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bbgky_hierarchy [2013/11/10 18:04] nikolaj |
bbgky_hierarchy [2013/11/10 18:18] nikolaj |
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| @#88DDEE: $\langle \mathcal M,H\rangle $ ... classical Hamilonian system | | | @#88DDEE: $\langle \mathcal M,H\rangle $ ... classical Hamilonian system | | ||
| @#DDDDDD: $3N\equiv \text{dim}(\mathcal M)$ | | | @#DDDDDD: $3N\equiv \text{dim}(\mathcal M)$ | | ||
- | | @#88DDEE: $H({\bf q}^1,\dots,{\bf q}^N,{\bf p}_1,\dots,{\bf q}_N)=\sum_{i=1}^N \left(T({\bf p}_i)+\Phi_\text{ext}({\bf q}^i)+\sum_{j<i}\Phi_\text{int}(|{\bf q}^i-{\bf q}^j|)\right)$ | | + | | @#DDDDDD: $ {\bf q} \in \mathcal M $ | |
+ | | @#DDDDDD: $ {\bf p} \in T^*\mathcal M $ | | ||
+ | |||
+ | $q^i,p_i$ denote tupples of three components. | ||
+ | |||
+ | | @#88DDEE: $H({\bf q},{\bf p})=\sum_{i=1}^N \left(T(p_i)+\Phi_\text{ext}(q^i)+\sum_{j<i}\Phi_\text{int}(|q^i-q^j|)\right)$ | | ||
| $s\in\text{range}(N)$ | | | $s\in\text{range}(N)$ | | ||
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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. | + | 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: |
+ | |||
+ | $\frac{\partial A}{\partial q^i}\frac{\partial B}{\partial p_i} \equiv \sum_j \frac{\partial A}{\partial (q^i)_j}\frac{\partial B}{\partial (p_i)_j}$. | ||
| @#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} $ | |