Power-efficiency-fluctuations trade-off in steady-state heat engines: The role of interactions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Power-efficiency-fluctuations trade-off in steady-state heat engines:The role of interactions
Giuliano Benenti,
1, 2, 3
Giulio Casati,
1, 4 and Jiao Wang Center for Nonlinear and Complex Systems, Dipartimento di Scienza e AltaTecnologia, Universit`a degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy NEST, Istituto Nanoscienze-CNR, I-56126 Pisa, Italy International Institute of Physics, Federal University of Rio Grande do Norte, CampusUniversit´ario - Lagoa Nova, CP. 1613, Natal, Rio Grande Do Norte 59078-970, Brazil Department of Physics and Key Laboratory of Low Dimensional Condensed Matter Physics(Department of Education of Fujian Province), Xiamen University, Xiamen 361005, Fujian, China
We consider the quality factor Q , which quantifies the trade-off between power, efficiency, andfluctuations in steady-state heat engines modeled by dynamical systems. We show that the non-linear scattering theory, both in classical and quantum mechanics, sets the bound Q = 3 / Q = 1 /
2, which is the universal upper bound in linearresponse. This result shows that interactions are necessary to achieve the optimal performance of asteady-state heat engine.
Introduction.-
Understanding the bounds that a heatengine must obey is of key importance both for basicscience and for technological development. Ideally, aheat engine should work with efficiency η close to theCarnot efficiency η C , deliver large power P , and havesmall power fluctuations ∆ P . The Carnot limit is intu-itively associated with infinitely slow engines, so that theoutput power vanishes. On the other hand, the secondlaw of thermodynamics by itself does not forbid an en-gine operating at the Carnot efficiency with a finite out-put power [1]. Such a dream engine was denied in mod-els with inelastic scattering [2–9] and for two-terminalsystems on the basis of symmetry considerations for theOnsager kinetic coefficients [10]. Moreover, for systemsdescribed as Markov processes, the bound P ≤ A ( η C − η )was proven [11], with η C = 1 − T R /T L , T L and T R ( T L > T R ) being the temperatures of the hot and the coldreservoir, respectively, and A a system-specific constant.On the other hand, A may diverge when approaching theCarnot efficiency [12–17], for instance, when the engineworking fluid is at the verge of a phase transition, andtherefore the Carnot efficiency may be approached at fi-nite power. However, fluctuations make impractical suchengines [18].On the base of thermodynamic uncertainty rela-tions [19–22] for the work current (i.e., for the powerdelivered by the engine), a trade-off encompassing effi-ciency, power, and fluctuations has been proven by Piet-zonka and Seifert [23], for a large class of steady-stateclassical stochastic heat engines with time-reversal sym-metry. Such class includes engines with a discrete setof internal states described by thermodynamically con-sistent rate equations, and continuous systems modeledwith an overdamped Langevin dynamics. The bound reads Q ≡
P ηη C − η k B T R ∆ P ≤ , (1)where k B is the Boltzmann constant and the power fluc-tuations are measured by [24]∆ P = lim t →∞ [ P ( t ) − P ] t, (2)where P ( t ) is the mean delivered power up to time t .In this paper, at difference from the above stochas-tic thermodynamics approach, we examine for purely dy-namical models the upper bound to the quality factor Q .We focus on the most desirable regime for a heat engine,i.e., when approaching the Carnot efficiency at the largestpossible output power. We first find a general solution tothis problem for systems that can be modeled by the non-linear scattering theory. That is, for noninteracting sys-tems or more generally for systems in which interactionscan be treated at a mean-field Hartree level. In this casewe prove that the quality factor Q < /
8, and that thelimit value Q = 3 / < / η → η C .Scattering theory sets therefore a stronger bound to thequality factor Q than Eq. (1). We stress that the aboveresults are valid both in classical and in quantum me-chanics. We then consider the class of interacting, non-integrable momentum-conserving systems. This is, toour knowledge, the only class of interacting dynamicalsystems which is known to achieve, at the thermody-namic limit, the Carnot efficiency [25–27]. We show ina concrete example of a nonintegrable gas of elasticallycolliding particles that such systems saturate the bound Q = 1 /
2. This value is achieved in the tight-couplinglimit, where the Onsager matrix of kinetic coefficientsbecomes singular.
Bound from scattering theory.-
For concreteness, here-after we consider thermoelectric transport, even thoughour results could be equally applied to other examplesof steady-state conversion of heat to work, like thermod-iffusion. In the Landauer-B¨uttiker quantum scatteringtheory, the electrical current, flowing from the left to theright reservoir, reads J e = eh Z ∞−∞ dǫ T ( ǫ ) [ f L ( ǫ ) − f R ( ǫ )] , (3)where e is the electron charge, h the Planck constant, T ( ǫ ) the transmission probability for a particle with en-ergy ǫ to transit from one end to another of the system(0 ≤ T ( ǫ ) ≤ f α ( ǫ ) = { ǫ − µ α ) /k B T α ] } − is the Fermi distribution function for reservoir α ( α = L, R ), at temperature T α and electrochemical potential µ α [28]. The heat current that flows into the system fromreservoir α is J h,α = 1 h Z ∞−∞ dǫ ( ǫ − µ α ) T ( ǫ ) [ f L ( ǫ ) − f R ( ǫ )] . (4)The output power P = (∆ V ) J e , where ∆ V = ∆ µ/e isthe applied voltage, with ∆ µ = µ R − µ L >
0. The effi-ciency of heat to work conversion is given by η = P/J h,L ,with
P, J h,L >
0. The transmission function which max-imizes the efficiency for a given power is a boxcar func-tion, T ( ǫ ) = 1 for ǫ < ǫ < ǫ and T ( ǫ ) = 0 other-wise [29, 30]. Here ǫ = ∆ µ/η C is obtained from thecondition f L ( ǫ ) = f R ( ǫ ) and ǫ can be determined nu-merically by solving the equation ǫ = ∆ µJ ′ h,L /P ′ , wherethe prime indicates the derivative over ∆ µ for fixed T (this equation is transcendental since J h,L and P dependon ǫ ). The maximum achievable power P max is obtainedwhen ǫ → ∞ and is given by P max ≈ . k B (∆ T ) /h ,with ∆ T = T L − T R >
0. In the limit ǫ → ǫ , known asdelta-energy filtering [31–33], P → η → η C .The power fluctuations can be computed from theLevitov-Lesovik cumulant generating function [34–36].For the above boxcar function, we obtain∆ P = (∆ µ ) h Z ǫ ǫ dǫ [ f L ( ǫ ) + f R ( ǫ ) − f L ( ǫ ) − f R ( ǫ )] . (5)Using the above defined expressions for η , P , and ∆ P ,we can compute the quality factor Q . As detailed in thesupplemental material [38], we can expand Q close tothe Carnot efficiency, that is, for 1 − η/η C ≪
1. We thenobtain the analytical result Q = 38 − T L + T R T R (cid:18) − ηη C (cid:19) + O "(cid:18) − ηη C (cid:19) . (6)When going far from the Carnot limit, the dependenceof Q on efficiency can be computed numerically. Theresults are shown in Fig. 1, for the optimal boxcar func-tion, which maximizes efficiency for any value of power.We can see that Q < / η , the value Q = 3 / η = η C (correspond-ingly, P = 0). A similar analysis can be performed inthe classical case [38], and for the optimal boxcar func-tion [37] expansion (6) is still valid. As shown in Fig. 1,classical and quantum Q differ at higher orders, with thequantum quality factor slightly larger than the classicalone. Momentum-conserving systems.-
The above resultsraise two interesting questions: (i) Is it possible to findinteracting systems which may overcome the scatteringtheory bound Q = 3 /
8? (ii) Is it possible to approach oreven overcome bound (1), Q = 1 /
2, when approachingthe Carnot efficiency? To address this question, we con-sider non-integrable momentum-conserving systems, forwhich the Carnot limit can be achieved at the thermo-dynami limit [25–27]. We perform nonequilibrium simu-lations, with the momentum-conserving system (specifi-cally, a classical one-dimensional diatomic gas of elasti-cally colliding particles [25]) in contact with two reser-voirs at different temperatures and electrochemical po-tentials, which maintain stationary coupled energy andparticle flows. In our simulations, particles are absorbedwhenever they hit a reservoir, while the two reservoirsinject particles with rates and energy distributions de-termined by their temperatures and electrochemical po-tentials [42].From our numerical simulations, we can determine thecharge and heat currents, and consequently power, ef-ficiency, and fluctuations. In Fig. 2, we compute thetrade-off Q for different system sizes L . Note that these C classical quantum analytical FIG. 1. Quantum and classical quality factor Q based on thescattering theory. For η close to η C , numerical results arein excellent agreement with the analytical formula (6) up tothe linear term (red solid line). The dashed line indicates thebound Q = 3 / T L = 1, T R = 0 . µ L = 0. For any value ofpower, µ R = ∆ µ is determined in the optimization proceduredescribed in the text. Here we use units such that k B = e = h = 1. C L = L = L = L = FIG. 2. Quality factor versus efficiency for a one-dimensionaldiatomic gas of elastically colliding particles of masses m = 1and M = 3 with T L = 1 . T R = 0 . µ L = − ∆ µ/
2, and µ R = ∆ µ/
2. The system size L is equal to the mean numberof particles inside the system. The dashed and the dot-dashedline indicate the bound Q = 3 / Q = 1 /
2, respectively.We set k B = e = 1. curves have two values for a given value of efficiency η ,as they are obtained by changing ∆ µ from zero, where η = 0, up to the stopping value, where the electrochemi-cal potential difference becomes too high to be overcomeby the temperature difference, and again η = 0. Thebranch with higher values of Q corresponds to the lowervalues of ∆ µ . Note that the scattering theory bound Q = 3 / Q = 1 /
2, is approached closer and closer.
Linear response results.-
For a given temperature dif-ference ∆ T = T L − T R , the gradient |∇ T | = ∆ T /L de-creases with the system size L . We can then apply thelinear response theory in the large- L regime, which is themost interesting one in our model, as the Carnot effi-ciency is achieved when L → ∞ . Within linear response,the charge and heat currents (from the left to the rightreservoir) are given by [43–45] (cid:18) J e J h (cid:19) = (cid:18) L ee L eh L he L hh (cid:19) (cid:18) F e F h (cid:19) , (7)where F e = −∇ [ µ/ ( ek B T )] and F h = ∇ [1 / ( k B T )] arethe thermodynamic forces and L ij ( i, j = e, h ) the On-sager kinetic coefficients, which obey, for systems withtime-reversal symmetry, the Onsager reciprocal relation L eh = L he . The matrix of the kinetic coefficients isknown as the Onsager matrix L . The second law of ther-modynamics imposes L ee ≥ L hh ≥
0, and det L ≥ J c = P α c α J α = c T L F , where F is the vec-tor of thermodynamic forces. Thermodynamic uncer-tainty relations are saturated when c k F on the or- thogonal complement of the kernel of L [46]. In nonin-tegrable momentum-conserving systems, L becomes sin-gular in the thermodynamic limit, where the Onsagermatrix becomes singular (a condition known as tight-coupling limit, for which the Carnot efficiency is achiev-able). In this limit, the orthogonal complement of thekernel of L is one-dimensional, and therefore the thermo-dynamic uncertainty relations are saturated for all c . Inparticular, bound (1) is saturated. In contrast, for finitesystem sizes the Onsager matrix is positive and the onlycurrent such that c k F is the entropy production rate˙ S = F e J e + F h J h . Since power fluctuations ∆ P are pro-portional to charge fluctuations and not to ˙ S , it followsthat bound (1) is saturated only at the thermodynamiclimit.The validity of using linear response to interpret ournumerical data is numerically confirmed by Fig. 3. Fordifferent system sizes, η/η C is shown as a function of P/P max , where P max is the maximum power obtainedwhen ∆ µ is varied from zero, where P = 0, up to thestopping value, where again P = 0. In the same fig-ure, we also show the linear response result in the tight-coupling limit [45]: ηη C = P/P max ± p − P/P max ) . (8)Moreover, we rewrite bound (1) as ηη C ≤
11 + 2
P k B T R / ∆ P (9)and in the same figure we show the right-hand side ofthis inequality for the numerically computed values ofpower and fluctuations at the largest available systemsize, L = 3200. Numerical results, however, suggest thatthe upper bound (9) does not depend on the system size.The tendency to saturate this bound when increasingthe system size is clearly seen in Fig. 3. The excellentagreement between linear response expectations in thetight-coupling limit and the numerically computed up-per bound on efficiency for given power and fluctuations,shows that linear response theory provides a satisfactoryexplanation of our results. Discussion and conclusions.-
We have shown thatfor steady-state heat engines the power-efficiency-fluctuations trade-off
Q ≤ / Q = 3 / Q = 1 /
2. This value is obtained in the tight-coupling limit, when the Onsager matrix becomes singu-lar and the Carnot limit can be achieved. From the view-point of thermoelectric transport, our results confirm therelevance of the figure of merit ZT = L eh / det L [45]: Not max C L = L = L = P/P L = FIG. 3. The same as Fig. 2 but for efficiency versus power forthe diatomic gas model. The linear response prediction (8) inthe tight-coupling limit (green dotted curve) and the upperbound (9) (blue dashed curve) are also shown. only the Carnot efficiency can be achieved at the tight-coupling limit ZT → ∞ , but also the power-efficiency-fluctuations bound Q = 1 / Q could be investigated experimen-tally in the context of cold atoms, where a thermoelec-tric heat engine with high ZT has already been demon-strated, both for weakly [47] and strongly interacting par-ticles [48].Our analysis does not include the effects of a mag-netic field. However, for two-terminal systems our con-clusions would not change. Indeed, the scattering-theorybound discussed in this paper is the same, irrespectiveof whether time-reversal symmetry is broken by an ex-ternal magnetic field or not. Moreover, the Onsager ma-trix obeys the reciprocal relation L eh = L he for inter-acting systems, even in presence of a generic magneticfield [10]. The effects of a magnetic field on thermoelec-tric efficiency were investigated in systems with three ormore terminals, which mimic inelastic scattering events,but in that case it was not possible to achieve the Carnotefficiency [4–9]. It remains therefore as an interestingquestion whether the linear-response bound Q = 1 / Supplemental Material
Here we provide more details on the derivation of thequality factor Q from the scattering theory, for efficiency close to the Carnot efficiency, Eq. (6) of the main text.Hereafter we set k B = e = h = 1. In the limit δ ≡ ǫ − ǫ → + , the power P ∼ δ , the deviation from theCarnot efficiency η C − η ∼ δ , the fluctuations ∆ P ∼ δ ,so that the trade-off factor Q goes to a constant when δ → + . More precisely, for the power we obtain P = (∆ V ) (cid:20) sech (cid:18) ∆ V T (cid:19)(cid:21) ∆ T T L T R δ + O ( δ ) , (10)where in the limit δ → + the voltage ∆ V = α ∆ T [30],with α ≈ .
24, the root of the transcendental equation α tanh (cid:16) α (cid:17) = 3 . (11)We can then rewrite the power as P = α (∆ T ) α ) T L T R δ + O ( δ ) . (12)Similarly, we obtain the heat current that flows from thehot reservoir, J h,L = α ∆ T α ) T R δ , + O ( δ ) , (13)and the efficiency η = η C − η C αT L δ + O ( δ ) . (14)To compute power fluctuations, we use the formula [36]∆ P = (∆ V ) Z + ∞−∞ dǫ ( T ( ǫ ) { f L ( ǫ )[1 − f L ( ǫ )]+ f R ( ǫ )[1 − f R ( ǫ )] } + T ( ǫ )[1 −T ( ǫ )][ f L ( ǫ ) − f R ( ǫ )] ) , (15)which reduces to Eq. (5) of the main paper for the boxcartransmission function we are considering. Upon expan-sion for small δ , we obtain∆ P = α (∆ T ) α δ + O ( δ ) . (16)Finally, we derive Q = 38 − T L + T R T L T R α δ + O ( δ ) , (17)which reduces to Eq. (6) of the main text after invertingthe relation between efficiency and width of the trans-mission window: δ = 32 αT L (cid:18) − ηη C (cid:19) + O "(cid:18) − ηη C (cid:19) . 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