Hidden Entropy Production and Work Fluctuations in an Ideal Active

Hidden entropy production and work fluctuations in an ideal active gas Suraj Shankara,b∗ and M. Cristina Marchettia,b† aPhysics Department and Syracuse Soft and Living Matter Program, Syracuse University, Syracuse, NY 13244, USA. bKavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. (Dated: September 5, 2018) Collections of self-propelled particles that move persistently by continuously consuming free energy are a paradigmatic example of active matter. In these systems, unlike Brownian “hot colloids”, the breakdown of detailed balance yields a continuous production of entropy at steady state, even for an ideal active gas. We quantify the irreversibility for a non-interacting active particle in two dimensions by treating both conjugated and time-reversed dynamics. By starting with underdamped dynamics, we identify a hidden rate of entropy production required to maintain persistence and prevent the rapidly relaxing momenta from thermalizing, even in the limit of very large friction. Additionally, comparing two popular models of self-propulsion with identical dissipation on average, we find that the fluctuations and large deviations in work done are markedly different, providing thermodynamic insight into the varying extents to which macroscopically similar active matter systems may depart from equilibrium. Introduction. What is irreversible in active matter? ∆s ˙ Overdamped Underdamped These systems are driven out of equilibrium by the con- h i 2 tinuous and sustained consumption of free energy at the v0 γDR TRS odd propulsion 0 microscopic scale [1–3], but quantifying such irreversibil- T (γ + DR) 2 2 2 ity is challenging. The persistent motion of E. coli per- v0 γ v0 γ TRS even propulsion forming run and tumble [4, 5] or of synthetic active col- T T (γ + DR) loids propelled by auto-phoresis [6, 7] are classic exam- ples of motion that breaks microscopic detailed balance TABLE I: A summary of the average entropy production rate by virtue of self-propulsion [8], yet is diffusive on large ∆s ˙ for various cases, applicable to both non-interacting h i 2 scales. The detailed balance violations due to persistence ABP and AOUP (using Ta = v0 γ/2DR). The difference often do not survive coarse-graining (even in the presence between the results obtained with underdamped and over- of weak external fields). This restores an effective equi- damped dynamics represents the hidden entropy production. librium picture on large scales, thereby allowing a dilute gas of self-propelled particles to be essentially treated as a gas of “hot colloids” [9] with an effective temperature unit, or as odd under TRS [19–21], corresponding to the [10–13]. In characterizing detailed balance violations on so-called conjugated dynamics [22]. Previous work has a coarse-grained scale, even manifestly non-equilibrium used both prescriptions, as well as techniques that leave phenomena, such as condensation in the absence of at- the sign under TRS unspecified [23–26], all with differing traction [14, 15], may then be understood by comparing and sometimes conflicting notions of dissipated heat and them to the “nearest” equilibrium like model at the same its relation to entropy production. Additionally, a single scale [16]. active particle has often been found to have vanishing entropy production [21, 23–26], seemingly suggesting equi- To quantify irreversibility of an ideal active gas, we librium behavior. We show that some of these issues can examine here the microscopic dynamics of an individual be clarified by using underdamped dynamics along with active particle and evaluate the entropy production rate thermal noise and taking the large friction limit only at arXiv:1804.03099v3 [cond-mat.soft] 2 Sep 2018 ∆s ˙ in two popular simple models of self-propelled par- the end, because for both TRS prescriptions the fast mo- ticlesh i in two dimensions (2d): Active Brownian particles menta degrees of freedom are responsible for a finite hid- (ABPs) where the propulsive force has fixed magnitude den entropy production [27–30], thereby demonstrating and its direction is randomized by rotational noise, and that a single active particle is thermodynamically irre- active Ornstein-Uhlenbeck particles (AOUPs) where self- versible. This is most evident for the case of conjugated propulsion is modeled as a Gaussian colored noise. En- dynamics where the hidden ∆s ˙ is the only contribution, tropy production provides a direct measure of the break- while it is subdominant ath largei friction for TRS even down of time-reversal symmetry (TRS) at steady state. propulsive forces (see Table I). If, in contrast, inertia is We show below that it crucially hinges on whether the neglected from the outset, a single active particle behaves propulsive force is treated as even under TRS [17, 18], as a passive colloid pulled by an external force (TRS even appropriate for active phoretic colloids, vibrated rods, propulsion) or as a colloid moving at the velocity of the or swimming bacteria, where the direction of motility solvent in a sheared fluid [21, 31] (propulsion here is the encodes a physical asymmetry of the microscopic active solvent velocity, which is TRS odd), with ∆s ˙ = 0. This h i 2 −A t result holds for both ABP and AOUP, thereby not dis- written as P [x(t) x(0)] e τ=0 δ(∂τ r p), where tinguishing the two models on the average. [x(τ)] is the Onsager| Machlup∝ functional [−41] (neglect- We then show that the non-equilibrium nature of ac- ingA unimportant additive constantsQ [55]), given by tive particles becomes evident in the fluctuations of ther- t modynamic quantities. By comparing the ABP and the 1 2 = dτ [∂τ p + γp fp] . (3) AOUP models, we find that even though they have the A 4Tγ ˆ0 − same long-time dynamics and dissipate identically on average, their work fluctuations are vastly different. We For non-interacting particles, the Hamiltonian of the system only involves the kinetic energy ( = p2/2) and the demonstrate in a precise fashion that the AOUP gas is H always further away from equilibrium compared to the first law takes the form (in Stratanovich convention) [42] ABP gas, for the same motility and persistence. Specif- d = p dp = d¯w d¯q , (4) ically, the variance of the cumulative work done to pro- H · − pel the particles, corresponding to the Fano factor, is where d¯w is the propulsive work done and d¯q is the strongly enhanced by activity over its linear response heat dissipated into the reservoir. The sign convention value for the AOUP, but not for the ABP. Our work used is that both heat dissipated into the bath and work can be extended to thermodynamic quantities of inter- done by the environment on the system are taken to acting active systems along with their fluctuations that be positive. Requiring the Clausius relation, we equate are beginning to be accessible experimentally [32–36]. d¯q(t)= T ∆s(t), which as we will see below is consistent The models. We consider an underdamped active par- with Sekimoto’s [42] definition of heat only for the TRS ticle and set the mass and Boltzmann factor to unity. The even case. It is clear from Eq. 2 that, as discussed in the r p particle velocity ˙ = obeys a Langevin equation, Introduction, entropy production depends on whether the propulsion is treated as a force (hence TRS even) p˙ = γp + fp + 2Tγ ξ(t) , (1) − or as a velocity (hence TRS odd). We discuss both cases where γ is the friction, T thep temperature of the en- here, although the TRS even prescription is more directly vironment providing a heat bath, and ξ(t) a delta- relevant to physical realizations. Also, the calculation of correlated Gaussian white noise. For ABP the propulsive the mean entropy production is outlined here for ABP. force fp = γv0eˆ has fixed magnitude, with v0 the self- The result turns out to be the same for AOUP. propulsion speed, and direction randomized by rotational TRS odd propulsion. The prescription of conjugated noise, eˆ(t) eˆ(0) = e−|t|DR . For AOUP the propul- dynamics (r†(τ) = r(t τ), p†(τ) = p(t τ) and − − − sive forceh is· an Ornstein-Uhlenbecki process, D−1f˙ = f †(τ) = f (t τ) on a time interval τ [0,t], see R p p − p − ∈ fp + √2γTaη(t)[η(t) white noise and Ta an active tem- Fig. 1(a)) most clearly illustrates the importance of re- − −|t|DR perature], so that fp(t) fp(0) =2γTaDRe . Both taining the fast momenta degrees of freedom and the types of particles areh diffusive· i at long times, with diffu- associated hidden entropy production. Considering from 2 sivity D = (T + Ta)/γ, where for ABP, Ta = v0 γ/(2DR). the outset overdamped dynamics and treating motility as It has been shown that the large-scale phenomenology of a TRS odd velocity seems to lead identically to ∆s ˙ = 0, the two models is similar even in the presence of strong in the absence of interactions [21, 23], wrongly suggesting interactions [37, 38] where they both exhibit motility- that the system is in equilibrium [56]. Working instead induced phase separation. Yet, as we shall show below, with the underdamped equations, we obtain the entropy their thermodynamic fluctuations are markedly different production rate to be ∆s ˙ = p˙ (p v0eˆ)/T . Averaging − · − even at the single particle level. over noise, in steady state, we get Mean entropy production. Irreversibility can be quan- v2γD v2 D tified through dissipation and entropy production, which ∆s ˙ = 0 R = 0 D + R . (5) h i T (γ + D ) T R O γ can be calculated within the framework of stochastic R thermodynamics [22]. At steady state, the total en- This demonstrates a hidden entropy production in ac- tropy production of the system equals the entropy flux to the environment (also called entropy production of the tive matter arising from the entropic cost to maintain a finite persistence and evade thermalization of the fast medium [39]).

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