  We study the thermodynamic properties of binary Bose mixtures, by developing
a beyond mean-field Popov theory which properly includes the effects of quantum
and thermal fluctuations in both the density and spin channels. Results for key
thermodynamic quantities, such as the isothermal compressibility and the
magnetic susceptibility, are derived from a perturbative calculation of the
grand-canonical potential. We find that thermal fluctuations can play a crucial
role on the miscibility condition of a binary mixture, favoring phase
separation at finite temperature even if the mixture is soluble at zero
temperature, as already anticipated in a previous work [Ota et al., Phys. Rev.
Lett. 123, 075301 (2019)]. We further investigate the miscibility condition for
binary mixtures in the presence of asymmetry in the intra-species interactions,
as well as in the masses of the two components. Furthermore, we discuss the
superfluid behavior of the mixture and the temperature dependence of the
Andreev-Bashkin effect.
