It is true that classical thermodynamic equations emerge from statistical mechanics. And that the increase in entropy depends on the increase in the number of microstates.
Decays also increase the number of microstates. They are irreversible because decay releases energy and the thermodynamic system cannot deliver enough energy and combination of particles to get back to the original state, as it cannot go back to any original microstate either. If a uranium breaks up, there is a probability if the right fragments with the correct energy collide to bind up again if the correct quantized energy is supplied to the fragments by fortuitous collisions , but the probability is very very small.
The question is, WHAT pushes a particle (e.g. an alpha article) out of the parent nucleus? WHY doesn't the alpha remain forever in the parent nucleus?
Nuclear decay happens because nuclei are bound by the strong force but there is the repulsive force of the protons, which is only balanced by the neutrons along the diagonal in this plot of isotopes. The higher the number of protons the more neutrons proportionately are needed for binding the isotope. Too many neutrons allow the instability of the neutron ( it decays when free) a probability of decay. Decay and fission release binding energy, because the system is no longer bound quantum mechanically and it breaks into fragments, creating more microstates.
Is the irreversibility of the nuclear decay connected with the 2nd principle of the thermodynamics? Or, is there some similarity between them? The configuration of daughter-nucleus + emitted particle, represent a system with MORE states? (Quantum-mechanically this system is described by a ONE SINGLE quantum state).
This system was described by one quantum mechanical state function before it decayed. After it decayed it is no longer in a single quantum state once the fragments interact in the heat bath of the environment.
Or, do the decay and the 2nd principle of the thermodynamics stem from a common, more fundamental principle?
The decay happens because the system has a quantum mechanical probability of decaying, a half life. It is computable with quantum mechanical models, not thermodynamic models( i.e. statistical mechanics). Potentials enter and energy levels and the Pauli exclusion principle, the whole artillery. Thermodynamics is an emergent phenomenon from the underlying quantum mechanical framework, certainly for materials with nuclear decays but also in general, as atoms and molecules are also quantum mechanical entities.
Edit after rereading next day
When one continues studies in disciplines that depend on physics, one should keep in mind that in describing natural phenomena, the appropriate framework should be considered. Also that there exists a hierarchy in physical frameworks, starting from the microscopic range of elementary particles going to nuclei, to atoms/molecules to solid/liquid/gas states . Each framework has its region of validity, models and computational tools.
Mathematically in the models as the hierarchy rises, at the confluence of two frameworks, the larger in centimeters framework emerges. It is a many body result of the fact that everything is composed of elementary particles and their bindings. Thus thermodynamics is an emergent theory and the second law is a law for large dimensions, with respect to the quantum mechanical framework on which it is founded in nature. It emerges from the probabilistic nature of quantum mechanics.
This became very clear with the black body problem and its solution, that thermodynamics with classical statistical mechanics was inadequate to describe the situation.
In cosmic dimensions, the force of gravity is postulated, and the classical theories described the motions on earth and of planets etc very well; the present view is that it is the highest framework of General Relativity which in the limiting case turns into the Newtonian mechanics gravitational theory. So in this case, Newton's laws are dependent on General Relativity laws , from the large frame to the lower one. Thermodynamics is not such a case.