这是一份uwa西澳大学MATH3024的成功案例

复杂系统|MATH3024 Complex Systems代写 UWA代写

Inserting for the $J_{i j}$ and cosidering $i \in I_{x}$ we get
$$
m_{i}=\tanh \left[\beta \sum_{\boldsymbol{x}^{\prime}} Q\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right) p_{\boldsymbol{x}^{\prime}} m_{\boldsymbol{x}^{\prime}}+\beta h_{i}\right]
$$
where we have introduced sub-lattice (equilibrium) magnetizations $m_{x}$ via
$$
m_{x}=\frac{1}{\left|I_{\boldsymbol{x}}\right|} \sum_{i \in I_{\boldsymbol{x}}} m_{i}
$$
Inserting this into (5.13) we get
$$
m_{\boldsymbol{x}}=\frac{1}{\left|I_{\boldsymbol{x}}\right|} \sum_{i \in I_{\boldsymbol{x}}} \tanh \left[\beta \sum_{\boldsymbol{x}^{\prime}} Q\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right) p_{\boldsymbol{x}^{\prime}} m_{\boldsymbol{x}^{\prime}}+\beta h_{i}\right]=\left\langle\tanh \left[\beta \sum_{\boldsymbol{x}^{\prime}} Q\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right) p_{\boldsymbol{x}^{\prime}} m_{\boldsymbol{x}^{\prime}}+\beta h\right]\right\rangle_{h},
$$

MATH3024 COURSE NOTES ：

Introducing the overlap vector
$$
m=\sum_{\xi} p_{\xi} \xi m_{\xi}=\left\langle\xi m_{\xi}\right\rangle_{\xi}
$$
we see that the fpe’s can be written as
$$
m_{\boldsymbol{\xi}}=\tanh [\beta \boldsymbol{\xi} \cdot \boldsymbol{m}]
$$
which after multiplying by $\boldsymbol{\xi}$ and averaging over $\boldsymbol{\xi}$ gives
$$
\boldsymbol{m}=\langle\boldsymbol{\xi} \tanh [\beta \boldsymbol{\xi} \cdot \boldsymbol{m}]\rangle_{\boldsymbol{\xi}},
$$
or, in components
$$
m_{\mu}=\left\langle\xi^{\mu} \tanh \left[\beta \sum_{\nu} \xi^{\nu} m_{\nu}\right]\right\rangle_{\xi},
$$
Note that $m_{\mu}$ is nothing but the overlap of the equilibrium spin-configuration with the pattern $\xi_{i}^{\mu}$,
$$
m_{\mu}=\frac{1}{N} \sum_{i} \xi_{i}^{\mu}\left\langle S_{i}\right\rangle=\sum_{\xi} p_{\xi} \xi^{\mu} m_{\xi}
$$