Table Of ContentUnderdamped stochastic heat engine at maximum efficiency
Andreas Dechant,1 Nikolai Kiesel,2 and Eric Lutz1
1Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-Nu¨rnberg, 91058 Erlangen, Germany
2Vienna Center for Quantum Science and Technology (VCQ),
Faculty of Physics, University of Vienna, A-1090 Vienna, Austria
We investigate the performance of an underdamped stochastic heat engine for a time-dependent
harmonic oscillator. We analytically determine the optimal protocol that maximizes the efficiency
atfixedpower. ThemaximumefficiencyreducestotheCurzon-Ahlbornformulaatmaximumpower
and the Carnot formula at zero power. We further establish that the efficiency at maximum power
6 is universally given by the Curzon-Ahlborn efficiency in the weakly damped regime. Finally, we
1 show that even small deviations from operation at maximum power may result in a significantly
0 increased efficiency.
2
b
e The universal upper bound on the efficiency of any to maximal power output and derived the correspond-
F heat engine operating between two equilibrium baths at ing efficiency at maximum power. Interestingly, they
temperaturesT andT (T <T )isgivenbytheCarnot foundthatη∗ maysurpasstheCurzon-Ahlbornefficiency
1 c h c h
formula,ηC =1−Tc/Th[1]. TheCarnotefficiencyishow- ηCA. Experimental realizations of microscopic Brown-
] ever only reachable in the reversible limit, for ideal en- ian engines in colloidal systems have been reported in
h
ginesthatruninfinitelyslowlyandthushavezeropower. Refs. [8, 9]. A proposal to implement a stochastic heat
c
e It is hence customary to consider the efficiency at maxi- engine using an optically levitated nanoparticle in the
m mum power η∗, popularized by Curzon and Ahlborn [2], opposite limit of weak damping and slow thermalization
- whichyieldsabetterestimatefortheperformanceofreal has recently been put forward in Ref. [10]. Optimization
t
a engines. Forlow-dissipationmachines,genericboundson in this underdamped regime is significantly more diffi-
t η∗ havebeenobtainedinRefs.[3,4]. Theyreducetothe cult, owing to the coupling between position and veloc-
s
p
. Curzon-Ahlborn efficiency, η =1− T /T , for sym- ity dynamics [11–13]. It has for this reason been little
t CA c h
a metriccouplingtothecoldandhotbaths. Theexistence explored so far. In the following, we optimize the heat
m
of these general bounds is of fundamental importance. engine for maximum efficiency at constant power in the
- Yet, they do not address the crucial question of how to underdamped limit. We compute the optimal frequency
d
actually reach them, as they do not provide information driving protocols and show that they exhibit discontinu-
n
o about optimal engine cycles. It has recently been shown itiesatthechangefromthehottothecoldbathandvice
c that seeking the efficiency at maximum power may not versa. Weobtainanexplicitexpressionforthemaximum
[
always be the best strategy [5, 6]. To achieve engines efficiency at given power that contains both the Curzon-
1 withhighefficiencyandhighpoweronemayratherwant Ahlborn efficiency at maximum power and the Carnot
v to maximize the efficiency at a given power. A motor efficiency at zero power as limiting cases. We further
2 working at maximum efficiency may, for instance, attain demonstrate that the Curzon-Ahlborn formula provides
9
higherefficiencyandpowerthanmotorsworkingattheir a universal upper bound to the efficiency at maximum
3
maximum power. Devices that supply a fixed power, power in the underdamped regime, in contrast to the
0
0 with the highest possible efficiency, are moreover often overdamped limit. We finally show that it is possible
. required in practical engineering applications. The effi- to considerably enhance the efficiency of the engine by
2
0 ciency at given power output, as opposed to maximum sacrificing relatively little power.
6 poweroutput,additionallyoffersadirectmethodtocom- Brownian heat engine. We consider a stochastic heat
1 pare the performance of different engines. engine consisting of a harmonically bound particle cou-
:
v In this paper, we compute the maximum efficiency at pled to a heat bath. We assume that its dynamics is
i fixed power η˜ for an underdamped harmonic stochastic described by the Langevin equations,
X
heat engine and determine the corresponding optimal
ar driving protocols. Such a system may be viewed as a mv˙ =−mγv−mω2x+p2mγkBT ξ, (1a)
minimal model for a microscopic piston engine: Increas- x˙ =v. (1b)
inganddecreasingthestrengthoftheharmonicpotential
corresponds to the compression and expansion phase of Here m, v and x are the respective mass, velocity and
the cycle, whereas heating and cooling is accomplished positionoftheBrownianparticle. Theheatbathischar-
bychangingthetemperatureofthesurroundingmedium. acterized by the damping coefficient γ and the temper-
In the overdamped limit that describes strong particle- ature T. In the following, we set k = 1 for simplic-
B
medium coupling and fast thermalization, the optimiza- ity. The quantity ξ(t) is a centered Gaussian white noise
tion problem has been solved by Schmiedl and Seifert force with hξ(t)ξ(t0)i=δ(t−t0). In order to operate this
[7]: they have calculated the optimal protocols that lead system as a heat engine, we cyclically vary the temper-
2
ature T and the frequency ω of the harmonic potential whereσ (t)isthesolutionofEq.(4)withtheappropriate
v
as a function of time. The harmonic confinement may boundary conditions. Here we define heat such that it is
berealized, forexample, viaopticaltweezers[8,9]orvia positiveiftheparticleabsorbsenergyfromthebath. The
a standing light wave in a cavity [10]. Since Eq. (1) is energy change, ∆U = Q+W, vanishes over one cycle
linear in x and v, the phase-space distribution P(x,v,t) andtheextractedworkisequaltothetotalheat, −W =
will always be Gaussian, provided that the initial state Q +Q . Asaresult, poweroutputP andefficiencyη of
h c
P(x,v,0) is Gaussian. We may accordingly describe the the Brownian heat engine may be written as,
system with an equivalent set of deterministic equations
Q +Q Q +Q
of motion for the variances, σx =hx2i and σv =hv2i, P = h c, η = h c. (6)
τ +τ Q
h c h
2γT
σ˙ +2γσ +ω2σ˙ = , (2a) Optimal protocols. We now fix the power at a given
v v x m
value, P = P , and seek the optimal protocol ω(t) that
σ¨ −2σ +2ω2σ +γσ˙ =0. (2b) 0
x v x x maximizestheefficiencyη. AccordingtoEq.(6),weneed
to simultaneously minimize −Q , the heat dissipated to
Wewilldiscusstwodifferentregimesforthedynamicsof c
the cold bath, and maximize Q , the heat uptake from
the particle [14]. In the overdamped limit, γ (cid:29) ω, the h
thehotbath. Inafirststep, wefixthecouplingtimesτ
velocity distribution relaxes to its thermal equilibrium h
and τ . In the underdamped regime, we have to maxi-
form on time scales that are short compared to the slow c
mize Eq. (5) with the constraint (4) imposed on σ . The
spatialmotioninthepotential;itmaythusbeassumedto v
effective Lagrangian for this optimization problem is,
be always thermal, P (v) = Z−1exp(−mv2/(2T)). The
v
velocityvariablemaythenbeeliminatedbysettingσv = h (cid:16) ω˙ (cid:17) γTi
L(σ ,σ˙ ,ω,ω˙,α)=σ +α σ˙ + γ− σ − . (7)
T/m in Eq. (2b), leading to a single equation for σx: v v v v ω v m
2ω2 2T The corresponding Euler-Lagrange equations read,
σ˙ + σ = . (3)
x γ x mγ
(cid:16) ω˙ (cid:17)
1+α γ− −α˙ =0 (8a)
We have here assumed that the second derivative of σ ω
x
1 d
is small in the long-time limit. In the opposite, under- (ασ )=0 (8b)
damped limit, ω (cid:29)γ, the particle completes many oscil- ωdt v
(cid:16) ω˙ (cid:17) γT
lations in the potential before its energy is significantly
σ˙ + γ− σ − =0. (8c)
changedthroughtheinteractionwiththebath. Byvirtue v ω v m
of the virial theorem, we may hence take kinetic and po-
We provide a detailed solution of the above equations
tentialenergytobeequal,mσ /2=mω2σ /2. Replacing
v x in the Supplemental Material. In particular, we estab-
σ byσ /ω2 inEq.(2a)andkeepinginmindthatω may
x v lish that a nontrivial solution of the optimization prob-
be time-dependent, we find to leading order in γ/ω,
lemrequiresdiscontinuitiesinbothω(t)andσ (t)atthe
v
transitions between cold and hot baths. We note that
(cid:16) ω˙ (cid:17) γT
σ˙ + γ− σ = . (4) intheunderdampedlimit, adiscontinuousvarianceσ is
v ω v m v
permitted,sinceweassumethatthesysteminstantlyad-
Whereas the system adjusts instantly to the bath in the juststothepotential. Fromtheconditionσ+ω− =σ−ω+
overdamped limit, it adjusts immediately to changes of at t=τh and a similar one at t=τh+τc, we find,
the external potential in the underdamped limit.
(cid:0) (cid:1)
pleDdutroingthaechyoctleboafththaethteeamtpeenrgaitnuer,ethTehpfaorrtiacletimisecoτuh-, ω(t)=ωc σσhce(cid:0)γh 1−m(cid:1)Tσhh (t−τh) for0<t<τh,
and then to the cold bath at temperature Tc < Th for eγc 1−mTσcc (t−τh) for0<t−τh <τc.
a time τc. We assume the switch between the baths to (9a)
be quasi-instantaneous, as in the experiment [8, 9] . We
(cid:16) T (cid:17) (cid:16) T (cid:17)
furtherallowfordifferentbathcouplingsγ andγ . Dur- γ τ 1− h +γ τ 1− c =0. (9b)
h c h h mσ c c mσ
ing this procedure, the frequency ω is varied with time h c
to compress or expand the system. After time τh +τc, The above equations completely specify the optimal fre-
the particle is again coupled to the hot bath and the cy- quencyprotocolω(t)thatleadstoextremalvaluesofQ
h
cle starts anew. In the underdamped regime, the heat and Q for fixed τ and τ and arbitrary ω . They con-
c c h c
exchanged with any of the two baths (i = h,c) is given stitute the first main result of this paper.
by [15, 16], Maximum efficiency at fixed power. In order to deter-
mine the maximum efficiency of the stochastic engine,
h Z τi i
Qi =γi τiTi−m dt σv(t) , (5) η =1+[γcτc(Tc−mσc)]/[γhτh(Th−mσh)], we eliminate
0 σh and σc using Eqs. (6), with P = P0, and Eq. (9b).
3
FIG.1. Maximumefficiencyη˜atconstantpowerP =yP∗as FIG. 2. (color online) Optimal frequency protocols for an
0
a function of y for a temperature ratio of α=T /T =0.25. underdamped stochastic heat engine. The parameters are
c h
Thelineistheanalyticunderdampedresult(11),thedotsare T /T = 2 and symmetric coupling γ = γ . The red (blue)
h c h c
obtained by numerically solving the exact dynamics (2) for shadedbackgroundshowsthecouplingtothehot(cold)bath.
ω/γ =1000(fullsymbols)andω/γ =2000(emptysymbols). Thesolidlineistheprotocolleadingtomaximumpowerout-
At maximum power, P = P∗, the efficiency reduces to the put P∗ =0.043 and corresponding efficiency η∗ =0.29. The
0
Curzon-Ahlborn formula η = 1−pT /T . Even slightly dashed line corresponds to P =0.8P∗ =0.034 and η˜=0.39.
CA c h
reducing the power yields a significantly enhanced efficiency. Remarkably, we have η˜/η∗ = 1.34; the increase in efficiency
is therefore larger than the reduction in power.
Maximizingtheefficiencyη withrespecttoτ , ∂ η =0,
h τh result is independent of the details of γ in the small γ
we obtain a relation between the two coupling times,
limit [18]. The Curzon-Ahlborn formula therefore pro-
τ rγ vides a generic upper bound to the efficiency at maxi-
h = c. (10)
τ γ mum power in the underdamped regime. For intermedi-
c h
ate power, 0 < P < P∗, Eq. (11) interpolates between
0
UsingEq.(10),wearriveaftersomealgebraatanexplicit the Carnot and Curzon-Ahlborn efficiencies, see Fig. 1.
expression for the maximum efficiency η˜, Close to maximum power, we may expand around y =1
√
" #," to find η˜ ’ η∗(1 + (T /T )1/4 1−y). The efficiency
√ √ c h
η˜= 2y(1− α)2 1−α+y(1− α)2 is thus nonanalytic with a diverging slope at this point.
The physical consequence of this behavior is that even
r (cid:16) √ (cid:17)# slightlyreducingthepowerbelowthemaximalvaluewill
− (1−y) y(1− α)4+(1−α)2 , (11) yield disproportionately large gains in efficiency. This
effect is most pronounced for small temperature differ-
where we have defined α=Tc/Th and y =P0/P∗ . The ences, Tc/Th ≈ 1. This situation, where the overall ef-
maximum power P∗ is given by, ficiency is small, is often encountered in practice, since
large temperature gradients are hard to maintain in mi-
P∗ = (√γ γ+hγ√c γ )2hTh+Tc−2pThTci. (12) croscopic systems. In this regime, it is thus possible to
h c significantly increase the efficiency of the heat engine by
Equation (11)isoursecondmainresult. Itexpressesthe sacrificing a relatively small amount of power.
maximum efficiency η˜as a function of the output power Figure 2 presents the optimal protocols (9a) for fixed
P (relative to the maximal value P∗) and the temper- power P and maximum power P∗. Their qualitative
0 0
ature ratio α. For vanishing power, y = 0, η˜ reduces behavior may be easily understood. During the cou-
to the Carnot efficiency, η = 1−T /T . On the other pling to the hot bath (red shaded area), the frequency
C c h
hand,formaximalpower,y =1,weobtaintheefficiency, ω of the harmonic potential is reduced exponentially.
p
η∗ =1− T /T . This is precisely the Curzon-Ahlborn This step allows the particle to explore a larger space,
c h
efficiency η found for an endoreversible heat engine similar to the expansion phase of a standard heat en-
CA
withlinearheatconduction[2]. TheunderdampedBrow- gine. The continuous exponential expansion is followed
nian heat engine is an ideal realization of this type of by an instantaneous expansion step (jump) immediately
engine: Due to the small dissipation, the engine is en- after the coupling to the cold bath. During the cold
doreversible, and the expansion for small γ ensures lin- phase (blue shaded area), the frequency increases expo-
ear coupling to the heat bath. We emphasize that this nentially,againfollowedbyaninstantaneouscompression
4
step. Thisbehavioristhesameirregardlessofthepower.
However, the precise values ω of the trap frequency do
i
depend on the power. The discontinuities in the opti-
mal protocols also occur in the overdamped regime [7].
They have their physical origin in the sudden tempera-
ture change when switching between the heat baths.
Discussion. Within the underdamped limit, the parti-
cleisabletoadjustinstantaneouslytochangesinthetrap
frequency. Thisisofcourseanapproximationforanyreal
physicalsystem. Inordertoverifyitsvalidity,wenumer-
ically solve the full dynamics (2), for arbitrary γ, using
theoptimalprotocolsobtainedintheunderdampedlimit,
as shown in Fig. 3. For comparison, we also include the
results from the overdamped case obtained by Schmiedl
and Seifert [7] (see also Supplemental Material). In the
underdamped regime, the optimal protocol (9) leads to FIG. 3. (color online) Efficiency (top) and power (bottom)
values of efficiency and power that are close to Eqs. (11) as a function of the ratio ω/γ. The dashed lines indicate
and (12) for ω (cid:38) 10γ. For any finite frequency ω, the thetheoreticalresultsEq.(11)fortheefficiencyatmaximum
underdamped description (4) requires that the particle power η∗ for y = 1 and Eq. (12) for the maximum power
P∗ (for the corresponding overdamped results, see Ref. [7]
is able to follow variations in the protocol. As a result,
and Supplemental Material). The symbols are obtained by
the rate of change of the frequency cannot be arbitrarily
numerically solving Eq. (2) using the underdamped (black
fast. This leads to the condition ω˙/ω (cid:28) ω, which we
squares, Eq. (9)) respectively overdamped (red circles, [7])
take into account by replacing the instantaneous jumps optimal protocols. For the maximum power, the theoretical
at τh and τh +τc with a linear variation of length τ∆, value is obtained both for the underdamped ω (cid:29) γ and the
such that ω˙/ω < ω/10. This provides results in good overdamped ω (cid:28) γ limit. While the efficiency is correctly
agreement with the analytical predictions. As the ratio predictedbyEq.(11)intheunderdampedlimit,theobserved
value in the overdamped case is lower than predicted in the
ω/γ decreases,bothpowerandefficiencystarttodropas
usual overdamped limit [7].
the underdamped approximation is no longer valid. For
ω/γ <1, the overdamped optimal protocols yield higher
power output which approaches the predicted maximum
valueforω (cid:46)0.1γ. Theefficiencyatmaximumpowerfor protocols [10]. In the overdamped regime, however, only
verysmalltemperaturedifferencesareachievable. Apos-
the full simulated dynamics, however, remains smaller
sible trick to reach fast switching with a large tempera-
than predicted in the overdamped limit, even at very
ture difference is to control the effective temperature via
small ratios of ω/γ. The reason for this deviation is the
an artificially applied force noise [9]. For the purpose
factthattherelaxationofthevelocity,whichisneglected
of feedback cooling, external optical or electric drivings
in the overdamped definition of heat, leads to increased
havebeenusedinavarietyofexperimentswithlevitated
dissipation which does not vanish when taking the over-
nanoparticles [24–28]. An experimental demonstration
damped limit [7, 17]. In the intermediate regime, γ ∼ω,
may thus be readily implemented in these setups.
wherethecouplingtothebathisneitherweaknorstrong,
our numerical analysis shows that the optimal protocols Conclusions. We have optimized an underdamped
derivedinbothunderdampedandoverdampedlimitfail. stochastic harmonic heat engine for maximum efficiency
Theexplicitoptimalprotocolsareherenotknowndueto at fixed power. We have analytically determined the op-
the increased complexity of the optimization procedure timal driving protocols for all values of the power. They
[11–13]. differ from those obtained in the overdamped limit, but
Experimental considerations. Our theoretical find- like the former exhibit jumps that mirror the sudden
ingsmightbeverifiedusingamicromechanicalharmonic change in temperature. We have demonstrated that the
oscillator with controllable spring constant and tem- Curzon-Ahlborn efficiency is the universal efficiency at
perature. A possible candidate system in the under- maximumpowerduetothelinearheattransportimplicit
dampedregimeislevitatedcavityoptomechanics[19–23] in the weak coupling limit. We have further shown that
where the parameters can essentially be tuned indepen- reducing the power slightly below its maximum value
dently.The environement determines the inital tempera- leads to disproportionately large gains in efficiency. Our
ture and Stokes friction for the particle. The effective findings confirm that maximizing the efficiency at maxi-
temperatureofthebathsisthencontrolledviatheintra- mumpowerisnotalwaysthebestwaytodesignmachines
cavity light field and the spring constant via the power with both high efficiency and high power.
of the optical trap. The variation of the latter two pa- Acknowledgments. This work was partially supported
rameters is sufficiently fast to implement nearly optimal bytheEUCollaborativeProjectTherMiQ(GrantAgree-
5
ment 618074) and the COST Action MP1209. A.D. was Overdamped optimization
partially employed as an International Research Fellow
of the Japan Society for the Promotion of Science. Intheoverdampedlimit, themaximumpowerandthe
corresponding efficiency have been obtained in Refs. [7].
The optimal protocols are, in terms of σ ,
x
SUPPLEMENTAL MATERIAL
(cid:16) (cid:17)2
Underdamped optimization σh 1+ τthσhσ+c√−σσhhσc for0<t<τh
σ (t)=
x (cid:16) (cid:17)2
The central insight for solving the Euler-Lagrange σc 1+ t−τcτhσcσ+h√−σσhcσc for0<t−τh <τc.
(16)
equations(8)inthemaintextcomesfromEq.(8b),which
shows that ασv is constant. Since neither α ≡ 0 nor Here σx is a continuous function, whereas the resulting
σv ≡0 are viable solutions, we may set α=−C/σv with frequencyprotocolω(t),determinedfromEq.(3),isgen-
some C 6= 0. We then find from Eqs. (8a) and (8c), erallydiscontinuous. Maximizingthepowerwithrespect
p
σv = CγT/m, implying that σv is constant. The only to τh and τc yields the equations [7],
waytohaveaconstantσ fromEq.(8c)iswithanexpo-
Anecnctoiradlipnrgolyt,ocol ω(t)=ωv0eκt with κ=γ(1−T/(mσv)). P∗ = 16(cid:0)γ +γ +ln22√(cid:0)γσσhcγ(cid:1)(T(cid:1)(cid:0)hσ−+Tcσ)2 −2√σ σ (cid:1) (17)
h c h c h c h c
ω(t)=ωωhceeγγch(cid:0)(cid:0)11−−mmTTσcσhch(cid:1)(cid:1)(tt−τh) ffoorr00<<tt−<ττhh<τc. η∗ = 2(cid:16)(cid:16)11++qqγγγγhchc(cid:17)(cid:17)−(cid:16)1(cid:16)−1−TThcTT(cid:17)hc(cid:17). (18)
(13)
Contrary to the underdamped limit, there exist no opti-
The exponential form of the protocols (13) has been ob- malvaluesforσh andσc. ThiscanbeseenfromEq.(17);
tainedinRef.[29]. However,thecrucialpointfortheop- themaximumpowerdivergesifoneofthetwoparameters
timization of a periodically operating engine, overlooked tends to zero while the other one stays constant. Phys-
in Ref. [29], is the implementation of the proper bound- ically, this limit corresponds to having a very large fre-
aryconditions. Theparametersω andσ areindeednot quency,atwhichpointtheoverdampedlimitisnolonger
i i
independent, but are fixed by the boundary conditions valid. Thus within the overdamped approximation, σh
at t=τh and t=τh+τc, in contrast to the overdamped and σc are free parameters that have to be chosen such
case (see below) In principle, the protocol ω(t) can be that the condition ω (cid:28)γ remains valid.
continuous or discontinuous at these points. Multiplying ComparingthemaximalvaluesforthepowerEqs.(12)
Eq. (4) by ω and integrating the resulting equation from and (17), we see that in the underdamped limit, the
τ −(cid:15)/2 to τ +(cid:15)/2, with (cid:15) small, we find, power is proportional to the overall damping strength,
h h
whereas it is inversely proportional in the overdamped
Z τh+2(cid:15) (cid:2) (cid:3) regime. Since the damping is assumed to be small in the
dt σ˙ ω−σ ω˙ =O(δ). (14)
v v
τh−2(cid:15) underdampedlimitandlargeintheoverdampedone,the
power itself is small in both limits, owing to the asymp-
If we allow discontinuities, the frequency ω jumps from
totic nature of the treatment. Physically, the small re-
the value ω = ω exp(γ (1−T /(mσ ))τ ) at t = τ−
− h h h h h h sultantpowerhasdifferentreasons: Intheunderdamped
to ω =ω at t=τ+. At the same time, σ jumps from
+ c h v limit, the particle exchanges energy with the bath at a
σ =σ toσ =σ . Closetot=τ ,wemaythuswrite,
− h + c h small rate, limiting the total transferred energy per cy-
ω(t)=ω +(ω −ω )θ(t−τ ), (15) cle. In the overdamped limit, the energy exchange with
− + − h
⇒ω˙(t)=(ω −ω )δ(t−τ ), the bath is fast, however the reaction to changes in the
+ − h
external potential is slow.
and similarly for σ . Plugging the above expressions
v
into Eq. (14), we obtain the condition σ ω = σ ω .
+ − − +
By contrast, if we demand that ω should be continuous,
ω =ω ,thesamehasalsotobetrueforσ . However,if
+ − v
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