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| jy:results [2018/07/30 06:37] – [7.30.18 Inverse Problem data] yiling | jy:results [2019/09/16 09:22] (current) – [2019-09-16 :Case N1: long tail antiferromagentic] yiling | ||
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| ====== 2018-07-2 :The solutions to the inverse Ising problem | ====== 2018-07-2 :The solutions to the inverse Ising problem | ||
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| ====== 7.19.18 Inverse Problem | ====== 7.19.18 Inverse Problem | ||
| - | The mean field theory (nMF): | + | {{page>dynstat:mean-field}} |
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| - | $\mathbf{J}^\text{nMF}=-\mathbf{C}^{-1}, | + | |
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| - | $h_{i}^\text{nMF}=\tanh^{-1}\left< | + | |
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| - | The independent-pair theory (IP): | + | |
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| - | $J_{ij}^\text{pair}=0.25*\ln\left[ \frac{(1+m_{i}+m_{j}+C_{ij}^{*}) (1-m_{i}-m_{j}+C_{ij}^{*})} {(1-m_{i}+m_{j}-C_{ij}^{*})(1+m_{i}-m_{j}-C_{ij}^{*})} \right]$, with $C_{ij}^{*}=C_{ij}+m_{i}m_{j} , (i \neq j)$ | + | |
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| - | $h_{i}^\text{pair}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{pair} m_{j}$. | + | |
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| - | The Thouless-Anderson-Palmer theory (TAP): | + | |
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| - | $\mathbf{C}^{-1}+J_{ij}^\text{TAP}+2(J_{ij}^\text{TAP})^{2}m_{i}m_{j}=0 , (i \neq j)$ | + | |
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| - | $h_{i}^\text{TAP}=h_{i}^\text{nMF}-m_{i}\sum_{i=1}^{N} (J_{ij}^\text{TAP})^{2} (1-m_{j})^{2}$. | + | |
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| - | The Sessak and Monasson theory (SM): | + | |
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| - | $J_{ij}^\text{SM}=J_{ij}^\text{nMF}+J_{ij}^\text{pair}-\frac{C_{ij}}{(1-m_{i}^{2})(1-m_{j}^{2})-C_{ij}^{2}} , (i \neq j)$ | + | |
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| - | $h_{i}^\text{SM}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{SM} m_{j}$. | + | |
| The coefficient of $R^{2}$ is: | The coefficient of $R^{2}$ is: | ||