Abstract
We consider a Brownian particle which, in addition to being in contact with a thermal bath, is driven by fluctuating forces which stem from active processes in the system, such as self-propulsion or collisions with other active particles. These active fluctuations do not fulfill a fluctuation-dissipation relation and therefore play the role of a non-equilibrium environment, which keeps the system permanently out of thermal equilibrium even in the absence of external forces. We investigate how the out-of-equilibrium character of the active matter system and the associated irreversibility is reflected in the trajectories of the Brownian particle. Specifically, we analyze the log-ratio of path probabilities for observing a certain particle trajectory forward in time versus observing its time-reversed twin trajectory. For passive Brownian motion, it is well-known that this path probability ratio quantifies irreversibility in terms of entropy production. For active Brownian motion, we show that in addition to the usual entropy produced in the thermal environment the path probability ratio contains a contribution to irreversibility from mutual information production between the particle trajectory and the history of the non-equilibrium environment. The resulting irreversibility measure fulfills an integral fluctuation theorem and a second-law like relation. When deriving and discussing these relations, we keep in mind that the active fluctuations can occur either due to 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. We obtain explicit expressions for active fluctuations modeled by an Ornstein-Uhlenbeck process. Finally, we illustrate our general results by analyzing a Brownian particle which is trapped in a static or moving harmonic potential.
Highlights
We investigate how the out-of-equilibrium character of the active matter system and the associated irreversibility is reflected in the trajectories of the Brownian particle
For active Brownian motion, we show that in addition to the usual entropy produced in the thermal environment, the path probability ratio contains a contribution to irreversibility from mutual information “production” between the particle trajectory and the history of the nonequilibrium environment
Here we address the above questions from a fundamental nonequilibrium statistical mechanics viewpoint by developing a trajectorywise thermodynamic description of active particle systems as a natural generalization of stochastic energetics [30,31] and thermodynamics [32,33,34,35,36] of passive particles in a purely thermal environment
Summary
Active particle systems consist of individual entities (“particles”) which have the ability to perform motion by consuming energy from the environment and converting it into a self-propulsion drive [1,2,3,4,5,6,7]. We analyze the path probability ratio based on particle trajectories, i.e., from the evolution of the particle positions in time, without explicitly modeling the (system-specific) microscopic processes behind the active fluctuations This approach allows for deriving general results on the nonequilibrium character and thermodynamic content of the dynamical behavior emerging in active matter systems independent of the specific processes generating the active fluctuations but with the caveat that dissipation and irreversibility occurring in these processes cannot be assessed [41]. Generalizes the trajectorywise stochastic thermodynamics of passive Brownian particles to active matter systems It shows that entropy production in the thermal bath, which fully quantifies the path probability ratio for passive Brownian motion, is complemented by the difference in mutual information accumulated with the active (nonequilibrium) bath in the time-forward versus the time-backward direction. The only requirement is that the particles are identical in the sense that they have the same coupling coefficients D and Da , and that the active fluctuations ηi ðtÞ of the individual constituents are independent but share identical statistical properties
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