  Ultracold atomic systems have been of great research interest in the past,
with more recent attention being paid to systems of mixed species. In this work
we carry out non-perturbative Path Integral Monte Carlo (PIMC) simulations of N
distinguishable particles at finite temperature, which can be thought of as an
ultracold atomic system containing N distinct species. We use the PIMC approach
to calculate thermodynamic properties of particles interacting via hard-sphere
and hard-cavity potentials. The first problem we study is a two-particle system
interacting via a hard-sphere and hard-cavity interaction in order to test the
effectiveness of two approximations for the thermal density matrix
corresponding to these potentials. We then apply the PIMC method to a system of
many hard-sphere particles under periodic boundary conditions at varying
temperature in order to calculate the energy per particle, pressure, and
specific heat of the system. We examine how finite-size effects impact the
results of PIMC simulations of hard-sphere particles and when the thermodynamic
limit has been reached. Our results provide microscopic benchmarks for a system
containing distinguishable particles, which can be thought of as a limiting
case for ultracold atomic systems of mixed species.
