  A stable real scalar provides one of the simplest possibilities to account
for dark matter. We consider the regime where its coupling to the Standard
Model fields is negligibly small. Due to self-coupling, the scalar field can
reach thermal or at least kinetic equilibrium, in which case the system is
characterized by its temperature and effective chemical potential. We perform a
fully relativistic analysis of dark matter evolution, thermalization conditions
and different freeze-out regimes, including the chemical potential effects. To
this end, we derive a relativistic Bose-Einstein analog of the Gelmini-Gondolo
formula for a thermal averaged cross section. Finally, we perform a
comprehensive parameter space analysis to determine regions consistent with
observational constraints. Dark matter can be both warm and cold in this model.
