6.5 The Fokker-Planck equation

The Fokker-Planck equation, which describes the time evolution of the nonequilibrium probability distribution $P(\mathbf{M}, t)$ of a set of Langevin equations like (), in the Stratonovich interpretation is given by [39]

$$\frac{\partial P}{\partial t} = -\frac{\partial}{\partial ... ...j} \left[ \left( D B_{ik}B_{jk} \right) P \right] \quad,$$ (6.22)

where Stratonovich calculus has been used to treat the multiplicative fluctuating terms in (). We can transform it to a continuity equation for the probability distribution by taking the $M_j$ derivatives of the second term on the right-hand side

$$\frac{\partial P}{\partial t} = -\frac{\partial}{\partial ... ...B_{jk}\frac{\partial}{\partial M_j} \right) P \right] \quad.$$ (6.23)

On using expression () we find

$$\frac{\partial B_{ik}}{\partial M_j}= -\gamma'\varepsilon ... ...s}} (\delta_{ij}M_k + \delta_{jk}M_i - 2\delta{ik}M_j) \quad.$$ (6.24)

Thus,

$$\frac{\partial B_{jk}}{\partial M_j} = -\gamma'\varepsilon _{jjk} -\frac{\alpha \gamma'}{M_\mathrm{s}} (\delta_{jj}M_k + \delta_{jk}M_j - 2 M_k)$$

$$= -\frac{\alpha \gamma'}{M_\mathrm{s}} 2 M_k$$ (6.25)

and

$$B_{ik}\frac{\partial B_{jk}}{\partial M_j} = \left[ -\gamma'\varepsilon _{ijk} M_j +\frac{\alpha \gamma'}{M_\m... ...delta_{ik}M^2) \right] \left( -2\frac{\alpha \gamma'}{M_\mathrm{s}} M_k \right)$$

$$= \frac{\alpha \gamma'}{M_\mathrm{s}} (M_i M_k M_k- \delta_{ik}M^2 M_k) \left( -2\frac{\alpha \gamma'}{M_\mathrm{s}} \right)$$

$$= 0 \quad.$$ (6.26)

We find, that the second term on the right hand side of () vanishes identically. For the third term we find

$$B_{ik}B_{jk}\frac{\partial P}{\partial M_j} = \gamma'^2 \Biggl[ -\varepsilon _{ilk} M_l -\frac{\alpha}{M_\mathrm{s}} (M_i M_k - \delta_{ik}M^2) \Biggr] \cdot$$

$$\quad \,\, \Biggl[ -\varepsilon _{jpk} M_p -\frac{\alpha}{M_\mathrm{s}} (M_j M_k - \delta_{jk}M^2) \Biggr] \frac{\partial P}{\partial M_j}$$

$$= \gamma'^2 \Biggl[ (\delta_{ij}\delta_{lp}-\delta_{ip}\delta_{jl}) M_l M_p$$

$$+\frac{\alpha}{M_\mathrm{s}} \left( -M^2 \varepsilon _{jpi}M_p+\v... ...}M_i M_p M_k - M^2 \varepsilon _{ilj}M_l +\varepsilon _{ilk}M_l M_j M_k \right)$$

$$+\frac{\alpha^2}{M^2_\mathrm{s}} \left( M^4 \delta_{ik}\delta_{jk... ...delta_{ik}M_j M_k)+ M_i M_j M^2 \right) \Biggr] \frac{\partial P}{\partial M_j}$$

$$= \gamma'^2 \Biggl[ \delta_{ij}M^2-M_i M_j + \frac{\alpha}{M_\mathrm{s}} \left( -M^2 \varepsilon _{jpi} M_p -M^2 \varepsilon _{ilj} M_l \right)$$

$$\quad \quad +\frac{\alpha^2}{M^2_\mathrm{s}} \left( M^4\delta_{ij}-M^2 M_i M_j \right) \Biggr] \frac{\partial P}{\partial M_j}$$

$$= \gamma'^2 \left[ (\alpha^2+1)(M^2 \delta_{ij}-M_i M_j) \frac{\partial P}{\partial M_j} \right]$$

$$= -\gamma'^2 (\alpha^2+1) \left[ \mathbf{M} \times \left( \mathbf{M} \times \frac{\partial P}{\partial M_j} \right) \right]_i \quad.$$ (6.27)

Our result for the Fokker-Planck equation is

$$\frac{\partial P}{\partial t} = -\frac{\partial}{\partial \mathbf{M}} \cdot \Biggl\{ \Biggl[ -\ga... ...a'}{M_\mathrm{s}} \mathbf{M} \times (\mathbf{M} \times \mathbf{H}_\mathrm{eff})$$

$$+ \frac{1}{2 \tau_N} \mathbf{M} \times \left( \mathbf{M} \times \frac{\partial}{\partial \mathbf{M}} \right) \Biggr] P \Biggr\} \quad,$$ (6.28)

where $P(\mathbf{M}, t)$ is the nonequilibrium probability distribution for $\mathbf{M}$ at time $t$, and $\frac{\partial}{\partial \mathbf{M}}\cdot$ stands for the divergence operator

$$\frac{\partial}{\partial \mathbf{M}} \cdot \mathbf{A} = \frac{\partial A_i}{\partial M_i}$$

and

$$\frac{1}{\tau_N}=2 D \gamma'^2 (1+\alpha^2)$$ (6.29)

is the Néel (free-diffusion) time.

Finally, we have to ensure, that the stationary properties of the stochastic Landau-Lifshitz equation (), supplemented by the statistical properties of the thermal field () and (), coincide with the appropriate thermal-equilibrium properties. Therefore, the stationary solution of the Fokker-Planck equation $P_0$, for which

$$\partial P_0/\partial t=0$$

is forced to be the Boltzmann distribution

$$P_0(\mathbf{M}) \propto \exp(-\beta \mathcal H(\mathbf{M})) \quad.$$ (6.30)

Since

$$\mu_0 v \mathbf{H}_\mathrm{eff}= -\frac{\partial \mathcal H}{\partial \mathbf{M}} \quad,$$

where $v$ denotes the discretization volume (the volume of a computational cell) we find

$$\frac{\partial P_0}{\partial \mathbf{M}}= \beta\mu_0 v \mathbf{H}_\mathrm{eff} P_0 \quad.$$

Hence,

$$\left[ \frac{\partial}{\partial \mathbf{M}} \cdot \left( \mathbf{M} \times \frac{\partial P_0}{\partial \mathbf{M}} \right) \right]_i =$$

$$\partial_{M_i}(\varepsilon _{ijk} M_j \partial_{M_k} P_0) = \varepsilon _{ijk} ( \delta_{ij}\partial_{M_k} P_0 + M_j \partial_{M_i} \partial_{M_k} P_0 ) =$$

$$= \varepsilon _{iik} \partial_{M_k} P_0 + M_j \varepsilon _{ijk} \partial_{M_i} \partial_{M_k} P_0$$

$$= 0$$ (6.31)

and the first term on the right hand side of the Fokker-Planck equation () vanishes.

Thus, the Fokker-Planck equation with the stationary solution $P_0$ reads

$$0=\Biggl[ -\frac{\alpha \gamma'}{M_\mathrm{s}} \mathbf{M}... ...es \beta \mu_0 v \mathbf{H}_\mathrm{eff} P_0 \right) \Biggr]$$

from which we find

$$\tau_N=\frac{1}{\alpha}\frac{\mu_0 v M}{2 \gamma' k_B T} \quad.$$

By comparison with () we arrive at

$$D=\frac{\alpha}{1+\alpha^2}\frac{k_B T}{\mu_0 v \gamma' M} \quad,$$ (6.32)

which was defined in () and determines the variance of the thermal field.