1.1.8 The Free Energy Functional

Now we are able to write the complete expression for the free energy of the ferromagnetic body. In fact, by collecting Eqs. (1.38), (1.40), (1.50) and (1.51), one has:

$$G(\textbf{M},\mathbf{H}_ ) =F_ +F_ +F_ +G_ =$$

$$=\int_\Omega \bigg\{ A[(\nabla m_x)^2+ (\nabla m_y)^2+ (\nabla m_z)^2] + f_ +$$

$$- \frac{1}{2} \mu_0\textbf{M}\cdot{\mathbf{H}_\text{m}}- \mu_0 \textbf{M}\cdot\mathbf{H}_\text{a}\bigg\} dV \quad,$$ (1.52)

which can be put in the compact form by expressing the exchange interaction energy density as $A(\nabla\textbf{{m}})^2$:

$$G(\textbf{M},\mathbf{H}_ )=\int_\Omega \bigg[A(\nabla \textbf{{m}})^2 + f_ + - \frac{1}{2} \mu_0\textbf{M}\cdot{\mathbf{H}_\text{m}}- \mu_0 \textbf{M}\cdot\mathbf{H}_\text{a}\bigg] dV \quad,$$ (1.53)