How far from equilibrium is active matter?
Étienne Fodor, Cesare Nardini, Mike E. Cates, Julien Tailleur, Paolo Visco, Frédéric van Wijland
HHow far from equilibrium is active matter? ´Etienne Fodor, Cesare Nardini,
2, 3
Mike E. Cates,
2, 3
Julien Tailleur, Paolo Visco, and Fr´ed´eric van Wijland Universit´e Paris Diderot, Sorbonne Paris Cit´e, MSC, UMR 7057 CNRS, 75205 Paris, France SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom DAMTP, Centre for Mathematical Sciences, University of Cambridge,Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Dated: April 5, 2016)Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relationbetween injection and dissipation of energy at the microscopic scale. We consider such a system of interactingparticles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics stillsatisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that,for short persistent times, the entropy production rate vanishes. This endows such systems with an effectiveFluctuation-Dissipation theorem akin to that of thermal equilibrium systems. Last we show how interactingparticle systems with viscous drags and correlated noises can be seen as in equilibrium with a visco-elastic bathbut driven out of equilibrium by non-conservative forces, hence providing an energetic insight on the departureof active systems from equilibrium.
Active matter systems comprise large assemblies of indi-vidual units that dissipate energy, often stored in the environ-ment, to produce mechanical work [1]. From the collectivemotion of self-propelled particles [2, 3] to the existence of aliquid phase in the absence of attractive forces [4–6] many in-triguing phenomena have generated a continuously growinginterest for active matter over the past decades [1]. Since ac-tive systems break detailed balance at the microscopic scale,they cannot be described by equilibrium statistical mechan-ics. However, it is often difficult to pinpoint precisely thesignature of non-equilibrium physics in their emerging prop-erties. For instance, motility-induced phase separation, whichleads to the liquid-gas coexistence of repulsive self-propelledparticles, is not associated to the emergence of steady-statemass currents. A number of works have actually proposed thatits large scale physics can be captured by an equilibrium the-ory [4, 7–9], the limits of which are heavily debated [10–12].Even for systems where steady currents arise the connectionto equilibrium physics can sometimes be maintained, as forthe transition to collective motion which amounts, for simplesystems, to a liquid-gas phase transition [13, 14]. More andmore approaches to active matter thus partly rely on the intu-ition built for equilibrium systems [4, 7, 8, 12, 15–18].Building a thermodynamic approach for active matter thusfirst require understanding how active systems depart fromthermal equilibrium. Insight into this question was gainedby studying how the Fluctuation Dissipation Theorem (FDT)breaks down in active matter [19–22]. At short time and spacescales, the persistent motion of active particles typically pre-cludes the existence of effective temperatures while at largerscales FDTs can sometime be recovered. In living systems,the violation of FDT is used to characterize the forces gener-ated by intracellular active processes [23–28]. The informa-tion extracted from violations of the FDT is however ratherlimited and non-equilibrium statistical mechanics offers moreelaborate tools to quantify the departure from equilibrium. Inparticular, the entropy production rate quantifies the break-down of time-reversal symmetry, whence probing the irre- versibility of the dynamics [29]. Hard to compute, and evenharder to measure experimentally, it has been little studied inactive systems [30, 31], hence the need for ‘simple but notsimpler’ systems which offer a natural way to establish theo-retical frameworks.In this letter we study a model system of active matterwhich has recently attracted lots of interest [9, 32–34]. It com-prises overdamped ‘self-propelled’ particles whose dynamicsread ˙ r i = − µ ∇ i Φ + v i , (1)where i refers to the particle label, µ to their mobility and Φ is an interaction potential. The self-propulsion veloci-ties v i , rather than having fixed norms as in models of Ac-tive Brownian Particles [5] (ABPs) or Run-and-Tumble Par-ticles [35] (RTPs), are zero-mean persistent Gaussian noisesof correlations (cid:104) v iα ( t ) v jβ (0) (cid:105) = δ ij δ αβ Γ( t ) , with greek in-dices corresponding to spatial components. In the simplestof cases, the v i ’s are Ornstein-Uhlenbeck processes, solutionsof τ ˙ v i = − v i + √ D η i , with η i ’s zero-mean unit-varianceGaussian white noises, so that Γ( t ) = D e −| t | /τ /τ . Here D controls the amplitude of the noise and τ its persistence time.Since the temporal correlations of the noise are not matchedby similar correlations for the drag, this system does not sat-isfy the standard generalization of the Stokes-Einstein relationto systems with memory [36]. Consequently, the system isout of thermal equilibrium and its stationary measure is notthe Boltzmann weight P B ≡ Z − exp( − β Φ) . This model, towhich we refer in the following as Active-Ornstein-UhlenbeckParticles (AOUPs), shares the essential features of active sys-tems: it correctly reproduces the behavior of passive tracersin bacterial baths [32, 37], leads to the standard accumulationof active particles close to confining walls [33], and shows ashifted onset of the glass transition [34]. As for many otherself-propelled particle systems [21, 38], the limit of vanish-ing persistence time of AOUPs correspond to an equilibriumBrownian dynamics, since v i reduces to a Gaussian whitenoise. a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r In the following, we characterize how the AOUPs de-part from thermal equilibrium. First, we compute perturba-tively their steady-state at small but finite persistence time τ .Surprisingly, we show that the small τ limit yields a non-Boltzmann distribution with which the system still respectsdetailed-balance : The entropy production, which we com-pute, can indeed be shown to vanish at order τ . In this regime,to which we refer as effective equilibrium , we also show thatAOUPs satisfy a generalized FDT. Finally, we close this arti-cle by providing an energetic interpretation of the breakdownof detailed-balance for AOUPs.We consider N particles, propelled by Ornstein Uhlenbeckprocesses, interacting through a potential Φ . For illustrationpurposes, we use pairwise repulsive forces in 2D Φ = 12 (cid:88) i,j V ( r i − r j ) , V ( r ) = A exp (cid:34) − − ( r/σ ) (cid:35) , (2)for which Fig. 1 shows that AOUPs exhibit Motility-InducedPhase Separation (MIPS) [4, 6], extending this phenomenonbeyond the reported cases of RTPs [4, 22] and ABPs [5, 9, 39].Our analytical results, however, are valid beyond this exam-ple, and hold for general potentials and dimensions. Introduc-ing the velocities p i , the dynamics (1) become τ ˙ p i = − p i − (1 + τ p k · ∇ k ) ∇ i Φ − √ T η i , (3)where the mobility µ is set to one. Here and in what follows,repeated indices are implicitly summed over.We have introduced D ≡ µT in Eq. (3) to make the equi-librium limit τ = 0 transparent. Surprisingly, it suffices totake either τ = 0 in the r.h.s. or in the l.h.s. of Eq. (3) to mapAOUPs onto (different) equilibrium dynamics. Suppressingthe non-linear damping in the r.h.s. indeed maps Eq. (3) ontoan underdamped Kramers-Langevin equation. Conversely,neglecting τ ˙ p i in the l.h.s. corresponds to the Unified Col-ored Noise Approximation [33, 40] which has been shown tosatisfy detailed balance [33]. Here, we propose to determineperturbatively the steady-state of AOUPs in the small τ limit,retaining both contributions of τ in Eq. (3). Rescaling time as t = √ τ ˜ t and introducing rescaled velocities ˜ p i = √ τ p i , thesteady-state distribution satisfies L P ( { r i , ˜ p i } ) = 0 , with L = − ˜ p iα ∂∂r iα + 1 √ τ ∂∂ ˜ p iα (cid:20) ˜ p iα + τ ∂ Φ ∂r iα r jβ ˜ p jβ (cid:21) + ∂∂ ˜ p iα ∂ Φ ∂r iα + T √ τ ∂ ∂ ˜ p iα . (4)Using the ansatz P ∝ exp (cid:34) − Φ T − ˜ p i T + ∞ (cid:88) n =2 τ n/ ψ n ( { r i , ˜ p i } ) (cid:35) , (5)we obtain a set of equations at every order in √ τ which can FIG. 1. AOUPs interacting via the repulsive potential (2) exhibitMIPS in a 2d box of size L with periodic boundary conditions. Pa-rameters: A = 100 , σ = 2 , N = 10 000 , D = 1000 , τ = 20 be solved recursively to yield P ∝ e − Φ+˜ p i / T (cid:110) − τ T (cid:104) ( ∇ i Φ) + (˜ p i · ∇ i ) Φ − T ∇ i Φ (cid:105) + τ / T (˜ p i · ∇ i ) Φ − τ / p i · ∇ i ) ∇ j Φ + O (cid:0) τ (cid:1) (cid:111) . (6)The distribution of positions can then be deduced by integrat-ing (6) over velocities; we define an effective potential ˜Φ byanalogy with the Boltzmann measure: P ( { r i } ) ∝ exp( − β ˜Φ) ,with β ≡ T − and ˜Φ ≡ Φ + τ [( ∇ i Φ) / − T ∇ i Φ] + O (cid:0) τ (cid:1) . (7)In the limit of vanishing τ , one recovers the standardMaxwell-Boltzmann distribution. The joint distribution of po-sition and velocities (6) beyond this regime is our first impor-tant result. First, it shows how, for finite τ , positions and ve-locities are correlated, in agreement with the UCNA approx-imation [41] but at contrast to thermal equilibrium where theenergy can be separated between kinetic and potential parts.In particular, this leads to a modified equipartition theorem: (cid:10) ˜ p iα (cid:11) = T − τ (cid:10) ( ∇ i Φ) (cid:11) B + O ( τ ) , (8)where the average (cid:104)· · · (cid:105) B is taken with respect to the Boltz-mann weigth P B . Second, the effective potential ˜Φ predictsthat repulsive pairwise potentials lead to effective attractive in-teractions, consistently with other approximation schemes [9,33]. This explains why purely repulsive interactions can trig-ger MIPS. Note also how a pairwise potential leads to ef-fective three-body interactions through the term ( ∇ i Φ) . Atthis stage, our controlled expansion allows us to describe thestatic properties of AOUPs in terms of an effective Boltzmannweight (7). Interestingly, for the evolution operator (4), theasymmetry in ˜ p i of the steady-state measure (6) implies thatthe dynamics is out-of-equilibrium [42]. This asymmetry isnot captured by UCNA approximation [41] which cannot de-scribe the non-equilibrium properties of AOUPs.To better measure the degree of irreversibility of the dynam-ics, we derive its entropy production rate σ [29]. It is obtainedby comparing the probability weights associated with a giventrajectory and its time-reversed counterpart, respectively de-noted by P and P R : σ ≡ lim t →∞ t ln PP R . (9)To keep the scaling in τ explicit, we work for now withthe rescaled variables ˜ t and ˜ p i and use the fact that σ is intensive in time. Using standard path-integral formal-ism [43], the trajectory weight can be written as P [ { r i (˜ t ) } ] ∼ exp( −S [ { r i (˜ t ) } ]) with S = √ τ T (cid:90) ˜ t d u (cid:104) ˙˜ p i + ˜ p i √ τ + (1 + √ τ ˜ p k · ∇ k ) ∇ i Φ (cid:105) . (10)The time-reversed trajectories are then given by t R = − t , r R i ( t ) ≡ r i ( − t ) and p R i ( t ) ≡ − p i ( − t ) so that P R is simplyobtained by injecting these expressions in (10). The entropyproduction rate is thus given by σ ∼ δ S /t where δ S is the dif-ference between the forward and backward actions [44]. Allin all, the entropy production rate reads σ = − lim ˜ t →∞ √ τT ˜ t (cid:90) ˜ t d u ( ˙˜ p i ·∇ i )(˜ p j ·∇ j )Φ = √ τ T (cid:104) (˜ p i ·∇ i ) Φ (cid:105) , (11)where the last equality follows from integrating by parts andusing the equality between time and ensemble averages insteady-state [45]. Interestingly, the entropy production rateexactly vanishes when Φ is quadratic in the particle displace-ments, hence showing that AOUPs are in this case an equilib-rium model. Their steady-state is however not the Boltzmannmeasure P B , which explains the difficulty of defining a tem-perature in this case [21]. As a result, the anharmonicity ofthe potential acts as a control parameter for the nonequilib-rium nature of AOUPs.The entropy production rate can also be computed in thesmall τ limit, using the stationary distribution (6) to evaluatethe correlation function appearing in Eq. (11). Going back tothe initial variables, the entropy production rate is given by σ = T τ (cid:68) ( ∇ i ∇ j ∇ k Φ) (cid:69) B + O (cid:0) τ (cid:1) . (12)The first non-vanishing contribution to σ comes from the τ / correction in the steady-state measure (6). At order τ , wethus have a non-Boltzmann steady-state given by the first lineof (6), or equivalently by (7) in position space, with a van-ishing entropy production rate. In such a regime, the AOUPsare effectively a non-thermal equilibrium model which is thecentral result of this letter.Let us now discuss the practical consequences of this ef-fective equilibrium dynamics. Oscillatory shear experimentshave become an increasingly standard procedure to sample themicrorheology of active systems [23, 24, 46, 47]. In this con-text, the violation of the equilibrium FDT has proven a nat-ural measure of the distance to equilibrium [19, 20, 48]. Letus consider that an external operator perturbs the dynamicsby applying a small constant force f j on the particle j , hence modifying the potential Φ as Φ → Φ − f i · r i . We define theresponse function R as R iαjβ ( t, s ) ≡ δ (cid:104) r iα ( t ) (cid:105) δf jβ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) f =0 . (13)Following standard procedures [49], we can use the dynamicaction formalism and the fact that δ P = − δ S . P to rewrite theresponse as R iαjβ ( t, s ) = − (cid:28) r iα ( t ) δ S δf jβ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) f =0 (cid:29) . (14)The perturbed dynamics of the AOUPs is readily given by τ ˙ p i = − p i − (1 + τ p k · ∇ k ) ∇ i Φ+ f i + τ ˙ f i −√ T η i , (15)so that the dynamical action S becomes S = 14 T (cid:90) t d u (cid:20)(cid:16) τ dd u (cid:17) ( p i + ∇ i Φ − f i ) (cid:21) . (16)The response function is then given by R iαjβ ( t, s ) = (cid:18) − τ d d t (cid:19) (cid:20) − T dd t (cid:104) r iα ( t ) r jβ ( s ) (cid:105) (17) + 12 T ( (cid:104) r iα ( t ) ∇ jβ Φ | t = s (cid:105) − (cid:104) r iα ( s ) ∇ jβ Φ | t (cid:105) ) (cid:21) . In the effective equilibrium regime, the vanishing entropy pro-duction tells us that the dynamics is symmetric under timereversal so that the second line of Eq. (17) vanishes and theresponse function finally reads: R iαjβ ( t, s ) = − T dd t (cid:10) r iα ( t ) r jβ ( s )+ τ p iα ( t ) p jβ ( s ) (cid:11) . (18)We have thus derived a generalized FDT which holds in thesmall τ limit where the AOUPs are effectively in equilibrium,though not with respect to the Boltzmann measure P B . Thisexplains the atypical form of the correlation function enter-ing, which involves the position autocorrelation function, asin thermal equilibrium, along with the velocity autocorrelationfunction. Note that, as in equilibrium, this FDT is completelyindependent of the interaction potential Φ , so that it shouldbe measurable without knowledge of the intimate details ofparticle interactions.To test whether a finite τ regime exists where our gener-alized FDT can indeed be measured, we consider a perturba-tion Φ → Φ − f ε i x i where ε i is a random variable equalto ± with equal probability [48]. We measure the suscep-tibility χ ( t ) ≡ (cid:82) t d sR ixix ( t, s ) /N in simulations of AOUPsinteracting with the repulsive potential (2). Our modified FDTpredicts that N T χ ( t ) = (cid:104) [ x i (0) − x i ( t )] x i ( t ) (cid:105) + τ (cid:104) [ ˙ x i (0) − ˙ x i ( t )] ˙ x i ( t ) (cid:105) , (19)which is shown to be valid at small τ in Figure 2a. .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . χ ( t ) C eff ( t ) FIG. 2. Parametric plot between the susceptibility χ ( t ) and the cor-relation function C eff ( t ) = (cid:104) x i ( t )( x i ( t ) − x i (0)) + τ ˙ x i ( t )( ˙ x i ( t ) − ˙ x i (0)) (cid:105) for N AOUPs interacting via the potential (2). The particlesexperience a stiff harmonic potential when they try to exit a box oflinear size L . Parameters: L = 30 , N = 720 , τ = 0 . , A = 20 .Blue, red and cyan dots correspond to T = 2 , , . and the solidline correspond to the theoretical prediction (19). Note that an entropy production rate σ of order τ meansthat trajectories of length ∝ τ − lead to an overall entropyproduction of order one. Since we are working in the small- τ -but-finite- D limit, diffusive equilibration times (cid:96) /D remainof order one, which legitimates the claim of an effective equi-librium regime. Nevertheless, we expect our FDT to breakdown in the long time limit.To get more physical insight into our effective equilibriumregime and its breakdown as τ increases, let us now discussthe energetics of AOUPs. Active matter is traditionnaly re-garded as a non-equilibrium medium because injection anddissipation of energy are uncorrelated. Indeed the formerstems from the conversion of some form of stored energywhile the latter results from the friction with the surround-ing medium. Consequently, fluctuations and dissipations arenot constrained by any form of Stokes-Einstein relations. Fordriven Langevin processes, the non-equilibrium nature of thedynamics can be measured as a mean heat transfer betweenparticles and thermostat [50, 51]. This leads to a standard def-inition of dissipation J as the imbalance between the powerinjected by the thermal noise and the one dissipated via thedrag force. This definition furthermore provides an energeticinterpretation of the entropy production since J = T σ . Anaive generalization of this reasoning to AOUPs would leadto the definition of dissipation through J = µ − (cid:104) p i · ( p i − v i ) (cid:105) . (20)It is however straightforward to see that J = (cid:104) p i · ∇ i Φ (cid:105) = d (cid:104) Φ (cid:105) / d t which necessarily vanishes in steady-state.The breakdown of detailed balance for AOUPs is indeednot linked to a mean heat flux extracted from an equilibratedbath but from the apparent lack of generalized FDT betweendamping and fluctuations in (1). To get more insight on theentropy production rate σ , we remark that this dynamics isequivalent to K ∗ ˙ r i = ξ i − µK ∗ ∇ i Φ , (21)where K ( t ) = [1 − τ ( d / d t ) ] δ ( t ) , ∗ denotes time convo- lution, and we have introduced the noise term ξ i ≡ K ∗ v i .The lhs of (21) corresponds to the damping of a visco-elasticfluid with memory kernel K . The first term on the rhs is afluctuating force whose variance is: (cid:104) ξ iα ( t ) ξ jβ (0) (cid:105) = δ ij δ αβ K ( t ) , (22)since by definition ( K ∗ Γ)( t ) = δ ( t ) . The damping andfluctuating forces appearing in (21) thus satisfy a generalizedStokes-Einstein relation [36]. They correspond to the connec-tion of particles with an equilibrated visco-elastic bath, forwhich the standard definition of the dissipation applies: J = µ − (cid:104) p i · ( K ∗ p i − ξ i ) (cid:105) . (23)From there, simple algebra shows that J = T σ , which yieldsa physical interpretation to σ as the dissipation in an equili-brated bath for the dynamics (21).Interestingly, this shows that the breakdown of detailed bal-ance in AOUPs can be seen equivalently as resulting from alack of generalized Stokes-Einstein relation between damp-ing and fluctuations or from the fact that K ∗ ∇ i Φ is not aconservative force. In this second interpretation, the entropyproduction rate now has a standard energetic interpretation.The existence of an effective equilibrium regime for small τ is then due to the fact that K ∗ ∇ i Φ behaves as a con-servative force ∇ i ˜Φ in this limit. The dynamics (21) with K ∗ ∇ i Φ replaced by ∇ i ˜Φ can be regarded as a dynamicalequilibrium approximation of AOUPs; one indeed checks, forinstance, that (cid:104) ˜Φ (cid:105) − (cid:104) K ∗ Φ (cid:11) = O ( τ ) or that our generalizedFDT corresponds to perturbing this equilibrium dynamics as ˜Φ → ˜Φ − r i · ( K ∗ f i ) .In this article we have thus shown that, as their persistencetime increases, Active Ornstein-Uhlenbeck Particles do notimmediately leave the realm of equilibrium physics. At shortpersistent time, they behave as an equilibrated visco-elasticmedium with effective Boltzmann weight P ∝ exp( − β ˜Φ) which differs from the thermal equilibrium P B ∝ exp( − β Φ) .In this regime, the fact that repulsive forces lead to effectiveattractive interactions can directly be read in ˜Φ . Beyond thisstatic result, the existence of an effective equilibrium regimeenforces a generalized fluctuation dissipation theorem, akinto its thermal counterpart though different correlators are in-volved. The breakdown of this FDT for larger persistencetimes can be linked to a non-zero entropy production ratewhose expression we have computed analytically. Last, wehave shown how to extend the notion of dissipation to under-stand the breakdown of detailed balance in AOUPs.Most of the results presented in this letter have been de-rived for the particular choice of noise correlator Γ( t ) = D e −| t | /τ /τ . Many of our results, such as the discussion ondissipation, however extends to more general correlators. Fur-thermore, it has recently been shown that static approxima-tions derived for the steady-state of AOUPs capture very wellthe physics of ABPs [9]. It would thus be very interesting toknow whether our effective equilibrium regime also extendsto this system. More generally, our study suggests that whensystems are driven out of thermal equilibrium by the conver-sion of some form of stored energy, an effective equilibriumregime may remain when the drive is moderate. This wouldbe a first step towards a thermodynamics of Active Matter. Acknowledgements.
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