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bbgky_hierarchy [2013/11/10 18:17] nikolaj |
bbgky_hierarchy [2013/11/10 18:18] nikolaj |
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| | @#DDDDDD: $L_s \equiv -\sum_{i=1}^s \left(\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i}\frac{\partial}{\partial p_i}\right)$ | | | @#DDDDDD: $L_s \equiv -\sum_{i=1}^s \left(\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i}\frac{\partial}{\partial p_i}\right)$ | | ||
| - | This is a Liuville-like operator/Poisson bracket, which only takes coordinates up to $s$ into account. Notice that the index runs over particles, i.e. a summation over three components is implied: $\frac{\partial}{\partial q^i}A\frac{\partial}{\partial p_i}B = \sum_j \frac{\partial}{\partial (q^i)_j}A\frac{\partial}{\partial (p_i)_j}B$. | + | This is a Liuville-like operator/Poisson bracket, which only takes coordinates up to $s$ into account. Notice that the index runs over particles, i.e. a summation over three components is implied: |
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| + | $\frac{\partial A}{\partial q^i}\frac{\partial B}{\partial p_i} \equiv \sum_{j=1}^3 \frac{\partial A}{\partial (q^i)_j}\frac{\partial B}{\partial (p_i)_j}$. | ||
| | @#55EE55: $ (\frac{\partial}{\partial t}-L_s)f_s = (N-s)\sum_{i=1}^s \frac{\partial }{\partial p_i}\int \frac{\partial \Phi_\text{int}(|q^i-q^{s+1}|)}{\partial q^i}f_{s+1}\ \mathrm d^3q^{s+1}\mathrm d^3p_{s+1} $ | | | @#55EE55: $ (\frac{\partial}{\partial t}-L_s)f_s = (N-s)\sum_{i=1}^s \frac{\partial }{\partial p_i}\int \frac{\partial \Phi_\text{int}(|q^i-q^{s+1}|)}{\partial q^i}f_{s+1}\ \mathrm d^3q^{s+1}\mathrm d^3p_{s+1} $ | | ||
