The BBGKY Hierarchy

Consider the first two equations associated with the BBGKY hierarchy [13] which describe the time evolution of single and two particle distribution functions $P_1(\vec{r},\vec{k},t)$ and $P_2(\vec{r}_1,\vec{k}_1,\vec{r}_2,\vec{k}_2,t)_1$.

$$\partial_t P_1 + \vec{k}_1 \! \cdot \! \nabla P_1 + \vec{F} \! \cdot \! \nabla_{\!\! k} P_1 =\Omega \! , \\$$ (1)

$$\partial_t P_2 + (\frac{\vec{k}_1}{m} \cdot \frac {\partial... ...\vec{r}_1,\vec{k}_1,\vec{r}_2,\vec{k}_2,\vec{r}_3,\vec{k}_3,t)$$ (2)

where,

$$\Theta_{ij}= \frac{\partial V(r_{ij})}{\partial \vec{r}_i} \f... ...{ij})}{\partial \vec{r}_j} \frac{\partial}{\partial \vec{k}_j}$$ (3)

P₃ is the three particle distribution function, F is an external force, k₁ is the microscopic momentum, t is time, V is the interparticle potential, and the collision term $\Omega$is

$$\Omega=-\int d^3k_2d^3r_2 \frac{\partial P_2(\vec{r}_1,\vec{r... ...k}_1} \cdot \frac{\partial V(r_{12})}{\partial \vec{r}_1} \,$$ (4)

with $\vec{r}_{12}= \vec{r}_1-\vec{r}_2$.

In the BBGKY hierarchy of equations the time evolution of a single particle distribution is expressed in terms of the two particle distribution function and so forth. Solving for the one through nparticle distribution functions involves approximating the form of the n+1 particle distribution function to obtain closure of the hierarchy. A variety of techniques and approximations have been developed to obtain closure [14], some of which are mentioned later in this paper.