Irreversibility in Active Matter Systems: Fluctuation Theorem and Mutual Information

PHYSICAL REVIEW X 9, 021009 (2019)

Lennart Dabelow Fakultät für Physik, Universität Bielefeld, 33615 Bielefeld, Germany

Stefano Bo and Ralf Eichhorn Nordita, Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

(Received 13 June 2018; revised manuscript received 22 February 2019; published 15 April 2019)

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 nonequilibrium 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 nonequilibrium 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.

DOI: 10.1103/PhysRevX.9.021009 Subject Areas: Statistical Physics

I. INTRODUCTION AND MAIN RESULTS and swarming [13–18], bacterial turbulence [19], or motility-induced phase separation [20,21] to name but a few. A. Introduction Microorganisms and mircoswimmers are usually dis- Active particle systems consist of individual entities persed in an aqueous solution at room temperature and (“particles”) which have the ability to perform motion by therefore experience thermal fluctuations which give rise to consuming energy from the environment and converting it a diffusive component in their self-propelled swimming – into a self-propulsion drive [1 7]. Prototypical examples motion. In addition, the self-propulsion mechanism is are collections of macroorganisms, such as animal herds, typically noisy in itself [2], for instance, due to environ- – schools of fish, flocks of birds, or ant colonies [8 10], and mental factors or intrinsic stochasticity of the mechanisms suspensions of biological microorganisms or artificial creating self-propulsion. These “active fluctuations” microswimmers, such as bacteria and colloidal particles exhibit two essential features. First, a certain persistence with catalytic surfaces [2,3,5,11,12]. Systems of this kind in the direction of driving over length- and timescales exhibit a variety of intriguing properties, e.g., clustering comparable to observational scales. Second, an inherent nonequilibrium character as a consequence of permanently converting and dissipating energy in order to fuel self- propulsion. Interestingly, similar active fluctuations with Published by the American Physical Society under the terms of the same characteristics can be observed in a complemen- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to tary class of active matter systems, namely, a passive the author(s) and the published article’s title, journal citation, colloidal “tracer” particle which is suspended in an aqueous and DOI. solution of active swimmers. The collisions with the active

2160-3308=19=9(2)=021009(31) 021009-1 Published by the American Physical Society DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) particles in the environment entail directional persistence mechanics viewpoint by developing a trajectorywise and nonequilibrium features in the motion of the passive thermodynamic description of active particle systems as tracer particle [22–24]. The sketch in Fig. 1 illustrates these a natural generalization of stochastic energetics [30,31] and two types of active particle systems. thermodynamics [32–36] of passive particles in a purely Despite the inherent nonequilibrium properties of active thermal environment. In particular, we study the breakdown matter systems, they appear to bear striking similarities to of time-reversal symmetry in the motion of particles which equilibrium systems [21,25–27], for instance, the dynamics are driven by thermal and active fluctuations. We quantify of individual active particles at large scales often looks like the associated irreversibility of individual particle trajecto- passive Brownian diffusion [3] (i.e., at scales beyond at ries as a measure for nonequilibrium by comparing the least the “persistence length” of the particle motion). In probability of a specific particle trajectory to occur forward connection with the ongoing attempts to describe active in time to that of observing its time-reversed counterpart. matter systems by thermodynamic (-like) theories, this We link the corresponding probability ratio to functionals observation raises the important question, “How far from over the forward trajectory and provide a thermodynamic equilibrium is active matter?” [28,29]. More specifically, interpretation of the resulting extensive quantities in terms the questions are (i) in which respects do emergent proper- of entropy production and mutual information. ties of active systems resemble the thermodynamics of In the case of passive particles in contact with a single thermal equilibrium systems, (ii) in which respects do they purely thermal bath, the path probability ratio is a well- not, and (iii) how do these deviations manifest themselves established fundamental concept in stochastic thermody- in observables describing the thermodynamic character of namics for assessing irreversibility [32–35]. It is known to the active matter system, like, e.g., the entropy production? provide relations between thermodynamic quantities, such Such questions should be answered independently of the as entropy production, work, or heat, and dynamical specific microscopic processes creating the active driving properties encoded in path probability ratios. Many such because, while these processes maintain the active particle relations have been found over the last two decades system out of equilibrium, they contain little or no [33,34,37–39], which, in their integrated forms, typically information on whether the emergent system behavior yield refinements of the second law of thermodynamics appears equilibriumlike or not. It is, for instance, of little [33,38,40]. In particular, the fluctuation theorem for the relevance for the system properties emerging on the coarse- total entropy production of a passive particle in a thermal grained scale of particle trajectories (e.g., the “diffusion” of environment reinforces the fundamental interpretation of active particles) and their similarities or dissimilarities to entropy as a measure of irreversibility (we give a brief thermal equilibrium, if an active particle is self-propelled summary of the results most relevant to our present work in by chemical processes occurring at its surface or by a Sec. III). In the presence of active fluctuations, however, the beating flagellum. In this spirit, here we address the above identification of dissipated heat and entropy production is questions from a fundamental nonequilibrium statistical less straightforward and is connected to the problem of how to calculate and interpret the path probability ratio in a physically meaningful way [41,42]. In the present work, 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 FIG. 1. Illustration of the two different kinds of active matter cannot be assessed [41]. The derived quantitative irrevers- systems. Left: Passive Brownian particle (red) in a thermal bath ibility and thermodynamic measures are easily accessible in (blue) with a suspension of active self-propelled particles standard experiments, e.g., from recording particle trajec- (yellow); collisions of the active particles with the passive one tories using video microscopy. In order to account for the are modeled by a fluctuating force η t . Right: Self-propelled ð Þ active fluctuations, which either stem from an active bath particle (yellow) in a thermal bath (blue); the fluctuating self- propulsion velocity is modeled to result from a fluctuation force the (passive) particle is dispersed in or from active self- ηðtÞ. In both cases, thermal fluctuations are described by unbiased propulsion (see Fig. 1 and Sec. IVA for specific examples δ-correlated Gaussian white-noise sources ξðtÞ, and the position of both cases), without resolving the microscopic processes of the particle of interest is denoted by xðtÞ; see Eqs. (5)–(7) in the that govern the interactions between active and passive main text. particles or drive self-propulsion, we follow the common

021009-2 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) approach to include stochastic “active forces” in the active diffusion coefficient Da, and active correla- equations of motion of the individual particle of interest tion time τa (implicit in Γτ). It is, in particular, not (see, e.g., Refs. [2,5] and references therein). When necessary to make any assumptions on the behavior calculating the path probabilities, we treat these active of the active fluctuations under time reversal to nonequilibrium forces in the same way as thermal equi- arrive at the above result for ΔΣ. librium noise, namely, as a “bath” the particle is exposed to Combined with the change in system entropy Δ − x τ x 0 with unknown microscopic details but known statistical Ssys ¼ kB ln½pð τ; Þ=pð 0; Þ, which compares properties (which are correlated in time and break detailed system probabilities at the beginning (t ¼ 0) and end τ ΔΣ Δ balance [43]). Our main results following this approach (t ¼ ) of the trajectory [34,38], þ Ssys fulfills and their implications are briefly summarized in the next an integral fluctuation theorem [Sec. IV C, Eq. (44)] section. valid for any duration τ of the particle trajectory,

−ðΔΣþΔS Þ=k B. Brief summary of the main results he sys B i¼1; ð2Þ Our central findings are the following: the average h·i is over all possible trajectories (i) Using a path-integral approach, we calculate the advancing from t ¼ 0 to t ¼ τ. As a consequence, probability density p½xjx0 of particle trajectories x¯ x τ x ∪ x τ x ∪ x they obey a second-law-like relation [Eq. (45)], ¼f ðtÞgt¼0 ¼ 0 f ðtÞgt>0 ¼ 0 as a result of the statistical properties of thermal and active ΔΣ x¯ ΔS x0; xτ ≥ 0: fluctuations simultaneously affecting the dynamics h ½ þ sysð Þi ð3Þ of the particle [Sec. IV B, Eqs. (38) and (40)]. Here, the coordinate xðtÞ may either refer to the position of In case of the equal sign, the active matter system a passive tracer particle in an active bath or to the appears reversible on the level of particle trajectories x position of an active self-propelled particle. When ðtÞ. From Eq. (1b), we see that in the stationary Δ 0 “ calculating these path weights, we consider the state (with Ssys ¼ ), the trajectories for free ” f ≡ 0 general situation in which the particle is also subject active diffusion without external force ( t ) to deterministic external (or interaction) forces always look reversible and thus equilibriumlike. f ¼ fðxðtÞ;tÞ, which may explicitly depend on In the presence of external forces, we have t ΔΣ Δ 0 ΔΣ Δ time via externally controlled driving protocols. We h þ Ssysi > in general, where h þ Ssysi summarize the mathematical details of the derivation becomes larger for more irreversible particle dy- as well as various known limiting cases in Appen- namics, indicating that the system appears to be dixes A–C. further away from an equilibriumlike behavior. ˜ (ii) We then find [Sec. IV C, Eqs. (42)] that the log ratio The path probability ratio p½x˜jx˜ 0=p½xjx0 is an of the probabilities for particle trajectories to occur important fundamental concept in stochastic thermo- forward in time versus backward in time (time- dynamics because it contains information on the backward trajectories are marked by a tilde symbol) thermodynamics of the system, such as second-law- can be expressed as a functional along the forward like bounds on entropy production as a consequence 0 path with a nonlocal “memory kernel” Γτðt; t Þ: of the fluctuation theorems, features of heat, work, and efficiency distributions in systems that represent p˜ x˜ x˜ heat engines, or violations of the fluctuation- ½ j 0 −ΔΣ x¯ =k ¼ e ½ B ; ð1aÞ response relation due to nonequilibrium drivings. p½xjx0 In the present work, we derive its most immediate with implications, namely, fluctuation theorems and second-law-like relations. Further insights into our Z Z 1 τ τ D active matter system that may be derived in future ¯ 0 _ T 0 a 0 ΔΣ½x¼ dt dt xt f t0 δðt − t Þ − Γτðt;t Þ : work from the explicit knowledge of the path T 0 0 D probability ratio are briefly discussed in the Con- ð1bÞ clusions, Sec. VII. (iii) By keeping track of the specific realization of the The functional ΔΣ therefore quantifies irreversibility active fluctuations which participated in generating in our active matter systems. With the explicit form the particle trajectory, we can identify the two 0 of Γτðt; t Þ given in Eq. (40), it can be calculated individual contributions to ΔΣ stemming from the for any particle trajectory xðtÞ. Information on the thermal environment and the active bath. The ΔΣ Γ 0 Δ x¯ η¯ thermal and active baths enters into and τðt; t Þ thermal part is the usual entropy Stot½ j produced only via their statistical parameters, like the thermal in the thermal environment and in the system along diffusion coefficient D, thermal bath temperature T, the particle trajectory (for the given active noise

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η¯ η τ – realization, which we denote ¼f ðtÞgt¼0). [22 24,49] and active motion driven by self-propulsion The part associated with the active bath is given [7,15–18,21,28,42,50–61]. The details of the model class by the difference ΔI½x¯; η¯ in the amount of corre- we consider are given in the next section. lations (measured in terms of pathwise mutual information) that are built up between the particle II. MODEL trajectory and the active noise in the time-forward We consider a colloidal particle at position x in direction as compared to the time-backward direc- d 1, 2, or 3 dimensions, which is suspended in an tion (Sec. V). This splitting provides the thermody- ¼ aqueous solution at thermal equilibrium with temperature namic interpretation T. The particle diffuses under the influence of deterministic external forces, in general consisting of potential forces ΔΣ½x¯þΔS ðx0; xτÞ¼ΔS ½x¯jη¯ − k ΔI½x¯; η¯ sys tot B −∇Uðx;tÞ [with the potential Uðx;tÞ] and nonconservative ð4Þ force components Fðx;tÞ. In addition, it experiences fluctuating driving forces due to permanent energy con- for our irreversibility measure ΔΣ derived from the version from active processes in the environment or the path probability ratio of particle trajectories; see particle itself. The specific examples we have in mind are a Eq. (1). It is valid for any active matter model which passive tracer in an active nonthermal bath (composed, e.g., includes the active fluctuations as forces in the of bacteria in aqueous solution; see left panel of Fig. 1) particle equations of motion, no matter if we assume [22–24,49] or a self-propelled particle (e.g., a colloidal the active fluctuations to be even [Eq. (58)] or odd microswimmer or bacterium; see right panel of Fig. 1) [Eq. (64)] under time reversal (denoted later in the [7,15–18,21,28,42,50–61]. text by sub- or superscripts). The interpretation (4) Neglecting inertia effects [64], we model the over- generalizes the trajectorywise stochastic thermody- damped Brownian motion of the colloidal particle by the namics of passive Brownian particles to active Langevin equation [47,48,65,66] matter systems. It shows that entropy production pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi _ in the thermal bath, which fully quantifies the path xðtÞ¼vðxðtÞ;tÞþ 2DaηðtÞþ 2DξðtÞ: ð5Þ probability ratio for passive Brownian motion, is complemented by the difference in mutual informa- The deterministic forces are collected in tion accumulated with the active (nonequilibrium) 1 bath in the time-forward versus the time-backward v x f x direction. From the fluctuation theorem for ΔΣ,we ð ;tÞ¼γ ð ;tÞ; ð6aÞ directly obtain an integral fluctuation theorem and a second-law relation for the mutual information where γ is the hydrodynamic friction coefficient of the difference ΔI [Eqs. (59), (65) and (60), (66)]. particle and We illustrate all these findings in Sec. VI by discussing the example of a colloidal particle subject to thermal and fðx;tÞ¼−∇Uðx;tÞþFðx;tÞ: ð6bÞ active fluctuations, which is trapped in a (static or moving) harmonic potential. For this simple linear system, all Thermal fluctuations are described by unbiased Gaussian relevant quantities can be calculated explicitly. We give white-noise sources ξðtÞ with mutually independent 0 the most important mathematical details in Appendix D. δ-correlated components ξiðtÞ, i.e., hξiðtÞξjðt Þi ¼ 0 We finally remark that the explicit expression obtained δijδðt − t Þ, where the angular brackets denote the average 0 under (i) and (ii) for Γτðt; t Þ [see Eq. (40)] is valid for a over many realizations of the noise and i; j ∈ f1; …;dg. Gaussian Ornstein-Uhlenbeck process [47,48] as a model The strength of the thermal fluctuations is given by the for the active fluctuations, while result (iii) is valid particle’s diffusion coefficient D, which is connected for general types of active fluctuating forces η with to the temperature T and the friction γ by the fluctuation- arbitrary (but well-defined) statistical properties. The dissipation relation D ¼ kBT=γ [62,67–69] (kB is Ornstein-Uhlenbeck process has become quite popular Boltzmann’s constant), as a consequence of the equilibrium – and successful in describing active fluctuations [7,15 properties of theffiffiffiffiffiffiffiffiffi thermal bath. – – p 18,21 24,28,42,49 61] because it constitutes a minimal The term 2DaηðtÞ in Eq. (5) represents the active force model for persistency in the active forcings due to its components with ηðtÞ¼½η1ðtÞ; …; ηdðtÞ being unbiased exponentially decaying correlations with finite correlation mutually independent noise processes, and Da being an time, and it can easily be set up to break detailed balance by effective “active diffusion” characterizing the strength of a “mismatched” damping term which does not validate the the active fluctuations. The model (5) does not contain fluctuation-dissipation relation [62,63]. Moreover, it is able “active friction,” i.e., an integral term over x_ðtÞ with a to describe both situations of interest mentioned above, friction kernel modeling the damping effects associated namely, passive motion in a bath of active swimmers with the active forcing. It thus represents the situation of

021009-4 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) low activity or particle density, where active fluctuations environment as a source of nonequilibrium fluctuations do not dominate over thermal ones and such friction effects ηðtÞ. Our main aim in this paper is to investigate in detail are negligibly small [24,70]. Despite this approxi- the role these nonequilibrium fluctuations play for the mation, the model (5) has been applied successfully to stochastic energetics [31] and thermodynamics [34] of the describe various active particle systems [15,17,21– Brownian particle. While we adopt the single-particle 24,49,61] and has become quite popular in an even more picture for simplicity, all our results hold for multiple simplified variant which neglects thermal fluctuations interacting Brownian particles as well. In this case, the (D ¼ 0) [7,16,18,28,42,50–59]. However, for thermody- symbol x in Eqs. (5) and (6) denotes a supervector namic consistency, it is necessary to consider both noise collecting the positions xi (i ¼ 1; 2; …;N) of all N par- sources [42], especially when assessing the flow of heat and ticles, i.e., x ¼ðx1; x2; …; xNÞ, and similar for the forces, entropy, which is in general produced in the thermal velocities, and so on. The only requirement is that the environment as well as in the processes fueling the active particles are identical in the sense that they have the same fluctuations [41,71,72]. coupling coefficients D and Da, and that the active As active fluctuations are the result of perpetual energy fluctuations ηiðtÞ of the individual constituents are inde- conversion (e.g., by the bacteria in the bath or by the pendent but share identical statistical properties. propulsion mechanism of the particle), their salient feature is that they do not fulfill a fluctuation-dissipation relation. III. THE IDEAL THERMAL BATH: Da =0 In our model (5), this violation is particularly obvious, as active friction effects are absent by the assumption of being In order to set up the framework and further establish negligibly small; in general, the active friction kernel would notation, we start with briefly recalling the well-known not match the active noise correlations [73,74]. In that case of a Brownian particle in sole contact with a thermal sense, the active fluctuations ηðtÞ characterize a nonthermal equilibrium reservoir, environment or bath for the Brownian particle with intrinsic pffiffiffiffiffiffiffi nonequilibrium properties. We emphasize that ηðtÞ is not a x_ðtÞ¼vðxðtÞ;tÞþ 2DξðtÞ; ð9Þ quantity directly measurable, but rather it embodies the net effects of the active components in the environment or the where the deterministic driving vðxðtÞ;tÞ is defined as in self-propulsion mechanism of the particle, similar to the Eq. (6). The Brownian particle can be prevented from white noise ξðtÞ representing an effective description of equilibrating with the thermal bath by time variations in the innumerable collisions with the fluid molecules in the the potential Uðx;tÞ or by nonconservative external aqueous solution. forces Fðx;tÞ. Although several results in the present paper are generally valid for any noise process ηðtÞ (with finite A. Energetics moments), our main aim is to study the specific situation of exponentially correlated Gaussian nonequilibrium fluc- Following Sekimoto [30,31], the heat δQ that the particle tuations [28,42,50–53,55,75] generated from an explicitly exchanges with the thermal bath while moving over an solvable Ornstein-Uhlenbeck process, infinitesimal distance dxðtÞ during a time step dt from t to t þ dt is quantified as the energy the thermal bath transfers 1 1 to the particle along this displacement due to friction η_ − η ζ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðtÞ¼ ðtÞþ ðtÞ; ð7Þ −γx_ t and fluctuations 2k Tγξ t , i.e., τa τa ð Þ B ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi _ i.e., the so-called Gaussian colored noise [76]. Here, δQðtÞ¼ð−γxðtÞþ 2kBTγξðtÞÞ · dxðtÞ; ð10Þ ζðtÞ¼½ζ1ðtÞ;…;ζdðtÞ are mutually independent, unbiased, δ-correlated Gaussian noise sources, just like ξðtÞ, where the product needs to be interpreted in the but completely unrelated to them. The characteristic Stratonovich sense [77]. With the definition (10), heat is time τa quantifies the correlation time of the process counted as positive if received by the particle and as (i; j ∈ f1; …;dg), negative when dumped into the environment (this sign convention is thus the same as Sekimoto’s original one δ ij 0 [30,31]). From the equation of motion (9), we immediately 0 −jt−t j=τa hηiðtÞηjðt Þi ¼ e ; ð8Þ 2τa see that the heat exchange can be equivalently written as as can be verified easily from the explicit steady-state δQðtÞ¼−f−∇UðxðtÞ;tÞþFðxðtÞ;tÞg · dxðtÞ: ð11Þ solution of Eq. (7). It is thus a measure for the persistence of the active fluctuations. The change of the particle’s “internal” energy dUðtÞ The model (5), (6) describes a single Brownian particle over the same displacement dxðtÞ is given by the total in simultaneous contact with a thermal bath and an active differential

021009-5 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) Z ∂ x τ _ 2 Uð ðtÞ;tÞ ðxt − vtÞ ∇ · vt dUðtÞ¼∇UðxðtÞ;tÞ · dxðtÞþ dt: ð12Þ p½xjx0 ∝ exp − dt þ ; ð16Þ ∂t 0 4D 2

Hence, if the potential does not vary over time, the only where we condition on a fixed starting position x0, and, τ contribution to dUðtÞ comes from the change in the accordingly, introduce the notation x ¼fxðtÞg 0 to denote x t> potential energy associated with the displacement d . a trajectory which starts at fixed position xð0Þ¼x0.In On the other hand, even if the particle does not move contrast to x¯ from above, the set of points x does therefore within dt, its internal energy can still change due to not contain the initial point x0 (i.e., x¯ ¼ x0 ∪ x). The path variations of the potential landscape by an externally probability (16) is to be understood as a product of applied time-dependent protocol. Prototype examples are transition probabilities in the limit of infinitesimal time- intensity or position variations of optical tweezers, which step size using a midpoint discretization rule. The diver- serve as a trap for the colloidal particle [23,78–81]. Being gence of v in the second term represents the path-dependent imposed and controlled externally, these contributions are part of normalization [86,87], while all remaining normali- interpreted as work performed on the particle. A second zation factors are path-independent constants and thus source of external forces which may contribute to such omitted. For convenience, we introduce the shorthand F x work are the nonconservative components ð ;tÞ in Eq. (9) notation xt ¼ xðtÞ and vt ¼ vðxðtÞ;tÞ. (originally not considered by Sekimoto [30] but system- We now consider the time-reversed version x˜ðtÞ of this atically analyzed later by Speck et al. [82]). The total trajectory, which is traced out backward from the final point work applied on the particle by external forces is therefore xðτÞ¼xτ to the initial point x0 when advancing time, given by x˜ðtÞ¼xðτ − tÞ; ð17Þ ∂UðxðtÞ;tÞ δWðtÞ¼ dt þ FðxðtÞ;tÞ · dxðtÞ: ð13Þ x˜ x˜ τ ∂t and ask how likely it is that ¼f ðtÞgt>0 is generated by the same Langevin equation (9) as x, with the same Combining Eqs. (11)–(13), we obtain the first law deterministic forces v acting at identical positions along the path. The latter requirement implies that in case of dU ¼ δQ þ δW ð14Þ explicitly time-dependent external forces, the force protocol has to be time inverted; i.e., vðx;tÞ is replaced by for the energy balance over infinitesimal displacements vðx; τ − tÞ in Eq. (9) to construct the Langevin equation for dxðtÞ valid at any point in time t. After integration along a the time-reserved path [88]. From that Langevin equation, τ p˜ x˜ x˜ specific trajectory fxðtÞg 0 of duration τ, which starts at we can deduce the probability ½ j 0 for observing the t¼ x˜ xð0Þ¼x0 and ends at some xðτÞ¼xτ [which is different backward trajectory conditioned on its initial position x˜ x τ x for every realization of the thermal noise ξðtÞ in Eq. (9), 0 ¼ ð Þ¼ τ, in analogy to Eq. (16). Using Eq. (17),we p˜ x˜ x˜ x even if xð0Þ¼x0 is kept fixed], we find the first law at the can then express ½ j 0 in terms of the forward path so trajectory level [35] that we find for the path probability ratio p˜ x˜ x˜ Δ x¯ x τ − x 0 x¯ x¯ ½ j 0 −Δ x¯ U½ ¼Uð τ; Þ Uð 0; Þ¼Q½ þW½ : ð15Þ ¼ e S½ =kB ; ð18Þ p½xjx0 Here, ΔU½x¯, Q½x¯, and W½x¯ denote the integrals of the with the quantity ΔS x¯ being a functional of the forward expressions (11) [or, equivalently, Eq. (10)], (12), and (13), ½ τ path only, respectively, along the trajectory x¯ ¼fxðtÞg ; the nota- t¼0 Z tion ½x¯ explicitly indicates the dependence on that 1 τ trajectory. ΔS½x¯¼ dt x_ðtÞ · fðxðtÞ;tÞ T Z0 1 τ B. Path probability and entropy ¼ fðxðtÞ;tÞ · dxðtÞ: ð19Þ T 0 The next step towards a thermodynamics characterization of the Brownian motion (9) is to introduce an entropy As a quantitative measure of irreversibility, ΔS½x¯ is change or entropy production associated with individual identified with the entropy production along the path x trajectories [38]. From the viewpoint of irreversibility, such with given initial position x0. an entropy concept has been defined as the log ratio of the For an infinitesimal displacement dxðtÞ, the correspond- probability densities for observing a certain particle tra- ing entropy change reads jectory and its time-reversed twin [83]. We express the probability density of a particle trajectory by the standard 1 δQðtÞ δSðtÞ¼ fðxðtÞ;tÞ · dxðtÞ¼− : ð20Þ Onsager-Machlup path integral [84–87], T T

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The last equality follows from comparison with Eq. (11) total forces acting on the Brownian particle, i.e., by pffiffiffiffiffiffiffiffiffi [see also Eq. (6)] and states that the entropy production replacing vðx;tÞ with vðx;tÞþ 2DaηðtÞ. However, there along dxðtÞ is given by the heat −δQðtÞ dissipated into the is another, maybe less obvious way of turning Eq. (9) into environment during that step divided by the bath temper- the model (5) with activeffiffiffiffiffiffiffiffiffi fluctuations, namely, by sub- δ Δ x¯ _ _ p ature. For that reason, SðtÞ and S½ are more accurately stituting x with x − 2DaηðtÞ. In the following, we argue called entropy production in the environment. that these two approaches correspond to two different The entropy of the Brownian particle itself (i.e., the physical situations (depicted in Fig. 1) with different entropy associated with the system degrees of freedom trajectorywise energy balances. (d.o.f.) x) is defined as the state function [34,38] 1. Active bath x − x Ssysð ;tÞ¼ kB ln pð ;tÞ; ð21Þ Examples for an “active bath” are the intracellular matrix – x of a living cell [89 95] or an aqueous suspension contain- where pð ;tÞ is the time-dependent solution of the Fokker- ing swimming bacteria [5,23]. These active environments Planck equation [47,48,66] associated with Eq. (9), for the then drive a passive Brownian particle, e.g., a colloid, via same initial distribution p0ðx0Þ¼pðx0; 0Þ from which the “ ” x x interactions or collisions, which are included in the initial value 0 for the path is drawn. The change in equation of motion for the passive tracer as effective system entropy along the trajectory is therefore given by nonequilibrium fluctuations ηðtÞ. In this case, the fluctuations η t can indeed be interpreted as additional time- Δ x x − x τ x 0 ð Þ Ssysð 0; τÞ¼ kB ln pð τ; ÞþkB ln pð 0; Þ: ð22Þ dependent forces from sources external to the Brownian particle. Hence, the total external force acting on the Combining ΔS½x¯ and ΔS ðx0; xτÞ, we define the total sys Brownianffiffiffiffiffiffiffiffiffi particle at time t is given by γfvðxðtÞ;tÞþ entropy production along a trajectory x¯ [38], p 2DaηðtÞg. This modification of the external force obviously does not affect the basic definition (10) of heat Δ x¯ Δ x¯ Δ x x Stot½ ¼ S½ þ Ssysð 0; τÞ: ð23Þ exchanged with the thermal bath. Using the force balance expressed in the Langevin equation (5) to replace It fulfills the integral fluctuation theorem [32,34] pffiffiffiffiffiffiffi −γx_ðtÞþγ 2DξðtÞ, we obtain −Δ Stot=kB 1 pffiffiffiffiffiffiffiffiffi he i¼ ð24Þ δ − f x 2 γη x QþðtÞ¼ ½ ð ðtÞ;tÞþ Da ðtÞ · d ðtÞ: ð25Þ as a direct consequence of Eq. (18) [see also Eq. (22)]. The η average in Eq. (24) is over all trajectories with a given but The active fluctuations ðtÞ formally play the role of additional nonconservative force components and thus arbitrary distribution p0ðx0Þ of initial values x0 [38]. affect the heat which is dissipated into the thermal environment in order to balance all acting external forces. IV. THE NONEQUILIBRIUM ENVIRONMENT However, they cannot be controlled to perform work on the We now focus on the full model (5)–(7) for Brownian particle due to their inherent fluctuating character as an motion subject to active fluctuations ηðtÞ. Our main goal is active bath, such that the definition (13) of the work to develop a trajectorywise thermodynamic description as a remains unchanged. Indeed, this standard definition of natural generalization of stochastic energetics and thermo- work from stochastic thermodynamics has been used to dynamics in a purely thermal environment (see previous analyze a microscopic heat engine operating between two section). We thus treat the active forces ηðtÞ in the same active baths [96]. Finally, the change of internal energy (12) x way as the thermal noise ξðtÞ, namely, as a source of over a time interval dt is determined by the potential Uð ;tÞ fluctuations whose specific realizations are not accessible only and thus is not altered by the presence of the active η but whose statistical properties determine the probability fluctuations ðtÞ either. Combining Eqs. (12), (13), and (25), we find the energy p½xjx0 for observing a certain particle trajectory x. We are interested in how the nonequilibrium characteristics of balance (first law) the active fluctuations affect the irreversibility measure δ δ δ encoded in the ratio between forward and backward path dU ¼ Qþ þ W þ Aþ ð26Þ probabilities and how this measure is connected to the energetics of the active Brownian motion. for Brownian motion in an active bath. Here, we introduce the energy exchanged with the active bath, pffiffiffiffiffiffiffiffiffi A. Energetics and entropy δ 2 γη x Aþ ¼ Da ðtÞ · d ðtÞ: ð27Þ Comparing Eq. (5) with Eq. (9), we may conclude that the stochastic energetics associated with Eq. (5) can be It might be best interpreted as “heat” in the sense of obtained from the energetics for Eq. (9) by adjusting the Sekimoto’s general definition, that any energy exchange

021009-7 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) with unknown or inaccessible d.o.f. may be identified as actualpffiffiffiffiffiffiffiffiffi velocity deviates from the self-propulsion velocity heat [31]. In our setup, the active fluctuations ηðtÞ represent 2D η t −γ x_ t − pffiffiffiffiffiffiffiffiffia ð Þ, resulting in hydrodynamic friction ½ ð Þ an effective description of the forces from the active 2DaηðtÞ between the fluid volume comprising the environment and are thus in general not directly measurable particle and its active self-propulsion mechanism on in an experiment. the one hand and the surrounding fluid environment on Based on the heat the Brownian particle exchanges with ptheffiffiffiffiffiffiffi other hand. Together with the thermal fluctuations the thermal environment, we can identify the entropy 2DγξðtÞ, these friction forces are the forces exchanged production in the thermal environment as with the thermal environment, entailing an energy transfer between particle and thermal bath. It is this dissipation δQ ðtÞ δS ðtÞ¼− þ associated with the deviations of the particle trajectory from þ T the self-propulsion path which we can quantify. The 1 pffiffiffiffiffiffiffiffiffi f x 2 γη x corresponding heat exchange with the thermal bath for a ¼ ½ ð ðtÞ;tÞþ Da ðtÞ · d ðtÞ; ð28Þ dx t dt T displacement ð Þ taking place overffiffiffiffiffiffiffiffiffi a timep intervalffiffiffiffiffiffiffi −γ x_ −p2 η 2 γξ at time t isffiffiffiffiffiffiffiffiffi given by ð ½ ðtÞ Da ðtÞþ D ðtÞÞ· in analogy to Eq. (20). We refrain, however, from defining p ½dxðtÞ− 2DaηðtÞdt, or, using that hydrodynamic friction an entropy production in the active bath. Such an envi- and thermal fluctuations are exactly balanced by the ronment, itself being in a nonequilibrium state due to external force f [see Eq. (5)], continuous dissipation of energy, does not possess a well- pffiffiffiffiffiffiffiffiffi defined entropy so that the heat dissipated into the active δQ−ðtÞ¼−fðxðtÞ;tÞ · ðdxðtÞ − 2D ηðtÞdtÞ: ð29Þ bath cannot be associated with a change of bath entropy. a

2. Self-propulsion This definition of heat is equivalent to the one suggested in Ref. [72] for active particles which are self-propelled by Self-propulsion occurs if a particle is able to locally chemical conversions. convert external fuel on its own into directed motion; such a The work δWðtÞ, on the other hand, performed on or by fuel may, e.g., be nutrients for a bacterium or some kind of the particle during the time step dt, which can be controlled chemical components like hydrogen peroxide for a colloi- or harvested by an external agent, is exactly the same as dal Janus particle; see Fig. 1 in the review [5] for an without active propulsion given in Eq. (13) because from the extensive collection of self-propelled (Brownian) particles. operational viewpoint of the external agent, it is irrelevant This self-propulsion entails that incessant particle motion what kind of mechanisms propel the particle. Accordingly, it can occur already without any external forces being has been used in Ref. [97] for connecting work in active applied. It is represented by the active fluctuations ηðtÞ particle systems to pressure and in Ref. [98] for measuring in the equations of motion (5) for the self-propelled the work performed by a colloidal heat engine with an active Brownian particle. For f x;t ≡ 0 and without thermal pffiffiffiffiffiffiffi ð Þ particle as the working medium. Finally, the definition (12) fluctuations 2Dξ t ≡ 0, the momentary particle velocity ð Þ pffiffiffiffiffiffiffiffiffi for the change in internal energy dUðtÞ is independent of x_ðtÞ is exactly equal to the active velocity 2D ηðtÞ. In that how particle motion is driven as well and thus remains a pffiffiffiffiffiffiffiffiffi sense, self-propulsion is force-free. More precisely, the unaffected by our interpretation of 2DaηðtÞ as active driving force, which is created locally by the particle for propulsion. self-propulsion, is compensated according to actio est Combining Eqs. (12), (13), and (29), we find the first- reactio, such that the total force acting on a fluid volume law-like relation comprising the particle and its active self-propulsion mechanism is zero. The corresponding dissipation (and dU ¼ δQ− þ δW þ δA− ð30Þ entropy production) inside such a fluid volume which results from the conversion of energy or fuel to generate for active self-propelled Brownian motion described by the the self-propulsion drive cannot be quantified by our Langevin equation (5). Here, we balance the different effective description of the active propulsion as fluctuating energetic contributions by introducing forces η because it does not contain any information on the underlying microscopic processes. In order to quantify such pffiffiffiffiffiffiffiffiffi δ −f x 2 η an entropy production, a specific model for the self- A−ðtÞ¼ ð ðtÞ;tÞ · Da ðtÞdt: ð31Þ propulsion mechanism is required [41,71,72]. In other words, our “coarse-grained” description (7) does not allow This quantity represents those contributions to the heat us to assess how much energy the conversion process exchange which are contained only in the active component pffiffiffiffiffiffiffiffiffi behind the self-propulsion drive dissipates. 2DaηðtÞdt of the full particle displacement dxðtÞ; see On the other hand, if the force-free motion of the self- Eq. (29). We can therefore interpret it as the heat transferred propelled particle is disturbed by the presence of external pffiffiffiffiffiffiffi from the active fluctuations to the thermal bath via the forces fðx;tÞ ≠ 0 or thermal fluctuations 2DγξðtÞ, its Brownian particle. Again, δA−ðtÞ is a quantity which, in

021009-8 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) rffiffiffiffiffi general, cannot be measured in an experiment, since it is τa −τ η2 η a 0 produced by the inaccessible active fluctuations ηðtÞ. psð 0Þ¼ π e : ð35Þ Finally, we can relate the heat exchange δQ− with the thermal bath to dissipation and define a corresponding At t ¼ 0, we place the Brownian particle into the fluid with entropy production in the environment, an initial distribution p0ðx0Þ of particle positions which can be prepared arbitrarily and start measuring immediately. δQ−ðtÞ On the other hand, we may let the Brownian particle adapt δS−ðtÞ¼− T to the active and thermal environments before performing 1 pffiffiffiffiffiffiffiffiffi measurements and assume that the system ðx0; η0Þ is in a ¼ fðxðtÞ;tÞ · ðdxðtÞ − 2D ηðtÞdtÞ: ð32Þ x η η x x 0 T a joint steady state psð 0; 0Þ¼p0ð 0j 0Þp0ð 0Þ at t ¼ .In that case, control over the distribution p0ðx0Þ of initial particle positions is limited, as it is influenced by the active B. Path probability fluctuations. The form of psðx0; η0Þ depends on the par- x F x We now calculate the probability p½xjx0 for observing a ticular setup, i.e., the specific choices for Uð ;tÞ and ð ;tÞ x x τ x in Eq. (6). certain path ¼f ðtÞgt>0 starting at 0 generated under the combined influence of thermal and active fluctuations. In the following, we perform the path integration (34) for We treat these two noise sources on equal terms, namely, as the first option, starting from independent initial conditions η x η fluctuating forces with unknown specific realizations but p0ð 0j 0Þ¼psð 0Þ; the calculation for the second option known statistical properties. Because of the memory of the can be carried out along the same lines if the joint steady x η η active noise ηðtÞ, the system (5) is non-Markovian. state psð 0; 0Þ is Gaussian in the active noise 0 [100]. Therefore, the standard Onsager-Machlup path integral Plugging Eqs. (33) and (35) into Eq. (34) and performing a η_2 [84–86] cannot be applied directly to obtain p½xjx0. partial integration of the term proportional to t in the However, for the Ornstein-Uhlenbeck model (7) of ηðtÞ, exponent, we obtain the combined set of variables ðx; ηÞ is Markovian, and we Z τ x_ − v 2 ∇ v can easily write down the path probability p½x; ηjx0; η0 for ð t tÞ · t p x x0 ∝ dt − − τ ½ j exp 4 2 the joint trajectory ðx; ηÞ¼f½xðtÞ; ηðtÞg 0 conditioned on 0 D t> Z Z pffiffiffiffiffiffiffiffiffi an initial configuration ðx0; η0Þ: τ 2 Dη¯ Da ηT x_ − v × exp dt 2 t ð t tÞ Z pffiffiffiffiffiffiffiffiffi 0 D τ x_ − v − 2 η 2 Z Z ð t t Da tÞ 1 τ τ p½x; ηjx0; η0 ∝ exp − dt − 0ηT ˆ 0 η 0 4D dt dt t Vτðt; t Þ t0 ; ð36Þ 2 0 0 ðτ η_ þ η Þ2 ∇ · v þ a t t þ t : ð33Þ 2 2 where we use the abbreviation η¯ ¼ η0 ∪ η for the full path including the initial point η0 in order to write Dηdη0 ¼ Dη¯. To calculate the path weight for the particle trajectories The differential operator p½xjx0, we have to integrate out the active noise history, ˆ 0 0 ˆ ˆ ˆ Vτ t; t δ t − t V t V0 t Vτ t 37a Z ð Þ¼ ð Þ½ ð Þþ ð Þþ ð Þ ð Þ p½xjx0¼ Dη dη0 p½x; ηjx0; η0p0ðη0jx0Þ; ð34Þ consists of an ordinary component

Vˆ ðtÞ¼−τ2∂2 þð1 þ D =DÞð37bÞ where p0ðη0jx0Þ characterizes the initial distribution of the a t a active fluctuations at t ¼ 0. Since the variables ηðtÞ represent an effective description and two boundary components of the active fluctuations which are not experimentally ˆ δ τ − τ2∂ accessible, in general it is not possible to set up a specific V0ðtÞ¼ ðtÞð a a tÞ; ð37cÞ initial state for ηðtÞ. As a consequence, there are basically ˆ δ − τ τ τ2∂ two physically reasonable choices for p0ðη0jx0Þ. VτðtÞ¼ ðt Þð a þ a tÞ; ð37dÞ On the one hand, we can assume that the active bath has reached its stationary state before we immerse the which include the boundary terms picked up by the partial _2 Brownian particle, such that the bath’s initial distribution integration of ηt and from the initial distribution (35) of η0. ˆ is independent of x0 and given by the stationary distribu- The subscript τ in Eq. (37a) indicates that the operator Vτ is tion of ηðtÞ, i.e., p0ðη0jx0Þ¼psðη0Þ. For the Ornstein- acting on trajectories of duration τ. Uhlenbeck process (7), the stationary distribution reads Since the path integral over the active noise histories ηðtÞ [47,99] is Gaussian, we can perform it exactly [102]. We find

021009-9 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) Z Z 1 τ τ indices t, t0. The path integral (36) is then a continuum 0 _ T p½xjx0 ∝ exp − dt dt ðxt − vtÞ 4D 0 0 generalization of an ordinary Gaussian integral for finite- dimensional matrices and can be performed by “completing 0 Da 0 × δðt − t Þ − Γτðt; t Þ ðx_ 0 − v 0 Þ the square.” We provide a rigorous derivation of Eqs. (38) D t t Z and (39) in Appendix A. 1 τ ’ − ∇ v In order to obtain the explicit form of the Green s dt · t ; ð38Þ Γ 0 2 0 function τðt; t Þ, we need to solve the integrodifferential ˆ 0 equation (39). In our case, the operator Vτðt; t Þ is propor- Γ 0 ’ where τðt; t Þ denotes the operator inverse or Green s tional to δðt − t0Þ [see Eq. (37)] and thus has a “diagonal” ˆ 0 function of Vτðt; t Þ in the sense that Z structure, such that Eq. (39) turns into an ordinary linear τ 0 ˆ 0 0 00 00 differential equation. We can solve it by following standard dt Vτðt; t ÞΓτðt ;t Þ¼δðt − t Þ: ð39Þ 0 methods [103,104]; details are given in Appendix B.We Roughly speaking, this Gaussian integration can be under- obtain ˆ stood by thinking of Vτ and Γτ as matrices with continuous

1 κ2 −λjt−t0j κ2 −λð2τ−jt−t0jÞ − κ κ −λðtþt0Þ −λð2τ−t−t0Þ 0 þe þ −e þ −½e þ e Γτ t; t ; ð Þ¼ 2τ2λ κ2 − κ2 −2λτ ð40aÞ a þ −e with We finally remark that our general expression (38) with (40a) reduces to known results in the three limiting cases 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D → 0 (passive particle), τ → 0 (white active noise), and λ 1 a a ¼ þ Da=D; ð40bÞ D → 0 (no thermal bath); details of the calculations can be τa found in Appendix C. Without active fluctuations → 0 and (Da ), we trivially recover the standard Onsager- Machlup expression (16) for passive Brownian motion. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the white-noise limit for ηðtÞ (τ → 0), the equation of κ 1 λτ 1 1 a ¼ a ¼ þ Da=D: ð40cÞ motion (5) involves two independent Gaussian white-noise processes with vanishing means and variances 2D and 2Da, 0 With this expression for Γτðt; t Þ, Eq. (38) represents the respectively. Their sum is itself a white-noise source with exact path probability density for the dynamics of the zero mean but variance 2ðD þ DaÞ. Accordingly, as Brownian particle (5) under the influence of active τa → 0, we obtain from Eq. (38) an Onsager-Machlup Ornstein-Uhlenbeck fluctuations (7). This is our first main path weight of the form (16) but with the diffusion result. We see that the active fluctuations with their colored coefficient D being replaced by D þ Da. In the third noise character lead to correlations in the path weight via limiting case of vanishing thermal fluctuations, we are left 0 the memory kernel Γτðt; t Þ. They relate trajectory points at with a pure colored noise path weight [105,106], different times by an exponential weight factor similar to Z τ 1 τ ∂ x_ − v 2 ∇ v the active noise correlation function (8) but with a corre- p x x ∝ − ½ð þ a tÞð t tÞ − · t −1=2 ½ j 0D→0 exp dt lation time which is a factor of ð1 þ D =DÞ smaller. We 0 4D 2 a a emphasize again that we assumed independent initial x_ − v 0 ffiffiffiffiffiffiffiffiffi0 conditions for the particle’s position x0 and the active bath × ps p ; ð41Þ 2Da variables η0, p0ðη0jx0Þ¼psðη0Þ [see also the discussion around Eq. (35)]. A different choice for the initial distri- where ps denotes the steady-state distribution (35) of the bution, for instance, the joint stationary state for x0 and η0, 0 colored noise. would result in a modified Γτðt; t Þ, whose precise form in general depends on the specific implementation of the deterministic forces f. The correlations in the path weight C. Fluctuation theorem 0 we measure via the memory kernel Γτðt; t Þ are thus With the explicit form (38) of the path probability, we influenced by our choice of the time instance at which can now derive an exact expression for the log ratio of path we start observing the particle trajectory. In that sense, the probabilities which relates time-forward and -backward system “remembers its past” even prior to the initial time paths. Following exactly the same line of reasoning as point t ¼ 0 because of the finite correlation time in the described in Sec. III B for the case of a passive Brownian active fluctuations. particle, we consider the ratio of probabilities for observing

021009-10 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) a specific trajectory x and its time-reversed twin x˜ created In Eq. (45), equality is achieved if and only if the under a time-reversed protocol vðx; τ − tÞ [88]. With the dynamics is symmetric under time reversal. The setting definition (17) of the time-reversed trajectory, we can considered here, however, is generally not symmetric express its probability in terms of the forward path. under time reversal because of our choice of particle 0 0 Using the property Γτðτ − t; τ − t Þ¼Γτðt; t Þ of the position and active fluctuations being independent initially. memory kernel [see Eq. (40a)], we then obtain the path The approach to a correlated (stationary) state is irrevers- probability ratio ible, such that we inevitably pick up transient contributions ΔΣ Δ to þ Ssys, which are strictly positive on average, even p˜ x˜ x˜ ½ j 0 −ΔΣ x¯ if the external forces f are time independent and ¼ e ½ =kB ; ð42aÞ p½xjx0 conservative. For the same reason, namely, the build up of correlations with between the particle trajectory xðtÞ and the active fluc- Z Z tuation ηðtÞ, the quantity ΔΣ is nonadditive, i.e., ΔΣ½x¯ ≠ τ τ 1 D 0 τ ΔΣ x¯ 0x_ Tf δ − 0 − a Γ 0 ΔΣ x t ΔΣ x ½ ¼ dt dt t t0 ðt t Þ τðt; t Þ : ½f ðtÞgt¼0þ ½f ðtÞgt¼t0 in general, for any inter- T 0 0 D mediate time 0

021009-11 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019)

Similar “memory forces” have already been found to affect p˜½x˜¯ p˜½x˜¯jη˜¯ ΔI½x¯; η¯¼ln − ln : ð51bÞ irreversibility by contributing to dissipation in Langevin p½x¯ p½x¯jη¯ systems with colored noise, which does not obey the fluctuation-dissipation relation [107]. This expression represents the pathwise mutual information Σ From Eq. (47), we can read off a production rate for , difference between a combined forward process ðx¯; η¯Þ and its backward twin. If ΔI½x¯; η¯ is positive, the pathwise 1 I x¯ η¯ στðtÞ¼x_ðtÞ · φτ½x¯;tð48aÞ mutual information ½ ; along the combined forward T path is larger than I˜ ½x˜¯; η˜¯ for the time-reversed path [see or Eq. (51a)], implying that correlations between the particle trajectory x¯ and the active fluctuation η¯ are stronger in the 1 time-forward direction. Intuitively, we may thus say that δΣτðtÞ¼dxðtÞ · φτ½x¯;t: ð48bÞ T they are more likely to occur together than their time- reversed twins, making the combined forward process Even though this expression for the Σ production is ðx¯; η¯Þ the more “natural” one of the two processes in terms analogous to the actual entropy productions in the envi- of pathwise mutual information. ronment as identified in Eqs. (28) or (32) [see also Eq. (20)] To make the connection to irreversibility more rigorous, and even contains a term fdxðtÞ · fðxðtÞ;tÞg=T, which we rewrite in Eq. (51b) the pathwise mutual information quantifies dissipation due to the external force f, its difference ΔI½x¯; η¯ explicitly as path probability ratios physical meaning beyond formally defined memory forces between time-forward and time-backward processes. We is unclear. In the following, we argue that a valid inter- now see that ΔI via the term − ln ðp˜½x˜¯=p½x¯Þ is directly pretation is provided by the mutual information between related to the irreversibility measure ΔΣ for the actively the active fluctuations ηðtÞ and the particle trajectory xðtÞ. driven Brownian particle and its change in system entropy Δ p˜ x˜¯ η˜¯ p x¯ η¯ Ssys. The additional log ratio ln ð ½ j = ½ j Þ involves V. MUTUAL INFORMATION the probability of the forward path x¯ being generated by the η¯ The pathwise mutual information [108] between the specific realization of the active fluctuations for which we measure the mutual information content with x¯ in ΔI, and, particle trajectory x¯ (starting at x0) and a realization η¯ of the likewise, the path probability of the time-reversed twin active fluctuations [with ηð0Þ¼η0] is given as being generated by the time-reversed fluctuation. We can p½x¯; η¯ p½x¯jη¯ rewrite this term by splitting off the contributions from the I½x¯; η¯¼ln ¼ ln ; ð49Þ p½x¯p½η¯ p½x¯ initial densities, ˜ ¯ ¯ ˜ ¯ ¯ p x¯ η¯ p x η x η x η p x¯ p x x x p½x˜jη˜ p½x˜jx˜ 0; η˜ pðx˜ 0jη˜Þ where ½ ; ¼ ½ ; j 0; 0pð 0; 0Þ, ½ ¼ ½ j 0pð 0Þ, ln ¼ ln þ ln : ð52Þ p η¯ p η η η x η p x¯ η¯ p x x0; η¯ p x0 η0 and ½ ¼R ½ j 0pð 0Þ include the initialR densities pð 0; 0Þ, ½ j ½ j ð j Þ pðx0Þ¼ dη0pðx0;η0Þ,andpðη0Þ¼ dx0pðx0;η0Þ of the Brownian particle and the active fluctuations (see also the Here we keep the possibility that the initial particle position discussion in Sec. IV B). This pathwise mutual information is conditioned on the initial state of the active fluctuations x η¯ x η quantifies the reduction in uncertainty about the path x¯ when pð 0j Þ¼pð 0j 0Þ, which is more general than the sit- x η¯ x ΔΣ we know the realization η¯ and vice versa, and can therefore, uation pð 0j Þ¼pð 0Þ for which we calculated in loosely speaking, be seen as a measure of the correlations Sec. IV B [see also the discussion around Eq. (35)]. between x¯ and η¯. Note that it can become negative if According to Eq. (17), the time-reversed initial position x˜ x p½x¯jη¯ < p½x¯, while its average is always positive. is given by the final point of the forward path, 0 ¼ τ.It Likewise, the pathwise mutual information between the therefore depends on the complete history of the active η˜¯ η¯ time-reversed trajectory x˜¯ from Eq. (17) and a suitably fluctuations, which is captured equivalently by or , for chosen time-reversed realization η˜¯ of the active fluctuations any reasonable choice of the time-reversed fluctuations in terms of the forward realization [see also Eq. (55) below]. is ¯ With pðx˜ 0jη˜Þ¼pðxτjη¯Þ, we can identify the boundary term p˜½x˜¯jη˜¯ in Eq. (52) as the change in system entropy I˜ ½x˜¯; η˜¯¼ln : ð50Þ p˜½x˜¯ jη¯ ¯ ¯ ¯ ΔSsys½xjη¼−kB ln pðxτjηÞþkB ln pðx0jη0Þ; ð53Þ For their difference which occurs along the trajectory x¯ under the specific ΔI½x¯; η¯¼I½x¯; η¯ − I˜ ½x˜¯; η˜¯; ð51aÞ realization η¯ of the active fluctuations (we explicitly indicate the conditioning on the realization of the active we thus find fluctuation by adding “jη¯” as a superscript).

021009-12 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) Z To quantify the first term in Eq. (52), we exploit that a τ τ ΔS ½x¯jη¯¼ dt δS ðtÞ prescribed active fluctuation η¯ ¼fηðtÞg acts like an þ þ t¼0 0Z additional driving force in Eq. (5). Without needing to 1 τ pffiffiffiffiffiffiffiffiffi x f x 2 γη know any further details about ηðtÞ (e.g., of how it is ¼ d ðtÞ ·½ ð ðtÞ;tÞþ Da ðtÞ: ð56bÞ T 0 generated), we can thus derive the conditioned forward path ¯ probability p½xjx0; η directly from Eq. (5) as a standard Together with Eq. (53), we thus obtain the total entropy Onsager-Machlup path integral, production along the particle trajectory x¯ which is gen- Z pffiffiffiffiffiffiffiffiffi η¯ τ _ 2 erated by a given realization of the active fluctuations ðxt − vt − 2DaηtÞ ∇ · vt p½xjx0; η¯ ∝ exp − dt − : assumed to be even under time inversion. It is defined in 0 4D 2 analogy to Eq. (22) as ð54Þ Δ þ x¯ η¯ Δ x¯ η¯ Δ jη¯ x¯ η¯ Stot½ j ¼ Sþ½ j þ Ssys½ j : ð57Þ In order to evaluate the time-reversed counterpart, we have to specify the behavior of the active fluctuation signal ηðtÞ By construction [see Eqs. (53), (56a), and also Eq. (52)], under inversion of the direction of time. There are essen- Δ þ x¯ η¯ − p˜ x˜¯ η˜¯ p x¯ η¯ Stot½ j is identical to the log ratio kB ln ð ½ j þ= ½ j Þ. tially two choices, Combining this result with Eqs. (52), (43), and (51),we then infer that ΔΣ½x¯þΔS ðx0; xτÞ is a combination of η˜ ðtÞ¼ηðτ − tÞ; ð55Þ sys the conditioned entropy production and the difference in corresponding to ηðtÞ being an even or odd process. While mutual information between time-forward and time- both these options a priori appear to be equally valid, they backward processes, are connected to different interpretations of the active ΔΣ x¯ Δ x x Δ þ x¯ η¯ − ΔI x¯ η¯ fluctuations. We argue below (beginning of Sec. VA) that ½ þ Ssysð 0; τÞ¼ Stot½ j kB þ½ ; ; ð58Þ for passive tracer particles which are suspended in an active ΔI x¯ η¯ η˜¯ η˜¯ bath, the active fluctuations η t bear the character of an where þ½ ; is given by Eq. (51) with ¼ þ ¼ ð Þ η˜ τ external force and therefore should be considered even f þðtÞgt¼0. Note that the individual terms on the right- η¯ under time reversal. In contrast, for an active particle with hand side both depend on the active noise with a self-propulsion (see beginning of Sec. VB), the active dependence that is compensated exactly because the left- η¯ fluctuations ηðtÞ are internal to the particle and thus should hand side is independent of . change sign when time is reversed (odd under time As an immediate consequence of Eq. (44), the total reversal), otherwise, the particle trajectory would not be conditioned entropy production together with the mutual reversed under inversion of the direction of time in a setup information difference fulfill the integral fluctuation in which only self-propulsion is acting (no thermal noise theorem and no external forces) [109]. −ðΔSþ =k −ΔI Þ he tot B þ ix¯ ¼ 1; ð59Þ

A. Active bath and the second-law-like relation If we interpret ηðtÞ as fluctuations coming from an active Δ þ x¯ η¯ − ΔI x¯ η¯ ≥ 0 bath, they are external to the particle and as such are similar h Stot½ j ix¯ kBh þ½ ; ix¯ : ð60aÞ to externally applied forces. The question of irreversibility under a prescribed realization η¯ is then related to the These results (59) and (60) are valid for any realization η¯ of question of how likely the backward particle trajectory x˜ðtÞ the active fluctuations because the original average h·i in [see Eq. (17)] is to be observed when exactly the same Eq. (44) is just over the distribution of all particle forces act on the particle at identical positions during the trajectories x¯. We accentuate this fact here by the subscript x¯ forward and backward motion. Hence, this case is at the averages h·ix¯ to distinguish them from the standard described by active fluctuations which are even under time meaning of the brackets h·i as an average over all stochastic reversal, i.e., by the plus sign in Eq. (55). quantities present. Since the terms in Eq. (60a) taken η¯ Using η˜ ðtÞ¼þηðτ − tÞ as the time-reversed fluc- together are independent of [see Eq. (58)], we can even þ ¯ ¯ tuation, we find average over the joint distribution p½x; η to obtain a further second-law-like relation p˜ x˜ x˜ η˜¯ ½ j 0; þ −Δ x¯ η¯ ln ¼ Sþ½ j =kB ð56aÞ Δ þ − ΔI ≥ 0 p½xjx0; η¯ h Stoti kBh þi : ð60bÞ Δ x¯ η¯ – for the path probability ratio, where Sþ½ j is given by We emphasize that Eqs. (58) (60), being a direct conse- the entropy production in the thermal environment from quence of the structure of the equation of motion (5), are Eq. (28), completely independent of the specific model behind the

021009-13 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) active fluctuations. Moreover, they are formally similar to −δQ−ðtÞ dissipated into the thermal bath during a displace- the fluctuation theorem with information exchange derived ment of the active self-propelled particle, which deviates by Sagawa and Ueda [110] for a completely different from the “force-free” path that would be carved out by the physical setup. Sagawa and Ueda considered the difference self-propulsion drive alone; see also Eq. (32) and the between initial and final correlations of a system of interest discussion above Eq. (29). In analogy to Eq. (22), we finally with an “information reservoir,” which provides correlations combine ΔS−½x¯jη¯ with Eq. (53) to define the total entropy as a resource of entropy changes. production under a given realization η¯ of the active pro- For the Ornstein-Uhlenbeck representation (7), we can pulsion as use our result (42b) for ΔΣ½x¯ to derive an explicit Δ − x¯ η¯ Δ x¯ η¯ Δ jη¯ x¯ η¯ expression for the mutual information difference, Stot½ j ¼ S−½ j þ Ssys½ j : ð63Þ x η¯ x η ΔI x¯ η¯ pð τj Þ − pð 0j 0Þ Note that with Eqs. (53) and (62a) [see also Eq. (52)], we can þ½ ; þ ln ln pðxτÞ pðx0Þ write this total entropy production as the log ratio pffiffiffiffiffiffiffiffiffi Z ˜ ¯ ¯ 1 τ −k ln ðp½x˜jη˜−=p½x¯jη¯Þ of path weights. ¼ 2D dt x_ Tη B a t t Hence, we find again that ΔΣ½x¯þΔS ðx0; xτÞ consists kBT 0 sys Z Z of conditional total entropy production and mutual infor- D τ τ a 0 _ T 0 þ dt dt xt f t0 Γτðt; t Þ mation difference [compare Eqs. (51)), (43), and (63)], D 0 0 1 ΔΣ x¯ Δ x x Δ − x¯ η¯ − ΔI x¯ η¯ Δ x¯ η¯ ½ þ Ssysð 0; τÞ¼ Stot½ j kB −½ ; : ð64Þ ¼ Aþ½ ; kBT Z Z This identity provides exactly the same interpretation of D τ τ a 0 _ T 0 ΔΣ x¯ Δ x x þ dt dt xt f t0 Γτðt; t Þ ; ð61Þ ½ þ Ssysð 0; τÞ as we found earlier for the case of an D 0 0 R active bath [cf. Eq. (58)] but with different expressions for Δ x¯ η¯ τ δ the entropy production and the mutual information differ- where in the last equality we use Aþ½ ; ¼ 0 dt AþðtÞ “ ” δ ence. The mutual information in the backward process is with the heat AþðtÞ exchanged with the active bath as identified in Eq. (27). now measured with respect to the sign-inverted active fluctuation process η˜−ðtÞ¼−ηðτ − tÞ, i.e., ΔI−½x¯; η¯ in η˜¯ η˜¯ η˜ τ Eq. (64) is given by Eq. (51) with ¼ − ¼f −ðtÞgt¼0.As B. Self-propulsion in the case of an active bath (58), the dependence of the Δ − ΔI If the active fluctuations represent an effective model for individual terms Stot and − in Eq. (64) on the active a self-propulsion mechanism of the particle, we choose ηðtÞ forcing η¯ cancel exactly to result in the η¯-independent ΔΣ x¯ Δ x x to be odd under time reversal for a proper assessment of expression ½ þ Ssysð 0; τÞ. irreversibility; i.e., we choose the minus sign in Eq. (55). From Eq. (44), we immediately obtain the integral Otherwise, we would relate irreversibility to the likelihood fluctuation theorem that thermal fluctuations make the particle move a certain − −ðΔS =k −ΔI −Þ path backward against its internal self-propulsion rather he tot B ix¯ ¼ 1; ð65Þ than to the probability that a self-propelled particle traces out a given trajectory backward in time driven by the same and the second-law-like relation propulsion forces [109]. ¯ Δ − x¯ η¯ − ΔI x¯ η¯ ≥ 0 Calculating the path probability ratio p½x˜jx˜ 0; η˜=p½xjx0; η¯ h Stot½ j ix¯ kBh −½ ; ix¯ ð66aÞ for odd active fluctuations η˜ðtÞ¼η˜−ðtÞ¼−ηðτ − tÞ,we obtain for active self-propulsion. Like in Eq. (59), we use the subscript x¯ to explicitly indicate that the averages are over p˜ x˜ x˜ η˜¯ ½ j 0; − ¯ ¯ the distribution p½x¯ alone and that the resulting quantities ln ¼ −ΔS−½xjη=kB: ð62aÞ p½xjx0; η¯ are still functionals of the active fluctuations and are valid for any realization η¯ of the self-propelling forces. After Δ x¯ η¯ The quantity S−½ j is given by extending to the average over p½x¯; η¯ to include these active Z τ fluctuations, we end up with the bound ΔS−½x¯jη¯¼ dt δS−ðtÞ 0 Δ − − ΔI ≥ 0 Z h Stoti kBh −i : ð66bÞ 1 τ pffiffiffiffiffiffiffiffiffi x − 2 η f x ¼ ðd ðtÞ Da ðtÞdtÞ· ð ðtÞ;tÞ: ð62bÞ – T 0 Again, these findings (64) (66) are independent of the specific model for the active fluctuations because they are a It thus represents the entropy production in the environment direct consequence of the particle’s equation of motion (5). along the trajectory x¯ because δS−ðtÞ results from the heat Moreover, they are formally similar to the Sagawa-Ueda

021009-14 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) fluctuation theorem with information [110]. If the active distinct two probability densities are. It is non-negative and propulsion ηðtÞ is modeled by Eq. (7), we can write the equals zero if and only if the two probabilities are identical mutual information difference in the explicit form [111,112]. The result (69) shows that the average mutual informa- p x η¯ p x η ΔI x¯ η¯ ΔI x¯ η¯ ð τj Þ − ð 0j 0Þ tion difference h ½ ; i is the difference between the −½ ; þ ln ln x¯ pðxτÞ pðx0Þ Kullback-Leibler divergence of the particle trajectory Z ¯ pffiffiffiffiffiffiffiffiffi τ relative to its time-reversed twin x˜ and the Kullback-Leibler 1 T ¼ − 2Da dt ηt f t divergence of the combined path ðx¯; η¯Þ relative to the time- kBT 0 x˜¯ η˜¯ Z Z reversed realization ð ; Þ. We can therefore interpret it to D τ τ a 0 _ T 0 measure how much harder it is to discriminate between þ dt dt xt f t0 Γτðt; t Þ D 0 0 time-forward and time-backward realizations if only the 1 particle trajectory is known rather than the full dynamics ¼ ΔA−½x¯; η¯ including the active noise realization η¯. Since x¯ can be seen k T B Z Z as a “coarse-graining projection” of ðx¯; η¯Þ, we expect the D τ τ p x¯ p˜ x˜¯ a 0 T 0 discrimination to become harder, i.e., DKLð ½ jj ½ Þ to þ dt dt x_ f 0 Γτðt; t Þ ð67Þ t t p x¯ η¯ p˜ x˜¯ η˜¯ D 0 0 become smaller compared to DKLð ½ ; jj ½ ; Þ (see Ref. [113] for a similar discussion). This intuition is ¯ ¯ Rby using Eq. (42b) and by defining ΔA−½x; η¼ corroborated by the so-called data processing inequality τ 0 dt δA−ðtÞ [see Eq. (31)] as the total “heat” transferred [111,112], which, applied to our situation, proves from the active propulsion to the thermal bath along the ¯ p x¯ η¯ p˜ x˜¯ η˜¯ ≥ p x¯ p˜ x˜¯ trajectory x. DKLð ½ ; jj ½ ; Þ DKLð ½ jj ½ Þ: ð71Þ

C. Relation to information theory As a direct consequence, we find from Eq. (69) the bound The second-law-like relations (60) and (66) involve the ΔI x¯; η¯ ≥ 0 72 ΔI ΔI h ½ i ð Þ averages h þi and h −i of the mutual information difference. As we show in the following, these averages are on the average mutual information difference. Exploiting closely related to central concepts in information theory Eqs. (60b) and (66b), respectively, we infer that entailing additional bounds and an interpretation in terms of Δ ≥ 0 h Stoti . This follows also from the fact that the hypothesis testing. Δ Stot are given as log ratios of path probabilities, which Because of the unbiased Gaussian character of the active are conditioned on η¯ [see the discussion around Eqs. (56) fluctuations, the probabilities for observing a time-forward and (62)]. Accordingly, they obey an integral fluctuation and its time-reversed realization are the same, i.e., theorem when averaged over the conditioned density p½x¯jη¯ p˜ η˜¯ p η¯ ½ ¼ ½ . We can therefore rewrite Eq. (51b) as but then also when averaged over the full density p½x¯; η¯. More interestingly, we can also use Eq. (72) to equip the p˜½x˜¯ p˜½x˜¯; η˜¯ ΔI ½x¯; η¯¼ln − ln ; ð68Þ second-law-like relations for the total irreversibility mea- p x¯ p x¯ η¯ ΔΣ Δ ½ ½ ; sure þ Ssys with an upper bound. Taking the average x¯ η¯ ¯ ¯ of Eqs. (58) and (64) over ð ; Þ, the non-negativity (72) of where we insert our two options η˜ ¼ η˜ from Eq. (55) for ΔI h i implies the time-reversed active fluctuation. Performing the average over ðx¯; η¯Þ with density p½x¯; η¯, we then find Δ x¯ η¯ ≥ ΔΣ x¯ Δ x x ≥ 0 h Stot½ j i h ½ þ Ssysð 0; τÞi : ð73Þ ΔI x¯ η¯ p x¯ η¯ p˜ x˜¯ η˜¯ − p x¯ p˜ x˜¯ h ½ ; i ¼ DKLð ½ ; jj ½ ; Þ DKLð ½ jj ½ Þ: The total average irreversibility of a particle trajectory ΔΣ Δ ð69Þ measured as h þ Ssysi is thus alwaysR smaller than (or Dx¯p x¯ η¯ Δ x¯ η¯ equal to) the mean entropy change ½ j Stot½ j , Here, we use the definition which would occur for a given realization η¯ of the active Z fluctuations if treated as a known (and measurable) external p½y¯ force contributing to the dissipation in the thermal envi- D ðp½y¯jjp˜½y˜¯Þ ¼ Dy¯ p½y¯ ln ð70Þ KL p˜½y˜¯ ronment averaged over theR distributionR p½η¯ of all possible Dη¯p η¯ Dx¯p x¯ η¯ Δ x¯ η¯ RactiveR fluctuations, i.e., ½ ½ j Stot½ j ¼ of the Kullback-Leibler divergence between a probability Dη¯ Dx¯p½x¯; η¯ΔS ½x¯jη¯¼hΔS ½x¯jη¯i. For this bound ¯ tot tot density p½y¯ for a process y¯ and another density p˜½y˜ for a to be valid, it does not matter whether these forcings come second process y˜¯; in our case, the two processes are related from an active bath and thus are considered even under time by time reversal. The Kullback-Leibler divergence is a reversal (“þ” sign), or from active self-propulsion, which is standard concept in information theory to measure how odd under time reversal (“−” sign). The difference between

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Δ x¯ η¯ ΔΣ Δ η h Stot½ j i and h þ Ssysi is compensated by the build flagellum), it may be possible to infer the direction of . up of mutual information. Moreover, we could think of an experimental setup in We point out that the upper bound for the total average which the active fluctuations η¯ are realized artificially by an irreversibility in Eq. (73) is stronger than the usual coarse- external colored noise source, like in Ref. [117], then all graining inequality obtained when integrating out “hidden” relevant quantities may be accessible. Δ d.o.f. (see, e.g., Ref. [114]). In the present notation, such The entropy productions Stot for an active bath and for Δ x¯ η¯ ≥ ΔΣ x¯ an inequality would read h Stot½ ; i h ½ þ active self-propulsion, respectively, as obtained from irre- Δ x x Δ x¯ η¯ Ssysð 0; τÞi. Here, h Stot½ ; i would denote the total versibility arguments in Eqs. (56) and (62) are consistent average entropy production in the thermal environment of with the energetics derived in Secs. IVA 1 and IVA 2 the combined processes ðx¯; η¯Þ, a quantity which is not well [compare with Eqs. (25) and (29)]. They are thus directly defined, however, because we cannot quantify the entropy related to the heat dissipated into the thermal part of the production associated with the auxiliary process η¯. environment. Hence, the appearance of the pathwise mutual Finally, we remark that the inequality (71) has an information difference in the fluctuation theorem is a interesting interpretation in the context of hypothesis test- consequence of the active fluctuations being present as a ing [112,115] when estimating the direction of the arrow of nonequilibrium bath in addition to the usual thermal bath. ΔI time in the system [33,116]. It states that the probability of Indeed, we can easily see from Eqs. (61) and (67) that misclassifying a specific path as having been generated by vanish identically in the absence of active fluctuations, 0 Δ a forward dynamics when, in reality, it was generated by a Da ¼ . Accordingly, S reduce to the standard entropy backward one decreases faster with the number of obser- production in the thermal environment in this limit [see vations if more detailed information on the dynamics of the Eqs. (56), (62), and (19)]. system is available by additionally monitoring the realiza- We emphasize that the pathwise mutual information tions η¯ of the active fluctuations. quantifies how the active fluctuations contribute to the irreversibility of the particle trajectory but does not capture the unavoidable dissipation connected with maintaining the D. Discussion active nature of the nonequilibrium environment itself. In The fluctuation theorems and second-law-like relations fact, our effective description (7) of the active fluctuations (59), (60) and (65), (66) for an active bath and for active as a time-correlated Gaussian noise source does not have self-propulsion, respectively, and their interpretation in any knowledge about the microscopic details generating terms of mutual information differences are our third main this noise so that it obviously cannot assess the associated result; see also Eq. (73). For both cases, we find that the dissipative processes. On the one hand, we may therefore ΔΣ x¯ Δ x x total irreversibility measure ½ þ Ssysð 0; τÞ consists suspect that the appearance of the pathwise mutual infor- of two contributions: First, the “usual” entropy production mation is a consequence of this coarse-grained description ΔS , which we would measure for the motion of the of the active environment and might be replaced by a tot “ ” Brownian particle under a given realization of the active finer measure once all the details are known [41,71,72]. fluctuations (as if the active fluctuations were just some On the other hand, we may argue that these microscopic additional known “external” driving force), and second, the details behind the active fluctuations are irrelevant if we are ΔI interested only in characterizing the irreversibility of the difference in mutual information accumulated between the particle trajectory and the active nonequili- particle trajectory and related thermodynamiclike proper- brium environment along the forward versus the backward ties of the system emerging on this level of coarse graining. path. We note that even though the individual contributions Then, knowledge of the statistical properties of the active Δ − ΔI η¯ fluctuations, as provided by Eq. (7), is sufficient, and the in Stot kB depend on the specific realization of the active fluctuations, their sum does not [see Eqs. (58) pathwise mutual information between particle trajectory and active noise realization emerges as a natural and and (64)]; i.e., entropy production and change in mutual adequate irreversibility measure. This situation may be information always compensate their dependence on the comparable to the one for entropy production in a thermal realization of the active fluctuations. However, if we want bath: We quantify entropy production solely from the to have access to these individual contributions ΔS and tot statistical properties of the thermal noise without having ΔI directly, we would have to measure the active to rely on a detailed microscopic description of the bath. fluctuations η¯, i.e., the direction and magnitude of the forces representing the activity in the system. In general, VI. EXAMPLE: HARMONIC POTENTIAL this may be a challenging task in typical experiments with active Brownian particles, as one would have to separate To illustrate our results, we consider the example of a thermal from active fluctuations. We can imagine, however, one-dimensional Brownian particle trapped in a harmonic that at least partial information about η¯ could be obtained potential Uðx; tÞ¼ðk=2Þðx − utÞ2, whose center is either experimentally. For example, tracking the orientation of an held at fixed position (u ¼ 0) or else is displaced at active self-propelled particle (e.g., a bacterium with a constant velocity u ≠ 0. Such a setup can be readily

021009-16 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) implemented in experiment with state-of-the-art optical (a) tweezers and in fact has been used to study various aspects of stochastic thermodynamics and active matter, for instance, in Refs. [23,49,79,118–120]. The Langevin equation of motion (5) for this specific setup is linear,

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi _ − k − 2 ξ 2 η xðtÞ¼ γ ½xðtÞ utþ D ðtÞþ Da ðtÞ: ð74Þ

We can therefore solve the associated Fokker-Planck equation analytically [47,99,121] to obtain the propagator in closed form. From that, we can explicitly compute the (b) averages of the irreversibility measure ΔΣ, the mutual ΔI information difference , and the change in system Δ entropy Ssys for a Gaussian initial distribution p0ðx0Þ of 2 particle positions with zero mean and variance c0. Although the calculations are straightforward (we present some details in Appendix D), the resulting formulas for ΔΣ Δ ΔI h i, h Ssysi, and h i are lengthy and bulky so that we discuss them mostly in graphical form (see Figs. 2 and 4) and give explicit expressions only in some limiting cases [see Eqs. (75), (76), and (79) below]. For the initial variance 2 c0, two specific cases are of particular physical relevance. 2 First, c0 ¼ kBT=k corresponding to a Gaussian distribution that would be created as an equilibrium state by thermal FIG. 2. Average contributions to irreversibility hΔΣi and Δ fluctuations only. Starting from this distribution, the time average change in system entropy h Ssysi as a function of the evolution of our irreversibility measures includes the observation time τ. We compare the cases of a particle trapped in 0 1 transient relaxation from the thermal state towards a static (u ¼ , orange lines) or moving (u ¼ , blue lines) harmonic potential having an initial distribution which is Gaus- the steady state which develops due to the presence of 2 2 2 sian with variance (a) c0 ¼ kBT=k ¼ Dγ=k and (b) c0 ¼ the active fluctuations. Second, c0 ¼ð1=kÞfkBT þ½ðγDaÞ= ð1=kÞfkBT þ½ðγDaÞ=(1 þðkτa=γÞ)g. Parameter values are (1 þðkτa=γÞ)g corresponding to a Gaussian distribution k ¼ 0.1, γ ¼ 1, τa ¼ 0.2, D ¼ Da ¼ 0.2. Note that the blue with a variance which is exactly the same as the one the and orange lines for the change in system entropy are on top of particle distribution has when particle and active fluctua- each other in both figures, as hΔSsysi does not depend on u. tions are in their joint steady state. In that case, the particle distributions at the beginning and end of the process are 1 Δ kB 1 Da identical so that any contributions to the irreversibility lim h Ssysi¼ ln þ τ : ð75Þ τ→∞ 2 D 1 þ k a measure ΔΣ that are not associated with the displacement u γ of the trap are solely due to the build up of correlations In contrast, the irreversibility measures hΔΣi for the between particle position and active d.o.f. In the following, static trap u ¼ 0 and the moving trap u ¼ 1 are similar only we focus on the first alternative and briefly come back to during a short transient phase but then become qualitatively the second alternative at the end of Sec. VI A and in different. In the static trap, hΔΣi becomes constant at large Sec. VI B. times τ,

lim hΔΣij 0 A. Irreversibility τ→∞ u¼ qffiffiffiffiffiffiffiffiffiffiffiffi ΔΣ Δ ΔΣ Δ τ τ In Fig. 2, we compare h i, h Ssysi, and h þ Ssysi 1 − k a 2 D 1 k a 1 1 Da ð γ Þþ ð þ γ Þ þ þ 2 0 1 τ Da D for u ¼ and u ¼ as a function of the duration of the − kB Da ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 2 qffiffiffiffiffiffiffiffiffiffiffiffi 2 ; trajectories. We find that the average change in system 2 τ D 1 1 Da k a 1 Da Δ þ þ γ þ þ entropy h Ssysi is independent of the driving velocity u, D D because it depends only on the variance of the initial and ð76Þ final Gaussian distributions but not on their centers. For long trajectories, i.e., large τ, it approaches the constant in accordance with the system reaching a currentless value equilibriumlike steady state. In the moving trap, the particle

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τ 2 is transported continuously by permanent dissipation k k a ðDaÞ ΔΣ Bγ D of energy so that hΔΣi increases with the length τ of the lim h iju¼0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 2 qffiffiffiffiffiffiffiffiffiffiffiffi 2 : ð79Þ τ→∞ τ τ 1 1 Da k a 1 Da trajectories. For large , the growth rate is given by the þ þ D γ þ þ D ensemble averageR of the time-averaged production rate τ σ ¼ limτ→∞ð1=τÞ dt στðtÞ, where στðtÞ is defined in 0 The origin of this positive contribution is our choice of Eq. (48a). Explicitly, we obtain independent initial conditions in the form of a product η η hΔΣi u2 density p0ðx0; 0Þ¼p0ðx0Þpsð 0Þ [see also the discussion hσi¼lim ¼ k : ð77Þ of Eq. (35)]. Hence, during the initial transient, there must τ→∞ τ B D þ D a be an irreversible build up of correlations between the We thus see that the total “irreversibility production” particle and the active fluctuations becoming manifest in a ΔΣ Δ ΔΣ Δ h þ Ssysi in the moving trap grows extensively with nonzero total h þ Ssysi. It turns out that the associated ΔΣ Δ the observation time as it would without active fluctuations, change in h þ Ssysi is nonmonotonic [see Fig. 2(b)], too. The growth rate, however, is reduced by the additional indicating that the variance of the particle distribution presence of active fluctuations, as the combined environ- departs over some (transient) time period, even though the 2 ment can be interpreted to have a higher “effective temper- initial variance c0 ¼ð1=kÞfðkBTÞþ½ðγDaÞ=(1þðkτa=γÞ)g ature.” Remarkably, the average growth rate (77) is is identical to the final one. In order to obtain a vanishing ΔΣ Δ independent of the relevant system timescales, i.e., the average h þ Ssysi, we would have to start from a joint τ correlation time a of active fluctuations and the relaxation stationary state psðx0; η0Þ of particle positions and active time γ=k in the harmonic potential. fluctuations instead of a factorized one. It can even be In the static trap u ¼ 0, we find that no such extensive shown that forward and backward paths are equally likely ΔΣ Δ growth occurs for h þ Ssysi. Hence, the system reaches in that case (see Ref. [100]), implying that ΔΣ½x¯þ Δ 0 a steady state, which appears equilibriumlike from the Ssysðx0;xτÞ¼ holds already on the level of individual viewpoint of particle trajectories without access to the trajectories. It is unsolved, however, if the latter property is microscopic processes generating the active driving. generic for stationary states of trapped active particles (in ΔΣ Δ 0 Nevertheless, we observe limτ→∞h þ Ssysi > the absence of symmetry-breaking forces), or if it is specific regardless of the initial distribution p0ðx0Þ. For the case to the harmonic trap, to the Ornstein-Uhlenbeck realization 2 of a Gaussian with “thermal variance” c0 ¼ kBT=k, the of the active fluctuations, or to the dimensionality one. relevant results are given in Fig. 2 and Eqs. (75) and (76). We emphasize again that these correlations are the very Δ ΔΣ ΔΣ Both limτ→∞h Ssysi and limτ→∞h i depend on just two reason that the irreversibility measure is nonadditive dimensionless parameters: the ratio of the two noise [see also the discussion below Eq. (45)]. The curves in intensities D =D and the ratio of the two system timescales Fig. 2 apply only to trajectories evolving over the complete a 0 τ ½ðkτ Þ=γ. The limits of large and small correlation time and time interval ½ ; . We cannot split a trajectory at an a ΔΣ noise amplitudes can be easily computed and present no intermediate time, calculate the individual for the difficulties. As an interesting example, we consider the case two parts of the trajectory from Eq. (42b), and then add ΔΣ of vanishing correlation time of the active fluctuations, them up to obtain for the full time interval because the “initial” state for the second part will inevitably depend on kB Da kB Da correlations between particle and active fluctuations accu- lim lim hΔΣ þ ΔS ij 0 ¼ ln 1 þ − : τ →0 τ→∞ sys u¼ 2 2 mulated during the first part. Such correlations are not a D D þ Da taken into account in Eq. (42b) which is based on the ð78Þ assumption of an initial product state. This expression is exactly the entropy which would be produced by a thermal white-noise process with diffusion B. Fluctuation theorem constant D þ Da relaxing from an initial Gaussian distri- To illustrate the integral fluctuation theorem (44) sat- γ ΔΣ Δ bution with variance D =k ¼ kBT=k to its equilibrium isfied by þ Ssys, we show in Fig. 3 the probability state, a Gaussian with variance ðD þ DaÞγ=k. densities for the path probability ratio (43) for the same two 2 It is quite obvious that for a variance c0 ¼ kBT=k of the situations of a static and a moving harmonic trapping initial Gaussian distribution, there must be some change in potential already analyzed in Fig. 2(b). The probability system entropy and a build up of irreversibility because the densities are obtained from simulating 105 trajectories of Gaussian particle distribution approached at long times has length τ ¼ 1, well within the transient regime of the system a larger variance ð1=kÞfðkBTÞþ½ðγDaÞ=ð1 þ kτa=γÞg evolution (see Fig. 2). While the distribution for − ΔΣ Δ [see Eq. (D2d)]. But even if we start with the initial exp½ ð þ SsysÞ=kB is almost symmetric about 1 in 2 variance c0 ¼ð1=kÞfðkBTÞþ½ðγDaÞ=(1 þðkτa=γÞ)g,we the static trap u ¼ 0, the most probable value lies visibly find a positive hΔΣi while reaching the steady state at below 1 for the moving trap u ¼ 1, indicating that τ ΔΣ Δ large , trajectories with a positive þ Ssys are more likely

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(a) thus are readily accessible in experiments and simulations. The specific role of the active fluctuations, on the other hand, is nicely captured by the splitting of irreversibility, i.e., the log ratio of path probabilities, into total conditional entropy Δ ΔI production Stot and mutual information , as described Sec. VAfor an active bath (þ sign) and in Sec. VBfor self- propulsion (− sign). Since for the present example of a particle trapped in a harmonic potential we have analytical expressions at hand for the combined propagator of particle position and active fluctuations (see Appendix D), we can calculate the conditional entropy production from Eqs. (56) and (62) and the mutual information from Eqs. (61) and (67) explicitly. Only the quantity ln pðxτjη¯Þ is not easily accessible because it is conditioned on the full realization η¯, such (b) that we add it to the change in mutual information in the form jη¯ of the change in (conditional) system entropy ΔSsys [see jη¯ Eq. (53)]. Note that ΔSsys is nonextensive with τ, and thus ΔI only a small correction to which becomes negligible for long times. In Fig. 4, we show the average conditional entropy production hΔSi and the average mutual information ΔI − Δ jη¯ kBh i h Ssysi for an active bath [Fig. 4(a)] and active self-propulsion [Fig. 4(b)]. The system parameters are the same as before in Figs. 2 and 3; in particular, we again compare a static trap u ¼ 0 with a moving trap u ¼ 1.We Δ ΔI can see that now in all cases, both h S i and h i grow τ FIG. 3. Probability densities of the path probability ratio (43) linearly with time for large . This conforms nicely with our Δ for a harmonically trapped particle (a) in a static trap u ¼ 0 and previous findings: For the conditional total entropy Stot, (b) in a moving trap u ¼ 1. The densities are obtained from the active fluctuations are treated like an external numerically simulating 105 sample trajectories of duration τ ¼ 1. time-dependent forcing. Such a force is then naturally Parameters are k ¼ 0.1, γ ¼ 1, τa ¼ 0.2, D ¼ Da ¼ 0.2, expected to produce entropy extensively. For the ensemble- 2 c0 ¼ðγ=kÞfD þ½ðDaÞ=ð1 þ kτa=γÞg. The insets show the cor- and time-averaged rate of total entropy production Δ ΔΣ Δ σ Δ τ responding densities of Stot ¼ þ Ssys. The average values h toti¼limt→∞h Stoti= , we find ΔΣ Δ ΔΣ Δ h i, h Ssysi, and h þ Ssysi as obtained from the numerical ð1 þ DaÞu2 D 1 1 simulations are consistent with the corresponding theoretical σþ D a h toti¼kB þ τ ; ð80aÞ predictions. (a) u ¼ 0, simulation hΔΣi¼0.010ð4Þ, hΔS i¼ k a τ sys D þ Da D 1 þ a −0 0090 13 ΔΣ Δ 0 00139 17 0 γ . ð Þ, h þ Ssysi¼ . ð Þ; u ¼ , theory ΔΣ 0 00975 Δ −0 00828 ΔΣ Δ Da 2 h i¼ . , h Ssysi¼ . , h þ Ssysi¼ ð1 þ Þu D 1 k σ− D a 0 00148 1 ΔΣ 0 017 4 h toti¼kB þ τ : ð80bÞ . . (b) u ¼ , simulation h i¼ . ð Þ, k a γ D þ Da D 1 þ γ hΔSsysi¼−0.0073ð13Þ, hΔΣ þ ΔSsysi¼0.0104ð5Þ; u ¼ 1, theory hΔΣi¼0.01883, hΔSsysi¼−0.00828, hΔΣ þ ΔSsysi¼ 0 01056 The first terms in these expressions contain the production . . rate of hΔΣi from Eq. (77). The second terms quantify the additional contributions from the active fluctuations. They than those with a negative value (see insets in Fig. 2). are balanced by the mutual information production rates σ ΔI τ The sample mean for the path probability ratio lies well h I i¼limτ→∞kBh i= (note that we include a factor of within 1 standard deviation of 1 in both cases, in accor- kB in this definition of the rates so that they have units of dance with the exact result (44). entropy/time), which explicitly read D u2 1 1 σþ a C. Mutual information h I i¼kB þ τ ; ð81aÞ k a τ D D þ Da 1 þ γ a The quantities ΔΣ and ΔS analyzed in the previous sys two sections as a measure for the irreversibility in the 2 1 σ− Da u k system evolution depend only on the particle trajectories h I i¼kB þ τ : ð81bÞ D D þ D 1 k a γ but not on the realizations of the active fluctuations, and a þ γ

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(a) correlation time of fluctuations τa, whereas it is the system’s relaxation time in the harmonic potential γ=k for the self-propelled particle.

VII. CONCLUSIONS AND DISCUSSION Our present work contributes to assessing the out-of- equilibrium character of active matter [28,29,42,50,51, 61,122,123] based on its observable dynamical behavior. Having in mind that in a typical experiment the central observables are particle trajectories, we quantify irreversibility in active matter systems based on particle trajectories alone, without resolving the microscopic mechanisms and (b) associated dissipation underlying the active fluctuations which drive particle motion. For the thermodynamiclike features emerging from the dynamical behavior, it is presumably of little relevance what kind of microscopic processes dissipate how much entropy in generating the active fluctuations in the system (e.g., via self-propulsion), as long as these processes are not altered due to the motion of the particle. We are therefore interested in how (ir)reversible a specific particle trajectory is, out of the set of all possible trajectories which can be generated by the combined influence of thermal and active fluctuations, but not in how (ir)reversible the processes which underly the active fluctuations are. In that spirit, we treat the active fluctuations as an active (nonthermal) bath the particle is in FIG. 4. Average mutual information and conditional entropy contact with in addition to the thermal bath, with statistical production as a function of the observation time τ for a particle properties which are completely independent of the internal trapped in a static (u ¼ 0, orange lines) or moving (u ¼ 1, blue 0 1 γ 1 dynamical state of the particle. lines) harmonic potential. Parameters are again k ¼ . , ¼ , We implement this active bath by following the common τa ¼ 0.2, D ¼ Da ¼ 0.2. (a) Interpretation of the active fluctuations as an active bath; i.e., they are considered being even under approach to include stochastic active forces into the time reversal. (b) Interpretation of the active fluctuations as self- equations of motion, which emulate the directional per- propulsion; i.e., they are considered being odd under time sistence and the nonequilibrium character of the active reversal. Note that even for u ¼ 0 (orange lines), the curves fluctuations [124]. A particularly successful description of ¯ ¯ Δ þ ΔI − Δ jη Δ − ΔI − Δ jη active fluctuations along these lines consists in a colored for h S i, kBh þi h Ssysi and h S i, kBh −i h Ssysi, respectively, are not identical, but just appear very similar on the noise model, more specifically, a Gaussian Ornstein- shown scale. Uhlenbeck process [7,15–18,21–24,28,42,49–61]. For this model class, we calculate the exact expression for the probability density of particle trajectories by integrating σ σ − In total, we therefore find in both cases that h i¼h toti over all possible realizations of the active fluctuations [see σ σ h I i holds, where h i is given in Eq. (77). Eq. (38)]. Since the colored noise renders the particle’s As we discuss in Sec. IVA, the two interpretations of the dynamics non-Markovian, the standard Onsager-Machlup active fluctuations as active bath or as self-propulsion path integral [84–87] cannot be applied directly to obtain mechanism correspond to measuring the mutual informa- the path probabilities. While expressions for a single tion with respect to, respectively, even or odd time reversal colored noise source exist in the literature [105,106], here of the active forcing so that the rate of “mutual information we derive for the first time the path weight for the production” is different in the two cases; compare superposition of colored Ornstein-Uhlenbeck noise (the Eqs. (81a) and (81b). Their difference reads active fluctuations) and white thermal noise. Building on this result, we then relate the probabilities of D 1 1 k time-forward and time-backward processes and establish σþ − σ− a − h I i h I i¼kB τ : ð82Þ – k a τ γ an integral fluctuation theorem [32 35] for their log ratio D 1 þ a γ ΔΣ, a functional over the forward particle trajectory [see Eqs. (44) and (42b)] which quantifies the time (ir)revers- We conclude that for the active bath, the relevant timescale ibility of the particle dynamics. From the integral fluc- the trajectory duration τ is measured against is the tuation theorem, we directly obtain a corresponding

021009-20 IRREVERSIBILITY IN ACTIVE MATTER SYSTEMS: … PHYS. REV. X 9, 021009 (2019) second-law-like relation for ΔΣ [see Eq. (45)]. With ΔΣ, specific realization η¯ of the active noise as a fluctuating we therefore provide an explicit expression to (exactly) force affecting the particle dynamics, we find that ΔΣ can evaluate the fluctuation theorem for a Brownian particle in be split into two parts which have a direct physical contact not only with a thermal bath but at the same time interpretation: the usual entropy production in the thermal also with an active nonequilibrium bath. In particular, it environment and a complementary dissipative component applies to trajectories of arbitrary finite duration. We expect in the active bath, which is expressed as the difference of that it can be tested experimentally with state-of-the-art pathwise mutual information ΔI accumulated along the micro(fluidic) technology for biological or synthetic active time-forward process ðx¯; η¯Þ versus its time-backward twin matter systems, as used, e.g., in Refs. [5,22–24,127–131]. process [see Eqs. (58) and (64)]. This partition of ΔΣ is We point out again that our explicit results for the path valid for any particle trajectory x¯ and any realization of the weight and the fluctuation theorem are based on the active noise fluctuation η¯ but with process-dependent assumption that the active fluctuations are Gaussian with amounts of dissipation in the two baths. exponential correlations in time generated by an Ornstein- All these general results and interpretations are inde- Uhlenbeck process. pendent of how we choose the active fluctuations to behave Like in the case of usual Brownian motion in contact under time reversal, even [Secs. IVA 1 and VA] or odd with a thermal bath only, the path probability ratio (42a) is [Secs. IVA 2 and VB]. The quantitative details, however, equal to the identity if there are no external forces acting on are different; compare, e.g., Eq. (61) with Eq. (67). These the particle. Therefore, for f ¼ 0 any time-backward quantitative differences are a consequence of the different trajectory is equally likely to occur as its time-forward amounts of heat exchanged with the thermal environment twin so that the particle dynamics looks reversible and along a displacement dx in the two cases; see Eqs. (25) and equilibriumlike. Indeed on a coarse-grained timescale (29). When we interpret the active fluctuations ηðtÞ as an (beyond τa) they appear very similar to free Brownian active environment the particle is moving through, the diffusion in thermal equilibrium. Accordingly, in the viscous friction forces from the thermal bath which balance absence of external forces, the probability ratio of particle these active fluctuations are included in the heat exchange. trajectories does not reveal the nonequilibrium nature of the In contrast, they are not counted as contributing to heat in system, even though the whole system is out of equilibrium the case of self-propulsion because self-propelled motion due to the active fluctuations. This observation should be occurs without external forces; i.e., on the coarse-grained true for any stationary (and unbiased) nonequilibrium bath, level of description based on particle d.o.f., self-propulsion not just the Ornstein-Uhlenbeck implementation consid- appears to be force- and thus dissipation-free. We argue in ered here. In order to detect the irreversibility connected Sec. IVA that the interpretation of the active fluctuations with the active fluctuations, we would have to resolve the ηðtÞ as an active environment requires ηðtÞ to be even under corresponding d.o.f. and analyze their behavior under time time reversal, while their interpretation as self-propulsion reversal [41,71,72]. corresponds to ηðtÞ being odd [109]. However, as we can see from comparing Eqs. (19) and In any case, the corresponding stochastic energetics for (42b), for nonvanishing forces f, the irreversibility measure active systems related to displacements of the particles, i.e., ΔΣ is distinctively different from the entropy production of the definition of heat and work based on particle trajecto- purely Brownian motion because it contains the nonlocal ries, appears as a natural and consistent generalization of 0 memory kernel Γτðt; t Þ. Driving the particle by an external Sekimoto’s stochastic energetics of passive Brownian force thus reveals the non-Markovian and nonequilibrium motion [31] (see Sec. IVA), with a clear-cut connection character of the environment; even in the limit τa → 0 of to the irreversibility measure ΔΣ as obtained from the 0 δ-correlated active fluctuations the kernel Γτðt; t Þ yields a path probability ratio. The consistency and unambiguity of nontrivial contribution (see also Appendix C), which these results rely on fully incorporating the thermal renders ΔΣ different from the entropy production in the fluctuations from the equilibrium heat bath into our actual thermal bath. We leave for future exploration how description [cf. Eqs. (5)]. A number of previous works this observation that external forces may reveal the non- [28,42,50,51,122,123] attempted to assess irreversibility in equilibrium character of the environment may be used to active particle systems and its connection to heat and entropy probe properties of the active bath. Preliminary results production using the model class (5)–(7), including, how- indicate that the external force f has to be nonlinear or time ever, only viscous friction effects from the heat bath and dependent, as a simple linear f leads to ΔΣ ¼ 0 already on neglecting thermal fluctuations [D ¼ 0 in Eq. (5) in our the level of individual trajectories [100]; see also Ref. [28]. notation]. In this way, they disregarded an essential part of Our irreversibility measure ΔΣ from Eq. (42) quantifies the energy exchange between system and heat bath [42], the combined “dissipation” into the thermal and the active leading to ambiguities in the definition of heat and entropy bath, which occurs along a (time-forward) particle trajec- production. Likewise, no consensus could be reached on tory x¯ but cannot be interpreted easily as entropy produc- the thermodynamic interpretation of the path probability tion or dissipated heat. However, if we keep track of a ratio in terms of entropy production (see Ref. [42] for a

021009-21 DABELOW, BO, and EICHHORN PHYS. REV. X 9, 021009 (2019) clear and enlightening summary, as well as the comment by the Deutsche Forschungsgemeinschaft under Grant [122] and the corresponding reply [123]). We believe that No. RE 1344/10-1 and within the Research Unit FOR here we resolve these problems in a unique and consistent 2692 under Grant No. 397303734, as well as by the way by incorporating the thermal fluctuations in our analysis Stiftung der Deutschen Wirtschaft. L. D. also thanks [see model (5)–(7)]. In particular, this leads to a clear Nordita for the hospitality and support during an internship interpretation of the path probability ratio in terms of entropy in 2016. We moreover thank Erik Aurell, Aykut Argun, production in the thermal bath (connected to the actual heat Giovanni Volpe, and Jan Wehr for illuminating discussions. dissipated in the bath) and mutual information “production” with the active bath [see Eqs. (58) and (64)]. In the present work, our focus is on deriving the path APPENDIX A: INTEGRATING OVER THE ˜ probability ratio p½x˜jx˜ 0=p½xjx0 as a measure of irrevers- ACTIVE FLUCTUATIONS η(t) ibility in active matter systems and on establishing its Here we show that the path integral (36) evaluates to interpretation in thermodynamic terms. As a fundamental expression (38). Gaussian path integrals of this form are concept in stochastic thermodynamics, it contains essential ubiquitous in field theories [101,102] with well-established information about the system and entails many interesting pffiffiffiffiffiffiffiffiffi results. Using the abbreviation w ¼½ð 2D Þ=ð2DÞðx_ −v Þ, implications. With the fluctuation theorems and the second- t a t t the relevant terms in Eq. (36) involving the active noise law-like relations, we derive the most immediate ones here. variable η become Further potential applications of our results include the t ΔΣ ΔI Z Z Z Z exploration of features in and which are character- 1 τ τ τ 0 T ˆ 0 T istic for the different phases of active matter [20], the Dη¯ − η η 0 η w exp 2 dt dt t Vτðt; t Þ t þ dt t t : analysis of linear response under small external perturba- 0 0 0 tions and its potential for probing properties of the active ðA1Þ bath [132], the connection between violations of the fluctuation-response relation and irreversibility [132,133], Completing the square, we wish to write this as the characterization of universal statistics of infima, stop- Z Z Z ΔΣ ΔI 1 τ τ ping times, and passage probabilities of and based Dη¯ 0 wTΓ 0 w ˜ exp dt dt ½ t τðt; t Þ t0 on the properties of p½x˜jx˜ 0=p½xjx0 [134], the connection 2 0 0 of ΔΣ and ΔI to the arrow of time [116,135] in active T ˆ 0 matter systems, and universal properties in the efficiency −ðηt þ εtÞ Vτðt; t Þðηt0 þ εt0 Þ ; ðA2Þ fluctuations of stochastic heat engines operating between active baths [96,136,137]. 0 where ε and Γτðt; t Þ are yet unknown functions. We can Here, we analyze the model (5), (6) in great detail from t then shift the active noise histories and integrate over η0 ¼ the viewpoint of a Brownian particle moving under the t η ε instead of η . Since the path integral effectively influence of active fluctuations, which are represented by t þ t t integrates over all possible states of η from −∞ to ∞ the Ornstein-Uhlenbeck process (7). However, most of t þ for any point in time t, this shift of trajectories does not alter our results, in particular, the path weight (38) and the associated integral fluctuation theorem (44), as well as its the domain of integration. Moreover, the Jacobian associated formulation in information-theoretic terms in Sec. V, are a with the transformation is the identity. Performing the η¯0 direct consequence of the mathematical structure of the remaining functional integral over , expression (A2) then model and are therefore valid in a much broader physical reduces to Z Z context. In principle, our results can be applied to any 1 τ τ ˆ −1=2 0 0 Brownian dynamics that is driven by an additional w Γ w 0 ðDetVτÞ exp 2 dt dt t τðt; t Þ t ; ðA3Þ Gaussian Ornstein-Uhlenbeck process. For instance, in 0 0 Ref. [138], an information-theoretic analysis similar to ours has been conducted for a colored-noise-driven assuming proper normalization of the integration measure Dη¯0 Brownian model. Other examples are Brownian motion . Absorbing the path-independent functional determinant ˆ −1=2 in a harmonic trap with fluctuating location [78,139–141] ðDetVτÞ into the normalization, we obtain the path weight and thermodynamic nonequilibrium processes using exter- stated in Eq. (38). Γ 0 nal (artificial) colored noise sources [117,142]. However, we still have to show that τðt; t Þ is the ˆ 0 operator inverse of Vτðt; t Þ; i.e., we have to verify Eq. (39). ACKNOWLEDGMENTS Requiring equality of Eqs. (A1) and (A2), we find that Z Z Z S. B. and R. E. acknowledge financial support from the τ ! 1 τ τ T 0 T 0 T ˆ 0 dt ηt wt¼ dt dt ½wt Γτðt; t Þwt0 − εt Vτðt; t Þεt0 Swedish Research Council (Vetenskapsrådet) under the 0 2 0 0 Grants No. 621-2013-3956, No. 638-2013-9243, and − 2ηT ˆ 0 ε No. 2016-05412. L. D. acknowledges financial support t Vτðt; t Þ t0 : ðA4Þ

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ˆ 0 ˆ 0 Note that we use Vτðt; t Þ¼Vτðt ;tÞ. We now observe that This is a linear second-order ordinary differential equation the term on the left-hand side as well as the last term on the with a δ inhomogeneity. We calculate its solution in two ’ Γ¯ 0 right-hand side are of first order in ηt, whereas the steps. First, we compute the Green s function ðt; t Þ of the remaining terms on the right-hand side do not contain ordinary component Vˆ ðtÞ with vanishing boundary con- ¯ 0 ¯ 0 ηt. Therefore, these two types of expressions must cancel ditions, i.e., Γð0;tÞ¼Γðτ;tÞ¼0. Then we add a solution 0 individually, i.e., Γ0;τðt; t Þ of the corresponding homogeneous problem ˆ 0 Z Z Z Z VðtÞΓ0;τðt; t Þ¼0 that fixes the two boundary terms. τ τ τ τ 0 ¯ 0 0 0 T ˆ 0 0 T 0 Their sum Γτðt;t Þ¼Γðt;t ÞþΓ0 τðt;t Þ satisfies Eq. (B1) dt dt εt Vτðt; t Þεt0 ¼ dt dt wt Γτðt; t Þwt0 ; ; 0 0 0 0 and thus gives the desired solution. ¯ 0 0 ðA5aÞ We can construct both these parts Γðt; t Þ and Γ0;τðt; t Þ from the solution of the homogeneous problem associated Z Z Z ˆ τ τ τ with Vτðt; tÞ, which reads [see Eq. (37)] T T 0 ˆ 0 dt ηt wt ¼ − dt ηt dt Vτðt; t Þεt0 : ðA5bÞ 0 0 0 2 2 ½−τa∂t þð1 þ Da=DÞΓðtÞ¼0: ðB2Þ From Eq. (A5b), we immediately infer Z We make an exponential ansatz ΓðtÞ ∼ eλt and easily obtain τ 0 ˆ 0 wt ¼ − dt Vτðt; t Þεt0 : ðA6Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 D ΓðtÞ¼αþeλt þ α−e−λt; λ ¼ 1 þ a; ðB3Þ τ D Substituting this result into the left-hand side of Eq. (A5a), a we obtain with constants α to be determined by the boundary Z Z Z τ τ τ conditions or the δ inhomogeneity. T T 0 0 − dt εt wt ¼ dt wt dt Γτðt; t Þwt0 ; ðA7Þ There exists a standard recipe [103,104] for the con- 0 0 0 struction of Green’s functions of boundary value problems for ordinary differential equations, which we follow here to implying calculate Γ¯ ðt; t0Þ. Splitting the interval ½0; τ at t ¼ t0,we Z τ write 0 0 εt ¼ − dt Γτðt; t Þwt0 : ðA8Þ 0 ¯ 0 0 ¯ 0 0 ¯ 0 Γðt; t Þ¼Θðt − tÞΓ<ðt; t ÞþΘðt − t ÞΓ>ðt; t Þ; ðB4Þ

This intermediate result gives the shift εt required to ¯ 0 complete the square in Eq. (A1) [see also Eq. (A2)]asa where Θ is the Heaviside step function and both Γ<ðt; t Þ 0 Γ¯ 0 function of Γτðt; t Þ and wt. Moreover, substituting Eq. (A8) and >ðt; t Þ satisfy the homogenous problem; i.e., they are back into Eq. (A6), we find of the form (B3) with constants α< and α>, respectively, to Z Z be determined. The constants are fixed by the boundary τ τ 0 00 ˆ 0 0 00 conditions, wt ¼ dt dt Vτðt; t ÞΓτðt ;t Þwt00 ; ðA9Þ 0 0 ¯ 0 ¯ 0 Γ<ð0;tÞ¼0 and Γ>ðτ;tÞ¼0; ðB5aÞ 0 which implies Eq. (39) and thus establishes that Γτðt; t Þ is ’ indeed the Green s function of the differential opera- and the requirements that Γ¯ is continuous at t ¼ t0, ˆ 0 tor Vτðt; t Þ. Γ¯ ðt0;t0Þ¼Γ¯ ðt0;t0Þ; ðB5bÞ APPENDIX B: CONSTRUCTION OF THE < > 0 GREEN’S FUNCTION Γτ(t;t ) 2 0 and has a jump of size −1=τa in its first derivative at t ¼ t , In this Appendix, we construct the Green’s function (40a) by solving its defining equation (39) with the 1 ¯ 0 ¯ 0 ˆ 0 ∂ Γ t; t − ∂ Γ t; t 0 − : differential operator Vτðt; t Þ from Eq. (37). Exploiting ½ t >ð Þ t <ð Þt¼t ¼ τ2 ðB5cÞ ˆ 0 a the “diagonal” structure of Vτðt; t Þ, we can directly evaluate the integral and obtain The last two conditions ensure the desired δ discontinuity at t ¼ t0 in the second derivative. Solving the resulting system ˆ ˆ ˆ 0 0 ½VðtÞþV0ðtÞþVτðtÞΓτðt; t Þ¼δðt − t Þ: ðB1Þ þ − þ − of linear equations for α<, α<, α>, and α>, we obtain

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1 eλτ e−λðt−t0Þ − e−λðtþt0Þ e−λτ eλðt−t0Þ − eλðtþt0Þ Γ¯ 0 ½ þ ½ >ðt; t Þ¼ 2 λτ −λτ ðB6Þ 2τaλ e − e

¯ 0 ¯ 0 and Γ<ðt; t Þ¼Γ>ðt ;tÞ. With Eq. (B4), we can thus write the Green’s function of the ordinary component on the entire interval ½0; τ in the form

1 e−λjt−t0j − e−λðtþt0Þ e−λð2τ−jt−t0jÞ − e−λð2τ−t−t0Þ Γ¯ 0 þ ðt; t Þ¼ 2 −2λτ : ðB7Þ 2τaλ 1 − e

We can immediately check that indeed Vˆ ðtÞΓ¯ ðt; t0Þ¼ probability is just the standard Onsager-Machlup expres- δðt − t0Þ as well as Γ¯ ð0;t0Þ¼0 and Γ¯ðτ;t0Þ¼0. Moreover, sion (16) for a Brownian particle in a thermal bath. we note that Γ¯ ðt; t0Þ is symmetric in its arguments, ¯ ¯ Γðt; t0Þ¼Γðt0;tÞ. 2. Vanishing correlation time τa → 0 As announced earlier, we now add a homogeneous In the limit τa → 0, the correlator (8) of the active noise 0 þ λt − −λt solution Γ0 τðt; t Þ¼α e þ α e to the ordinary inverse 0 0 ; approaches a δ distribution, i.e., hηiðtÞηjðt Þi → δijδðt − t Þ. Γ¯ 0 ˆ Γ 0 0 Γ¯ 0 ðt; t Þ. Since VðtÞ 0;τðt; t Þ¼ and ðt; t Þ vanishes at This means that ηðtÞ becomes just another Gaussian white 0 τ both boundaries of the interval ½ ; , the differential noise with diffusion constant Da. The equation of motion 0 ¯ 0 0 equation (B1) for the sum Γτðt; t Þ¼Γðt; t ÞþΓ0;τðt; t Þ (5) thus contains two independent Gaussian white noises becomes with zero mean and variance 2D and 2Da, respectively. Their sum is itself a Gaussian white noise with zero mean δ − τ τ2∂ Γ¯ 0 αþτ κ λτ α−τ κ −λτ 2 ðt Þ½ a t >ðt; t Þjt¼τ þ a þe þ a −e but variance ðD þ DaÞ. Therefore, we expect Eq. (38) to δ −τ2∂ Γ¯ 0 αþτ κ α−τ κ 0 reduce to þ ðtÞ½ a t <ðt; t Þjt¼0 þ a − þ a þ¼ ; Z τ _ 2 ðB8Þ ðxt − vtÞ ∇ · vt p x x0 ∝ exp dt − − C1 ½ j τa→0 ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 4ðD þ DaÞ 2 κ 1 λτ 1 1 with ¼ a ¼ þðD=DaÞ. We require the two terms in square brackets to vanish individually and thus as τa → 0. obtain two linear equations for the two unknowns αþ and We first note that λ diverges in this limit so that we have α−. Solving them and combining the results for Γ¯ðt; t0Þ and to take more care when analyzing the Green’s function 0 0 ¯ 0 0 Γ0 τðt; t Þ into Γτðt; t Þ¼Γðt; t ÞþΓ0 τðt; t Þ, we finally (40a). Since the limit is expressed more compactly as ; ; λ → ∞ τ find the Green’s function (40a). , we rewrite all occurrences of a in Eq. (40a) in terms of λ, D, and Da. By definition [see Eqs. (40b) and 2 2 (B3)], τa ¼½1 þðDa=DÞ=λ . Thus, the leading-order APPENDIX C: LIMITING CASES 0 behavior of Γτðt; t Þ is FOR THE PATH WEIGHT −λjt−t0j − κ− −λðtþt0Þ −λð2τ−t−t0Þ Here we analyze three relevant limiting cases of the path λ e κ ½e þ e 0 þ → 0 Γτðt; t Þ ∼ : ðC2Þ weight (38), namely, Da (usual Brownian particle 2 1 Da þ D without active fluctuations), τa → 0 (memoryless active → 0 fluctuations), and D (no thermal fluctuations). All We observe that three limits reduce the Langevin equation (5) to simpler setups, for which the path probabilities are already known λ lim e−λjt−t0j ¼ δðt − t0Þ; ðC3aÞ in the literature. We recover all these results when perform- λ→∞ 2 ing the respective limits in our general expression (38). λ 1 lim e−λtΘðtÞ¼ δðtÞ; ðC3bÞ 1. Usual Brownian motion Da → 0 λ→∞ 2 2 Removing the effects of the active fluctuations from the such that the exponentials become δ distributions as η 0 system amounts to setting ðtÞ¼ or, equivalently, Da ¼ λ → ∞: 0 in the equation of motion (5). Performing this limit for the path weight is straightforward: In this case, λ ¼ 1=τ is 0 κ− 0 0 a δðt − t Þ − 2κ ½δðt þ t Þþδð2τ − t − t Þ 0 þ well behaved, so that the term containing the Green’s Γτðt; t Þ ∼ : 1 Da function simply drops out in Eq. (38). The remaining path þ D

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However, the last two δ distributions map either of the two We rewrite the limit D → 0 again as λ → ∞; i.e., we 0 times t and t outside of the interval of integration when express all occurrences of D in terms of λ, Da, and τa 2 2 integrating over the other. Therefore, they do not contribute according to ðDa=DÞ¼λ τa − 1 [see Eqs. (40b) and (B3)]. when integrating over both t and t0. We can see this also For the leading-order behavior of the Green’s function 0 from partially integrating the corresponding term in Γτðt; t Þ, we then find Eq. (C2) with two test functions f and g, −λjt−t0j 2 −2 −λðtþt0Þ −λð2τ−t−t0Þ Z Z e þ½1− τ λ þOðλ Þ½e þe τ τ 0 a λ 0 0 Γτðt;t Þ∼ ; 0 0 −λðtþt Þ −λð2τ−t−t Þ 2τ2λ dt dt fðtÞgðt Þ 2 ½e þ e a 0 Z 0 τ 1 τ ðC6Þ −λ 0 −λ 2τ− − 0 ¼ dtfðtÞ gðt0Þ ½−e ðtþt Þ þ e ð t t Þ 0 2 0 Z t ¼0 which implies τ 1 Z Z − dt0g_ðt0Þ ½−e−λðtþt0Þ þ e−λð2τ−t−t0Þ 1 τ τ 2 A x¯ ∼ 0wTw λ2τ2 − 1 δ − 0 0 ½ D→0 dt dt t t0 ð a Þ ðt t Þ 4D 0 0 → 0 λ → ∞ a as : λ3τ2 − a − λ ½e−λjt−t0j þ e−λðtþt0Þ þ e−λð2τ−t−t0Þ Γ 0 ∼ δ − 0 1 τ → 0 2 Therefore, τðt; t Þ ðt t Þ=½ þðDa=DÞ as a . Substituting this finding into the path weight (38) leads λ2τ O λ −λðtþt0Þ −λð2τ−t−t0Þ precisely to the expected limit (C1). þ½ a þ ð Þ½e þ e :

3. No thermal fluctuations D → 0 We divide our further analysis into three parts by writing A½x¯ ∼ ½1=ð4D ÞðB1½x¯þB2½x¯þB3½x¯Þ with The limit of vanishing thermal white noise is a little more a Z Z involved. If we let D → 0 in the equation of motion (5),we τ τ ¯ 0 T 2 2 0 are left with a system driven by a single colored noise B1½x¼ dt dt wt wt0 ðλ τa − 1Þδðt − t Þ 0 0 source. The resulting path probability density is known in 3 2 the literature [105,106] and given by Eq. (41). However, it λ τ 0 − a − λ e−λjt−t j ; is not immediately obvious how we can obtain this result 2 ðC7aÞ 1 4 from Eq. (38) because both the prefactor =ð DÞ and the Z Z exponent λ diverge as D → 0. τ τ λ3τ2 ¯ 0 T a B2½x¼− dt dt wt wt0 − λ We first rewrite Eq. (41) so that the action can be 0 0 2 expressed in the form of Eq. (38). To this end, we introduce −λ 0 −λ 2τ− − 0 _ × ½e ðtþt Þ þ e ð t t Þ; ðC7bÞ the abbreviation wt ¼ xt − vt in Eq. (41) and remember that Z Z we choose ps to be the stationary distribution (35) of the τ τ w_ 2 ¯ 0 T 2 colored noise. After partial integration of the t term, we B3½x¼ dt dt wt wt0 ½λ τa þ OðλÞ find 0 0 0 0 Z × ½e−λðtþt Þ þ e−λð2τ−t−t Þ: ðC7cÞ 1 τ p x x ∝ − −τ2wTẅ w2 ½ j 0D→0 exp dtð a t t þ t Þ −λt −λt 4D 0 Upon repeated partial integration using λe ¼ −∂ e to a t remove powers of λ, and upon performing the λ → ∞ 2 _ T τ 2 τ 2 2 þ τawt wtj0 þ τawt j0 þ 2τaw0 limiting procedure for well-behaved terms, we find for the Z first part τ ∇ · vt Z − dt : ðC4Þ τ λτ2 0 2 ¯ 2 T ̈ 2 2 2 a B1½x ∼ dt½−τawt wt þ wt þðw0 þ wτ Þ : 0 2 We now show that the action Similarly, the second part reduces to Z Z 1 τ τ 2 0 T 0 Da 0 0 λτ A½x¯¼ dt dt w δðt − t Þ − Γτðt; t Þ w 2 2 a 2 T T t t B2 x¯ ∼− w wτ τ wτ w_ τ − w w_ 0 : 4D 0 0 D ½ ð 0 þ Þ 2 þ að 0 Þ ðC5Þ For the third part, we first remark that the terms of order λ R τ vanish here again under double time integrals because they −A½x¯− dt∇·v =2 of the full path weight p½xjx0 ∝ e 0 t as given become δ distributions that map one of the times outside the in Eq. (38) reduces to the form of the action in Eq. (C4) in interval of integration, similar to the case we have for the the limit D → 0. limit of vanishing correlation time. Hence,

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Z Z τ τ ¯ 0 T 2 −λðtþt0Þ −λð2τ−t−t0Þ B3½x ∼ dt dt wt wt0 λ τa½e þ e Z0 Z0 τ τ 0 T −λðtþt0Þ −λð2τ−t−t0Þ ¼ dt dt wt wt0 λτa∂t0 ½−e þ e Z0 0 τ wTw λτ − −λðtþt0Þ −λð2τ−t−t0Þ τ ¼ dt t t0 a½ e þ e jt0¼0 0Z Z τ τ 0 T _ −λðtþt0Þ −λð2τ−t−t0Þ − dt dt wt wt0 λτa½−e þ e Z 0 0 τ T −λðτþtÞ −λðτ−tÞ ∼ dt wt fwττaλ½−e þ e g 0 Z τ T −λt −λð2τ−tÞ − dt wt fw0τaλ½−e þ e g 0 2 2 2 ∼ τaðwτ − w0Þþ2τaw0:

Combining these three results, we finally find of the active fluctuations. This propagator represents the Z probability to be at “position” q at time t when having 1 τ q A x¯ −τ2wTẅ w2 been at 0 at an earlier time t0

T −ðt−t0ÞA −ðt−t0ÞA 1. Dynamics Cðtjt0Þ¼Cð∞Þ − e Cð∞Þe ðD2cÞ The equations of motion for the joint Markovian system of particle and active fluctuations combined from Eqs. (74) with u ¼ðu; 0Þ and the stationary covariance matrix and (7) (for d ¼ 1) read 0 h i qffiffiffiffiffi 1 γ Da Da 1 D þ τ τ x_t xt u ξt B k 1 k a 2 1 k a C ¼ −A − t þ B ðD1aÞ B þ γ þ γ C η_ η 0 ζ Cð∞Þ¼@ qffiffiffiffiffi A: ðD2dÞ t t t Da 1 1 2 kτa 2τ 1þ γ a with ffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p As we describe in the main text, we consider the situation k=γ − 2D 2D 0 A ¼ a and B ¼ : in which initially the distribution of particle positions is 01τ 01τ = a = a independent of the active fluctuations (which are in their ðD1bÞ stationary state) so that the initial distribution for the joint system factorizes as p0ðx0; η0Þ¼p0ðx0ÞpsðηÞ. We further Because of its Markovian character, all statistical properties assume that p0ðx0Þ is Gaussian with zero mean and 2 of this joint system are encoded in the propagator variance c0, whereas psðη0Þ is given in Eq. (35). The pðq;tjq0;t0Þ, where q ¼ðx; ηÞ, q0 ¼ðx0; η0Þ are “gener- initial probability density p0ðx0; η0Þ is thus also Gaussian alized coordinates” summarizing particle position and state with mean μð0Þ¼0 and covariance matrix

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2 c0 0 the Stratonovich convention throughout this work. For Cð0Þ¼ : ðD3Þ ΔΣ Γ 0 01=2τ evaluating c, we observe that τðt; t Þ is continuously a differentiable except at t ¼ t0 [see Eq. (B5c)]. Splitting the t 0 0 0 τ Because of the linearity of the system [see Eq. (D1)], the integral into the intervals ½ ;t and ½t ; , we can transfer _ distribution remains Gaussian for all later times t>t0, the time derivative from the trajectory xðtÞ to the memory with the expectation values and covariances evolving kernel by partial integration, so that according to Z 1 D τ ΔΣ a 0 Γ 0 0 − Γ τ 0 − A −1 c ¼ dt x0ft0 <ð ;tÞ xτft0 >ð ;tÞ μ − 1 − t A u T D 0 ðtÞ¼½t ð e Þ ; ðD4aÞ Z Z t0 τ T ∂ Γ 0 ∂ Γ 0 C t C ∞ e−tA C 0 − C ∞ e−tA : þ dt xtft0 t <ðt; t Þþ dt xtft0 t >ðt; t Þ ; ð Þ¼ ð Þþ ½ ð Þ ð Þ ðD4bÞ 0 t0 ΔΣ ΔI ðD7Þ In order to calculate averages of, e.g., or ,wehave to evaluate correlators of particle positions and/or active 0 0 where Γ<ðt; t Þ and Γ>ðt; t Þ are defined in analogy to fluctuations at two different time points (see next section). ΔΣ Using the propagator (D2) and the time-dependent prob- Eq. (B4). Now, the average h ci involves correlations − 0 ability density (D4), we find hxtft0 i¼khxt½xt0 ut i between different time points. Using the autocorrelation functions (D5) and the result 0 q − μ q 0 − μ 0 (40) for Γτðt; t Þ, it can thus be evaluated as an ordinary h½ ðtÞ ðtÞi½ ðt Þ ðt Þji integral. −ðt−t0ÞAC 0 ≥ 0 ½e ðt Þij for t t ; In its full general form, the resulting expression for D5 ¼ − 0− A ð Þ ΔΣ ΔΣ ΔΣ is rather lengthy so that we omit it ½e ðt tÞ CðtÞ for t

2. Evaluation of averages Δ ΔΣ We are interested in the averages h Ssysi, h i, and ΔI [1] S. Ramaswamy, The Mechanics and Statistics of Active h i, where the general, trajectorywise expressions of all these quantities are given in Eqs. (22), (42b), and (61), (67), Matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010). [2] P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, and L. respectively. While calculating hΔS i is relatively sys Schimansky-Geier, Active Brownian Particles, Eur. Phys. straightforward (and basically amounts to computing the J. Spec. Top. 202, 1 (2012). variances of the Gaussians at initial and final time), the [3] M. E. Cates, Diffusive Transport without Detailed Balance ΔΣ ΔI evaluation of h i and h i is more complicated. It in Motile Bacteria: Does Microbiology Need Statistical 0 involves the nonlocal memory kernel Γτðt; t Þ and the Physics?, Rep. Prog. Phys. 75, 042601 (2012). _ 0 − 0 ΔΣ ΔI averages hxðtÞk½xðt Þ ut i [for h i and h i, see [4] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Eqs. (42b) and (61), (67)], and hx_ðtÞηðt0Þi or hηðtÞk½xðt0Þ − Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydro- 0 ΔI dynamics of Soft Active Matter, Rev. Mod. Phys. 85, 1143 ut i [for h i, see Eqs. (61) and (67)]. Here, we use 0 0 0 0 (2013). ft0 ¼ fðxðt Þ;tÞ¼k½xðt Þ − ut ; see Eq. (74). It turns out to be convenient to move the time derivative from x_ðtÞ over [5] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, 0 G. Volpe, and G. Volpe, Active Particles in Complex and to Γτ t; t by partial integration. Then all the correlations ð Þ Crowded Environments, Rev. Mod. Phys. 88, 045006 0 η 0 reduce to hxðtÞxðt Þi and hxðtÞ ðt Þi, which we already (2016). calculated in Eq. (D5). [6] S. Ramaswamy, Active Matter, J. Stat. Mech. (2017) We exemplify the procedure in more detail for hΔΣi, 054002. ΔI h i can be evaluated in an analogous way. For the [7] É. Fodor and M. C. Marchetti, The Statistical Physics of explicit calculation, it is useful to split Eq. (42b) into a Active Matter: From Self-Catalytic Colloids to Living white-noise and a colored-noise contribution, Cells, Physica (Amsterdam) 504A, 106 (2018). Z [8] J. K. Parrish and W. M. Hamner, Animal Groups in 1 τ Three Dimensions: How Species Aggregate (Cambridge ΔΣ w ¼ dxtft; ðD6aÞ University Press, Cambridge, England, 1997). T 0 [9] A. Cavagna and I. Giardina, Bird Flocks as Condensed Z Z 1 τ τ Matter, Annu. Rev. Condens. Matter Phys. 5, 183 (2014). Da 0 0 ΔΣc ¼ − dt dt x_tft0 Γτðt; t Þ; ðD6bÞ [10] A. Cavagna, I. Giardina, and T. S. Grigera, The Physics of T D 0 0 Flocking: Correlation as a Compass from Experiments to Theory, Phys. Rep. 728, 1 (2018). ΔΣ ΔΣ ΔΣ ΔΣ such that ¼ w þ c. The white-noise part w is [11] J. Elgeti, R. G. Winkler, and G. Gompper, Physics of an ordinary stochastic integral, and its average can be Microswimmers–Single Particle Motion and Collective evaluated using standard techniques; we recall that we use Behavior: A Review, Rep. Prog. Phys. 78, 056601 (2015).

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