How the nature of an observation affects single-trajectory entropies.

Projection of a Markov process with constant rates of transition to a small number of observable aggregated states can result in complex kinetics with memory. Here, we define the entropy production along a single sequence of aggregated states and show that it obeys detailed and integral fluctuation theorems. More importantly, we prove that projection shifts the distribution of entropy production over the ensemble of paths for a nonequilibrium process toward one characteristic of a system at equilibrium. This statement represents an analog of the second law of thermodynamics for path-dependent entropies and thus a new form of constraint of irreversible systems. Numeric examples are presented to illustrate these ideas.