Motivated by this question.
Truncating the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy at the first order using the molecular chaos approximation:
$$f(v_1, v_2, r_1, r_2)=\chi f(v_1)f(v_2),$$
leads to the usual Boltzmann equation from which we can prove the H-theorem, namely that $dH/dt \leq 0$ with $H(t) = \int dv f(v, t)\log(f(v, t))$. Which is interpreted as a proof of the non decreasing nature of (some) entropy with time. This is a result general to Markovian process following detailed balance.
My question is: is there such result if the BBGKY hierarchy is truncated at a higher order?. I guess that it can depends on the closure or even on the choice of entropy. Since we could probably define $H'(t) = \int dv_1 dv_2 f(v_1, v_2, t) \log(f(v_1, v_2, t))$ as a different but (equally?) valid H function.
