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We consider a Brownian particle which, in addition to being in contact with athermal bath, is driven by fluctuating forces which stem from active processesin the system, such as self-propulsion or collisions with other activeparticles. These active fluctuations do not fulfill a fluctuation-dissipationrelation and therefore play the role of a non-equilibrium environment, whichkeeps the system permanently out of thermal equilibrium even in the absence ofexternal forces. We investigate how the out-of-equilibrium character of theactive matter system and the associated irreversibility is reflected in thetrajectories of the Brownian particle. Specifically, we analyze the log-ratioof path probabilities for observing a certain particle trajectory forward intime versus observing its time-reversed twin trajectory. For passive Brownianmotion, it is well-known that this path probability ratio quantifiesirreversibility in terms of entropy production. For active Brownian motion, weshow that in addition to the usual entropy produced in the thermal environmentthe path probability ratio contains a contribution to irreversibility frommutual information production between the particle trajectory and the historyof the non-equilibrium environment. The resulting irreversibility measurefulfills an integral fluctuation theorem and a second-law like relation. Whenderiving and discussing these relations, we keep in mind that the activefluctuations can occur either due to a suspension of active particles pushingaround a passive colloid or due to active self-propulsion of the particleitself; we point out the similarities and differences between these twosituations. We obtain explicit expressions for active fluctuations modeled byan Ornstein-Uhlenbeck process. Finally, we illustrate our general results byanalyzing a Brownian particle which is trapped in a static or moving harmonicpotential.

Irreversibility in active matter systems: Fluctuation theorem and mutual information