Is there any proof for the 2nd law of thermodynamics?
It's simple to "roughly prove" the second law in the context of statistical physics. The evolution $A\to B$ of macrostate $A$, containing $\exp(S_A)$ microstates, to macrostate $B$, containing $\exp(S_B)$ microstates, is easily shown by the formula for the probability "summing over final outcomes, averaging over initial states", to be $\exp(S_B-S_A)$ higher than the probability of the inverse process (with velocities reversed). Because $S_B-S_A$ is supposed to be macroscopic, such as $10^{26}$ for a kilogram of matter, the probability in the wrong direction is the exponential of minus this large difference and is zero for all practical purposes.
The more rigorous versions of this proof are always variations of the 1872 proof of the so-called H-theorem by Ludwig Boltzmann:
H-theorem
This proof may be adjusted to particular or general physical systems, both classical ones and quantum ones. Please ignore the invasive comments on the Wikipedia about Loschmidt's paradoxes and similar stuff which is based on a misunderstanding. The H-theorem is a proof that the thermodynamic arrow of time - the direction of time in which the entropy increases - is inevitably aligned with the logical arrow of time - the direction in which one is allowed to make assumptions (the past) in order to evolve or predict other phenomena (in the future).
Every Universe of our type has to have a globally well-defined logical arrow of time: it has to know that the future is being directly evolving (although probabilistically, but with objectively calculable probabilities) from the past. So any universe has to distinguish the future and the past logically, it has to have a logical arrow of time, which is also imprinted to our asymmetric reasoning about the past and the future. Given these qualitative assumptions that are totally vital for the usage of logic in any setup that works with a time coordinate, the H-theorem shows that a particular quantity can't be decreasing, at least not by macroscopic amounts, for a closed system.
It was first found empirically, and later derived from various more theoretical assumptions.
There is a proof in Section 7.2 of Chapter 7: Phenomenological Thermodynamics of Classical and Quantum Mechanics via Lie algebras, based on a few axioms for thermodynamics, and a proof in Chapter 9 that these laws follow from the standard assumptions in statistical mechanics.
The reversibility objections (Loschmidt's paradox) are unjustified since the Poincare recurrence theorem assumes that the system in question is bounded, which is (most likely) not the case for the real universe.
If we assume time evolution is unitary and hence reversible, and the total size of the phase space subject to constraints based upon the total energy and other conserved quantities is finite, then the only conclusion is Poincaré recurrences cycling ergodically through the entire phase space. Boltzmann fluctuations to states of lower entropy might occur with exponentially suppressed probabilities, but the entropy would increase both toward its past and future. This is so not the second law as Boltzmann's critics never tire of pointing out.
The H-theorem depends upon the stosszahlansatz assumption that separate events in the past are uncorrelated, but that is statistically exceedingly improbable assuming a uniform probability distribution.
If the total size of the phase space is infinite, Carroll and Chen proposed that in eternal inflation there can be some state with finite entropy with entropy increasing in both time directions.
To me, the most likely scenario is to drop the assumption of unitarity and replace that with time evolution using Kraus operators acting upon the density matrix.