How do physicists mathematically define entropy (for the second law of thermodynamics) and how is it related to statistical definitions of entropy?

Even though there are many questions on this site about entropy (such as this one), none I could find was mathematically rigorous or had a complete rigorous answer.

I am looking for precise answers that can be understood by mathematicians.

In mathematical statistics, we have many different definitions of the entropy of (or between) probability distributions. Notable ones are:

  • The $\alpha$-entropy $N(\alpha)$ of a distribution $\rho$ on the integers, which is defined as $$\log \sum_{n \in \mathbb{N}} \rho(n)^\alpha.$$ It can be extended to the entropy of a distribution defined on any separable metric space.
  • The Kullback-Leibler divergence (or relative entropy) $$D_{KL}(P, Q) = \int \log\frac{dP}{dQ} dP.$$

Note that transforms $T$ of the sample space can only increase relative entropy: $D_{KL}(PT^{-1}, QT^{-1}) \geq D_{KL}(P, Q)$, with equality iff $T$ is a sufficient statistic for $\{P, Q\}$ and where $PT^{-1}(A) = P(T^{-1}(A))$. That's all I know about increase of entropy and the impossibility to create information.

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