I will start out with a brief pedagogical introduction to the geometry of irreversible processes. Within this geometrical framework, I shall formulate and prove the following three assertions:
Under a Markovian process not only does entropy monotonically increase, indicating the approach equilibrium, but an even stronger statement holds:
Given two different initial macrostates undergoing the same Markovian process, their pairwise statistical distance as measured by their relative entropy can only decrease. Such convergence is known to hold for linear quantum operations; here I shall argue that - at least locally - it applies to nonlinear processes, too.
Given an experimentally observed congruence of integral curves of some unknown irreversible process, it is possible to reconstruct from these data the dynamical law governing the irreversible process. I shall provide a general prescription for doing so.
And finally, I shall argue that in principle every genuinely irreversible process can serve as a clock -- even if its underlying dynamics is not a priori known.