Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
bbgky_hierarchy [2013/11/10 18:18] nikolaj |
bbgky_hierarchy [2014/03/21 11:11] (current) |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== BBGKY hierarchy ===== | ===== BBGKY hierarchy ===== | ||
| ==== Theorem ==== | ==== Theorem ==== | ||
| - | | @#88DDEE: $\langle \mathcal M,H\rangle $ ... classical Hamilonian system | | + | | @#55CCEE: context | @#55CCEE: $\langle \mathcal M,H\rangle $ ... classical Hamilonian system | |
| - | | @#DDDDDD: $3N\equiv \text{dim}(\mathcal M)$ | | + | | @#DDDDDD: range | @#DDDDDD: $3N\equiv \text{dim}(\mathcal M)$ | |
| - | | @#DDDDDD: $ {\bf q} \in \mathcal M $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf q} \in \mathcal M $ | |
| - | | @#DDDDDD: $ {\bf p} \in T^*\mathcal M $ | | + | | @#DDDDDD: range | @#DDDDDD: $ {\bf p} \in T^*\mathcal M $ | |
| $q^i,p_i$ denote tupples of three components. | $q^i,p_i$ denote tupples of three components. | ||
| - | | @#88DDEE: $H({\bf q},{\bf p})=\sum_{i=1}^N \left(T(p_i)+\Phi_\text{ext}(q^i)+\sum_{j<i}\Phi_\text{int}(|q^i-q^j|)\right)$ | | + | | @#55CCEE: context | @#55CCEE: $H({\bf q},{\bf p})=\sum_{i=1}^N \left(T(p_i)+\Phi_\text{ext}(q^i)+\sum_{j<i}\Phi_\text{int}(|q^i-q^j|)\right)$ | |
| | $s\in\text{range}(N)$ | | | $s\in\text{range}(N)$ | | ||
| - | | @#88DDEE: $f_s$ ... symmetrized reduced distribution function | | + | | @#55CCEE: context | @#55CCEE: $f_s$ ... symmetrized reduced distribution function | |
| where $s$ is universally quantified and runs from 1 to the dimension of the associated Hamiltonian system. | where $s$ is universally quantified and runs from 1 to the dimension of the associated Hamiltonian system. | ||
| - | | @#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: range | @#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: | 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: | ||
| Line 21: | Line 21: | ||
| $\frac{\partial A}{\partial q^i}\frac{\partial B}{\partial p_i} \equiv \sum_{j=1}^3 \frac{\partial A}{\partial (q^i)_j}\frac{\partial B}{\partial (p_i)_j}$. | $\frac{\partial A}{\partial q^i}\frac{\partial B}{\partial p_i} \equiv \sum_{j=1}^3 \frac{\partial A}{\partial (q^i)_j}\frac{\partial B}{\partial (p_i)_j}$. | ||
| - | | @#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: postulate | @#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} $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| Line 34: | Line 34: | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/BBGKY_hierarchy|BBGKY hierarchy]] | Wikipedia: [[http://en.wikipedia.org/wiki/BBGKY_hierarchy|BBGKY hierarchy]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Symmetrized reduced distribution function]] | [[Symmetrized reduced distribution function]] | ||
