21, 064201 (2018).