Work and power fluctuations in a critical heat engine
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Work and power fluctuations in a critical heat engine
Viktor Holubec
1, 2, ∗ and Artem Ryabov Institut f¨ur Theoretische Physik, Universit¨at Leipzig, Postfach 100 920, D-04009 Leipzig, Germany Charles University, Faculty of Mathematics and Physics,Department of Macromolecular Physics, V Holeˇsoviˇck´ach 2, CZ-180 00 Praha, Czech Republic (Dated: July 31, 2018)We investigate fluctuations of output work for a class of Stirling heat engines with working fluidcomposed of interacting units and compare these fluctuations to an average work output. In par-ticular, we focus on engine performance close to a critical point where Carnot’s efficiency may beattained at a finite power as reported in [M. Campisi and R. Fazio, Nat. Commun. , 11895 (2016)].We show that the variance of work output per cycle scales with the same critical exponent as the heatcapacity of the working fluid. As a consequence, the relative work fluctuation diverges unless theoutput work obeys a rather strict scaling condition, which would be very hard to fulfill in practice.Even under this condition, the fluctuations of work and power do not vanish in the infinite systemsize limit. Large fluctuations of output work thus constitute inseparable and dominant element inperformance of the macroscopic heat engines close to a critical point. PACS numbers: 05.20.-y, 05.70.Ln, 07.20.Pe
I. INTRODUCTION
Heat engines and their performance trouble minds ofengineers and physicists for more than two centuries [1].One question is particularly exciting: Can efficiency of apractical heat engine working between temperatures T h and T c , T h > T c , reach the upper bound η C = 1 − T c /T h found by Sadi Carnot [2–8]? Recent surprising theo-retical breakthroughs suggest that this may be possi-ble. The long lasting conviction that Carnot’s bound canbe reached in a quasi-static (infinitely slow) limit only,and thus also at a vanishing power, has been proven un-founded. Two ways how to reach η C at a finite powerhas been suggested. First, the bound may be reachedin the limit of infinitely fast dynamics of the system [9–11]. Second, the bound may be saturated in heat enginesbased on systems working near critical point in case theheat capacity of the working medium scales in a suitablemanner with the system size measured by the number ofinteracting particles, N [12, 13].In this work, we will take a closer look on the secondseries of results. Near the second order phase transition,space correlation functions of the system diverge and thusthe system experiences large fluctuations. These fluctua-tions allow the system working close to the critical pointto reach Carnot’s bound at a finite power in the large N limit. It is natural to ask whether also work fluctuationsshow critical behavior and, if so, whether these fluctua-tions do not dominate the average work produced by theengine.In what follows, we derive concise formulas for workfluctuation, σ w , and for relative work fluctuation (rela-tive standard deviation), f w , for the class of Stirling heatengines introduced in the inspiring study [12] and dis-cuss their physical consequences. The results show that ∗ viktor.holubec@mff.cuni.cz in the leading order in the distance from Carnot’s effi-ciency, ∆ η = η C − η , the work variance is proportionalto the heat capacity σ w = (cid:10) w (cid:11) − h w i ∝ C. (1)Combining this result with the formula h w i / ∆ η ∝ C found by Campisi and Fazio [12] we obtain the followingscaling of the relative work fluctuation f w = σ w h w i ∝ √ C ∆ η . (2)Above, C denotes the heat capacity of the working sub-stance, h w i the average work performed by the engineand (cid:10) w (cid:11) the corresponding second moment.Using these formulas, we discuss performance of heatengines working during the whole cycle in the vicinity ofa critical point. Then the thermodynamic behavior ofthe engine is determined by a set of critical exponentsof the engine working fluid [12, 14]. We find that suchheat engines exhibit behavior which contradicts what isobserved for standard non-interacting heat engines awayfrom the phase transition. For the latter, the relativework fluctuation decreases as f w ∝ / √ N ensuring neg-ligible fluctuations in the macroscopic limit. We find thatthe output work fluctuations for the critical heat engineincrease with the number of interacting subsystems N and diverge in the large system limit unless the strictscaling condition h w i ∝ η ∝ √ C ∝ N / (2 − α ) , −∞ < α < , (3)is fulfilled. In this special case f w ∝ α T , λ λ T , λ λ adiabaticexpansionadiabaticcompression isochoricthermalizationisochoricthermalization FIG. 1. (Color online) The Stirling cycle. The engine ex-changes heat with reservoirs (performs work on the environ-ment) during the isochoric (adiabatic) branches only. exactly. Such a precision is impossible to achieve in prac-tical experiments or numerical simulations. Our findingsthus practically close the way to construct a standardmacroscopic critical heat engine operating at Carnot’sefficiency at a finite average power with finite fluctua-tion.This, however, does not mean that the idea of a crit-ical heat engine is completely useless. One can for ex-ample construct a non-standard heat engine with a feed-back mechanism which would harvest the positive workfluctuations from the system. On the other hand, thesensitivity of the critical heat engine to the scaling of themean output work can be used as a basis of a novel exper-imental technique for precise measurement of the criticalexponent α . Last but not least, if the scaling would beclose enough to Eq. (3), one can construct a heat engineworking at a large efficiency (although smaller than η C )and a large output power with acceptable fluctuation bychoosing a suitable finite value of N . II. MODEL
Consider the Stirling cycle performed by a mediumdescribed by the Hamiltonian H ( t ) = λ ( t ) K, (4)where K is an N -particle Hermitian operator and λ ( t ) is aparameter which can be varied in time by the experimen-talist. We assume that the Hamiltonian is confining sofor any reasonable λ = λ ( t ) > P eq ( βλ ) = exp( − βλK ) / Tr[exp( − βλK )], β = ( k B T ) − . By varying the parameter λ ( t ) we changethe “volume” of the whole N -particle system. We are in-terested in the collective response of the whole system tothe driving which may lead to an enhanced performanceof the heat engine as compared to the system composedof N non-interacting particles. This model was first in-troduced in Ref. [12].The considered Stirling cycle is depicted in Fig. 1.It is composed of two adiabatic processes during which the system distribution does not change and the systemperforms work on its surroundings and two very slowisochoric processes where the system eventually equili-brates. During the first adiabatic process, the Hamilto-nian changes from λ K to λ K , and the system is ini-tially in equilibrium with the bath at the temperature T . Along the second adiabatic process, the Hamilto-nian changes back from λ K to λ K and the systemstarts from equilibrium at the temperature T . We as-sume that λ < λ , and that T > T . Then the adia-batic process at T mimics adiabatic expansion (the con-finement λ changes form stronger to weaker) and viceversa for the adiabatic process at T . Assuming further λ /λ > T /T the system operating in a time-periodicsteady state performs on average more work during theexpansion than it consumes during the compression andthus it operates as a heat engine [12].Let us now investigate the output work of the engine.In order to do that, it is favorable to work in a specificbasis | x i , where x stands for a set of independent internaldegrees of freedom of the system. If we denote as K ( x ) = h x | K | x i the corresponding elements of the Hamiltonian(4), the stochastic work done by the system during onecycle reads w ( x , x ) = ( λ − λ )( K ( x ) − K ( x )) , (5)where x ( x ) is the state of the system during the first(second) adiabat. Because the isochoric processes allowthe system to relax to equilibrium, the variables x and x are independent and the probability density of thework w is given by the formula ρ ( w ) = Z Z dx dx δ [ w − w ( x , x )] × exp[ − β λ K ( x )] Z exp[ − β λ K ( x )] Z , (6)where δ ( • ) denotes the Dirac’s δ -function. The function(6) allows us to calculate all moments of the output work.The mean work along one cycle reads h w i = Z dw wρ ( w ) = ( λ − λ ) (cid:16) h K i β λ − h K i β λ (cid:17) (7)and for the work fluctuation we get σ w = ( λ − λ ) h(cid:16)(cid:10) K (cid:11) β λ − h K i β λ (cid:17) + (cid:16)(cid:10) K (cid:11) β λ − h K i β λ (cid:17)i . (8)Above, h K i β λ denotes average with respect to the dis-tribution exp[ − β λ K ( x )] /Z and similarly for h K i β λ .The work fluctuation determine the number of cycles forwhich the output work of the heat engine must be mea-sured in order to get a reliable prediction of the meanoutput work. According to the central limit theorem,the fluctuation of the predicted average work after a largeenough number N m of measurements behaves as σ w /N m .Large fluctuation σ w implies that one have to measure fora long time to get a reliable prediction of the mean work. III. NEAR CARNOT’S BOUND
The heat accepted by the system from the hot bathis given by h q in i = λ ( h K i β λ − h K i β λ ) and thus theefficiency of the Stirling engine reads [12] η = h w ih q in i = 1 − λ λ < − β β = η C . (9)As outlined in the Introduction, the main aim of thepresent paper is to investigate fluctuations of the out-put work of the critical heat engine operating close toCarnot’s bound. The distance from the Carnot’s boundis for the present model given by [12]∆ η = η C − η = λ λ − β β > . (10)In what follows, we will study the behavior of work fluc-tuation σ w and relative work fluctuation f w up to thefirst order in ∆ η .In Ref. [12] it was shown that for small ∆ η the ratioof mean work and the distance from Carnot’s bound isproportional to the heat capacity of the system:Π = h w i ∆ η ∝ C β λ , (11)where C β λ = − k B β dU β λ /dβ denotes the heat ca-pacity and U β λ = λ h K i β λ the internal energy of thesystem.From the formula (8) for the work fluctuation it followsthat σ w is determined by the equilibrium fluctuations ofthe internal energy of the system σ u βiλi = λ (cid:10) K (cid:11) β i λ i − U β i λ i , i = 1 ,
2, which are in turn proportional to the heatcapacities σ u βiλi = β − C β i λ i /k B , i = 1 ,
2. Altogether weget σ w = ( λ − λ ) k B (cid:18) β λ C β λ + 1 β λ C β λ (cid:19) . (12)From Eq. (10) it follows that β λ = β λ ∆ η + β λ , andthus, up to the leading order in ∆ η , the work fluctuationcan be further rewritten as σ w ≈ λ − λ ) k B β λ C β λ = 2 η k B β C β λ ∝ C β λ . (13)The work variance hence scales in the same way as Π.The relative work fluctuation then scales as f w ∝ p C β λ ∆ η . (14)Introducing duration of one period of the cycle τ anddefining the fluctuating output power as p ( x , x ) = w ( x , x ) /τ we find that the formula (14) describes alsothe relative fluctuation of the output power: f p = q h p i − h p i / h p i = f w . Equations (13) and (14) consti-tute our first main result. The next sections emphasizetheir significance for the engine performance. IV. NEAR THE PHASE TRANSITION
The engine working fluid consists of N interacting sub-units. In Eq. (4), the interaction is incorporated in the N -particle operator K . Without the interaction, the ther-modynamic quantities in question would follow the wellknown scaling behavior: C β λ ∝ h w i ∝ σ w ∝ N and∆ η ∝
1. Thus, by enlarging the number of subunits,one would not get any interesting gain in the engine per-formance. The main idea of Ref. [12] is that the abovesimple scaling can change if the subunits interact and, forexample, the engine works in the neighborhood of a criti-cal point (phase transition). Let us now use the formulas(13) and (14) to uncover another side of this idea omittedin Ref. [12]. The following analysis shows that, togetherwith the enhanced scaling of the mean output work andefficiency found in Ref. [12], the engine working close to acritical point inevitably exhibits also unfavorable scalingof work fluctuations.Near the phase transition, the system behavior canbe described in a very general fashion using the finitesize scaling theory [12, 26, 27]. Let us denote as T c thetemperature of the phase transition in question, θ = | T − T c | /T c the distance from the critical temperature and d the dimension of the system. When reaching the criticaltemperature ( θ →
0) in the thermodynamic limit, theheat capacity C diverges with the exponent α > C ∝ θ − α and the correlation length ξ diverges with theexponent ν > ξ ∝ θ − ν . For finite size systemsoperating close to the critical point, the heat capacitydevelops a peak with height C and width δ which scale aslog C/ log N ∝ α/ ( dν ) = 2 / (2 − α ) and log δ/ log N = − / ( dν ) = − / (2 − α ), respectively. Here we have usedthe formula νd = 2 − α [28].As explained in Ref. [12], the scaling of the heat ca-pacity has important consequences for the validity of ap-proximations used in the derivation of Eqs. (11), (13)and (14). In deriving the formula (11) one assumesthat the internal energy U β λ can be approximated bythe first order Taylor expansion around β λ . Sim-ilarly, in the derivation of Eq. (13) we assume thatthe heat capacity C β λ can be approximated by ze-roth order expansion around β λ . More precisely,one assumes that Π = h w i / ∆ η ≈ ∂ h w i /∂ ∆ η | ∆ η =0 , U β λ = U β λ ∆ η + β λ ≈ U β λ + dU x /dx | x = β λ β λ ∆ η and C β λ = C β λ ∆ η + β λ ≈ C β λ . Assuming thatthe inverse critical temperature of the Hamiltonian K is given by β λ , the expression β λ ∆ η = β λ − β λ measures the distance from the critical point. The regionwhere the internal energy can be approximated by the lin-ear expression above for a given N is given by the width δ of the heat capacity peak. This means that β λ ∆ η must shrink with increasing N in the same manner as δ or faster in order to secure the validity of Eq. (11). Wearrive at the condition∆ η ∝ N − γ , γ ≥ / (2 − α ) ≥ β λ is “close” to β λ = β c . The size of this neighbor-hood shrinks with increasing number of subunits N pro-portionally to δ and hence the distance from the phasetransition, measured by ∆ η , must shrink in the samemanner or faster in order to keep the engine close to thephase transition for any N .Inserting the scaling (15) of ∆ η together with the scal-ing of heat capacity into the formula (14) for the relativework fluctuation f w , we find f w ∝ N − / (2 − α )+ γ . (16)In order to have finite f w in the large N limit we thusarrive at the condition 1 / (2 − α ) ≥ γ . Altogether, if oneasks both for the validity of the linear approximationdescribed above and for a nonzero relative work fluctua-tion, the critical exponent must obey the strict condition1 / (2 − α ) = γ which leads to the scaling∆ η ∝ √ C ∝ N − / (2 − α ) (17)of the distance from Carnot’s efficiency. This observationconstitutes our second main result. Let us stress that fornon-interacting subsystems the relative work fluctuationscales as f w ∝ / √ N , while for the present critical heatengine we obtain at best f w ∝ h w i ∝ √ C ∝ N / (2 − α ) . (18)Such scaling of output work can be achieved for exam-ple by fixing λ and carefully choosing the parameter λ for each value of N such that the formula h w i ∝ √ C is fulfilled [12]. In order to do this, one should howeverknow the exact value of the critical exponent α . Thisrepresents a thin bottle neck which may be really hardto surpass as discussed at the end of the present section.Nevertheless, let us now investigate the performance ofhypothetical heat engines where the scaling (18) can beachieved in practice.According to the dynamical finite-size scaling theory,close to a critical point the relaxation time to equilib-rium scales with the number of subsystems as τ R ∝ N z/d ,where z can be both positive (critical slowing down) andnegative (critical speeding up) [12]. It is thus reasonableto assume that the time needed to perform the isochoricbranches of the Stirling cycle, where the systems thermal-izes with the individual heat reservoirs, scales as N z/d .If we further assume that the adiabatic branches are per-formed infinitely fast, the total duration of the cycle τ would scale as τ ∝ N z/d . This is the shortest possibleduration of the cycle which maximizes the power output.Assuming that the duration of the adiabatic branchesscales with N , an arbitrary scaling of the cycle durationof the form τ ∝ N ζ/d ≥ N z/d can be obtained.Under the assumption of the fastest possible cycle ( τ ∝ N z/d ) the average output power of the engine scales as h p i = h w i τ ∝ √ Cτ ∝ N / (2 − α ) − z/d . (19)The formula dν = 2 − α implies that −∞ < α < α → −∞ ) as h w i ∝ h p i ∝ N − z/d ,respectively. The scaling of work is better than in thesituation when the N subsystems do not interact ( h w i ∝ N ) whenever α >
1. A gain in power is obtained only ifthe further condition 1 / (2 − α ) − z/d > zd < α − − α (20)is fulfilled. For example, for the 3D Ising model we have α ≈ .
12 [29] and z ≈ .
35 [30] leading to h w i ∝ N . and h p i ∝ N − . . Another example is the exotic sub-stance Dy Ti O which posses the rather large coeffi-cient α ≈ .
38 [31] and exhibits the critical speeding up( νz ≈ − . z ≈ − . h w i ∝ N . and h p i ∝ N . . In this case, the critical speeding up fur-ther allows to reduce the work fluctuation by averagingthe work output over a large number of cycles in a shorttime. Indeed, the amount of cycles per unit time scales as N m = 1 /τ ∝ N − z/d = N . and thus the relative vari-ance f w / √ N m of the output work averaged over the unittime vanishes with N as f w / √ N m ∝ N z/ (2 d ) = N − . .Let us note that in a more realistic situation where thetotal cycle time scales as τ ∝ Ti O still reachesuseful output power which scales as h p i ∝ N . (scalingof other variables do not change).Our analysis of work fluctuations thus leaves openthe way to construct a classical macroscopic heat en-gine working at Carnot’s efficiency at a nonzero aver-age power suggested by Campisi and Fazio as long asthe scaling (17) of the output work can be experimen-tally achieved. However, to do this, one should knowthe exact value of the critical exponent α . Obtainingan exact value of a physical parameter is impossible toachieve both in experiments and in numerical simula-tions and thus it is our opinion that it would be nearlyimpossible to achieve the scaling (17) in practice. Evenvery small discrepancy in the used α would lead to adifferent scaling. Whenever the work would scale slowerthan √ C the heat engine would not work in the criti-cal regime as discussed above and thus the analysis us-ing the critical exponents would not be correct. On theother hand, whenever the work would scale faster than √ C the output work and power would be dominated byfluctuations rendering the average work/power output ofthe machine useless (unmeasurable) for large N , whereCarnot’s efficiency is achieved. One may argue that thescaling τ ∝ N z/d of the total cycle time may allow usto average out the large work fluctuations in finite timeas mentioned above for Dy Ti O . However, also thisscaling is beyond the reach of experiments where the to-tal cycle time is always bounded from bellow and thus itscales rather as τ ∝ N limit, one should rather focuson harvesting the positive power fluctuations as can beachieved in feedback driven systems. The strong sensitiv-ity of the critical heat engine to the scaling of the outputwork can be also used as a basis of a novel experimentaltechnique for precise measurements of the critical expo-nent α .To conclude, the critical heat engines can be in practiceused as standard heat engines only if one does not pursuethe large N limit too far. In such a case, choosing thescaling of the output work close enough to (18) will re-sult in a weak divergence of the relative power fluctuationwith N (for example taking γ = 1 / (2 − α ) + ǫ will lead tothe scaling f w ∝ N ǫ , the parameter ǫ > α ).One can then construct a heat engine operating close toCarnot’s efficiency at a large average power output bychoosing N small enough that the power fluctuations areacceptable for the respective application (for example themeasurement time available for averaging out the fluctu-ations). V. CONCLUSION AND OUTLOOK
We have studied the class of Stirling heat engines de-scribed in Fig. 1. In case the heat engine is based on N interacting subsystems and operates during the wholecycle close to a critical point, we have found the strictcondition (3) under which the engine can at the sametime reach Carnot’s efficiency and a finite relative workfluctuation in the large N limit. In the unlikely case whenthis condition can be fulfilled, the relative work fluctua-tion does not scale with N and thus it remains finite inthe large N limit. In all other cases, the work fluctuationsdominate the average output work of the engine and ren-ders the actual output work/power of such engines nearlyunpredictable. Thus the device can not work as a clas-sical heat engine. One should rather supplement the itwith a feedback mechanism which would harvest positivework fluctuations. The strong sensitivity of work fluctu- ations to the scaling of the output work in heat enginesworking close to a critical point may be used to a novelexperimental technique for precise measurements of thecritical exponent α .The restriction (3) results from the necessity to workwithin the linear regime around the critical point in thederivation of the results. The situation may change whendifferent Hamiltonians and/or thermodynamic cycles areconsidered. Furthermore, different results can be ob-tained outside the linear regime, i.e. for heat engineswhich do not operate close to the critical point duringthe whole cycle. One can also think about different waysof achieving suitable scaling of thermodynamic variablesin question, namely of the mean work, distance fromCarnot’s bound and relative work/power fluctuation (seefor example the Ref. [9] where such a scaling is discussedfor a quantum dot and Ref. [11] where a scaling lead-ing to Carnot’s efficiency at a nonzero power and finitepower fluctuation is presented for a Brownian heat en-gine). Finally, as already highlighted in Ref. [12], it isnot necessary to consider the limit of large N in order toget better performance with respect to non-interactingsystems.Although Carnot’s efficiency is achieved in the large N limit only, also the power fluctuations diverge only inthis limit. Hence, by choosing a finite N , it is possibleto achieve a suitable trade-off between large efficiency,average output power and acceptably large power fluc-tuation. For a small enough system, one can further indetail optimize interactions between the individual sub-units in order to get the desired engine behavior. Suchdetailed optimization would be indeed impossible for sys-tems composed of many subunits.The take-home message of the present work is verysimple: when studying performance of heat engines inexotic regimes of operation where thermodynamic quan-tities such as correlation functions diverge, one shouldalways address both the mean values and the fluctua-tions. Only then one can be sure that the average valuescan be observed experimentally and they wont get lostin a deep forest of randomness. ACKNOWLEDGMENTS
The authors thanks M. Campisi and N. Shiraishi forinspiring discussions. Support of the work by the COSTAction MP1209 and by the Czech Science Foundation(project No. 17-06716S) is gratefully acknowledged.VH in addition gratefully acknowledges the support byAlexander von Humboldt agency. [1] I. M¨uller,
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