Recently there has been renewed interest in the influence of fringe fields on particle dynamics, due to studies that revealed their importance in some cases, as, for example, the proposed Neutrino Factory and muon colliders. In this paper, we present a systematic study of generic fringe field effects. Using as an example a lattice of the proposed Neutrino Factory, we show that fringe fields influence the dynamics of particles at all orders, starting with the linear motion. It is found that the widely used sharp cutoff approximation leads to divergences regardless of the specific fall-off shape of the fields. The results suggest that a careful consideration of fringe field effects in the design stage of small machines for large emittances is always recommended.

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We investigate some generalizations of the most likely path formalism developed for proton-computed tomography. The stochastic path of a proton inside a homogeneous medium is replaced by a deterministic smooth path that maximizes the probability of the proton passing through the points on this curve, given measured entrance and exit parameters for each individual proton. We study various factors that influence this curve and the associated error envelopes. These factors are the influence of the energy loss, a logarithmic correction factor in the small angle Coulomb scattering and the importance of path length versus material thickness. We develop a method for further constraining the possible proton paths by including energy information in the derivation of the most likely path, utilizing an infinite-dimensional constrained functional analysis method. It is shown that while there is an additional uncertainty on the most likely path that is difficult to determine experimentally, the associated error envelopes are smaller, resulting in possibly slightly improved spatial resolution for proton-computed tomography.