Mathematical proof of the Second Law of Thermodynamics

I believe it was Boltzmann who first made the connection between entropy and micro states. chapter 12 of "Classical and Statistical Thermodynamics" by Ashley H. Carter discusses Boltzmann's arguments. To summarize from that book:

Entropy ($S$) corresponds to a particular configuration of an ensemble of particles called a macro state. A macro state can be achieved using a number of different micro states ($w$). Therefore, $S = f(w)$. Micro states represent the probability of being in the macro state (they just need to be normalized). If two systems are combined, the total entropy is just $S = S_A + S_B$, or $f(w) = f(w_A) + f(w_B)$. The likelihood of being in micro state $w$ is just $w_A \cdot w_B$, since independent probabilities are multiplicative. Therefore, $S(w_A w_B) = S(w_A) + S(w_B)$. Carter then says that the only function that satisfies this property is the natural logarithm, so $S\propto ln(w)$. The constant of proportionality is $k_B$. A lot more details in the text, and examples of micro states (discussion in the context of quantum as well, leading up to density of states).

The book has a good general treatment of statistical mechanics. It starts off with classical thermodynamics and then moves onto statistical mechanics and makes the connection to quantum mechanics.