I know the Boltzmann kinetic equation is invariant under T transformation ($t \rightarrow -t$). Also, I could derive the entropy's time evolution from H-theorem as:
$$\frac{\partial S}{\partial t} = -k_{B} \int \int \int \ln\Bigg(\frac{f_{2} f_{1}}{f^{'}_{2}f^{'}_{1}}\Bigg) (f^{'}_{2}f^{'}_{1}-f_{2}f_{1}) g \alpha_{1} d\mathbf{e}^{'}d\mathbf{v}_{2}d\mathbf{v}_{1}$$
Where $f_{2}$ and $f_{1}$ are the probabilities before collision for two particles distributions, $f^{'}_{2}$ and $f^{'}_{1}$ are the probabilities after collision, $g$ is the relative velocity of particle 1 and 2 $g = |\mathbf{v}_{2}-\mathbf{v}_{1}|$ before collision and because the collision is elastic: $g = g^{'}$. Also, $\alpha_{1}$ is the differential cross section of the collision.
According to this equation because terms $\ln\Bigg(\frac{f_{2}f_{1}}{f^{'}_{2}f^{'}_{1}}\Bigg)$ and $(f^{'}_{2}f^{'}_{1}-f_{2}f_{1})$ have always opposite signs the integral is always negative and as a result entropy will increase always or $\frac{\partial S}{\partial t} > 0$. I know it's called Loschmidt's paradox which basically says that it is impossible to extract a time asymmetric equation from time symmetric ones. The thing which is not quite clear to me is that no matter the Boltzmann equation is time symmetric or not, this equation for entropy time evolution holds true so always we could say $\frac{\partial S}{\partial t} > 0$. So why it is not possible to prove the second law of thermodynamics by using H-theorem?
