  We present a novel method to constrain the past collisional evolution of
observed globular cluster (GC) systems, in particular their mass functions. We
apply our method to a pair of galaxies hypothesized to have recently undergone
an episode of violent relaxation due to a strong galaxy-galaxy interaction,
namely NGC 1052-DF2 and NGC 1052-DF4. We begin by exploring the observational
evidence for a collisional origin for these two recently discovered
ultra-diffuse galaxies observed in the NGC 1052 group, posited in the
literature to be dark matter (DM)-free. We compute the timescales for infall to
the central nucleus due to dynamical friction (DF) for the GCs in these
galaxies, using the shortest of these times to constrain how long ago a
galaxy-galaxy interaction could have occurred. We go on to quantify the initial
GC numbers and densities needed for significant collisional evolution to occur
within the allotted times, and show that, if the hypothesis of a previous
galaxy-galaxy interaction is correct, a paucity of low-mass GCs should be
revealed by deeper observational surveys. If any are found, they should be more
spatially extended than the currently observed GC population. Finally, we apply
our method to these galaxies, in order to illustrate its efficacy in
constraining their dynamical evolution. Our results motivate more complete
observations of the GC luminosity functions in these galaxies, in addition to
future studies aimed at combining the method presented here with a suite of
numerical simulations in order to further constrain the origins of the curious
GC populations in these (and other) galaxies.
