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 $\{g_2(1,3,t),g_2(2,3,t)\}$. 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 $\mathit{\text{IJMPC}}$ $\mathbf{\text{10}}$ 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 $\mathit{\text{i.e.,}}$ ${\displaystyle \frac{dH(t)}{dt}=0}$ where $H(t)$ refers to the Boltzmann’s H-function.