Irreversibility in active matter systems: Fluctuation theorem and mutual information
We consider a Brownian particle which, in addition to being in contact with a thermal bath, is driven by active fluctuations. These active fluctuations do not fulfill a fluctuation-dissipation relation and therefore play the role of a non-equilibrium environment. Using an Ornstein-Uhlenbeck process as a model for the active fluctuations, we derive the path probability of the Brownian particle subject to both thermal and active noise. From the case of passive Brownian motion, it is well-known that the log-ratio of path probabilities for observing a certain particle trajectory forward in time versus observing its time-reserved twin trajectory quantifies the entropy production. We calculate this path probability ratio for active Brownian motion and derive a generalized “entropy production”, which fulfills an integral fluctuation theorem. We show that those parts of this “entropy production” which are different from the usual dissipation of heat in the thermal environment, can be associated with the mutual information between the particle trajectory and the history of the non-equilibrium environment. When deriving and discussing these results we keep in mind that the active fluctuations can occur due to either a suspension of active particles pushing around a passive colloid or due to active self-propulsion of the particle itself; we point out the similarities and differences between these two situations. Finally, we illustrate our general results by analyzing a Brownian particle which is trapped in a static or moving harmonic potential.
Active particle systems consist of individual entities (“particles”) which have the ability to perform motion by consuming energy from the environment and converting it into a self-propulsion drive Ramaswamy (2010); Romanczuk et al. (2012); Cates (2012); Marchetti et al. (2013); Bechinger et al. (2016); Ramaswamy (2017); Fodor and Marchetti (2018). Prototypical examples are collections of macro-organisms, such as animal herds, schools of fish, flocks of birds, or ant colonies Parrish and Hamner (1997); Cavagna and Giardina (2014); Cavagna et al. (2017), and suspensions of biological microorganisms or artificial microswimmers, such as bacteria and colloidal particles with catalytic surfaces Romanczuk et al. (2012); Cates (2012); Elgeti et al. (2015); Bechinger et al. (2016); Patteson et al. (2016). Systems of this kind exhibit a variety of intriguing properties, e.g. clustering and swarming Ben-Jacob et al. (2000); Peruani et al. (2006); Fily and Marchetti (2012); Maggi et al. (2015); Marini Bettolo Marconi and Maggi (2015); Paoluzzi et al. (2016), bacterial turbulence Dombrowski et al. (2004), or motility-induced phase separation Cates and Tailleur (2015); Farage et al. (2015), to name but a few.
Microorganisms and mircoswimmers are usually dispersed in an aqueous solution at room temperature and therefore experience thermal fluctuations which give rise to a diffusive component in their self-propelled swimming motion. In addition, the self-propulsion mechanism is typically noisy in itself Romanczuk et al. (2012), for instance, due to environmental factors or intrinsic stochasticity of the mechanisms creating self-propulsion. These “active fluctuations” exhibit two essential features. First, a certain persistence in the direction of driving over length and time scales comparable to observational scales. Second, an inherent non-equilibrium character as a consequence of permanently converting and dissipating energy in order to fuel self-propulsion. Interestingly, similar active fluctuations with the same characteristics can be observed in a complementary class of active matter systems, namely a passive colloidal “tracer” particle which is suspended in an aqueous solution of active swimmers. The collisions with the active particles in the environment entail directional persistence and non-equilibrium features in the motion of the passive tracer particle Maggi et al. (2014); Argun et al. (2016); Maggi et al. (2017).
Despite the inherent non-equilibrium properties of active matter systems they appear to bear striking similarities to equilibrium systems Tailleur and Cates (2008); Speck et al. (2014); Takatori et al. (2014); Farage et al. (2015), for instance, the dynamics of individual active particles at large scales often looks like passive Brownian diffusion Cates (2012) (i.e., at scales beyond at least the “persistence length” of the particle motion). In connection with the ongoing attempts to describe active matter systems by thermodynamic (-like) theories this observation raises the important question “How far from equilibrium is active matter?” Fodor et al. (2016); Nardini et al. (2017). More specifically, the questions are: in which respects do emergent properties of active systems resemble the thermodynamics of thermal equilibrium systems, in which respects do they not, and how do these deviations manifest themselves in observables describing the thermodynamic character of the active matter system, like, e.g, the entropy production?
We here address these questions from a fundamental non-equilibrium statistical mechanics viewpoint by quantifying the probability density of particle trajectories generated under the combined influence of thermal and active fluctuations, the latter stemming either from an active bath the (passive) particle is dispersed in or from active self-propulsion. In order to account for the active fluctuations without resolving the microscopic processes that drive self-propulsion or govern the interactions between active and passive particles, we follow the common approach to include stochastic “active forces” in the equations of motion of the individual particle of interest (see, e.g., Romanczuk et al. (2012); Bechinger et al. (2016) and references therein). When calculating the path probabilities, we treat these active non-equilibrium forces in the same way as thermal equilibrium noise, namely as a “bath” the particle is exposed to with unknown microscopic details, but known statistical properties (which are correlated in time and break detailed balance Note1 ()). Adopting the viewpoint of stochastic thermodynamics Seifert (2008); Jarzynski (2011); Seifert (2012); Van den Broeck and Esposito (2015); Seifert (2018), we study the breakdown of time-reversal symmetry in the motion of the particle and the associated irreversibility of particle trajectories as a measure for non-equilibrium. We do so by comparing the probability of a specific particle trajectory to occur forward in time to that of its time-reversed counterpart, and by linking the corresponding probability ratios to extensive quantities produced along the forward trajectory.
In the case of passive particles in contact with a single, purely thermal bath this procedure is well established, and is known to provide relations between thermodynamic quantities, such as entropy production, work, or heat, and dynamical properties encoded in path probability ratios. Many such relations have been found over the last two decades Crooks (1999); Seifert (2005); Chetrite and Gawedzki (2008); Jarzynski (2011); Seifert (2012), which, in their integrated forms, typically yield refinements of the second law of thermodynamics Jarzynski (1997); Seifert (2005); Jarzynski (2011). In particular, the fluctuation theorem for the total entropy production of a passive particle in a thermal environment reinforces the fundamental interpretation of entropy as a measure of irreversibility (we give a brief summary of the results most relevant to our present work in Sec. III). In the presence of active fluctuations, however, the identification of dissipated heat and entropy production is less straightforward, and is connected to the problem of how to calculate and interpret the path probability ratio in a physically meaningful way Pietzonka and Seifert (2018); Puglisi and Marini Bettolo Marconi (2017).
The results we present in this paper contribute to resolving these questions. Our main findings are the following: (i) Using a path-integral approach, we calculate the probability density of particle trajectories as a result of the statistical properties of thermal and active fluctuations simultaneously affecting the dynamics of the particle [Sec. IV.2, Eqs. (34) and (36)]. When calculating these path weights, we consider the general situation in which the particle is also subject to external (or interaction) forces. We summarize the mathematical details of the derivation as well as various known limiting cases in Appendices A-C. (ii) We then find that the log-ratio of the probabilities for particle trajectories to occur forward in time versus backward in time can be expressed as a functional along the forward path with a non-local “memory kernel” [Sec. IV.3, Eqs. (38)]. This functional (we denote it ) therefore quantifies irreversibility in our active matter systems. Combined with the change in system entropy, it fulfills an integral fluctuation theorem [Sec. IV.3, Eq. (40)], valid for any duration of the particle motion, and consequently obeys a second-law like relation [Eq. (41)]. (iii) By keeping track of the specific realization of the active fluctuations which participated in generating the particle trajectory, we can identify the two individual contributions to stemming from the thermal environment and the active bath. The thermal part is the usual entropy produced in the thermal environment along the particle trajectory (for the given active noise realization). The part associated with the active bath is given by the difference in the amount of correlations (measured in terms of path-wise mutual information), that are built up between the particle trajectory and the active noise in the time-forward direction as compared to the time-backward direction (Sec. V). This splitting is valid no matter if we assume the active fluctuations to be even [Eq. (54)] or odd [Eq. (60)] under time-reversal. From the fluctuation theorem for we then obtain an integral fluctuation theorem and a second-law relation for the mutual information difference [Eqs. (55), (61) and (56), (62)].
We illustrate all these findings in Sec. VI by discussing the example of a colloidal particle subject to thermal and active fluctuations, which is trapped in a (static or moving) harmonic potential. For this simple linear system all relevant quantities can be calculated explicitly. We give the most important mathematical details in Appendix D.
Our results (i) and (ii) rely on the specific model of a Gaussian Ornstein-Uhlenbeck process Gardiner (1985); Van Kampen (1992) for the active fluctuations, while result (iii) is valid for general types of active fluctuations. The Ornstein-Uhlenbeck process has become quite popular and successful in describing active fluctuations Fily and Marchetti (2012); Farage et al. (2015); Maggi et al. (2014); Argun et al. (2016); Maggi et al. (2017); Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017); Puglisi and Marini Bettolo Marconi (2017); Koumakis et al. (2014); Szamel (2014); Szamel et al. (2015); Maggi et al. (2015); Flenner et al. (2016); Paoluzzi et al. (2016); Marini Bettolo Marconi et al. (2016); Szamel (2017); Caprini et al. (2018); Fodor and Marchetti (2018); Berthier and Kurchan (2013); Marini Bettolo Marconi and Maggi (2015); Shankar and Marchetti (2018), because it constitutes a minimal model for persistency in the active forcings due its exponentially decaying correlations with finite correlation time, and it can easily be set up to break detailed balance by a “mismatched” damping term which does not validate the fluctuation-dissipation relation Kubo (1966); Marconi et al. (2008). Moreover, it is able to describe both situations of interest mentioned above, namely passive motion in a bath of active swimmers Maggi et al. (2014); Argun et al. (2016); Maggi et al. (2017) and active motion driven by self-propulsion Fily and Marchetti (2012); Farage et al. (2015); Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017); Puglisi and Marini Bettolo Marconi (2017); Koumakis et al. (2014); Szamel (2014); Szamel et al. (2015); Maggi et al. (2015); Flenner et al. (2016); Paoluzzi et al. (2016); Marini Bettolo Marconi et al. (2016); Szamel (2017); Caprini et al. (2018); Fodor and Marchetti (2018); Berthier and Kurchan (2013); Marini Bettolo Marconi and Maggi (2015); Shankar and Marchetti (2018). The details of our model are given in the next section.
We consider a colloidal particle at position in , or dimensions, which is suspended in an aqueous solution at thermal equilibrium with temperature . The particle diffuses under the influence of deterministic external forces, in general consisting of potential forces (with the potential ) and non-conservative force components . In addition, it experiences fluctuating driving forces due to permanent energy conversion from active processes in the environment or the particle itself. The specific examples we have in mind are a passive tracer in an active non-thermal bath (composed, e.g., of bacteria in aqueous solution) Maggi et al. (2014); Argun et al. (2016); Maggi et al. (2017), or a self-propelled particle (e.g., a bacterium or colloidal microswimmer) Fily and Marchetti (2012); Farage et al. (2015); Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017); Puglisi and Marini Bettolo Marconi (2017); Koumakis et al. (2014); Szamel (2014); Szamel et al. (2015); Maggi et al. (2015); Flenner et al. (2016); Paoluzzi et al. (2016); Marini Bettolo Marconi et al. (2016); Szamel (2017); Caprini et al. (2018); Fodor and Marchetti (2018); Berthier and Kurchan (2013); Marini Bettolo Marconi and Maggi (2015); Shankar and Marchetti (2018).
The deterministic forces are collected in
|where is the hydrodynamic friction coefficient of the particle and|
Thermal fluctuations are described by unbiased Gaussian white noise sources with mutually independent, delta-correlated components , i.e. , where the angular brackets denote the average over many realizations of the noise and . The strength of the thermal fluctuations is given by the particle’s diffusion coefficient , which is connected to the temperature and the friction by the fluctuation-dissipation relation Einstein (1905); Nyquist (1928); Callen and Welton (1951); Kubo (1966) ( is Boltzmann’s constant), as a consequence of the equilibrium properties of the thermal bath.
The term in (1) represents the active force components, with being unbiased, mutually independent noise processes, and being an effective “active diffusion” characterizing the strength of the active fluctuations. The model (1) does not contain “active friction”, i.e. an integral term over with a friction kernel modelling the damping effects associated with the active forcing. It thus represents the limiting case in which such friction effects are negligibly small. In spite of this approximation, the model (1) has been applied successfully to describe various active particle systems Fily and Marchetti (2012); Farage et al. (2015); Maggi et al. (2014); Argun et al. (2016); Maggi et al. (2017); Marini Bettolo Marconi and Maggi (2015); Shankar and Marchetti (2018), and has become quite popular in an even more simplified variant which neglects thermal fluctuations () Fodor et al. (2016); Marconi et al. (2017); Mandal et al. (2017); Puglisi and Marini Bettolo Marconi (2017); Koumakis et al. (2014); Szamel (2014); Szamel et al. (2015); Maggi et al. (2015); Flenner et al. (2016); Paoluzzi et al. (2016); Marini Bettolo Marconi et al. (2016); Szamel (2017); Caprini et al. (2018); Fodor and Marchetti (2018). However, for thermodynamic consistency it is necessary to consider both noise sources Puglisi and Marini Bettolo Marconi (2017), especially when assessing the flow of heat and entropy, which is in general produced in the thermal environment as well as in the processes fuelling the active fluctuations Gaspard and Kapral (2017); Pietzonka and Seifert (2018).
As active fluctuations are the result of perpetual energy conversion (e.g., by the bacteria in the bath, or by the propulsion mechanism of the particle), their salient feature is that they do not fulfill a fluctuation-dissipation relation. In our model (1), this is particularly obvious as active friction effects are absent by the assumption of being negligibly small; in general, the active friction kernel would not match the active noise correlations Zamponi et al. (2005); Ohkuma and Ohta (2007). In that sense, the active fluctuations characterize a non-thermal environment or bath for the Brownian particle with intrinsic non-equilibrium properties. We emphasize that is not a quantity directly measurable, but rather embodies the net effects of the active components in the environment or the self-propulsion mechanism of the particle, similar to the white noise representing an effective description of the innumerable collisions with the fluid molecules in the aqueous solution.
Although several results in the present paper will be generally valid for any noise process (with finite moments), our main aim is to study the specific situation of exponentially correlated, Gaussian non-equilibrium fluctuations Koumakis et al. (2014); Szamel (2014); Marconi et al. (2016, 2017); Fodor et al. (2016); Flenner et al. (2016); Mandal et al. (2017); Puglisi and Marini Bettolo Marconi (2017), generated from an explicitly solvable Ornstein-Uhlenbeck process,
i.e. so-called Gaussian colored noise Hänggi and Jung (1994). Here, are mutually independent, unbiased, delta-correlated Gaussian noise sources, just like , but completely unrelated to them. The characteristic time quantifies the correlation time of the process (),
as can be verified easily from the explicit steady-state solution of (3). It is thus a measure for the persistence of the active fluctuations.
The model (1), (2) describes a single Brownian particle in simultaneous contact with a thermal bath and an active environment as a source of non-equilibrium fluctuations . Our main aim in this paper is to investigate in detail the role these non-equilibrium fluctuations play for the stochastic energetics Sekimoto (2010) and thermodynamics Seifert (2012) of the Brownian particle. While we adopt the single particle picture for simplicity, all our results hold for multiple, interacting Brownian particles as well. In this case, the symbol in (1), (2) denotes a super-vector collecting the positions () of all particles, i.e. , and similarly for the forces, velocities, and so on. The only requirement is that the particles are identical in the sense that they have the same coupling coefficients and , and that the active fluctuations of the individual constituents are independent, but share identical statistical properties.
Iii The ideal thermal bath:
In order to set up the framework and further establish notation, we start with briefly recalling the well-known case of a Brownian particle in sole contact with a thermal equilibrium reservoir,
where the deterministic driving is defined as in (2). The Brownian particle can be prevented from equilibrating with the thermal bath by time-variations in the potential or by non-conservative external forces .
Following Sekimoto Sekimoto (1998, 2010), the heat that the particle exchanges with the thermal bath while moving over an infinitesimal distance during a time-step from to is quantified as the energy the thermal bath transfers to the particle along this displacement due to friction and fluctuations , i.e.
where the product needs to be interpreted in Stratonovich sense Note2 (). With the definition (6), heat is counted as positive if received by the particle and as negative when dumped into the environment (this sign convention is thus the same as Sekimoto’s original one Sekimoto (1998, 2010)). From the equation of motion (5), we immediately see that the heat exchange can be equivalently written as
The change of the particle’s “internal” energy over the same displacement is given by the total differential
Hence, if the potential does not vary over time, the only contribution to comes from the change in the potential energy associated with the displacement .
On the other hand, even if the particle does not move within , its “internal” energy can still change due to variations of the potential landscape by an externally applied, time-dependent protocol. Prototype examples are intensity- or position-variations of optical tweezers, which serve as a trap for the colloidal particle Gomez-Solano et al. (2010); Schmiedl and Seifert (2007); Argun et al. (2016); Ciliberto (2017); McGloin et al. (2016). Being imposed and controlled externally, these contributions are interpreted as work performed on the particle. A second source of external forces which may contribute to such work are the non-conservative components in (5) (originally not considered by Sekimoto Sekimoto (1998), but systematically analyzed later by Speck et al. Speck et al. (2008)). The total work applied on the particle by external forces is therefore given by
for the energy balance over infinitesimal displacements , valid at any point in time . After integration along a specific trajectory of duration , which starts at and ends at some (which is different for every realization of the thermal noise in (5), even if is kept fixed), we find the first law at the trajectory level Van den Broeck and Esposito (2015)
Here, , , and denote the integrals of the expressions (7) [or, equivalently, (6)], (8), and (9), respectively, along the trajectory ; the notation explicitly indicates the dependence on that trajectory.
iii.2 Path probability and entropy
The next step towards a thermodynamics characterization of the Brownian motion (5) is to introduce an entropy change or entropy production associated with individual trajectories Seifert (2005). From the viewpoint of irreversibility, such an entropy concept has been defined as the log-ratio of the probability densities for observing a certain particle trajectory and its time-reversed twin Andrieux et al. (2007). We express the probability density of a particle trajectory by the standard Onsager-Machlup path integral Onsager and Machlup (1953); Machlup and Onsager (1953); Chernyak et al. (2006); Cugliandolo and Lecomte (2017),
where we condition on a fixed starting position , and, accordingly, introduce the notation to denote a trajectory which starts at fixed position . In contrast to from above the set of points does therefore not contain the initial point (i.e. ). The path probability (12) is to be understood as a product of transition probabilities in the limit of infinitesimal time-step size, using a mid-point discretization rule. The divergence of in the second term represents the path-dependent part of normalization Chernyak et al. (2006); Cugliandolo and Lecomte (2017), while all remaining normalization factors are path-independent constants and thus omitted. For convenience, we have introduced the short-hand notation and .
We now consider the time-reversed version of this trajectory, which is traced out backward from the final point to the initial point when advancing time,
and ask how likely it is that is generated by the same Langevin equation (5) as , with the same deterministic forces acting at identical positions along the path. The latter requirement implies that in case of explicitly time-dependent external forces the force protocol has to be time-inverted, i.e. is replaced by in (5) to construct the Langevin equation for the time-reserved path Note3 (). From that Langevin equation we can deduce the probability for observing the backward trajectory , conditioned on its initial position , in analogy to (12). Using (13), we can then express in terms of the forward path , so that we find for the path probability ratio
with the quantity being a functional of the forward path only,
As a quantitative measure of irreversibility, is identified with the entropy production along the path with given initial position .
For an infinitesimal displacement the corresponding entropy change reads
The last equality follows from comparison with (7) [see also (2)] and states that the entropy production along is given by the heat dissipated into the environment during that step divided by the bath temperature. For that reason, , and , are more accurately called entropy production in the environment.
where is the time-dependent solution of the Fokker-Planck equation Gardiner (1985); Van Kampen (1992); Mazo (2002) associated with (5), for the same initial distribution from which the initial value for the path is drawn. The change in system entropy along the trajectory is therefore given by
Combining and , we obtain the total entropy production along a trajectory ,
Iv The non-equilibrium environment
We now focus on the full model (1), (2), (3) for Brownian motion subject to active fluctuations . Our main goal is to develop a trajectory-wise thermodynamic description as a natural generalization of stochastic energetics and thermodynamics in a purely thermal environment (see previous section). We thus treat the active forces in the same way as the thermal noise , namely as a source of fluctuations whose specific realizations are not accessible but whose statistical properties determine the probability for observing a certain particle trajectory . We are interested in how the non-equilibrium characteristics of the active fluctuations affect the irreversibility measure encoded in the ratio between forward and backward path probabilities, and how this measure is connected to the energetics of the active Brownian motion.
iv.1 Energetics and entropy
Comparing (1) with (5), we may conclude that the stochastic energetics associated with (1) can be obtained from the energetics for (5) by adjusting the total forces acting on the Brownian particle, i.e. by replacing with . However, there is another, maybe less obvious way of turning (5) into the model (1) with active fluctuations, namely by substituting with . In the following we argue that these two approaches correspond to two different physical situations, with different trajectory-wise energy balances.
iv.1.1 Active bath
In case the model non-equilibrium fluctuations from an active environment (consisting, e.g., of swimming bacteria), they indeed can be interpreted as additional time-dependent forces from sources external to the Brownian particle. Hence, the total external force acting on the Brownian particle at time is given by . This modification of the external force does obviously not affect the basic definition (6) of heat exchanged with the thermal bath. Using the force balance expressed in the Langevin equation (1) to replace we obtain
The active fluctuations formally play the role of additional non-conservative force components, and thus affect the heat which is dissipated into the thermal environment in order to balance all acting external forces. However, they cannot be controlled to perform work on the particle due to their inherent fluctuating character as an active bath, such that the definition (9) of the work remains unchanged. Finally, the change of “internal” energy (8) over a time-interval is determined by the potential only, and thus is not altered by the presence of the active fluctuations either.
for Brownian motion in an active bath. Here, we have introduced the energy exchanged with the active bath,
It might be best interpreted as “heat” in the sense of Sekimoto’s general definition, that any energy exchange with unknown or inaccessible degrees of freedom may be identified as “heat” Sekimoto (2010). In our setup, the active fluctuations represent an effective description of the forces from the active environment, and are thus in general not directly measurable in an experiment.
Based on the heat exchanged with the thermal environment, we can identify the entropy production in the thermal environment as
in analogy to (16). We refrain, however, from defining an entropy production in the active bath. Such an environment, itself being in a non-equilibrium state due to continuous dissipation of energy, does not possess a well-defined entropy, so that the “heat” dissipated into the active bath cannot be associated with a change of bath entropy.
In case the active fluctuations represent self-propulsion of the Brownian particle, systematic particle motion can occur already without any external forces being applied. For , and without thermal fluctuations, , the momentary particle velocity is exactly equal to the active velocity . In that sense, self-propulsion is force-free. More precisely, the driving force, which is created locally by the particle for self-propulsion, is compensated according to actio est reactio, such that the total force acting on a fluid volume comprising the particle and its active self-propulsion mechanism is zero. The corresponding dissipation (and entropy production) inside such a fluid volume, which results from the conversion of energy or fuel to generate the self-propulsion drive, can not be quantified by our effective description of the active propulsion as fluctuating forces , because it does not contain any information on the underlying microscopic processes. In order to quantify such entropy production, a specific model for the self-propulsion mechanism is required Gaspard and Kapral (2017); Pietzonka and Seifert (2018). In other words, our “coarse-grained” description (3) does not allow us to assess how much energy the conversion process behind the self-propulsion drive dissipates. We can only measure the dissipation associated with deviations of the particle trajectory from the self-propulsion path, which occurs if the particle velocity differs from due to the action of external forces or thermal fluctuations. Accordingly, the heat exchange with the thermal bath for a displacement taking place over a time-interval at time is given by , or, using (1),
The work , on the other hand, performed on or by the particle during the time-step , which can be controlled or harvested by an external agent, is exactly the same as without active propulsion, given in (9), because from the operational viewpoint of the external agent it is irrelevant what kind of mechanisms propel the particle. Likewise, the definition (8) for the change in “internal” energy is independent of how particle motion is driven, and thus remains unaffected by our interpretation of as active propulsion.
for active, self-propelled Brownian motion described by the Langevin equation (1). Here, we balance the different energetic contributions by introducing
This quantity represents the contribution to the heat exchange with the thermal bath, which is contained only in the active component of the full particle displacement , see (25). We can therefore interpret it as the “heat” transferred from the active fluctuations to the thermal bath via the Brownian particle. Again, this is a quantity which can in general not be measured in an experiment, since it is produced by the inaccessible active fluctuations .
Finally, we can relate the heat exchange with the thermal bath to dissipation and define a corresponding entropy production in the environment,
iv.2 Path probability
We now calculate the probability for observing a certain path starting at , generated under the combined influence of thermal and active fluctuations. We treat these two noise sources on equal terms, namely as fluctuating forces with unknown specific realizations but known statistical properties. Due to the memory of the active noise , the system (1) is non-Markovian. Therefore, the standard Onsager-Machlup path integral Onsager and Machlup (1953); Machlup and Onsager (1953); Chernyak et al. (2006) cannot be applied directly to obtain . However, for the Ornstein-Uhlenbeck model (3) of the combined set of variables is Markovian and we can easily write down the path probability for the joint trajectory , conditioned on an initial configuration :
To calculate the path weight for the particle trajectories , we have to integrate out the active noise history,
where characterizes the initial distribution of the active fluctuations at .
Since the variables represent an effective description of the active fluctuations which are not experimentally accessible, it is in general not possible to set up a specific initial state for . As a consequence there are basically two physically reasonable choices for .
On the one hand, we can assume that the active bath has reached its stationary state, before we immerse the Brownian particle, such that the bath’s initial distribution is independent of and given by the stationary distribution of , i.e. . For the Ornstein-Uhlenbeck process (3), the stationary distribution reads Risken (1984); Gardiner (1985)
At we place the Brownian particle into the fluid with an initial distribution of particle positions which can be prepared arbitrarily, and start measuring immediately.
On the other hand, we may let the Brownian particle adapt to the active and thermal environments before performing measurements, and assume that the system is in a joint steady state at . In that case, control over the distribution of initial particle positions is limited, as it is influenced by the active fluctuations. The form of depends on the particular set-up, i.e. the specific choices for and in (2).
In the following, we will perform the path integration (30) for the first option, starting from independent initial conditions ; the calculation for the second option can be carried out along the same lines, if the joint steady state is Gaussian in the active noise Dabelow et al. (2018). Plugging (29) and (31) into (30), and performing a partial integration of the term proportional to in the exponent, we obtain
where we have used the abbreviation for the full path including the initial point , in order to write . The differential operator
|consists of an ordinary component,|
|and two boundary components|
which include the boundary terms picked up by the partial integration of , and from the initial distribution (31) of . The subscript in (33a) indicates that the operator is acting on trajectories of duration .
Since the path integral over the active noise histories is Gaussian, we can perform it exactly Zinn-Justin (1996). We find
where denotes the operator inverse or Green’s function of in the sense that
Roughly speaking, this Gaussian integration can be understood by thinking of and as matrices with continuous indices , . The path integral (32) is then a continuum generalization of an ordinary Gaussian integral for finite-dimensional matrices, and can be performed by “completing the square”. We provide a rigorous derivation of (34) and (35) in Appendix A.
In order to obtain the explicit form of the Green’s function , we need to solve the integro-differential equation (35). In our case, the operator is proportional to [see (33a)] and thus has a “diagonal” structure, such that (35) turns into an ordinary linear differential equation. We can solve it by following standard methods Bender and Orszag (1999); Stakgold and Hols (2011), details are given in Appendix B. We obtain
With this expression for , (34) represents the exact path probability density for the dynamics of the Brownian particle (1), under the influence of active Ornstein-Uhlenbeck fluctuations (3). This is our first main result. We see that the active fluctuations with their colored noise character lead to correlations in the path weight via the memory kernel . They relate trajectory points at different times by an exponential weight factor similar to the active noise correlation function (4), but with a correlation time which is a factor of smaller. We emphasize again that we assumed independent initial conditions for the particle’s position and the active bath variables , [see also the discussion around Eq. (31)]. A different choice for the initial distribution, for instance the joint stationary state for and , would result in a modified , whose precise form in general depends on the specific implementation of the deterministic forces . The correlations in the path weight we measure via the memory kernel are thus influenced by our choice of the time instance at which we start observing the particle trajectory. In that sense, the system “remembers its past” even prior to the initial time point , because of the finite correlation time in the active fluctuations.
We finally remark that our general expression (34) with (36a) reduces to known results in the three limiting cases (passive particle), (white active noise), and (no thermal bath); details of the calculations can be found in Appendix C. Without active fluctuations (), we trivially recover the standard Onsager-Machlup expression (12) for passive Brownian motion. In the white noise limit for (), the equation of motion (1) involves two independent Gaussian white noise processes with vanishing means and variances and , respectively. Their sum is itself a white noise source with zero mean, but variance . Accordingly, as , we obtain from (34) an Onsager-Machlup path weight of the form (12), but with the diffusion coefficient being replaced by . In the third limiting case of vanishing thermal fluctuations, we are left with a pure colored noise path weight McKane et al. (1990); Hänggi (1989),
where denotes the steady-state distribution (31) of the colored noise.
iv.3 Fluctuation theorem
With the explicit form (34) of the path probability, we can now derive an exact fluctuation theorem which relates forward and backward paths, following exactly the same line of reasoning as described in Sec. III.2 for the case of a passive Brownian particle. We consider the ratio of probabilities for observing a specific trajectory and its time-reversed twin , created under a time-reversed protocol Note1 (). With the definition (13) of the time-reversed trajectory, we can express its probability in terms of the forward path. Using the property of the memory kernel [see Eq. (36a)], we then obtain the path probability ratio
As a stochastic integral, this is to be understood in the Stratonovich sense. Note that the procedure of time inversion does not involve the active fluctuations in any way, and thus does not require any assumptions on their properties under time reversal. In fact, the probability density for the trajectories of the Brownian particle is a result of integrating over all possible realizations of the active fluctuations, containing any pair of conceivable time-forward and time-backward twins with their natural weight of occurrence [see also Eq. (29)]. Hence, for the probability ratio of particle trajectories (34) the behavior of the active fluctuations under time inversion is irrelevant.
As described in Sec. III.2 a relation like (38a) based on path probability ratios entails an integral fluctuation theorem, if the entropy production in the system [see Eq. (18)] is taken into consideration. Explicitly, we find
By Jensen’s inequality we conclude
where equality is achieved if and only if the dynamics is symmetric under time reversal.
The setting considered here, however, is generally not symmetric under time-reversal because of our choice of particle position and active fluctuations being independent initially. The approach to a correlated (stationary) state is irreversible, such that we inevitably pick up transient contributions to , which are strictly positive on average, even if the external forces are time independent and conservative.
For the same reason, namely the build-up of correlations between the particle trajectory and the active fluctuation , the quantity is non-additive, i.e. in general, for any intermediate time . In order to establish such an additivity property, we would have to take into account the correlations between and in the initial distribution for the path integral (30) of the second part of the trajectory with that have been build up until the time point . However, it is not possible to specify this conditional distribution in the general situation considered here allowing for arbitrary forces and force protocols , see (2b) and the remark below (31).
The path probability ratio (38a), relating the time reversibility of trajectories to the extensive quantity , together with its corresponding integral fluctuation theorem (40), constitute our second main result. The interpretation of and its fluctuation theorems in physical terms is, however, not as straightforward as in the case of a passive Brownian particle, in which we could identify the logarithm of the path probability ratio as the heat dissipated into the thermal environment divided by the bath temperature Seifert (2005, 2012), see Eq. (16). Although we cannot make such a simple identification for the Brownian particle driven by active fluctuations, it is still possible to connect to physically meaningful quantities, as we will show in the following.
we can bring (38b) into the form
in obvious analogy to (III.2), but with the essential difference that the “force” at time depends not only on but rather on the full trajectory via the memory term . Similar “memory forces” have already been found to affect irreversibility by contributing to dissipation in Langevin systems with colored noise, which does not obey the fluctuation-dissipation relation Puglisi and Villamaina (2009).
From (IV.3) we can read off a production rate for ,
Even though this expression for the -production is analogous to the actual entropy productions in the environment as identified in (24) or (28) [see also (16)], and even contains a term , which quantifies dissipation due to the external force , its physical meaning beyond formally defined “memory forces” is unclear. In the following, we will argue that a valid interpretation is provided by the mutual information between the active fluctuations and the particle trajectory .
V Mutual information
The path-wise mutual information Parrondo et al. (2015) between the particle trajectory (starting at ) and a realization of the active fluctuations (with ) is given as
where , , and , include the initial densities , , and of the Brownian particle and the active fluctuations (see also the discussion in Sec. IV.2). This path-wise mutual information quantifies the reduction in uncertainty about the path when we know the realization and vice versa, and can therefore, loosely speaking, be seen as a measure of the correlations between and . Note that it can become negative if , while its average is always positive.
Likewise, the path-wise mutual information between the time-reversed trajectory from (13) and a suitably chosen time-reversed realization of the active fluctuations is
For their difference
|we thus find|
This expression represents the path-wise mutual information difference between a combined forward process and its backward twin. If is positive, the path-wise mutual information along the combined forward path is larger than for the time-reversed path [see (47a)], implying that correlations between the particle trajectory and the active fluctuation are stronger in the time-forward direction. Intuitively, we may thus say that they are more likely to occur together than their time-reversed twins, making the combined forward process the more “natural” one of the two processes in terms of path-wise mutual information.
To make the connection to irreversibility more rigorous, we rewrite in (47b) the path-wise mutual information difference explicitly as path probability ratios between time-forward and time-backward processes. We now see that , via the term , is directly related to the irreversibility measure for the actively driven Brownian particle and its change in system entropy . The additional log-ratio involves the probability of the forward path being generated by the specific realization of the active fluctuations for which we measure the mutual information content with in , and, likewise, the path probability of the time-reversed twin being generated by the time-reversed fluctuation. We can rewrite this term by splitting off the contributions from the initial densities,
We here keep the possibility that the initial particle position is conditioned on the initial state of the active fluctuations, , which is more general than the situation for which we calculated in Section IV.2 [see also the discussion around Eq. (31)]. According to (13), the time-reversed initial position is given by the final point of the forward path, . It therefore depends on the complete history of the active fluctuations, which is captured equivalently by or , for any reasonable choice of the time-reversed fluctuations in terms of the forward realization [see also Eq. (51) below]. With