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A new truncation scheme for BBGKY hierarchy: conservation of energy and time reversibility

We propose a new truncation scheme for Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We approximate the three particle distribution function $f_{3}(1,2,3,t)$ in terms of $f_{2}(1,2,t)$, $f_{1}(3,t)$ and two point correlation functions $\left\lbrace g_{2}(1,3,t), g_{2}(2,3,t)\right\rbrace $. Further $f_{2}$ is expressed in terms of $f_{1}(1,t)$ and $g_{2}(1,2,t)$ to close the hierarchy, resulting a set of coupled kinetic equations for $f_{1}$ and $g_{2}$. In this paper we show that, for velocity independent correlations, the kinetic equation for $f_{1}$ reduces to the model proposed by Martys[Martys N S 1999 \textit{IJMPC} \textbf{10} 1367-1382]. In the steady state limit, the kinetic equation for $g_{2}$ reduces to Born-Green-Yvon (BGY) hierarchy for homogeneous density. We also prove that the present scheme respects the energy conservation and under specific circumstances, time symmetry \textit{i.e.,} $\displaystyle \frac{dH(t)}{dt} = 0$ where $H(t)$ refers to the Boltzmann's H-function.