Stochastic Thermodynamics for Self-Propelled Particles

Stochastic thermodynamics for self-propelled particles

Grzegorz Szamel Department of Chemistry, Colorado State University, Fort Collins, CO 80523 (Dated: July 28, 2020) We propose a generalization of stochastic thermodynamics to systems of active particles, which move under the combined influence of stochastic internal self-propulsions (activity) and a heat bath. The main idea is to consider joint trajectories of particles’ positions and self-propulsions. It is then possible to exploit formal similarity of an active system and a system consisting of two subsystems interacting with different heat reservoirs and coupled by a non-symmetric interaction. The resulting thermodynamic description closely follows the standard stochastic thermodynamics. In particular, total entropy production, ∆stot, can be decomposed into housekeeping, ∆shk, and excess, ∆sex, parts. Both ∆stot and ∆shk satisfy fluctuation theorems. The average rate of the steady-state housekeeping entropy production can be related to the violation of the fluctuation-dissipation theorem via a Harada-Sasa relation. The excess entropy production enters into a Hatano-Sasa-like relation, which leads to a generalized Clausius inequality involving the change of the system’s entropy and the excess entropy production. Interestingly, although the evolution of particles’ self-propulsions is free and uncoupled from that of their positions, non-trivial steady-state correlations between these variables lead to the non-zero excess dissipation in the reservoir coupled to the self-propulsions.

Introduction. – Active matter systems [1–6] consist of systems onto equilibrium systems with non-conservative particles that move on their own accord by consuming en- interactions in which the effective temperature (which ergy from their environment. Since these systems are out characterizes the strength of the self-propulsions) plays of equilibrium, they cannot be described using standard the role of the temperature of the medium. It is not thermodynamics. Recently, there have been several at- clear how to generalize these studies to active systems tempts [7–16] to extend the formalism of stochastic ther- that are influenced by both self-propulsions and thermodynamics to describe active matter. mal noise [29]. In contrast, Speck [10, 11], Shankar and Stochastic thermodynamics [17] is the most successful Marchetti [12], and Dabelow et al. [13] followed the stan- framework for the description of an admittedly limited dard stochastic thermodynamics more closely and consid- class of non-equilibrium systems. In its standard form, ered active matter systems in contact with a heat bath. stochastic thermodynamics deals with systems which are Here we follow the spirit of Ref. [12] and consider joint in contact with a heat reservoir characterized by a con- trajectories of particles’ positions and self-propulsions stant temperature and are displaced out of equilibrium by [30], for an active system in contact with a heat bath external forces (in contrast to active systems that evolve [31]. This approach allows us to follow the standard under the influence of internal self-propulsions). It com- stochastic thermodynamics framework and to generalize bines stochastic energetics [18], which generalizes notions a number of its results to active matter systems. In par- of work and heat to the level of systems’ trajectories, with ticular, we divide the entropy production into the house- the notion of stochastic entropy [19]. It allows one to de- keeping part, which originates from the non-equilibrium rive a number of results characterizing out-of-equilibrium character of active matter, and the excess part, we de- states and processes. We shall mention here fluctuation rive a fluctuation theorem for the housekeeping entropy theorems for entropy production [20–22] which shed light production, and we relate the steady-state entropy pro- on the probability of rare, 2nd law violating fluctuations, duction to the experimentally measurable violation of a the Jarzynski relation [23] which expresses the free en- fluctuation-dissipation relation. We also obtain a gener- ergy difference between two equilibrium states in terms alized Clausius inequality which extends the 2nd law of of the average of the exponential of the work performed thermodynamics to active matter systems. in a non-equilibrium process between the same two states Model: Active Brownian particle in an external poten- and a generalization of the Jarzynski relation to transi- tial. – To illustrate our approach we use a minimal model tions between two stationary non-equilibrium states [24]. system consisting of a single active Brownian particle Several results obtained in the framework of stochastic (ABP) [32] under the influence of an external force, in thermodynamics were experimentally verified, see, e.g., two spatial dimensions. The equations of motion read Refs. [25–28]. ′ ′ Two types of a generalization of stochastic thermo- γtṙ = Fλ + γtv0e + ζ ⟨ζ(t)ζ(t )⟩ = 2IγtT δ(t − t ), dynamics to active matter systems have been proposed. (1) ′ ′ Fodor et al. [7] and Mandal et al. [8] considered model γrφ̇ = η ⟨η(t)η(t )⟩ = 2γrT δ(t − t ), athermal active matter systems and derived two differ- (2) ent expressions for the entropy production. The common feature of these two studies is that they map athermal where γt and γr are the translational and rotational fric- 2 tion coefficients [33], respectively, and e≡(cos(φ), sin(φ)) The rate of change of the systems entropy reads is the orientation vector. Next, Fλ is an external force, ∂ p(r, φ; t) ∂ p(r, φ; t) ∂ p(r, φ; t) which depends on a control parameter λ, and may in- ṡ(t) = − t − r · ṙ − φ φ̇ clude a non-conservative component, and v0 is the self- p(r, φ; t) p(r, φ; t) p(r, φ; t) propulsion speed. Finally, ζ and η are Gaussian white ∂tp(r, φ; t) jt(r, φ; t) noises representing thermal fluctuations. We use the sys- = − + · ṙ p(r, φ; t) (T/γt)p(r, φ; t) tem of units such that the Boltzmann constant kB = 1. F + γ v e j (r, φ; t) − t 0 · ṙ + r φ̇ . (7) The model system defined through Eqs. (1-2) is math- T (T/γr)p(r, φ; t) ematically equivalent to a system consisting of two subsystems, with each subsystem connected to its heat reser- The first term in the last line can be interpreted asthe voir, and with the subsystems coupled through a non- entropy production in the medium, symmetric, i.e. violating Newton’s 3rd law, interaction. −1 In the present case of a single ABP the reservoirs are ṡ m(t) = T (F + γtv0e) · ṙ. (8) at the same temperature. If we were to consider an active Ornstein-Uhlenbeck particle (AOUP) [7, 34, 35], the We will see in the remainder of this Rapid Communica- temperatures of the two reservoirs would be unrelated. tion that identification (8) leads to a consistent frame- Interestingly, the temperature of the reservoir coupled to work of active stochastic thermodynamics. Using Eqs. the self-propulsions, Eq. (2), is different from the so- (7-8) we can express the total stochastic entropy pro- called effective temperature of the self-propulsions [12], duction in terms of local velocities evaluated along the 2 trajectory, Ta = v0γtγr/(2T ). Systems consisting of subsystems coupled to different ∂ p(r, φ; t) 1 ṡ (t) = − t + (γ v · ṙ + γ v φ̇ ) , (9) heat reservoirs have been extensively studied, both the- tot p(r, φ; t) T t t r r oretically [36–42] and experimentally [28, 43, 44]. We note that in the present case, with a non-symmetric in- where vt(r, φ; t) = jt(r, φ; t)/p(r, φ; t) and vr(r, φ; t) = teraction between the two subsystems, even if the tem- jr(r, φ; t)/p(r, φ; t). peratures of the reservoirs are the same, the combined We note that averaging ṙ and φ̇ over all trajectories system is out of equilibrium. under the condition that the position and self-propulsion Stochastic entropy production. – To define a stochas- at time t are equal to r and φ gives the local translational tic entropy we follow Seifert [19]. First, we consider the and rotational velocity, respectively, Fokker-Planck equation for the joint probability density for the particle’s position and self-propulsion, which cor- ⟨ṙ|r, φ⟩ = vt(r, φ; t) (10) responds to Eqs. (1-2), ⟨φ̇ |r, φ⟩ = vr(r, φ; t). (11)

Here and in the following ⟨· · · ⟩ denotes averaging over ∂tp(r, φ; t) = −∂r · jt(r, φ; t) − ∂φjr(r, φ; t), (3) the trajectories. Combining Eqs. (9) and (10-11) allows us to calculate ̇ where current densities in the position and orientation the average total entropy production, S(t) = ⟨ṡ tot(t)⟩, spaces read ∫︂ γ v2(r, φ; t) + γ v2(r, φ; t) Ṡ (t) = drdφ t t r r p(r, φ; t). T −1 jt(r, φ; t) = γt (Fλ + γtv0e − T ∂r) p(r, φ; t), (4) (12)

jr(r, φ; t) = − (T/γr) ∂φp(r, φ; t). (5) Due to non-vanishing currents, the total entropy in- creases even in a steady state with a conservative force. We emphasize that due to the non-symmetric interaction Housekeeping entropy production and Harada-Sasa re- between r and φ sectors, in a steady state currents (4-5) lation – Oono and Paniconi [46] introduced the concept do not vanish [45], even if Fλ is conservative. of a housekeeping heat, i.e. the heat dissipated in a non- Next, we define the trajectory-level entropy for the sys- equilibrium steady state. Here we generalize this concept tem (i.e the particle), using the solution of the Fokker- into a housekeeping entropy production. Following the Planck equation (3) for a time-dependent control param- spirit of Hatano and Sasa [24] and of Speck and Seifert eter λ(t), evaluated along the stochastic trajectory of the [47] we define the housekeeping increase of the entropy, particle’s position and self-propulsion, ∆shk, as follows ∫︂ t −1 ′ ∆shk = T dt (γtvts · ṙ + γrvrsφ̇ ) . (13) s(t) = − ln p(r(t), φ(t); t). (6) 0 3

Here vts and vrs are steady-state local translational and of freedom, in spite of the fact that the rotational motion rotational velocities, respectively, is free and decoupled from the translational one. Further- more, as expected, in a steady state the total entropy vts(r, φ|λ(t)) = jts(r, φ|λ(t))/ps(r, φ|λ(t)) (14) production (9) and the housekeeping entropy production vrs(r, φ|λ(t)) = jrs(r, φ|λ(t))/ps(r, φ|λ(t)), (15) (13) coincide. evaluated along the stochastic trajectory. In Eqs. (14-15) jts(λ(t)), jrs(λ(t)) and ps(λ(t)) are the currents and the Using the regularization method described by Speck probability distribution in a steady state corresponding and Seifert [47] one can derive an equation of motion to a fixed instantaneous value of the control parameter, for the joint probability distribution of the particle’s po- λ(t). We note that the housekeeping entropy increase sition, orientation and housekeeping entropy increase, originates from both translational and rotational degrees ρ(r, φ, ∆shk; t),

−1 2 ∂tρ(r, φ, ∆shk; t) = −γt ∂r · [F + γtv0e − T ∂r] ρ(r, φ, ∆shk; t) + (T/γr)∂φρ(r, φ, ∆shk; t) (16) +T −1 [︁γ v2 + γ v2 ]︁ ∂2 ρ(r, φ, ∆s ; t) + [︁2∂ · v + 2∂ v − T −1 (︁γ v2 + γ v2 )︁]︁ ∂ ρ(r, φ, ∆s ; t) t ts r rs ∆shk hk r ts φ rs t ts r rs ∆shk hk

Equation of motion (16) allows us to show that the av- In an equilibrium Brownian system the time-dependent erage of exp(−∆shk) is time-independent, response function R(t) is related to velocity autocorrela- tion function, C(t) = ⟨v(t)v(0)⟩ through the fluctuation- d ∫︂ −∆shk dissipation relation, T R(t) = C(t). The velocity au- drdφd∆shke ρ(r, φ, ∆shk; t) = 0, (17) dt tocorrelation function has a part proportional to the δ function and a time-dependent part. For our system the which leads [47] to the integral fluctuation theorem for short-time limit of the latter part can be calculated as the housekeeping entropy production, ∫︂ ⟨exp(−∆shk)⟩ = 1. (18) + −2 C(0 ) = γt drdφ (F + γtv0e) (22)

We note that fluctuation theorem (18) is valid for any × (F + γtv0e − 2T ∂r) ps(r, φ) time dependence of the control parameter, including the time-independent order parameter, i.e the steady state. Combining Eqs.(21) and (22) and then using the equation The average steady state housekeeping entropy pro- for the steady-state probability distribution one can show duction, which can be calculated from Eq. (16) as that in a steady state −1 ⟨︁ 2 2 ⟩︁ ∂t ⟨∆shk⟩ = T γtv + γrv , (19) ts rs [︁ + + ]︁ Ṡ (t) = ∂t ⟨∆shk⟩ = (γt/T )Tr C(0 ) − T R(0 ) . (23) can be related to a violation of the fluctuation-response relation via an equality equivalent to the Harada-Sasa Equality (23) is the Harada-Sasa relation [48] written in relation [48]. To prove this equality we first consider a the time domain (note that the instantaneous part of perturbation of our system, initially in a steady state, T R(t) cancels the δ function part of C(t)). It expresses by a weak, constant in space, external force ϵfext(t). The the average steady state entropy production in terms of change of the average translational velocity of the particle the violation of the fluctuation-dissipation relation in- can be expressed in terms of response function R(t), volving the dynamics of the particle’s position. Inter- estingly, although the analogous fluctuation-dissipation ∫︂ t ′ ′ relation for the dynamics of the particle’s orientation is δ ⟨vt(t)⟩ = ϵ R(t − t ) · fext(t ). (20) −∞ not violated (in fact, rotational dynamics is free), the expressions for both the total and housekeeping entropy The response function (which geometrically is a second production include terms that originate from the rota- rank tensor) has an instantaneous part and a time de- tional motion. We note that a generalized Harada-Sasa layed part. The short-time limit of the time-delayed re- relation was also derived within a field-theoretical de- sponse can be calculated as scription of active matter [49]. ∫︂ + −2 Hatano-Sasa relation and a generalized Clausius in- R(0 ) = −γt drdφ (F + γtv0e) ∂rps(r, φ) (21) equality. – Using Eq. (8) we can write the entropy dissi- 4 pated into the medium as vanishing persistence time at constant effective temperature an AOUP becomes equivalent to a Brownian par- ∫︂ t −1 ′ ′ ′ ′ ticle. We note that in this limit, the joint steady-state ∆sm(t) = T dt (F(t ) + γtv0e(t )) · ṙ(t ) (24) 0 distribution factorizes a product of the distributions of ∫︂ t the position and of the self-propulsion, and, if the exter- ′ ̇ ′ = ∆shk(t) − ∆ϕ + dt λ(t )∂λϕ(r, φ; λ), nal force is conservative, the former distribution acquires 0 Gibbsian form. As a result, generalized Clausius inequal- where ϕ(t) = − ln ps(r(t), φ(t); λ(t)) and ∆ϕ = ϕ(t) − ity (27) becomes the standard Clausius inequality. ϕ(0) [50]. Eq. (24) allows us to express the excess entropy Integral fluctuation theorem for ∆stot.– To derive the production, ∆sex = ∆sm − ∆shk in terms of the last two above described framework we did not make any explicit terms in the second line. assumptions regarding the behavior of the self-propulsion Next, we note that Hatano and Sasa’s [24] derivation under the time-reversal symmetry [51]. We note that of their Eq. (11) can be easily adapted to our model the identification (8) can be derived from the assump- active system resulting in the following relation tion that the self-propulsion is even under the reversal of time [12]. If we do use this assumption, we can de- ⟨︃ [︃ ∫︂ t ]︃⟩︃ ′ ̇ ′ rive an integral fluctuation theorem for the total entropy exp − dt λ(t )∂λϕ(r, φ; λ) = 1. (25) 0 production following Seifert’s [19] derivation of the analogous theorem in standard stochastic thermodynamics. Combining Eqs. (24) and (25) we get a version of the We consider a time dependent control parameter, i.e. Hatano-Sasa relation, a protocol, λ(t′), and a reversed protocol λ˜(t′) ≡ λ(t−t′), and forward and reversed trajectories, x(t′) and x˜(t′) ≡ ⟨exp (−∆sex − ∆ϕ)⟩ = 1, (26) x(t − t′), where x ≡ [r, φ]. It can be shown that the and then using the Jensen inequality we obtain a gener- ratio of the probabilities of the forward and backward alized Clausius relation, trajectories, conditioned on their respective initial values, gives the stochastic entropy production in the medium, − ⟨∆sex⟩ ≥ ∆ ⟨ϕ⟩ . (27) Eq. (8),

The right-hand-side of Eq. (27) can be rewritten as the p[x(t′)|x(0)] ln = ∆s (t). (30) change of the average (Shannon) entropy of the system, p˜[x˜(t′)|x˜(0)] m

[︃ ∫︂ ]︃ Next, we combine the left-hand-side of Eq. (30) with ∆ ⟨ϕ⟩ = ∆ − drdφ p (r, φ; λ) ln p (r, φ; λ) . (28) s s normalized distributions of the initial values for the forward and reversed trajectories, p(x(0)) and p(x˜(0)), and It can be shown that for a quasi-static process Eq. (27) we note that the latter distribution is equal to the dis- becomes an equality. tribution of the final values of the forward trajectory, Recalling Eq.(24) and the definition of ∆shk we can p(x˜(0)) = p(x(t)). In this way we get write the stochastic excess entropy production as p[x(t′)|x(0)]p(x(0)) ∫︂ t ln = ∆stot(t), (31) ′ ′ ′ p˜[x˜(t′)|x˜(0)]p(x˜(0)) ∆sex(t) = − dt [∂rϕ(r, φ; λ) · ṙ(t ) + ∂φϕ(r, φ; λ)φ̇ (t )] . 0 (29)which leads to the integral fluctuation theorem fors ∆ tot,

∑︂ ′ −∆stot Here, the first term at the right-hand-side can be inter- ⟨exp(−∆stot)⟩ = p[x(t )|x(0)]p(x(0))e preted as the excess entropy production in the medium. x(t′),x(0) ∑︂ The second term, which has the form of the excess en- = p[x˜(t′)|x˜(0)]p(x˜(0)) = 1.(32) tropy production in the reservoir coupled to the self- x˜(t′),x˜(0) propulsions, originates from non-trivial correlations between the particle’s position and self-propulsion. Perspective. – We proposed a generalization of the Generalized Clausius inequality (27) extends the 2nd standard stochastic thermodynamics framework to active law of thermodynamics to active matter. For its deriva- matter. We showed that many previously derived results, tion it is essential to distinguish between the total and e.g. stochastic total and housekeeping entropies, their the excess entropy production, since for quasistatic pro- fluctuation theorems, Harada-Sasa and Hatano-Sasa re- cesses the former is not well defined [24]. lations, appear naturally in the new framework. This Physically, in the limit of very fast evolution of self- opens the way to experimental and computational stud- propulsions one expects that an active system should be ies of the thermodynamics of small active matter systems. governed by some kind of effective thermodynamic de- In our opinion, further work is needed in two direc- scription. In particular, one can show that in the limit of tions. First, the framework presented here is naturally 5 well adapted to describe systems in which both ther- [11] T. Speck, “Active Brownian particles driven by constant mal noise and self-propulsion are important. Indeed, affinity”, EPL, 123, 20007 (2018). the housekeeping entropy production quantifies the non- [12] S. Shankar and M.C. Marchetti, “Hidden entropy pro- equilibrium character of an active system, compared to duction and work fluctuations in an ideal active gas”, Phys. Rev. E 98, 020604(R) (2018) the corresponding equilibrium system without any activ- [13] L. Dabelow, S. Bo and R. Eichhorn, “Irreversibility in ity. On the other hand, athermal systems with rapidly Active Matter Systems: Fluctuation Theorem and Mu- varying self-propulsion (e.g. an AOUP in the limit of tual Information”, Phys. Rev. X 9, 021009 (2019). vanishing persistence time) are equivalent to equilibrium [14] L. Caprini, U.M.B. Marconi, A. Puglisi and A. Vulpiani, systems. It would be interesting to develop a criterion “The entropy production of Ornstein-Uhlenbeck active quantifying how close a given active matter system is to particles: a path integral method for correlations”, J. the equivalent equilibrium system. Second, it would be Stat. Mech. 053203 (2019). [15] For a special case of an active particle in a harmonic po- interesting to analyze the work done by both external tential see S. Chaki and R. Chakrabarti, “Entropy pro- and self-propulsion forces, and to investigate whether a duction and work fluctuation relations for a single parti- free energy-like quantity can be defined. We note that cle in active bath”, Physica A 511, 302 (2018). if an active system is close to the corresponding equi- [16] We should also mention papers by D. Chaudhuri and librium system without any activity, one would try to collaborators, who considered a different model of ac- follow Hatano and Sasa [24] and use the temperature of tive matter, in which activity originates from a nonlinear the medium to define the free energy. On the other hand, velocity-dependent force. See, e.g., C. Ganguly and D. Chaudhuri, “Stochastic thermodynamics of active Brow- for an active system with rapidly varying self-propulsion, nian particles”, Phys. Rev. E 88, 032102 (2013) and D. which is close to an effective equilibrium system, one Chaudhuri, “Active Brownian particles: Entropy produc- should probably use an effective temperature to define tion and fluctuation response”, Phys. Rev. E 90, 022131 the free energy. It is at present unclear how to interpo- (2014). late between these two cases. [17] U. Seifert, “Stochastic thermodynamics, fluctuation the- I thank Elijah Flenner for comments on the orems and molecular machines”, Rep. Prog. 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