$H$ theorem for a homogeneous gas when allowance is made for various approximations in the density parameter

It is shown that the truncated “free energy” $\Gamma[l]$ is extremal as a function of $g_1,g_2,\dots,g_l$ at the equilibrium point. This result is used in the framework of BBGKY theory to verify the necessary condition of nondecrease of the truncated free energy per unit volume. In addition, it is shown that such extremality is a “almost sufficient” condition for the validity of the $H$ theorem, namely, the “rate of change” of the truncated free energy differs from a nonnegative expression by terms of order $\varepsilon_0^{l+1}$ and higher, where $\varepsilon_0$ is the density parameter, provided surface effects are ignored.