Using Kernel-Based Statistical Distance to Study the Dynamics of Charged
  Particle Beams in Particle-Based Simulation Codes
  Measures of discrepancy between probability distributions (statistical
distance) are widely used in the fields of artificial intelligence and machine
learning. We describe how certain measures of statistical distance can be
implemented as numerical diagnostics for simulations involving charged-particle
beams. Related measures of statistical dependence are also described. The
resulting diagnostics provide sensitive measures of dynamical processes
important for beams in nonlinear or high-intensity systems, which are otherwise
difficult to characterize. The focus is on kernel-based methods such as Maximum
Mean Discrepancy, which have a well-developed mathematical foundation and
reasonable computational complexity. Several benchmark problems and examples
involving intense beams are discussed. While the focus is on charged-particle
beams, these methods may also be applied to other many-body systems such as
plasmas or gravitational systems.
