content_image/373643.jpg content_image/373649.jpg
<img src='content_image/373643.jpg'>

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. 

## A. Landau functional in spatially inhomogeneous case 

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]: 

<img src='content_image/373649.jpg'>

where φ ( M, P, T ) is the free energy functional which incorporates thermal fluctuation in equilibrium, i.e. 

