1.3.5.3 Classical treatment of dissipation

It is possible to give a generalized form for introducing the dissipation in magnetization dynamics, through the so-called Rayleigh dissipation function. We focus now the attention on the case when $\frac{\partial \textbf{h}_\text{a}}{\partial t}=\mathbf{0}$, without affecting the generality of the analysis. Let us suppose to assign the following function:

$$\mathcal{R}\left(\frac{\partial \textbf{{m}}}{\partial t}\right)=... ...l t}\cdot \mathcal{A} \cdot \frac{\partial \textbf{{m}}}{\partial t} dv \quad,$$ (1.104)

where $\mathcal{A}$ is a symmetric positive-definite second order tensor. Now, we can rewrite the Landau-Lifshitz-Gilbert equation (1.98) in the following way:

$$\frac{\partial \textbf{{m}}}{\partial t} = - \textbf{{m}}\times ... ...delta \mathcal{R}}{\delta \frac{\partial \textbf{{m}}}{\partial t}} \right) ,$$ (1.105)

where the variational derivative of the Rayleigh function determines the `viscous force' acting during magnetization motion. The important property of this formulation lies in the fact that equilibrium configurations remain unchanged after the introduction of the dissipation, as one can see from the observation of the Rayleigh function (1.104). Now, if we scalar multiply both sides of Eq. (1.105) by $\frac{\delta g}{\delta \textbf{{m}}}+\frac{\delta \mathcal{R}}{\delta \frac{\partial \textbf{{m}}}{\partial t}}$ and integrate over the volume $\Omega$, we end up with:

$$\frac{dg}{dt}=-\int_\Omega \frac{\delta \mathcal{R}}{\delta \fra... ...bf{{m}}}{\partial t}}\cdot \frac{\partial \textbf{{m}}}{\partial t} dv \quad.$$ (1.106)

By applying Euler's theorem on homogeneous functions, the latter equation becomes:

$$\frac{dg}{dt}=-2\mathcal{R}= -\int_\Omega \frac{\partial \textbf{... ... t}\cdot \mathcal{A} \cdot \frac{\partial \textbf{{m}}}{\partial t} dv \quad.$$ (1.107)

The choice of Gilbert damping corresponds to assume

$$\mathcal{A}=\alpha\mathcal{I} \quad,$$ (1.108)

where $\mathcal{I}$ is the identity tensor and $\alpha$ is the Gilbert damping constant. Such an approach can be generalized if $\mathcal{A}$ is a self-adjoint operator in a suitable function space. An example of this is considered in Ref. [82], where an additional term is considered in the Rayleigh function involving the time derivative of the spatial gradient of magnetization vector field. Moreover, in Ref. [82] the most general gyromagnetic precessional equation is reported, which includes both the cases of Landau-Lifshitz and Landau-Lifshitz-Gilbert equations.