How are irreversibility and probability related?

The second law of thermodynamics states that all irreversible processes (and such are practically all thermal processes, in any case, all naturally occurring processes) proceed in such a way that the entropy of the bodies participating in them increases, tending to a maximum value. The maximum value of entropy is reached when the system comes to equilibrium. At the same time, it was already noted above that the transition to an equilibrium state is much more probable than all other transitions. Therefore, only those state changes are observed in which the system passes from a less probable to a more probable state (the thermodynamic probability increases).
Noteworthy is the striking similarity in the behavior of two quantities – entropy and thermodynamic probability: both of them are involved in the transition of the system to equilibrium. In addition, experimental studies show that the macroscopic properties of a system are determined by its microscopic properties. Therefore, it is natural to assume the existence of a connection between entropy and thermodynamic probability.
s = f (w)
The connection between the thermodynamic probability of the state of the system and its entropy was established in 1875 by two famous scientists – D. Gibbs and L. Boltzmann. This connection is expressed by the Boltzmann formula, which has the form S = klnW
where k = R / N9A), R is the universal gas constant, Na is the Avogadro number

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