第737行: |
第737行: |
| | | |
| <math> | | <math> |
− | Eff=\frac{EI}{\ln L^n}\approx 1-\frac{m\ln\left(2\pi e\right)}{2n \ln L}+\frac{1}{2L^n n\ln L}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|^2 d\mathbf{x}, | + | Eff=\frac{EI}{\ln L^n}\approx 1-\frac{m\ln\left(2\pi e\right)}{2n \ln L}+\frac{1}{n\ln L}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\cdot \ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right| d\mathbf{x}, |
| </math> | | </math> |
| | | |
第755行: |
第755行: |
| | | |
| <math> | | <math> |
− | \mathcal{J}=\frac{EI}{n}\approx \ln L - \frac{1}{2}\ln (2\pi e)+\frac{1}{2nL^n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|^2 d\mathbf{x} | + | \mathcal{J}=\frac{EI}{n}\approx \ln L - \frac{1}{2}\ln (2\pi e)+\frac{1}{n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\cdot \ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right| d\mathbf{x} |
| </math> | | </math> |
| | | |
第761行: |
第761行: |
| | | |
| <math> | | <math> |
− | \Delta \mathcal{J}\equiv \mathcal{J_F}-\mathcal{J_f}=\frac{EI_F}{N}-\frac{EI_f}{n}\approx \frac{1}{n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|^2 d\mathbf{x}-\frac{1}{N}\int_{-[\frac{L}{2},\frac{L}{2}]^N}\frac{1}{L^N}\ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right|^2 dX | + | \Delta \mathcal{J}\equiv \mathcal{J_F}-\mathcal{J_f}=\frac{EI_F}{N}-\frac{EI_f}{n}\approx \frac{1}{n}\int_{[-\frac{L}{2},\frac{L}{2}]^n}\frac{1}{L^n}\cdot \ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right| d\mathbf{x}-\frac{1}{N}\int_{-[\frac{L}{2},\frac{L}{2}]^N}\frac{1}{L^N}\cdot \ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right| dX |
| </math> | | </math> |
| | | |
− | 注意,上式中的积分可以写成均匀分布下的期望,即[math]\int_{[-\frac{L}{2},\frac{L}{2}}\frac{1}{L^n}]\cdot=\mathbb{E}_{\mathbf{x}\sim \mathcal{U}[-\frac{L}{2},\frac{L}{2}]^n}\cdot[/math],继而上式化为: | + | 注意,上式中的积分可以写成均匀分布下的期望,即[math]\int_{[-\frac{L}{2},\frac{L}{2}}]\frac{1}{L^n}\cdot=\mathbb{E}_{\mathbf{x}\sim \mathcal{U}[-\frac{L}{2},\frac{L}{2}]^n}\cdot[/math],继而上式化为: |
| | | |
| | | |
| <math> | | <math> |
− | \Delta \mathcal{J}\approx \frac{1}{n}\mathbb{E}_{[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|^2-\frac{1}{N}\mathbb{E}_{-[\frac{L}{2},\frac{L}{2}]^N}\ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right|^2 | + | \Delta \mathcal{J}\approx \frac{1}{n}\mathbb{E}_{\mathbf{x}\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^n}\ln\left|\det\left(\frac{\partial_\mathbf{x} f(\mathbf{x})}{\Sigma^{1/2}}\right)\right|-\frac{1}{N}\mathbb{E}_{X\sim\mathcal{U}[-\frac{L}{2},\frac{L}{2}]^N}\ln \left|\det\left(\frac{\partial_\mathbf{X} F(\mathbf{X})}{\Sigma'^{1/2}}\right)\right| |
| </math> | | </math> |
| | | |