FIG. 9: (Color online). Temperature dependence of the specific heat for BiMnO$_{3}$: mean-field specific heat values - dashed line, experimental data [13] - solid line. The magnetic part of the specific heat data are obtained by subtracting the total specific heat of BiScO$_{3}$ from that of BiMnO$_{3}$ [13, 22], since the former is not magnetic. taken as the difference between the total specific heat of BiMnO$_{3}$ and that BiScO$_{3}$ [13]. In this manner, Δ C$_{p}$ ( T ) represents only the magnetic part of the specific heat. Therefore, we discuss the role of Gaussian fluctuations of M ( ⃗ r ) and P ( ⃗ r ) on the thermodynamic properties next. In previous Sections, we made a very crucial assumption namely, that the order parame- ters are spatially homogeneous. The following calculations are provided in order to improve the temperature dependence of the specific heat part Δ C$_{p}$ ( T ) obtained above by taking into account the spatial fluctuations of the order parameters. For that purpose, we introduce the effective free energy F as a functional of spatially inhomogeneous order parameters [23]: where φ ( M, P, T ) is the free energy functional which incorporates thermal fluctuation in equilibrium, i.e. 15 F = F$_{0}$ + ∫ φ ( M ( ⃗ r ) , P ( ⃗ r ) , T ) d $^{3}$r, (33) A. Landau functional in spatially inhomogeneous case