Consider an isolated system. We know that it will reach equilibrium after enough time. According to what I have read so far, the system should undergo irreversible processes to reach equilibrium, but is there any counterexample to this statement? i.e a system and a nonequilibrium initial condition such that processes used to reach equilibrium are all reversible?
Assume an isolated system. Start from an equilibrium state $0$ and remove some internal constraint so that the system spontaneously moves to another equilibrium state $1$. The difference in entropy $\Delta S = S_1 - S_0$ between the entropies of the final $S_1$ and initial $S_0$ is never negative (2nd law), and it is positive for an irreversible process the amount being characteristic and a measure of irreversibility.