  The question of macroscopic occupation and spontaneous emergence of coherence
for exciton ensembles has gained renewed attention due to the rise of van der
Waals heterostructures made of atomically thin semiconductors. The hosted
interlayer excitons exhibit nanosecond lifetimes, long enough to allow for
excitonic thermalization in time. Several experimental studies reported
signatures of macroscopic occupation effects at elevated exciton densities.
With respect to theory, excitons are composite particles formed by fermionic
constituents, and a general theoretical argument for a bosonic thermalization
of an exciton gas beyond the linear regime is still missing. Here, we derive an
equation for the phonon mediated thermalization at densities above the
classical limit, and identify which conditions favor the thermalization of
fermionic or bosonic character, respectively. In cases where acoustic,
quasielastic phonon scattering dominates the dynamics, our theory suggests that
transition metal dichalcogenide (TMDC) excitons might be bosonic enough to show
bosonic thermalization behaviour and decreasing dephasing for increasing
exciton densities. This can be interpreted as a signature of an emerging
coherence in the exciton ground state, and agrees well with the experimentally
observed features, such as a decreasing linewidth for increasing densities.
