Irreversibility in Active Matter Systems: Fluctuation Theorem and Mutual Information

We consider a Brownian particle which, in addition to being in contact with a thermal bath, is driven by active fluctuations. These active fluctuations do not fulfill a fluctuation-dissipation relation and therefore play the role of a non-equilibrium environment. Using an Ornstein-Uhlenbeck process as a model for the active fluctuations, we derive the path probability of the Brownian particle subject to both thermal and active noise. From the case of passive Brownian motion, it is well-known that the log-ratio of path probabilities for observing a certain particle trajectory forward in time versus observing its time-reserved twin trajectory quantifies the entropy production. We calculate this path probability ratio for active Brownian motion and derive a generalized "entropy production", which fulfills an integral fluctuation theorem. We show that those parts of this "entropy production" which are different from the usual dissipation of heat in the thermal environment, can be associated with the mutual information between the particle trajectory and the history of the non-equilibrium environment. When deriving and discussing these results we keep in mind that the active fluctuations can occur due to either a suspension of active particles pushing around a passive colloid or due to active self-propulsion of the particle itself, we point out the similarities and differences between these two situations. Finally, we illustrate our general results by analyzing a Brownian particle which is trapped in a static or moving harmonic potential.