Abstract
Active matter systems are driven out of equilibrium by conversion of energy into directed motion locally on the level of the individual constituents. In the spirit of a minimal description, active matter is often modeled by so-called active Ornstein-Uhlenbeck particles an extension of passive Brownian motion where activity is represented by an additional fluctuating non-equilibrium “force” with simple statistical properties (Ornstein-Uhlenbeck process). While in passive Brownian motion, entropy production along trajectories is well-known to relate to irreversibility in terms of the log-ratio of probabilities to observe a certain particle trajectory forward in time in comparison to observing its time-reversed twin trajectory, the connection between these concepts for active matter is less clear. It is therefore of central importance to provide explicit expressions for the irreversibility of active particle trajectories based on measurable quantities alone, such as the particle positions. In this technical note we derive a general expression for the irreversibility of AOUPs in terms of path probability ratios (forward vs. backward path), extending recent results from [PRX 9, 021009 (2019)] by allowing for arbitrary initial particle distributions and states of the active driving.
Highlights
Irreversible thermodynamic processes are characterized by a positive entropy change in their “Universe”, i.e., in the combined system of interest and its environment [1]
In small mesoscopic systems on the micro- and nanometer scale, such as a colloidal Brownian particle diffusing in an aqueous solution, it has been established within the framework of stochastic thermodynamics [2,3,4,5,6] that the total entropy change should be evaluated from the entropy produced in the system and in its thermal environment along the specific trajectory the system follows during the process
What can we learn about the non-equilibrium nature of an active system by observing particle trajectories, i.e., the evolution of particle positions over time? Within the framework of a minimal model for particulate active matter on the micro- and nanoscale, the active Ornstein-Uhlenbeck particle [15, 25,26,27,28,29,30,31], we here contribute an essential step toward exploring this question by deriving an exact analytical expression for the path weight (Eqs. 25–27), which is valid for any values of the model parameters, any external driving forces, arbitrary initial particle positions and configurations of the active fluctuations, and arbitrary trajectory durations
Summary
Irreversible thermodynamic processes are characterized by a positive entropy change in their “Universe”, i.e., in the combined system of interest and its environment [1]. We will provide a central step toward an understanding of the role of the path probability ratio in active matter by providing exact analytical expressions for a simple but highly successful and well-established [15, 25,26,27,28,29,30,31] model of active matter, namely the active OrnsteinUhlenbeck particle (AOUP) [16, 18,19,20,21, 24, 32,33,34,35,36,37,38,39,40,41,42,43,44] In this model, self-propulsion is realized via a fluctuating “driving force” in the equations of motion [7, 10] with Gaussian distribution and exponential time-correlation (see Section 2.1). For an AOUP, the active fluctuations follow a Gaussian process with exponential timecorrelations, which can be generated by a so-called Ornstein-
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