What is the connection between the non-reversibility of the decay of unstable nuclei (as Uranium, Plutonium) and the 2nd principle of thermodynamics? The 2nd principle of the thermodynamics says that if a system (e.g. an ideal gas) is left undisturbed, its number of microscopic states only increases. This is a statement of irreversibility of the process that the system undergoes. For instance, if we have two separated chambers, one with cold gas and one with hot gas of the same type, and bring the chambers into contact (by some pipe), the temperature of the two chambers will become equal after some time. It will not happen that the cold gas gets colder and the hot gas hotter.
In the nuclear decay, one particle (or more) get out from an unstable nucleus instead of remaining in it forever. And the process is irreversible, the particle (or particles) can be sent back (by some mirror) but the unstable nucleus won't be restored as is was before the decay, because while part of the emitted wave returns to the nucleus trying to restore the parent-nucleus, this nucleus keeps on emitting.
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, or, more generally, inside the volume delimited by the potential barrier?
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, represents a system with MORE states? (This idea seems non-plausible because quantum-mechanically, this pair is described by a ONE SINGLE composite quantum state as long as de-coherence is avoided - e.g. by keeping the system in vacuum).
Alternatively, do the decay and the 2nd principle of the thermodynamics stem from a common, more fundamental principle?
 A: 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.
