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.