  We study the statistical mechanics of quantum droplets in a one-dimensional
ring geometry. The relevant model of a modified two-component Gross-Pitaevskii
equation under symmetry considerations, i.e. same particle numbers and equal
intra-component interaction strengths, reduces to a single-component equation.
To determine the classical partition function thereof, we leverage the
semi-analytical transfer integral operator (TIO) technique. The latter predicts
a distribution of the observed wave function amplitudes and yields two-point
correlation functions providing insights into the emergent dynamics involving
quantum droplets. We compared the ensuing TIO results with the equilibrium
properties of suitably constructed Langevin dynamics and with direct numerical
simulations of the original system's modulational instability dynamics where
the generated droplets are found to coalesce. The results indicate good
agreement between the distinct methodologies aside from intermediate
temperatures in the special limit where the droplet widens, approaching a
plane-wave. In this limit, the distribution acquires a pronounced bimodal
character at the TIO level, not captured by the Langevin dynamics, yet observed
within the modified Gross-Pitaevskii framework.
