4.2.1 General properties of the effective field

By considering the variational derivative of Eq. (4.3) with respect to the vector field $\textbf{{m}}$ and by using Eqs. (4.4)-(4.5) and the boundary condition (4.6), one can readily derive that the effective field is constituted by the sum of four terms: the exchange field $\textbf{h}_$ex, the magnetostatic field $\textbf{h}_$m, the anisotropy field $\textbf{h}_$an and the applied field $\textbf{h}_$a:

$$\textbf{h}_{\text{eff}}(\textbf{{m}},t) = -\frac{\delta g}{\delta... ...xt{ex} + \textbf{h}_\text{m}+ \textbf{h}_\text{an}+\textbf{h}_\text{a}(t) ,$$ (4.7)

where the explicit dependence of $\textbf{h}_{\text{eff}}$ on time is related to the dependence on time of $\textbf{h}_$a. The first three terms in Eq. (4.7) can be related to the vector field $\textbf{{m}}$ through the following equations (sections 1.2.2 and 3.1):

$$\textbf{h}_ = \frac{2A}{\mu_0 M_s^2} \nabla^2 \textbf{{m}} ,$$ (4.8)

$$\textbf{h}_ = -\frac{1}{4\pi}\nabla\!\!\int_{\Omega} \!\!\nabla'\!\left(\fr... ...right)\cdot \textbf{{m}}(\textbf{r}^{\prime}) dV_{\textbf{r}^{\prime}} ,$$ (4.9)

$$\textbf{h}_ = \frac{2K_1}{\mu_0 M_s^2} \mathbf{e}_ (\textbf{r}) (\mathbf{e}_ (\textbf{r}) \cdot \textbf{{m}}(\textbf{r})) ,$$ (4.10)

From the Eqs. (4.4)-(4.5), (4.6) and (4.8)-(4.10), one can easily prove that the sum of the first three terms of the effective field (4.7) is a formally self-adjoint operator acting on the vector field $\textbf{{m}}$ in a suitable subspace of $\mathbb{L}^2(\Omega)$ with respect to the usual scalar product in $\mathbb{L}^2(\Omega)$:

$$\left(\textbf{v},\textbf{w}\right)_{\mathbb{L}^2(\Omega)}= \int_{\Omega} \textbf{v}(\textbf{r}) \cdot \textbf{w}(\textbf{r}) dV .$$ (4.11)

In other terms the effective field (4.7) can be written in the following form

$$\textbf{h}_{\text{eff}}(\textbf{r},t)= - \mathcal{C} \textbf{{m}}+\textbf{h}_\text{a}(t)$$ (4.12)

where $\mathcal{C}$ is a formally self-adjoint integro-differential operator in $\mathbb{L}^2(\Omega)$.