Efficiency and large deviations in time-asymmetric stochastic heat engines

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1 Fast Track Communication Efficiency and large deviations in time-asymmetric stocastic eat engines Todd R Gingric 1, Grant M Rotskoff 2, Suriyanarayanan Vaikuntanatan 3 and Pillip L Geissler 1,4 1 Department of Cemistry, University of California, Berkeley, CA 94720, USA 2 Biopysics Graduate Group, University of California, Berkeley, CA 94720, USA 3 James Franck Institute and Department of Cemistry, University of Cicago, Cicago, IL 60637, USA 4 Materials Sciences Division and Cemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA gingric@berkeley.edu, rotskoff@berkeley.edu, svaikunt@ucicago.edu and geissler@berkeley.edu Received 10 September 2014, revised 25 September 2014 Accepted for publication 8 October 2014 Publised 24 October 2014 New Journal of Pysics 16 (2014) doi: / /16/10/ Abstract In a stocastic eat engine driven by a cyclic non-equilibrium protocol, fluctuations in work and eat give rise to a fluctuating efficiency. Using computer simulations and tools from large deviation teory, we ave examined tese fluctuations in detail for a model two-state engine. We find in general tat te form of efficiency probability distributions is similar to tose described by Verley et al (2014 Nat. Commun ), in particular featuring a local minimum in te long-time limit. In contrast to te time-symmetric engine protocols studied previously, owever, tis minimum need not occur at te value caracteristic of a reversible Carnot engine. Furtermore, wile te local minimum may reside at te global minimum of a large deviation rate function, it does not generally correspond to te least likely efficiency measured over finite time. We introduce a general approximation for te finite-time efficiency distribution, P ( η), based on large deviation statistics of work and eat, tat remains very accurate even wen P ( η) deviates significantly from its large deviation form. Content from tis work may be used under te terms of te Creative Commons Attribution 3.0 licence. Any furter distribution of tis work must maintain attribution to te autor(s) and te title of te work, journal citation and DOI. New Journal of Pysics 16 (2014) /14/ $ IOP Publising Ltd and Deutsce Pysikalisce Gesellscaft

2 Keywords: non-equilibrium fluctuations in small systems, molecular motors, large deviations in non-equilibrium systems 1. Introduction As engineering capabilities reac molecular scales, design principles must account for te large fluctuations inerent in te beavior of nanoscale macines [1 7]. Tese macines, ubiquitous in biology [8 11] and increasingly relevant syntetically [12, 13], operate stocastically. As a consequence, familiar termodynamic quantities, suc as te eat absorbed from a ot bat ( ), work extracted from te system ( ), and efficiency ( η ), do not realize a single value. Instead, eac quantity varies from one measurement to te next according to a probability distribution and must be understood using stocastic termodynamics [14 18]. Fluctuations away from te mean beavior become insignificant in te long-time limit, but many molecular macines operate intermittently, performing teir function over a sort time. In order to analyze te termodynamic efficiency of nanoscale macines operating over a finite time, an understanding of te entire efficiency distribution is crucial. Here, we aim to explore generic features of tese efficiency distributions in non-equilibrium engines. A system can be out of equilibrium in a time-independent manner if it is eld in contact wit multiple reservoirs maintained at different termodynamic conditions (e.g. two unequal temperature bats can induce a temperature gradient across a system). Verley et al constructed a model of one suc system, a potoelectric cell, for wic te temperature of Eart serves as one bat and te temperature of te Sun as anoter [18]. Te non-equilibrium protocol driving tese systems is time independent and terefore time-reversal symmetric. Verley et al employed large deviation teory to argue tat in a long-time limit te efficiency distribution η would attain te form P () η e t obs J( ), were t obs denotes te observation time under tese conditions. Surprisingly, te large deviation rate function, J ( η), as a global minimum at te reversible efficiency, at wic sufficiently long trajectories produce no entropy [18]. In te case of a eat engine, te reversible efficiency is te Carnot efficiency, η C = 1 Tc T, te maximum tat can be attained by a eat engine on average [19, 20]. Systems can also be maintained out of equilibrium if tey are driven in a time-dependent manner. Standard engine protocols, including te Carnot and Stirling cycles, feature suc timedependent driving and generally lack time-reversal symmetry in bot macroscopic and microscopic realizations [12, 19]. It is terefore important to understand efficiency distributions under muc more generic cyclic driving protocols tan tose considered in [18]. Here we analyze engine performance fluctuations for tis general case, exemplified by a model two-level eat engine. Study of a similar model appeared sortly after te submission of our manuscript [21]. Specifically, we examine te statistics and large deviation scaling for te work, eat, and termodynamic efficiency for te dynamics sketced in figure 1. By computing a large deviation rate function for joint observations of and, we in turn calculate te long-time beavior of te probability distribution for η [22]. In particular, we determine te rate function for η, wic resembles rate functions analyzed in [18]. However, we sow tat time-asymmetric driving sifts te location of te minimum away from te Carnot efficiency. As suc, te primary result of Verley et al [18] does not generalize to encompass common engine protocols. 2

3 Figure 1. A scematic of te model stocastic eat engine. A single particle can occupy one of two energy levels wit termal transitions between te states. Te energy of te rigt state, E R, and te temperature, T or T c, are instantaneously switced in a cycle among four stages. Tese switces occur in multiples of τ 4, wereτ is te period of te cycle. Furtermore, we note an important caveat pertaining to te relationsip between te probability distribution of η and its asymptotic representation as tobsj() η e. As a consequence te efficiency distribution sampled over long but finite times may not reveal te minimum predicted by an analysis of te efficiency rate function. We obtain a general form of tese finite-time distributions, wic we expect to be significantly more relevant to te understanding of efficiency statistics in experiments. 2. A two-state model engine We begin by constructing a two-state model of a stocastic engine. Te temperature and energy levels are varied cyclically troug four consecutive stages (see figure 1), Stage 1: (1) T = Tc, EL = 0, E = 0 Stage 2: (2) T = Tc, EL = 0, ER = ΔE Stage 3: (3) T = T, EL = 0, ER = 2ΔE Stage 4: (4) T = T, EL = 0, ER = ΔE, wit E L and E R te energies of te left and rigt states. T and T c are te ig and low temperatures respectively acieved by alternately coupling te two-state system to ot or cold bats. Te superscript on temperatures and energies acts as an index for te stage of te cycle. We carry out eac cycle in time τ, wit eac stage lasting for τ 4 units of time. During eac stage te particle can op between te left and rigt states wit Arrenius rates given by a tunable barrier eigt, B. Te continuous time rate matrix for te it stage of te cycle is terefore = β B β() i i ( B E e e R () ), β B β() i i e e ( B ER ) (1) R (1) wit i β ( kt B () ) 1 and te Boltzmann constant, k B, set equal to unity trougout. 3

4 Work is extracted from te system wen te rigt energy level is occupied wile being instantaneously lowered. Eac transition between te energy levels requires eat absorbed from te reservoir equal to te energy difference between te levels. We adopt te conventions tat positive eat flow corresponds to eat flowing into te system and tat positive work is performed on te system [16, 23]. Te simplicity of our four stage, two-state model lends itself to formal analysis of tese fluctuating quantities as well as exaustive computational study. 3. Simulations Wile te particle dynamics of our model occurs in continuous time, te eat and work witin eac stage of a cycle are solely functions of te systemʼs states at te beginning and end of tat stage. We advance time in units of τ 4 by drawing te state at te end of eac stage in proportion to its exact probability, computed using te matrix exponential = e τ 4. We collected statistics on te work extracted, eat absorbed, and efficiency of steady state stocastic trajectories evolved over many repeated engine cycles, focusing on a set of parameters (Δ E = 2.375, Tc = 2, T = 14, B = 0.05, and τ = 10) for wic work is extracted on average. Direct sampling of trajectories numerically illustrates tat te eat and work distributions tend to a large deviation form as t obs [22, 24, 25]. Te probability of observing a total work and total eat absorbed from te ot bat in a trajectory of lengt t obs can tus be written as ( ) tobsi P, e ( tobs, tobs), (2) were denotes an equality in te asymptotic limit and I( tobs, tobs) is te large deviation rate function. Te large deviation scaling is robust even for a modest t obs of only 50 engine cycles. Te eat and work statistics of trajectories evolved for at least 50 engine cycles can terefore be well-described by a large deviation rate function. Te statistics of efficiency are also sown in figure 2(c) along wit te efficiency rate function, J ( η). Te sampled distributions for te reported finite-time measurements sow no η minimum. As expected, te efficiency distribution tends toward P () η e t obs J( ) at long times, but in practice even observation over 500 cycles is not sufficient for te large deviation rate function to predict te efficiency distribution. In marked contrast to te statistics of te timeadditive quantities and, te large deviation form is not predictive of te sampled efficiency distributions for te reported values of t obs. 4. Large deviation rate functions for ; ; and η Te large deviation function, I( tobs, tobs ), can be calculated using standard metods of large deviation teory [22, 24]. We introduce two fields, λ and λ and construct a scaled cumulant generating function for and, 4

5 Figure 2. (a) Work, (b) eat, and (c) efficiency sampling from trajectories of te four stage, two-state engine wit Δ E = 2.375, Tc = 2, T = 14, B = 0.05, and period τ = 10. Te large deviation rate functions are sown as a solid black line. Histograms of sampled values are sown as large circles. Te small circles in (c) plot te result of integrating out ΔS in equation (16). ( ) λ λ ψ λ λ = tobs tobs, lim 1 log e. (3) Applying a saddle point approximation, wic is exact in te long-time limit, reveals tat I( tobs, tobs) can be obtained from ψ ( λ, λ) by a Legendre transform 5. In te longtime limit te scaled cumulant generating function can be found as a maximum eigenvalue of te appropriate tilted operator, wic for our model must involve a product of tilted operators stemming from eac stage of te engine [22]. Recall tat work is performed instantaneously between stages and only if te particle is in te rigt state, so we define te tilted operator 1 0 ( λ ) =, (4) λ ΔE 0 e wose derivatives wit respect to λ provide statistical information about te work. ΔE denotes te cange in E R between stages of te protocol. Te eat absorbed from te ot bat differs from te entropy production only by a factor of β, so te tilted rate matrix for eat absorbed during tose stages is analogous to te entropy production tilted operator of Lebowitz and Spon [24]. Its matrix elements are given by () jk = ( λ ) i, j k, = λ λ (5) jk () ( kj i T () ) ( jk i T ) 1, j k. Te tilted rate matrix propagates in continuous time, but it is convenient to also define a tilted operator wic accounts for te complete stage, a time of τ 4: 5 Tis logic assumes an absence of dynamic pase transitions. Oterwise te Legendre transform returns a convex ull of te rate function. 5

6 Figure 3. (a) Te large deviation rate function, I, for te joint observations of and. Parameters for te engine protocol are te same as in figure 2. Te black lines ave slopes η* and η C, corresponding to te set of (, ) observations consistent wit te respective values of efficiency. (b) Te efficiency rate function evaluated at η is given by te maximum of I attained along a line troug te origin wit slope -η. Red and blue lines sow te rate function for te forward and reverse protocols, respectively. Note tat te minimal efficiency, η* corresponds to te slope of te tangent line troug te origin in (a). (c) Plot of te most likely value of entropy production in te long-time limit conditioned upon a given value of efficiency. Only in time-reversal-symmetric protocols will tis function be strictly non-negative. τ i λ = λ ( ) ( ) () exp. (6) 4 We now construct a tilted operator, cycle, for te entire cycle by forming a matrix product of te tilted operators for eac stage (and eac transition between stages) ( ) ( ) ( ) ( ) ( ) ( ) ( ) cycle (4) (4) (3) (3) (2) (2) (1) (1) λ λ = λ λ λ λ λ λ,. (7) Because we do not record eat absorbed from te cold bat, te cold stages involve time propagators rater tan tilted operators for. cycle raised to te N t power generates te statistics of work and eat after N cycles. In te limit of a large number of cycles tis matrix operator is dominated by te largest eigenvalue of cycle, wic we denote as ν ( λ, λ). Te scaled cumulant generating function can ten be expressed as (, λ) = ν( λ λ) ψ λ 1 log,. (8) τ A numerical Legendre transform gives te desired large deviation rate function, wic is sown in figure 3(a). Tese long-time joint statistics of and determine te statistics of efficiency. We employ te contraction principle to obtain te efficiency rate function [22]. Tis tecnique amounts to making a saddle-point approximation along eac line of constant efficiency on I, 6

7 J () η = max I t, η t. (9) ( obs obs) Te result of te contraction is sown in figure 3(b). As in te prior work of Verley et al [18], te minimum value obtained by eac curve corresponds to a class of trajectories for wic te average entropy production is zero, demonstrated in figure 3(c). Our results demonstrate, owever, tat tere can be two values of efficiency for wic te average entropy production vanises. One of tese values necessarily corresponds to η C. For our engine it is te oter efficiency value, η*, for wic J ( η) is minimal. 5. Te location of te minimum of J (η) Te key distinguising feature of our engine compared wit tose discussed in [18] is tat our protocol lacks time-reversal symmetry. To empasize te importance of time-reversal symmetry, we also plot te efficiency distribution of a time-reversed protocol, sown in red in figure 3. Te minimum in te rate function for te time-reversed protocol, denoted η *, is distinct from bot η* and η C. Indeed, we can relate η* to η * via te fluctuation teorem [16, 26], tereby illustrating tat η* = ηc if and only if te protocol is time-reversal symmetric. Entropy production is defined on a single trajectory level as ΔS = log Pxt [ () Λ()] t Pxt () Λ () t, (10) were x(t) is a trajectory subject to a time-dependent non-equilibrium protocol Λ()and t a tilde denotes te time reversal of a function [27]. Assuming tat te dynamics of te system is microscopically reversible, we can interpret te entropy production in terms of te eat absorbed from te termal reservoirs. In particular ΔS = β β + log c c pss ( x(0) Λ(0)) p x (0) Λ (0), (11) ss ( ) wit p ss denoting te steady-state probability [16]. Te eat is extensive in time; te contribution involving p ss and its time-reversed counterpart is subextensive for te two-state system and can terefore be neglected in te long-time limit. By te first law, c = ΔE W.Herewecan neglect ΔE wic also grows sub-extensively wit te lengt of te trajectory. Tus in te longtime limit ΔS = βc ηc η. It follows from equation (10) tat ( ) ( ) ( ) ( ) I t, t = I t, t + β η + t, (12) obs obs obs obs c C obs were Ĩ is te large deviation rate function for te time-reversed protocol. Geometrically, te set of work and eat values yielding efficiency η fall on a line in te ( tobs, tobs) plane passing troug te origin wit slope η. Lines corresponding to efficiencies η* and η C are drawn in figure 3. In te long-time limit te probability of observing a given efficiency is dominated by te most likely point on tis line. Tus J ( η) is extracted from te maximum of I( tobs, tobs) along te line wit slope η, as expressed in equation (9). Te minima of te efficiency rate functions terefore correspond to te lines tangent to te level curve of I at (, ) = (0, 0), requiring 7

8 I I ( ) ( ) (0,0) (0,0) η I I ( ) (0,0) ( ) (0,0) I ΔS= 0 and ΔS = ΔS η 0 η* = * =. (13) ( ) 0 Equation (13) implies = η= η* *, were te average is taken over te subensemble of trajectories wit efficiency η*. Differentiation of equation (12) wit respect to and implies, after some simplifying algebra, Note tat te derivative η* η = ( η* η * )t T C obs c Ĩ (0,0) I (0,0). (14) is non-zero because (i) I and Ĩ are convex, (ii) teir maxima locate te corresponding mean values of and, and (iii) a useful engine sould extract nonzero work on average. We terefore see tat η* = ηc precisely wen η* = η *. Te minimum of J ( η) occurs at te Carnot efficiency if te protocol is time-reversal symmetric since te symmetry enforces η* = η *. In te more generic case of time-asymmetric engines, owever, distinct values of η* and η * imply tat neiter of te minima occur at η C. 6. Finite time efficiency distributions As demonstrated by numerical simulation, tere is a significant regime of observation times for wic te efficiency distribution is not well-described by te large deviation form and no local minimum is evident. Te minimum in te efficiency rate function terefore may not be apparent for actual experimental measurement of efficiency statistics. Neverteless, we may leverage te large deviation form for work and eat, equation (2), to construct an approximation for P ( η) tat is muc more faitful to finite-time statistics. Consider te coordinate transformation from (, ) to ( η, ΔS), were η =, ΔS = βc( + ηc). (15) Te Jacobian for tis transformation contributes negligibly to te distribution P( η, ΔS) in te long-time limit since it does not vary exponentially wit t obs. At long but finite times, owever, it can strongly sape statistics of η and ΔS. Retaining tis Jacobian, wile exploiting te large deviation form of P (, ), we estimate P(, η ΔS) Tc ΔS S tobsi ( η η ) t C 2 C ηt ΔS c T Δ η η ( η η ) e, obs ( C ) tobs. (16) c 2 Equation (16) is a very general result for te joint distribution of efficiency and entropy production. Its sole underlying assumption is tat work and eat fluctuations are well-described by a large deviation form. For a model dynamics in wic and obey a large deviation principle at all times, suc as tat studied in [28], equation (16) is tus exact. Obtaining an efficiency distribution P ( η) from tis result requires marginalizing over ΔS. In work tat appeared after te submission of tis manuscript, Polettini et al in effect integrated 8

9 equation (16) over ΔS analytically for a linear response model wose work and eat statistics ave a large deviation form by construction [28]. More generally, te assumption leading to equation (16) will be valid only wen t obs is sufficiently large. In tis case a saddle point approximation is likely to be a well-justified and practical alternative to exact marginalization. For our two-state system tis alternative approac produces very accurate predictions for P ( η) at finite t obs (small dotted lines in figure 2(c)). Te form of P ( η) may terefore be reliably extracted from te large deviation form for work and eat fluctuations. Te prefactor in equation (16) attenuates te probability of observing very large positive and negative values of η wen t obs is finite. As a result, for any finite-time observation η* will not strictly be te least likely; η is a continuous variable wit infinite support, so no finite efficiency can be te least probable. Neverteless, at very long times, trajectories wit efficiency η* will be increasingly rare. 7. Discussion Efficiency is meant to provide an assessment of ow muc work can be extracted from a macine relative to te expense of operating it. For macroscopic eat engines, fluctuations in work and eat are vanisingly small compared to teir means suc tat work, eat, and efficiency can be reasonably replaced by teir average values. In contrast, fluctuations cannot be neglected for a microscopic engine. An understanding of te finite-time and long-time statistics of te efficiency provides a lens troug wic to assess te design of a microscopic engine. We ave extended te analysis of efficiency fluctuations [18] to include te common case of time-asymmetric driving, illustrating tat te Carnot efficiency does not minimize te efficiency rate function for most molecular macines. Under bot time-symmetric and timeasymmetric driving, long trajectories tat realize te Carnot efficiency necessarily ave zero entropy production by equation (15). However, in te case of time-asymmetric driving, te minimum in te efficiency rate function corresponds to a value η*, distinct from η C. Te subensemble of trajectories wit efficiency η* as a vanising entropy production on average as sown in figure 3. For bot η* and η C, ΔS η = 0, complicating te notion of a reversible efficiency. For sufficiently long observation times, a local minimum is evident in te efficiency distribution. A minimum in te efficiency large deviation rate function may provide teoretical insigt, but will be irrelevant to many experiments. We ave demonstrated tat finite-time efficiency distributions can be accurately captured using a large deviation form for joint fluctuations of eat and work. We anticipate tat te form of tese distributions will be a more relevant consideration for te design of microscopic engines tan te identification of η*. Acknowledgements We acknowledge support from te Fannie and Jon Hertz Foundation (T.R.G.) and te National Science Foundation Graduate Researc Fellowsip (G.M.R.). Tis work was supported in part by te Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences, and Engineering Division, of te U.S. Department of Energy under contract No. DE AC02-9

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