The entropy maximum postulate states that given a thermodynamic system there's a function $S$ of the extensive parameters called entropy which has the property that once a constraint is removed the equilibrium state the system goes to is the one which maximize $S$ among the available states.
I took a statistical mechanics course recently and in the statistical mechanics approach one considers the number of available microstates $\Omega$ corresponding to some macroscopic state characterized by certain extensive parameters. In that case we have $\Omega = \Omega(X_1,\dots,X_n)$ being $X_i$ the extensive parameters we are considering.
One particular case would be $\Omega = \Omega(E,V,N)$ with $E$ being energy, $V$ the volume and $N$ the number of particles.
In that approach we define the function $S(X_1,\dots,X_n)=k_B \ln \Omega(X_1,\dots,X_n)$. The main idea them is that we consider that on the thermodynamic limit, which is attained when we let the parameters grow with the densities fixed, this function is the thermodynamic entropy.
After that everything works as in thermodynamics and we them apply the entropy maximum postulate.
I've remained with one question, however. This tells how, from statistical mechanics, we get the entropy and so on, so that we can describe the system using thermodynamic theory. On the other hand, nothing justfied the idea of associating this $S$ to the thermodynamics $S$ and applying the entropy maximum postulate.
I always believed statistical mechanics could justify this postulate. In that case: what justifies identifying this entropy with the one found in thermodynamics? And more importantly, how does one justify, using statistical mechanics, the entropy maximum postulate?