BBGKY hierarchy
Theorem
| context | $\langle \mathcal M,H\rangle $ … classical Hamilonian system |
| range | $3N\equiv \text{dim}(\mathcal M)$ |
| range | $ {\bf q} \in \mathcal M $ |
| range | $ {\bf p} \in T^*\mathcal M $ |
$q^i,p_i$ denote tupples of three components.
| context | $H({\bf q},{\bf p})=\sum_{i=1}^N \left(T(p_i)+\Phi_\text{ext}(q^i)+\sum_{j<i}\Phi_\text{int}(|q^i-q^j|)\right)$ |
| $s\in\text{range}(N)$ | |
| context | $f_s$ … symmetrized reduced distribution function |
where $s$ is universally quantified and runs from 1 to the dimension of the associated Hamiltonian system.
| range | $L_s \equiv -\sum_{i=1}^s \left(\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q^i}\frac{\partial}{\partial p_i}\right)$ |
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:
$\frac{\partial A}{\partial q^i}\frac{\partial B}{\partial p_i} \equiv \sum_{j=1}^3 \frac{\partial A}{\partial (q^i)_j}\frac{\partial B}{\partial (p_i)_j}$.
| postulate | $ (\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
Schematically, the Liouville equation gives us the time evolution for the whole $N$-particle system in the form $(\frac{\partial}{\partial t}-L_s)f_N=0$, which expresses an incompressible flow of the probability density in phase space. We then define the reduced distribution functions incrementally by integrating out another particle's degrees of freedom $f_s \sim \int f_{s+1}$. An equation in the BBGKY hierarchy tells us that the time evolution for such a $f_s$ is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the N-s suppressed particles
$(\frac{\partial}{\partial t}-L_s) f_s \sim \text{div}_{\mathbf p} \langle \text{grad}_{\mathbf q}\Phi_\text{int}\rangle_{f_{s+1}}.$
The problem of solving the BBGKY hierarchy of equations is as hard as solving the original Liouville equation, but approximations for the BBGKY hierarchy (which allow truncation of the chain into a finite system of equations) can readily be made. The merit of these equations is that the higher distribution functions $f_{s+2},f_{s+3},\dots$ affect the time evolution of $f_s$ only implicitly via $f_{s+1}$. Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical or quantum kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum Boltzmann equations and the first order corrections to the Boltzmann equations. Other approximations, such as the assumption that the density probability function depends only on the relative distance between the particles or the assumption of the hydrodynamic regime, can also render the BBGKY chain accessible to solution.
For a derivation, see “Kinetic theory” by “Liboff”. On the way, one passes a set of equations for not necessarily symmetrized distrubutions. These, however, are not a hierarchy where $f_s$ is determined by only $f_s$. The higher functions $f_{s+2},f_{s+3},\dots$ only get rid of by taking the symmetrization into account.
Reference
Wikipedia: BBGKY hierarchy
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