5.2 The stochastic Landau-Lifshitz equation

Thermal activation is introduced in the Landau-Lifshitz equation () by a stochastic thermal field $\mathbf{H}_\mathrm{th}$, which is added to the effective field (). It accounts for the effects of the interaction of the magnetization with the microscopic degrees of freedom (eg. phonons, conducting electrons, nuclear spins, etc.), which cause fluctuations of the magnetization distribution. This interaction is also responsible for the damping, since fluctuations and dissipation are related manifestations of one and the same interaction of the magnetization with its environment.

Since a large number of microscopic degrees of freedom contribute to this mechanism, the thermal field is assumed to be a Gaussian random process with the following statistical properties:

$$\langle {H_{\mathrm{th},i}(t)} \rangle =0$$ (5.1)

This means, that the average of the thermal field taken over different realizations vanishes in each direction $i \in \{x,y,z\}$ of space. The second moment, or variance, is given by

$$\langle {H_{\mathrm{th},i}(t) H_{\mathrm{th},j}(t')} \rangle = 2 D \delta_{ij}\delta(t-t')$$ (5.2)

This equation is a manifestation of the fluctuation-dissipation theorem. It relates the strength of the thermal fluctuations (the thermal field) to the dissipation due to the damping of our system [38]. The Kronecker $\delta$ expresses the assumption, that the different components of the thermal field are uncorrelated, whereas the Dirac $\delta$ expresses, that the autocorrelation time of the thermal field is much shorter than the response time of the system (``white noise'', cf. section ).

After adding the thermal field we get the stochastic Landau-Lifshitz equation

$$\frac{d \mathbf{M}}{dt} = -\gamma'\mathbf{M} \times (\math... ...hbf{H}_\mathrm{eff} + \mathbf{H}_\mathrm{th}) \right) \quad.$$ (5.3)

Rearrangement to separate deterministic from stochastic contributions gives

$$\frac{d \mathbf{M}}{dt} = -\gamma'\mathbf{M} \times \mathbf{H}_\mathrm{eff} -\frac{\alpha \... ...a'}{M_\mathrm{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H}_\mathrm{eff})$$

$$-\gamma'\mathbf{M} \times \mathbf{H}_\mathrm{th} -\frac{\alpha \g... ...\mathrm{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H}_\mathrm{th}) \quad,$$ (5.4)

which reveals, that it is a Langevin type stochastic differential equation with multiplicative noise.

To keep the notation simple, we rewrite () by substituting

$$A_i(\mathbf{M}, t) = \left[ -\gamma'\mathbf{M} \times \ma... ... \times (\mathbf{M} \times \mathbf{H}_\mathrm{eff}) \right]_i$$ (5.5)

and

$$B_{ik}(\mathbf{M}, t) = -\gamma'\varepsilon _{ijk} M_j -\frac{\alpha \gamma'}{M_\mathrm{s}} \varepsilon _{ijn} M_j \varepsilon _{nmk} M_m$$

$$= -\gamma'\varepsilon _{ijk} M_j -\frac{\alpha \gamma'}{M_\mathrm{s}} (\delta_{im}\delta_{jk} - \delta_{ik}\delta_{jm}) M_j M_m$$

$$= -\gamma'\varepsilon _{ijk} M_j -\frac{\alpha \gamma'}{M_\mathrm{s}} (M_i M_k - \delta_{ik}M^2) \quad,$$ (5.6)

where we have written $M^2$ for $M_{jj} = M_\mathrm{s}$. We have used the Einstein summation convention and we will do so in the following. The outer products have been rewritten with the totally antisymmetric unit tensor $\varepsilon$ (Levi-Civita symbol).

Hence, we can simplify the stochastic Landau-Lifshitz equation () and get

$$\frac{d M_i}{dt} = A_i(\mathbf{M}, t) + B_{ik}(\mathbf{M}, t) H_{\mathrm{th},k}(t) \quad.$$ (5.7)

This is the general form of a system of Langevin equations with multiplicative noise, because the multiplicative factor $B_{ik}(\mathbf{M}, t)$ for the stochastic process $H_{\mathrm{th},k}(t)$ is a function of $\mathbf{M}$.