  The fluctuations in thermodynamic and transport properties in many-body
systems gain importance as the number of constituent particles is reduced.
Ultracold atomic gases provide a clean setting for the study of mesoscopic
systems; however, the detection of temporal fluctuations is hindered by the
typically destructive detection, precluding repeated precise measurements on
the same sample. Here, we overcome this hindrance by utilizing the enhanced
light--matter coupling in an optical cavity to perform a minimally invasive
continuous measurement and track the time evolution of the atom number in a
quasi two-dimensional atomic gas during evaporation from a tilted trapping
potential. We demonstrate sufficient measurement precision to detect atom
number fluctuations well below the level set by Poissonian statistics.
Furthermore, we characterize the non-linearity of the evaporation process and
the inherent fluctuations of the transport of atoms out of the trapping volume
through two-time correlations of the atom number. Our results establish coupled
atom--cavity systems as a novel testbed for observing thermodynamics and
transport phenomena in mesosopic cold atomic gases and, generally, pave the way
for measuring multi-time correlation functions of ultracold quantum gases.
