# Coupled differential equations metropolis algorithm

I would like to simulate the following coupled SDE. $$d\Omega=-\Omega H\left(\mathrm{n}\right)d\tau$$ $$dn=-Ad\tau+\sum_{i}B^{\left(i\right)}dW_{i}+\sum_{i}C^{\left(i\right)}dW’_{i}$$ provided $$A=\frac{1}{2}n\left(T-UM\right)\left(I-n\right)+\frac{1}{2}\left(I-n\right)\left(T-UM\right)n$$

$$B_{xy}^{\left(i\right)}=\sqrt{\frac{U}{2}}\sum_{\sigma,\sigma’}\sigma_{\sigma\sigma’}^{z}n_{x,\left(i\sigma’\right)}\left(\delta_{\left(i\sigma\right),y}-n_{\left(i\sigma\right),y}\right)$$ $$\sigma^{z}=\begin{bmatrix}1 & 0\ 0 & -1 \end{bmatrix}$$

$$C_{xy}^{\left(i\right)}=\sqrt{\frac{U}{2}}\sum_{\sigma,\sigma’}\sigma_{\sigma\sigma’}^{z}\left(\delta_{x,\left(i\sigma’\right)}-n_{x,\left(i\sigma’\right)}\right)n_{\left(i\sigma\right),y}$$

$$M_{\left(i\sigma\right),\left(j\sigma’\right)}=\delta_{ij}\sum_{\eta,\eta’}n_{\left(i\eta\right),\left(i\eta’\right)}\left(\sigma_{\sigma\sigma’}^{z}\sigma_{\eta\eta’}^{z}-\sigma_{\sigma\eta’}^{z}\sigma_{\eta\sigma’}^{z}\right)$$

Initial condition as $$n=\frac{1}{2}I$$ $$I$$ is a 2M X 2M matrix.$$M$$ is the mode number.

$$H(n)=\sum_{x,y=1}^{2M}T_{xy}n_{xy}+\frac{U}{2}\sum_{i=1}^{M}\sum_{\begin{array}{cc} \eta, & \eta’\ \sigma, & \sigma’ \end{array}\begin{array}{c} \ =1 \end{array}}^{2}\left[n_{\left(i\eta\right),\left(i\sigma’\right)}n_{\left(i\sigma\right),\left(i\eta’\right)}-n_{\left(i\eta\right),\left(i\eta’\right)}n_{\left(i\sigma\right),\left(i\sigma’\right)}+n_{\left(i\eta\right),\left(i\sigma’\right)}\delta_{\sigma\eta’}-n_{\left(i\eta\right),\left(i\sigma’\right)}\delta_{\left(i\sigma\right),\left(i\eta’\right)}\right]\sigma_{\eta\eta’}^{z}\sigma_{\sigma\sigma’}^{z}$$

For example if $$M=2$$ initial value of If M=2, then

Initial value of n matrix will be

$$n=.5\begin{bmatrix}1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Also $$T=\begin{bmatrix}-.1 & -2 & 0 & 0\ -2 & -.1 & 0 & 0\ 0 & 0 & -.1 & -2\ 0 & 0 & -2 & -.1 \end{bmatrix}$$

$$U/t=2$$

Is there a way to solve the above using the successive metropolis method as shown below? I found this in the following paper. https://arxiv.org/abs/0704.3792