nonuniversality of heat engine efficiency at maximum power

Nonuniversality of heat engine efficiency at maximum power

in�collaboration�with�엄재곤�(서울대학교)�&�박형규�(고등과학원)

2017년�가을��학술논문발표회�[D2.06]�@�10월�26일,�경주화백컨벤션센터

이상훈�고등과학원�물리학부�

http://newton.kias.re.kr/~lshlj82

quasi-static

(reversible) engine

the Carnot e�ciency ⌘C =Weng

|Qh|=

|Qh|� |Qc||Qh|

= 1� Tc

Thquasi-static

(the 1st law of thermodynamics)

0: cyclic process

the 2nd law of thermodynamics: �Stot = �Seng +�Sres = �Qh

Th+

Qc

Tc� 0

(per cycle)

! |Qc||Qh|

� Tc

Th! ⌘ = 1� |Qc|

|Qh| 1� Tc

Th= ⌘C

) ⌘ ⌘C in general, and ⌘C is the theoretically maximum e�ciency.

source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg

Carnotthe

Weng reaches the maximum value for given |Qh| in the Carnot engine,but the power P = Weng/⌧ ! 0 where ⌧ is the operating time ! 1

quasi-static

(reversible) engine

the Carnot e�ciency ⌘C =Weng

|Qh|=

|Qh|� |Qc||Qh|

= 1� Tc

Thquasi-static

(the 1st law of thermodynamics)

0: cyclic process

the 2nd law of thermodynamics: �Stot = �Seng +�Sres = �Qh

Th+

Qc

Tc� 0

(per cycle)

! |Qc||Qh|

� Tc

Th! ⌘ = 1� |Qc|

|Qh| 1� Tc

Th= ⌘C

) ⌘ ⌘C in general, and ⌘C is the theoretically maximum e�ciency.

source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg

Carnotthe

endoreversible engine

the (Chambadal-Novikov-)Curzon-Ahlborn e�ciency ⌘CA = 1�r

Tc

Th

“endoreversibility”

Th

Tc

hot reservoir

cold reservoir

Thw

Tcw

during t1

irreversible heat conduction

the input energy (linear heat conduction) Qh = ↵t1(Th � Thw)

the reversible engineoperated at Thw and Tcw!

Qh

Thw=

Qc

Tcw

during t2

irreversible heat conduction

the heat rejected (linear heat conduction) Qc = �t2(Tcw � Tc)

maximizing power P =Qh �Qc

t1 + t2

• P. Chambadal, Les Centrales Nuclaires (Armand Colin, Paris, 1957). • I. I. Novikov, Efficiency of an atomic power generating installation, At. Energy 3, 1269 (1957);

The efficiency of atomic power stations, J. Nucl. Energy 7, 125 (1958). • F. L. Curzon and B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43, 22 (1975).3/31/16, 12:03Endoreversible thermodynamics - Wikipedia, the free encyclopedia

Page 2 of 3https://en.wikipedia.org/wiki/Endoreversible_thermodynamics

Power Plant (°C) (°C) (Carnot) (Endoreversible) (Observed)West Thurrock (UK) coal-fired power

plant 25 565 0.64 0.40 0.36

CANDU (Canada) nuclear power plant 25 300 0.48 0.28 0.30Larderello (Italy) geothermal power

plant 80 250 0.33 0.178 0.16

As shown, the endoreversible efficiency much more closely models the observed data. However, such anengine violates Carnot's principle which states that work can be done any time there is a difference intemperature. The fact that the hot and cold reservoirs are not at the same temperature as the working fluidthey are in contact with means that work can and is done at the hot and cold reservoirs. The result istantamount to coupling the high and low temperature parts of the cycle, so that the cycle collapses.[7] In theCarnot cycle there is strict necessity that the working fluid be at the same temperatures as the heat reservoirsthey are in contact with and that they are separated by adiabatic transformations which prevent thermalcontact. The efficiency was first derived by William Thomson [8] in his study of an unevenly heated body inwhich the adiabatic partitions between bodies at different temperatures are removed and maximum work isperformed. It is well known that the final temperature is the geometric mean temperature so that

the efficiency is the Carnot efficiency for an engine working between and .

Due to occasional confusion about the origins of the above equation, it is sometimes named theChambadal-Novikov-Curzon-Ahlborn efficiency.

See alsoHeat engine

An introduction to endoreversible thermodynamics is given in the thesis by Katharina Wagner.[4] It is alsointroduced by Hoffman et al.[9][10] A thorough discussion of the concept, together with many applications inengineering, is given in the book by Hans Ulrich Fuchs.[11]

References1. I. I. Novikov. The Efficiency of Atomic Power Stations. Journal Nuclear Energy II, 7:125–128, 1958. translated from

Atomnaya Energiya, 3 (1957), 409.2. Chambadal P (1957) Les centrales nucléaires. Armand Colin, Paris, France, 4 1-583. F.L. Curzon and B. Ahlborn, American Journal of Physics, vol. 43, pp. 22–24 (1975)4. M.Sc. Katharina Wagner, A graphic based interface to Endoreversible Thermodynamics, TU Chemnitz, Fakultät für

Naturwissenschaften, Masterarbeit (in English). http://archiv.tu-chemnitz.de/pub/2008/0123/index.html5. A Bejan, J. Appl. Phys., vol. 79, pp. 1191–1218, 1 Feb. 1996 http://dx.doi.org/10.1016/S0035-3159(96)80059-66. Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed. ed.). John Wiley &

Sons, Inc.. ISBN 0-471-86256-8.7. B. H. Lavenda, Am. J. Phys., vol. 75, pp. 169-175 (2007)8. W. Thomson, Phil. Mag. (Feb. 1853)

! !

❄ ❄


Tc

Th

Q. How universal is this?

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)✓

* ⌘C = 1� T2

T1

◆


Tc

Th

Q. How universal is this?

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)✓

* ⌘C = 1� T2

T1

◆

“tight coupling” between the thermodynamics fluxes: the heat flux is directly proportional to the work-generating flux ref) C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005).

tight coupling + symmetry between the reservoirs (“left-right” symmetry) ref) M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 102, 130602 (2009).

quantum dotlead 1 lead 2

µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

the quantum dot heat engine model

T1 > T2

µ1 < µ2

Nonuniversality of heat engine e�ciency at maximum power

Sang Hoon Lee,1 Jaegon Um,2, 3 and Hyunggyu Park1, 2

1School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea2Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea

3CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea

We study the e�ciency of a quantum dot engine in the condition of the maximum power output. In contrast tothe quasi-statically operated Carnot engine whose e�ciency reaches the theoretical maximum, recent researchon more realistic engines operated in finite time has revealed other classes of e�ciency such as the Curzon-Ahlborn e�ciency maximizing the power. The linear coe�cient of such power-maximizing e�ciency as afunction of the reservoir temperature ratio has been argued to be universal as 1/2 under the tight-couplingcondition between thermodynamic fluxes. By taking the quantum dot heat engine, however, we show thatdepending on the constraint posed on the engine, the linear coe�cient can be unity, which implies that thee�ciency at the maximum power actually approaches the Carnot e�ciency in the equilibrium limit. As a result,we dismiss the notion of universal linear coe�cient of the e�ciency at the maximum power, and discuss theimplication of such a result in terms of entropy production and irreversible thermodynamics. We claim that theparticular scheme for the linear coe�cient of unity is actually more realistic and experimentally realizable, as itcorresponds to controlling the gate voltage of the quantum dot, for given temperatures and chemical potentialsof the leads connected to the quantum dot.

I. INTRODUCTION

The e�ciency of heat engines is a quintessential topic ofthermodynamics [1]. In particular, an elegant formula ex-pressed only by hot and cold reservoir temperatures for theideal quasi-static and reversible engine coined by Sadi Carnothas been an everlasting textbook example [2]. That ideal en-gine, however, is not the most e�cient engine any more whenwe consider its power output (the extracted work per unittime), which has added di↵erent types of optimal engine e�-ciency such as the Curzon-Ahlborn (CA) e�ciency for somecases [3–5]. Following such steps, researchers have taken sim-ple systems to investigate various theoretical aspects of under-lying principles of macroscopic thermodynamic engine e�-ciency in details [6–12].

In this paper, we take a quantum dot heat engine composedof a single quantum dot connected to two leads with charac-teristic temperatures and chemical potentials [13–15] to elu-cidate the condition for the maximum power in details. Weanalyze more general cases than the previous works and findan intriguing result: when one of the two chemical potentialsis given, the quadratic coe�cient deviates from the conven-tional value 1/8, and when the di↵erence between the chem-ical potentials of the leads is given, the linear coe�cient alsodeviates from the value 1/2 that has been believed to be “uni-versal” for any tight-coupling engine. The latter case of givenchemical potential di↵erence, in particular, is relevant for ex-perimental realization as it corresponds to adjusting the gatevoltage of the quantum dot [16–18], which we believe is morepractically realizable than other cases where one has to controlchemical potentials of the leads.

More precisely, as the linear coe�cient in fact becomesunity for the case of given chemical potential di↵erence, thee�ciency at the maximum power for a quantum dot enginewith this constraint actually exceeds the tight-coupling limit,while the achievable maximum power can still be a signifi-cant fraction of the globally optimized engine for reasonableranges of temperature di↵erence and chemical potential dif-

ference. To further investigate the origin of e�ciency in theparticular form, we consider the ratio of entropy productionto the heat absorption, and take the viewpoint of irreversiblethermodynamic.

The rest of the paper is organized as follows. We introducethe autonomous quantum dot heat engine model and its mathe-matically equivalent non-autonomous two-level model of oursin Sec. II. The case of global optimization of power with re-spect to the parameter is presented in Sec. III. In Secs. IV andV, we present our main contribution of the optimization withvariable constraints and its resultant nonuniversal behavior ofe�ciency at the maximum power, in particular, for the caseof fixed chemical potential di↵erence in details (Sec. V). Weconclude with the summary and a remark on future work inSec. VI.

II. HEAT ENGINE MODELS

A. Quantum dot heat engine model

We take a quantum dot heat engine introduced in Ref. [13],which is composed of a quantum dot with the energy levelEQD where a single electron can occupy, in contact with twoleads, denoted by R1 and R2 at di↵erent temperatures (T1 >T2) and chemical potentials (µ1 < µ2 < EQD), respectively, asshown in Fig. 1, where we introduce the di↵erence betweenthe chemical potentials �µ = µ2 � µ1. The transition rates ofthe electron to the quantum dot from R1 and R2 are given asthe following Arrhenius form,

q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)

respectively (we let the Boltzmann constant kB = 1 for no-tational convenience), thus the inequality 0 < ✏ < q < 1/2holds (✏ < q is essential to get the positive amount of net

! ❄

#

ref) M. Esposito, K. Lindenberg, and C. Van den Broeck, Europhys. Lett. 85, 60010 (2009);M. Esposito, N. Kumar, K. Lindenberg, and C. Van den Broeck, Phys. Rev. E 85, 031117 (2012);R. Toral, C. Van den Broeck, D. Escaff, and K. Lindenberg, Phys. Rev. E 95, 032114 (2017).

#


µ2

T1 T2

q

q

✏✏

EG

EQD�µµ1

the quantum dot heat engine model

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


! ❄

ref) M. Esposito, K. Lindenberg, and C. Van den Broeck, Europhys. Lett. 85, 60010 (2009);M. Esposito, N. Kumar, K. Lindenberg, and C. Van den Broeck, Phys. Rev. E 85, 031117 (2012);R. Toral, C. Van den Broeck, D. Escaff, and K. Lindenberg, Phys. Rev. E 95, 032114 (2017).

Vg

gate voltage

Vsd

source-drain voltage

experimental realization: L. P. Kouwenhoven et al., Electron transport in quantum dots, Kluwer Series E345, 105–214, in Proceedings of the NATO Advanced Study Institute on Mesoscopic Electron Transport (Curaçao, Netherlands Antilles, 1997).


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

the efficiency:

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ

FIG. 1. A schematic illustration of the quantum dot heat engine com-posed of the quantum dot whose energy level EG is in contact withthe leads, which plays the role and heat and particle reservoirs withthe temperatures T1 and T2, and the chemical potentials µ1 and µ2.

of the electron to the quantum dot from R1 and R2 are givenas the following Arrhenius form,

q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)

respectively (we let the Boltzmann constant kB = 1 for no-tational convenience), thus the inequality 0 < ✏ < q < 1/2holds (✏ < q is essential to get the positive amount of network). We denote the probability of occupation in the quan-tum dot by Po and its complementary probability (of absence)by Pe = 1 � Po. The probability vector |Pi = (Po, Pe)T isdescribed by the master equation

d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)

With the normalization q + q = ✏ + ✏ = 1, the steady-statesolution is

Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)

where the relation to the energy variables is

q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)

The probability currents from R1 to the quantum dot and thatfrom the quantum dot to R2 are then,

I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)

respectively, and they are identical to each other, which repre-sents the conservation of the particle flux. From now on, wedenote this particle flux carrying the energy current by

J ⌘12

(q � ✏) , (7)

and it corresponds to thermodynamic flux, a cornerstone ofthe irreversible thermodynamics approach later.

The heat production rate to the quantum dot from R1 andthat from the quantum dot to R2 are

Q1 = JEQD ,

Q2 = J�EQD � �µ

�.

(8)

The total entropy production rate is given by the net entropychange rate of the leads,

S = �Q1

T1+

Q2

T2= JX , (9)

which is the product of the thermodynamics flux J in Eq. (7)and thermodynamics force X involving the temperature andchemical potential gradients, given by

X ⌘EQD � �µ

T2�

EQD

T1, (10)

where we divide the flux and force terms further by consider-ing the thermal term caused by the temperature gradient andmechanical term caused by the chemical potential gradientapart later. The amount of net power extracted by movingthe electron from the hot lead R1 to the cold lead R2 is thengiven by

W = Q1 � Q2 = J�µ , (11)

by the first law of thermodynamics, and the chemical potentialdi↵erence �µ will play the role of mechanical force responsi-ble for the work.

The e�ciency of the engine is, therefore, given by the ratio

⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)

and ⌘ approaches the Carnot e�ciency [1, 2],

⌘C = 1 �T2

T1, (13)

when ✏ ' q, and meaningful only for q > ✏, or P > 0, whichcorresponds to the actual heat engine that converts the heatdi↵erence to the positive net work.

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


the heat production rates the total entropy production rate

2


EQD

µ1µ2

T1 T2

q

q

�

�

FIG. 1. A schematic illustration of the quantum dot heat engine com-posed of the quantum dot whose energy level EQD in contact with theleads, which plays the role and heat and particle reservoirs with thetemperatures T1 and T2, and the chemical potentials µ1 and µ2.

work). We denote the probability of occupation in the quan-tum dot by Po and its complementary probability (of absence)by Pe = 1 � Po. The probability vector |Pi = (Po, Pe)T isdescribed by the master equation

d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�(EQD�µ1)/T1

1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = J�EQD � µ1

�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


B. Two-level heat engine model

The autonomous quantum dot heat engine introduced inSec. II A is in fact equivalent to a simpler non-autonomoustwo-level heat engine described in Fig. 2. The two-level sys-tem is characterized by two discrete energy states composedof the ground state (E = 0) and the excited state (E = E1 orE = E2, depending on the reservoir of consideration). Thetransition rates from the ground state to the excited state aredenoted by q and ✏, respectively, and their reverse processesby q and ✏. We assume E1 > E2 and T1 > T2. The systemis attached to two di↵erent reservoirs: R1 with temperature T1

the net power

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


(the thermodynamic force)

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)




(the thermodynamic flux)

in the steady state . . .

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)



µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

the efficiency:

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


tunabletunable

the heat production rates the total entropy production rate

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)




the net power

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


(the thermodynamic force)

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)




(the thermodynamic flux)

the e�ciency at the maximum power, ⌘op

in the steady state . . .

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


the power maximization with respect to both EQD and �µ@W

@EQD

��EQD=E⇤

QD,�µ=�µ⇤

=@W

@�µ

��EQD=E⇤

QD,�µ=�µ⇤

= 0

E⇤QD,�µ⇤

reported in M. Esposito et al., EPL 85, 60010 (2009) and their follow-up studies, but we have also independently (re)discovered this with the equivalent non-autonomous two-level heat engine system.ref) SHL, J. Um, and H. Park, e-print arXiv:1612.00518.

4

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

q* a

nd ε

*

ηc = 1 − T2 / T1

q*ε*

q*(ηc→0) = ε*(ηc→0)q*(ηc=1)

ηc→1 asymptote

FIG. 3. Numerically found q⇤ and ✏⇤ values satisfying Eq. (18), as afunction of ⌘C = 1�T2/T1, along with the q⇤(⌘C ! 0) = ✏⇤(⌘C ! 0)and q⇤(⌘C = 1) values presented in Sec. III B 2. ✏⇤(⌘C = 1) = 0 (thehorizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34).

schematically . . .

�

q

� = q

no net work

as �C is increased

q�(�C � 0) = ��(�C � 0) � 0.083 221 720 199 517 7

q�(�C = 1) � 0.217 811 705 719 800��(�C = 1) = 0

FIG. 4. Illustration of the optimal transition rates (q⇤, ✏⇤) for the max-imum power output as the T2/T1 value varies.

2. Asymptotic behaviors obtained from series expansion

The upper bound for q⇤ is given by the condition ⌘C = 1,satisfying ln[(1 � q⇤)/q⇤] = 1/(1 � q⇤) and q⇤(⌘C = 1) '0.217 811 705 719 800 found numerically and ✏⇤(⌘C = 1) = 0exactly from Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re-gardless of q⇤ values, so finding the optimal q⇤ is meaningless(in fact, when ⌘C = 0, the operating regime for the engineis shrunk to the line q = ✏ and there cannot be any positivework). Therefore, let us examine the case ⌘C ' 0 using theseries expansion of q⇤ with respect to ⌘C , as

q⇤ = q0 + a1⌘C + a2⌘2C + a3⌘

3C + O

⇣⌘4

C

⌘. (22)

Substituting Eq. (22) into Eq. (18) and expanding the left-handside with respect to ⌘C again, we obtain

2 � (1 � 2q0) ln[(1 � q0)/q0]2q0 � 1

⌘C

+q0(1 � q0) � 2a1(1 � 2q0)

2(1 � q0)q0(1 � 2q0)3 ⌘2C

+ c3(q0, a1, a2)⌘3C + O

⇣⌘4

C

⌘= 0 ,

(23)

where c3(q0, a1, a2) = [10q60 + 3a2

1 � 6q0(a21 + a2) � 6q5

0(5 +6a1+8a2)�12q3

0(1+6a1+16a21+9a2)+q2

0(1+18a1+132a21+

42a2)+q40(31+90a1+96a2

1+120a2)]/[6(1�2q0)5(1�q0)2q20].

Letting the linear coe�cient to be zero yields

21 � 2q0

= ln

1 � q0

q0

!, (24)

from which the lower bound for q⇤(⌘C ! 0) = q0 =✏⇤(⌘C ! 0) ' 0.083 221 720 199 517 7 found numerically[lim⌘C!0 U(⌘C , q⇤) = 1 � 2q⇤, thus ✏⇤(⌘C ! 0) = q⇤(⌘C ! 0)by Eq. (16b)]. Figure 3 shows the numerical solution (q⇤, ✏⇤)as a function of ⌘C , where the asymptotic behaviors derivedabove hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ ismonotonically increased and ✏⇤ is monotonically decreased,as ⌘C is increased, i.e., q⇤min = q⇤(⌘C ! 0), q⇤max = q⇤(⌘C = 1),✏⇤min = 0, and ✏⇤max = ✏

⇤(⌘C ! 0). Figure 4 illustrates the situ-ation on the (q, ✏) plane. The linear coe�cient a1 in Eq. (22)can be written in terms of q0 when we let the coe�cient of thequadratic term in Eq. (23) to be zero, as

a1 =q0(1 � q0)2(1 � 2q0)

. (25)

Similarly, the coe�cient a2 in Eq. (22) can also be written interms of q0 alone, by letting c3(q0, a1, a2) = 0 in Eq. (23) andusing the relations in Eqs. (24) and (25), as

a2 =7q0(1 � q0)24(1 � 2q0)

. (26)

With the relations of coe�cients in hand, we find theasymptotic behavior of ⌘op in Eq. (19) by expanding it withrespect to ⌘C after substituting q⇤ as the series expansion of⌘C in Eq. (22). Then,

⌘op =1

(1 � 2q0) ln[(1 � q0)/q0]⌘C

+

a1q0�3q2

0+2q30+

[q20+2a1�q0(1+4a1)] ln[(1�q0)/q0]

(1�2q0)3

ln2[(1 � q0)/q0]⌘2

C

+ d3(q0, a1, a2)⌘3C + O

⇣⌘4

C

⌘,

(27)

where d3(q0, a1, a2) = {2(1 � 2q0)2a1[q20 + 2a1 � q0(1 +

4a1)] ln[(1�q0)/q0]+2[�2q40+a1�4a2

1�2a2+4q0(4a21+3a2)+

4q30(1+a1+4a2)�2q2

0(1+3a1+8a21+12a2)] ln2[(1�q0)/q0]+(1�

2q0)4{�2a2

1+[(1�2q0)a21�2(1�q0)q0a2] ln[(1�q0)/q0]}}/[(1�

q20)2q2

0]. Using Eqs. (24), (25) and (26), Eq. (27) becomes sim-ply

⌘op =12⌘C +

18⌘2

C +7 � 24q0 + 24q2

0

96(1 � 2q0)2 ⌘3C + O

⇣⌘4

C

⌘. (28)

No closed-form solution, but with the series expansion at

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)

✓* ⌘C = 1� T2

T1

◆

⌘C ! 0

where2

1� 2q0= ln

✓1� q0q0

◆

reported in M. Esposito et al., EPL 85, 60010 (2009) and their follow-up studies, but we have also independently (re)discovered this with the equivalent non-autonomous two-level heat engine system.ref) SHL, J. Um, and H. Park, e-print arXiv:1612.00518.

4

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

q* a

nd ε

*

ηc = 1 − T2 / T1

q*ε*

q*(ηc→0) = ε*(ηc→0)q*(ηc=1)

ηc→1 asymptote

FIG. 3. Numerically found q⇤ and ✏⇤ values satisfying Eq. (18), as afunction of ⌘C = 1�T2/T1, along with the q⇤(⌘C ! 0) = ✏⇤(⌘C ! 0)and q⇤(⌘C = 1) values presented in Sec. III B 2. ✏⇤(⌘C = 1) = 0 (thehorizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34).

schematically . . .

�

q

� = q

no net work

as �C is increased

q�(�C � 0) = ��(�C � 0) � 0.083 221 720 199 517 7

q�(�C = 1) � 0.217 811 705 719 800��(�C = 1) = 0

FIG. 4. Illustration of the optimal transition rates (q⇤, ✏⇤) for the max-imum power output as the T2/T1 value varies.

2. Asymptotic behaviors obtained from series expansion

The upper bound for q⇤ is given by the condition ⌘C = 1,satisfying ln[(1 � q⇤)/q⇤] = 1/(1 � q⇤) and q⇤(⌘C = 1) '0.217 811 705 719 800 found numerically and ✏⇤(⌘C = 1) = 0exactly from Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re-gardless of q⇤ values, so finding the optimal q⇤ is meaningless(in fact, when ⌘C = 0, the operating regime for the engineis shrunk to the line q = ✏ and there cannot be any positivework). Therefore, let us examine the case ⌘C ' 0 using theseries expansion of q⇤ with respect to ⌘C , as

q⇤ = q0 + a1⌘C + a2⌘2C + a3⌘

3C + O

⇣⌘4

C

⌘. (22)

Substituting Eq. (22) into Eq. (18) and expanding the left-handside with respect to ⌘C again, we obtain

2 � (1 � 2q0) ln[(1 � q0)/q0]2q0 � 1

⌘C

+q0(1 � q0) � 2a1(1 � 2q0)

2(1 � q0)q0(1 � 2q0)3 ⌘2C

+ c3(q0, a1, a2)⌘3C + O

⇣⌘4

C

⌘= 0 ,

(23)

where c3(q0, a1, a2) = [10q60 + 3a2

1 � 6q0(a21 + a2) � 6q5

0(5 +6a1+8a2)�12q3

0(1+6a1+16a21+9a2)+q2

0(1+18a1+132a21+

42a2)+q40(31+90a1+96a2

1+120a2)]/[6(1�2q0)5(1�q0)2q20].

Letting the linear coe�cient to be zero yields

21 � 2q0

= ln

1 � q0

q0

!, (24)

from which the lower bound for q⇤(⌘C ! 0) = q0 =✏⇤(⌘C ! 0) ' 0.083 221 720 199 517 7 found numerically[lim⌘C!0 U(⌘C , q⇤) = 1 � 2q⇤, thus ✏⇤(⌘C ! 0) = q⇤(⌘C ! 0)by Eq. (16b)]. Figure 3 shows the numerical solution (q⇤, ✏⇤)as a function of ⌘C , where the asymptotic behaviors derivedabove hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ ismonotonically increased and ✏⇤ is monotonically decreased,as ⌘C is increased, i.e., q⇤min = q⇤(⌘C ! 0), q⇤max = q⇤(⌘C = 1),✏⇤min = 0, and ✏⇤max = ✏

⇤(⌘C ! 0). Figure 4 illustrates the situ-ation on the (q, ✏) plane. The linear coe�cient a1 in Eq. (22)can be written in terms of q0 when we let the coe�cient of thequadratic term in Eq. (23) to be zero, as

a1 =q0(1 � q0)2(1 � 2q0)

. (25)

Similarly, the coe�cient a2 in Eq. (22) can also be written interms of q0 alone, by letting c3(q0, a1, a2) = 0 in Eq. (23) andusing the relations in Eqs. (24) and (25), as

a2 =7q0(1 � q0)24(1 � 2q0)

. (26)

With the relations of coe�cients in hand, we find theasymptotic behavior of ⌘op in Eq. (19) by expanding it withrespect to ⌘C after substituting q⇤ as the series expansion of⌘C in Eq. (22). Then,

⌘op =1

(1 � 2q0) ln[(1 � q0)/q0]⌘C

+

a1q0�3q2

0+2q30+

[q20+2a1�q0(1+4a1)] ln[(1�q0)/q0]

(1�2q0)3

ln2[(1 � q0)/q0]⌘2

C

+ d3(q0, a1, a2)⌘3C + O

⇣⌘4

C

⌘,

(27)

where d3(q0, a1, a2) = {2(1 � 2q0)2a1[q20 + 2a1 � q0(1 +

4a1)] ln[(1�q0)/q0]+2[�2q40+a1�4a2

1�2a2+4q0(4a21+3a2)+

4q30(1+a1+4a2)�2q2

0(1+3a1+8a21+12a2)] ln2[(1�q0)/q0]+(1�

2q0)4{�2a2

1+[(1�2q0)a21�2(1�q0)q0a2] ln[(1�q0)/q0]}}/[(1�

q20)2q2

0]. Using Eqs. (24), (25) and (26), Eq. (27) becomes sim-ply

⌘op =12⌘C +

18⌘2

C +7 � 24q0 + 24q2

0

96(1 � 2q0)2 ⌘3C + O

⇣⌘4

C

⌘. (28)

different' 0.077 492

= 0.0625

tight coupling between the thermodynamics fluxes: the heat flux is directly proportional to the work-generating flux ref) C. Van den Broeck, PRL 95, 190602 (2005).

tight coupling + symmetry between the reservoirs (“left-right” symmetry) ref) M. Esposito, K. Lindenberg, and C. Van den Broeck, PRL 102, 130602 (2009).

No closed-form solution, but with the series expansion at

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)

✓* ⌘C = 1� T2

T1

◆

⌘C ! 0

where2

1� 2q0= ln

✓1� q0q0

◆

the power maximization with respect to a single parameter

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc

EQD = 1up to the ηC

2 orderηC/2 + ηC

2/8

FIG. 4. The e�ciency at the maximum power ⌘op(EQD) for EQD = 1and T2 = 1 as the function of the Carnot e�ciency ⌘C . The blackthick curve represents the e�ciency at the maximum power from thenumerically found value of ✏ that maximizes W, and the red curveshows the asymptotic behavior at ⌘C ! 0 up to the quadratic orderin Eq. (26). For comparison, we also plot the ⌘C/2 + ⌘2

C/8 curve.

with the expansion form

⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)

when ⌘C ' 0. As a result, ⌘op and ⌘CA share a very similarfunctional form for ⌘C . 1/2, as shown in Fig. 3. The thirdorder coe�cient (' 0.077 492) in Eq. (20), however, is dif-ferent from 1/16 for the ⌘CA. In other words, the deviationfrom ⌘CA for ⌘op enters from the third order that has not beentheoretically investigated yet. Indeed, ⌘op deviates from ⌘CAfor ⌘C & 1/2, until they coincide at ⌘C = 1. The asymptoticbehavior of ⌘op for ⌘C ! 1 is given by

⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤

max) ln[q⇤max(1 � q⇤

max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)

where qmax is the solution of

11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.

IV. LOCAL OPTIMIZATION FOR GIVEN ONE OF THEENERGY VARIABLES

A. The e�ciency at the maximum power

For a given set of temperature values T1 and T2, supposethe quantum dot energy and one of the chemical potential aregiven. We take the case of the fixed EQD value (so the fixed q

value accordingly) without loss of generality. With the sameprocedure as in Appendix A but with the single-valued func-tion optimization with respect to ✏, we obtain ⌘op(q,T1,T2) or

equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.

(26)Therefore, the linear coe�cient 1/2 is expected from the tight-coupling condition [21], but the quadratic coe�cient is in gen-eral di↵erent from the value 1/8 for the optimized case withrespect to both parameters. One can of course find the con-dition for the quadratic coe�cient to actually become 1/8,which is

EQD

T2tanh

EQD

2T2

!= 2 . (27)

It means that a certain value of EQD satisfying Eq. (27) with agiven temperature results in the coe�cient 1/8. We will meetthis condition again in Sec. IV B.

B. The irreversible thermodynamics approach

Let us take this problem in the viewpoint of irreversiblethermodynamics [22, 23]. The total entropy production ratein Eq. (9) can be written as

S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)

where the entropy production rate is composed of the sum ofthe products of flux and force as followings: the thermal flux

Jt = Q1 = JEQD , (29)

the thermal force representing the temperature gradient,

Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)

and the mechanical force representing the chemical potentialgradient,

X1 =�µ

T22, (32)

where the extra terms are chosen for the unit consistencywhere the force variables have the reciprocal of energy or tem-perature (as we set kB ⌘ 1) and the flux variables have theenergy or temperature unit [24]. Obviously, the product ofmechanical flux and force leads to the power

J1X1 = �Q1 � Q2

T2= �

J�µ

T2

by the energy conservation, Eq. (11). The condition Xt =X1 = 0 corresponds to the thermal and mechanical equilib-rium state, and we take a perturbative approach from that equi-librium point.


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


tunable

@W

@�µ

��µ=�µ⇤

= 0

! a single-parameter (�µ) case

the fixed-EQD case: controlling the source-drain voltage, or �µ

for given T2: T1 = T2/(1� ⌘C)


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


the tight-coupling condition

6=18(no reserv

oir symmetry?)


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


tunable

@W

@�µ

��µ=�µ⇤

= 0

! a single-parameter (�µ) case

the fixed-EQD case: controlling the source-drain voltage, or �µ


the (linear) irreversible thermodynamics approach

the total entropy production rate

5

Finally, by substituting Eq. (30) to Eq. (27), we obtain thedi↵erence between the Carnot e�ciency at the e�ciency atthe maximum power from Eq. (26) as

⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)

which is of course consistent with Eq. (24).

C. The irreversible thermodynamics approach


S = JX = JtXt + J1X1 , (32)

where the entropy production rate is composed of the sum ofthe products of flux and force. Specifically, the thermal flux

Jt = J�EQD � µ1 � �µ

�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)

where the extra terms are chosen for the unit consistencywhere the force variables have the reciprocal of energy or tem-perature (as we set kB ⌘ 1) and the flux variables have theenergy or temperature unit [23].

For the linear regime where ⌘C ! 0 and �µ! 0,

e�(EQD�µ1��µ)/T2 'e�(EQD�µ1)/T1

⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)

where ⇠ = Jt/J1 = �T1(EQD � µ1 � �µ)/T 22 = �(EQD � µ1 �

�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �

µ1 � �µ)/T2 as ⌘C ! 0. Therefore, it can be described bythe linear irreversible thermodynamics with the tight-couplingcondition with ⇠ [21], which guarantees the linear coe�cient1/2 in Eq. (24). Let us explicitly show that here. If we applythe tight-coupling condition Jt / J1, i.e., Jt = ⇠J1 with theproportionality coe�cient ⇠, Eq. (32) is rewritten as

S = J1 (X1 + ⇠Xt) , (39)

where the stalling condition corresponds to X1 = T1⌘C , whichleads to the vanishing entropy production rate. The net poweroutput can also be written in terms of these coe�cients as

P = J�µ = �J1X1T1 . (40)

The basic assumption of the linear irreversible thermody-namics is the following Onsager relation [21, 22, 24]

JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)

with the Onsager reciprocity Lt1 = L1t. The aforementionedproportionality relation Jt = ⇠J1 implies Ltt/Lt1 = L1t/L11 =⇠, so

J1 = L (X1 + ⇠Xt) , (42)

with the proportionality constant L = q(1 � q) from Eq. (38).The optimal power output condition with respect to the me-chanical force X1 (introduced in Ref. [21] as well) is then

dPdX1

��X1=X⇤1

= 0 , (43)

With Eqs. (36) and (40), for a given Xt value,

X⇤1 = �12⇠Xt , (44)

and the e�ciency at the optimal power output

⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)

as ⌘C ! 0.As we have discussed in Sec. IV A, the condition for the

particular q value that actually makes the quadratic coe�cientis given by Eq. (25). It can also be shown that the conditionis equivalent to the “energy-matching condition” described inRef. [22], which states that if we expand J1 up to the quadraticterms as

J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)

the coe�cient � should be given by

�⇠

T1= 1 + O (⌘C) , (47)

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)



The autonomous quantum dot heat engine introduced inSec. II A is in fact equivalent to a simpler non-autonomoustwo-level heat engine described in Fig. 2. The two-level sys-tem is characterized by two discrete energy states composedof the ground state (E = 0) and the excited state (E = E1 orE = E2, depending on the reservoir of consideration). Thetransition rates from the ground state to the excited state aredenoted by q and ✏, respectively, and their reverse processesby q and ✏. We assume E1 > E2 and T1 > T2. The systemis attached to two di↵erent reservoirs: R1 with temperature T1the thermodynamic flux

the thermodynamic force

2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


the thermal flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)

the thermal force: the temperature gradient

the mechanical flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)

the mechanical force: the chemical potential gradient

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


Xt = X1 = 0: the thermal andmechanical equilibrium state

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


W = �T2J1X1 , (33)

as in Eq. (11). The condition Xt = X1 = 0 corresponds tothe thermal and mechanical equilibrium state, and we take aperturbative approach from that equilibrium point.

the power

ref) S. Sheng and Z. C. Tu, PRE 89, 012129 (2014); PRE 91, 022136 (2015).

heat part

work part



5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)





2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


the thermal flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


the mechanical flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2



4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


W = �T2J1X1 , (33)


the power

5

For the linear regime where Xt ! 0 and X1 ! 0, i.e., ⌘C !

0 and �µ! 0, expansions of

q =e�EQD/T e

EQDXt

1 + e�EQD/T eEQDXt

, ✏ =e�EQD/T e

T X1

1 + e�EQD/T eT X1,

lead to the flux Eq. (31), J1 = (1/2)(✏ � q)T , where we havedropped the subscript in the temperature T2, given by

J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (33)

where

L =T

2e�EQD/T

2�1 + e�EQD/T

�2 , (34)

⇠ = �EQD/T , (35)� = (T/2) tanh

⇥EQD/ (2T )

⇤. (36)

Since the fluxes satisfy the tight-coupling condition,

Jt/J1 = �EQD/T = ⇠ , (37)

one can construct the following Onsager matrix [25]

Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (38)

which guarantees the linear coe�cient 1/2 in Eq. (26) [21] .For further coe�cients, we optimize the power

W = �T LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤

with respect to the mechanical force X1 as

dW

dX1

��X1=X

⇤

1

= 0 ,

which gives the optimal X⇤

1 upto the second order of Xt

X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (39)

Since the e�ciency in terms of the thermodynamic fluxes andforces defined in Eq. (28) is given by

⌘ = �J1X1T

Jt

= �X1T

⇠, (40)

plugging X⇤

1 Eq. (39) to Eq. (40) yields

⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T⌘2

C. (41)

As we have discussed in Sec. IV A, the condition for the par-ticular energy value EQD that actually makes the quadratic co-e�cient is given by Eq. (27). It can also be shown that thecondition is equivalent to the “energy-matching condition”described in Ref. [23], which states that the coe�cient � inEq. (36) should be given by

�⇠

T= �1 + O (⌘C) , (42)

for the e�ciency at the maximum power to have the quadraticcoe�cient 1/8 with respect to ⌘C [26]. From Eqs. (35) and(36), one can easily see that the condition in Eq. (42) is equiv-alent to the one in Eq. (27).

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

ηop

(∆µ

)

ηc

∆µ = 1ηC

up to the ηC2 order

ηC/2

FIG. 5. The e�ciency at the maximum power ⌘op(�µ) for �µ = 1and T2 = 1 as the function of the Carnot e�ciency ⌘C . The blackthick curve represents the e�ciency at the maximum power from thenumerically found value of EQD that maximizes W, and the red curveshows the asymptotic behavior at ⌘C ! 0 up to the quadratic orderin Eq. (48). We also plot the ⌘C and ⌘C/2 lines for comparison.

V. OPTIMIZATION FOR FIXED CHEMICAL POTENTIALDIFFERENCE


Let us take another case, where the di↵erence of chemicalpotentials of the two leads �µ is given. This condition corre-sponds to controlling only the EQD value for fixed µ1 and µ2.The case is easily realizable for a quantum dot engine wherethe source-drain voltage is fixed while the gate voltage is ad-justed to maximize the power [17–20], in contast to the caseof given quantum dot energy in Sec. IV mhere the maximumpower is obtained by adjusting the source-drain voltage.

In this case, the system still has a single free parameter EQD,but we will show that the e�ciency at the maximum powertakes a completely di↵erent form, in contrast to the situationwhere we fix q in Sec. IV. For the sake of convenience werewrite the power Eq. (11) in terms of energies,

W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2

1 + e�EQD/T2 e�µ/T2

!�µ . (43)

By optimizing the power with respect to EQD, i.e.,

@W

@EQD

��EQD=E

⇤

QD

= 0 , (44)

we obtain the equation for E⇤

QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (45)

For the asymptotic behavior at ⌘C ! 0, or equivalentlyT2/T1 ! 1�, where accordingly E

⇤

QD ! 1, if we keep thelowest order terms of e

�E⇤

QD/T2 , the equation becomes⇣e�E⇤

QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (46)

the linear regime where ⌘C ! 0 and �µ ! 0(a perturbative approach from Xt = X1 = 0)


the linear irreversible thermodynamics with the tight-coupling condition with

5

We assume the situation of a given value of T2 as in Sec. IV Aand T1 is controlled by the thermal force term in Eq. (30). Theparticular choice of terms depending on T2 are chosen for theunit consistency where the force variables have the reciprocalof energy or temperature (as we set kB ⌘ 1) and the flux vari-ables have the energy or temperature unit [24]. Accordingly,the product of mechanical flux and mechanical force leads tothe power

W = �T2J1X1 , (33)

consistent with Eq. (11). The condition Xt = X1 = 0 corre-sponds to the thermal and mechanical equilibrium state, andwe take a perturbative approach from that equilibrium point.


0 and �µ! 0, the series expansions of the exact relations

q =e�EQD/T2 e

EQDXt

1 + e�EQD/T2 eEQDXt

, ✏ =e�EQD/T2 e

T2X1

1 + e�EQD/T2 eT2X1,

lead to the mechanical flux in Eq. (31), given by

J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)

one can construct the following Onsager matrix [25] for therelation,

Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)

which guarantees the linear coe�cient 1/2 in Eq. (26) as pre-sented in Ref. [21] .

For further coe�cients, we optimize the power

W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,


1 up to the quadratic order of Xt as

X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤

1 in Eq. (40) to Eq. (41) yields

⌘op(⇠, �,T2, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)




Let us take another case, where the di↵erence �µ of chem-ical potentials of the two leads is given. This condition cor-responds to controlling only the EQD value for fixed µ1 andµ2. The case is easily realizable for a quantum dot enginewhere the source-drain voltage is fixed, while the gate volt-age is adjusted to maximize the power [17–20], in contrast tothe case of given quantum dot chemical potential di↵erence inSec. IV where the maximum power is obtained by adjustingthe source-drain voltage.

In this case, the system has a single free parameter EQD asin �µ of Sec. IV, but we will show that the e�ciency at the

heat part

work part



5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)





2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


the thermal flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


the mechanical flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2



4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


W = �T2J1X1 , (33)


the power

5



q =e�EQD/T e

EQDXt


, ✏ =e�EQD/T e

T X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (33)

where

L =T

2e�EQD/T

2�1 + e�EQD/T

�2 , (34)

⇠ = �EQD/T , (35)� = (T/2) tanh

⇥EQD/ (2T )

⇤. (36)


Jt/J1 = �EQD/T = ⇠ , (37)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (38)


W = �T LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (39)


⌘ = �J1X1T

Jt

= �X1T

⇠, (40)

plugging X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T⌘2

C. (41)


�⇠

T= �1 + O (⌘C) , (42)


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2






W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (43)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (44)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (45)


⇤


�E⇤


QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (46)

the linear regime where ⌘C ! 0 and �µ ! 0(a perturbative approach from Xt = X1 = 0)


the linear irreversible thermodynamics with the tight-coupling condition with

5


W = �T2J1X1 , (33)




q =e�EQD/T2 e

EQDXt


, ✏ =e�EQD/T2 e

T2X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)



W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤


⌘op(⇠, �,T2, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)






heat part

work part

5



q =e�EQD/T e

EQDXt


, ✏ =e�EQD/T e

T X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (33)

where

L =T

2e�EQD/T

2�1 + e�EQD/T

�2 , (34)

⇠ = �EQD/T , (35)� = (T/2) tanh

⇥EQD/ (2T )

⇤. (36)


Jt/J1 = �EQD/T = ⇠ , (37)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (38)


W = �T LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (39)


⌘ = �J1X1T

Jt

= �X1T

⇠, (40)

plugging X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T⌘2

C. (41)


�⇠

T= �1 + O (⌘C) , (42)


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2






W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (43)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (44)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (45)


⇤


�E⇤


QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (46)

5


W = �T2J1X1 , (33)




q =e�EQD/T2 e

EQDXt


, ✏ =e�EQD/T2 e

T2X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)



W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤


⌘op(⇠, �,T2, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)






the maximum power output condition +O

⇣⌘3

C

⌘

the fixed-�µ case: controlling the gate voltage, or EQD


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtunable

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


! a single-parameter (EQD) case

5



q =e�EQD/T e

EQDXt


, ✏ =e�EQD/T e

T X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (33)

where

L =T

2e�EQD/T

2�1 + e�EQD/T

�2 , (34)

⇠ = �EQD/T , (35)� = (T/2) tanh

⇥EQD/ (2T )

⇤. (36)


Jt/J1 = �EQD/T = ⇠ , (37)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (38)


W = �T LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (39)


⌘ = �J1X1T

Jt

= �X1T

⇠, (40)

plugging X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T⌘2

C. (41)


�⇠

T= �1 + O (⌘C) , (42)


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2






W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (43)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (44)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (45)


⇤


�E⇤


QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (46)

6

leading to a closed form of solution in terms of ⌘C ,

E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)

Inserting Eq. (47) to the e�ciency, Eq. (12), finally we arriveat the e�ciency at the maximum power for given �µ,

⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)

In contrast to the case where EQD is given (Sec. IV), there-fore, even the linear coe�cient of unity deviates from 1/2,let alone the negative quadratic coe�cient, as numericallychecked in Fig. 5. This example clearly illustrates that the typeof restriction imposed in the two-parameter system to makethe system e↵ectively a one-parameter one is crucial, and thedi↵erence choice can make a completely di↵erent behavior ofe�ciency at the maximum power. In other words, the path tothe global optimization matters a lot. In particular, we havedemonstrated that the e�ciency at the optimal power outputapproximately approaches the theoretically maximal Carnote�ciency at the linear order for the fixed �µ case, in contrastto the half of Carnot e�ciency that has been believed to beuniversal for conventional tight-coupling engines. It wouldbe a crucial experiment to check if the quantum dot enginesuch as the one introduced in Refs. [13, 19, 20] actually shows⌘op ' ⌘C + O(⌘2

C) by optimizing the power only with respect

to the gate voltage.

B. The relation to the entropy production

The fact that ⌘op approaches ⌘C involves an important sin-gular behavior. Another way to express the second law ofthermodynamics is

⌘C � ⌘ =T2S

Q1� 0 , (49)

which demonstrates that the upper bound of the e�ciency ofany generic heat engine is given by ⌘C , from �S � 0 [27].If we substitute the explicit expressions Eq. (8) and (9) intoEq. (49), we obtain

T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)

The above relation indicates that if the energy ratio �µ/EQD isequal to ⌘C , the e�ciency of quantum dot engine can achievethe Carnot e�ciency, which means that the system is a re-versible engine.

Another possible way to get the Carnot e�ciency is puttingparticular energy values in order that T2X/EQD ! 0. Indeed,E⇤

QD(�µ) in Eq. (47) meets the condition as ⌘C ! 0 because

T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤

QD ⇡ �µ/⌘C , so that

⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)

Therefore, we confirm that ⌘op(�µ) ' ⌘C as ⌘C ! 0. Thisis the consequence of the vanishing scaled entropy S /J or theentropy production during 1/J as S /J ⇠ ⌘C SHL:' or /? andthe diverging of scaled heat as Q/J ⇠ 1/⌘C . SHL:' or /?

In contrast, for the case of a given value of EQD in Sec. IV,from Eq. (32) and (39), the series expansion form of �µ⇤ (thatachieves the maximum power) with respect to ⌘C is given by

�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)

Substituting Eq. (53) to Eq. (50), we obtain the di↵erence be-tween the Carnot e�ciency and the e�ciency at the maximumpower for this case from Eq. (49) as

⌘C � ⌘op(EQD) =T2S

Q1

��EQD,�µ=�µ⇤

'12⌘C , (54)

which is of course consistent with the previous result inSec. IV. In this case, since only the scaled entropy is decay-ing as S /J ⇠ ⌘C in the limit of ⌘C ! 0, ⌘op(EQD) has thecorrection in the linear order of ⌘C .


As in Sec. IV, let us consider this problem with irreversiblethermodynamics [22, 23]. The expression for the entropy pro-duction rate is again given by Eq. (28). However, in this case,we use di↵erent variables for mechanical flux and force, tobetter cope with this particular case of fixed �µ. For this, weuse the mechanical force defined as

X2 =1

EQD, (55)

which is a valid choice of control parameters here, and thecorresponding mechanical flux J2 reads

J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)

Using the same thermal flux Jt = J/X2 and force Xt = 1/T2 �

1/T1 in Eqs. (29) and (30), we get

S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)

Here, we take the limit of small force, Xt ! 0 and X2 ! 0like the case of fixed EQD. Note that in contrast to the casein Sec. IV B, the condition Xt = X2 = 0 does not correspondsto equilibrium because of the finite value of �µ. As Xt ! 0and X2 ! 0, equivalently T1 ! T2 = T and EQD ! 1,respectively, the entropy production rate Eq. (9) reads

S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)



the fixed-�µ case: controlling the gate voltage, or EQD


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtunable

T1 > T2

µ1 < µ2






I. INTRODUCTION









q/q = e�(EQD�µ1)/T1 ,

✏/✏ = e�(EQD�µ1��µ)/T2 ,(1)


! a single-parameter (EQD) case

5



q =e�EQD/T e

EQDXt


, ✏ =e�EQD/T e

T X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (33)

where

L =T

2e�EQD/T

2�1 + e�EQD/T

�2 , (34)

⇠ = �EQD/T , (35)� = (T/2) tanh

⇥EQD/ (2T )

⇤. (36)


Jt/J1 = �EQD/T = ⇠ , (37)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (38)


W = �T LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (39)


⌘ = �J1X1T

Jt

= �X1T

⇠, (40)

plugging X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T⌘2

C. (41)


�⇠

T= �1 + O (⌘C) , (42)


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2






W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (43)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (44)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (45)


⇤


�E⇤


QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (46)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

no “universal” linear and quadratic coefficients! what happens?! $



heat part

work part

the entropy production rate


7


The fact that ⌘op approaches ⌘C involves an important sin-gular behavior. In the fixed-�µ case, for the optimal valueof E2 = E⇤2, the relationship between the deviation from theCarnot e�ciency and the entropy production in Eq. (26) be-comes

T2SQ1=

E⇤2 � T2⇣E⇤2 + �µ

⌘/T1

E⇤2 + �µ. (57)

By using the condition for E⇤2 as ⌘C ! 0 given by Eq. (54),

E⇤2 'T2�µ

T1⌘C�

T2

⌘Cln (1 � ⌘C) , (58)

as ⌘C ! 0, so Eq. (57) becomes

⌘C � ⌘op =T2SQ1

��µ,E2=E⇤2

'T1⌘2

C(1 � ⌘C)�µ + T1⌘C(1 � ⌘C)

'T1

�µ⌘2

C ! 0 ,

(59)

as ⌘C ! 0 which is consistent with Eq. (56), so ⌘op ' ⌘C as⌘C ! 0 is confirmed. In other words, we achieve the Carnote�ciency at the maximum power by the vanishing �S anddiverging E⇤2 (and thus, diverging Q1) as ⌘C ! 0.


As in Sec. IV, let us consider this problem with irreversiblethermodynamics [21, 22]. The expression for the entropy pro-duction rate is again given by Eq. (32). However, in this case,we use di↵erent variables for mechanical flux and force, tobetter cope with this particular case of fixed �µ. For this, weuse the same thermal flux as in Sec. IV C,


�, (60)

the thermal force

Xt =⌘C

T2. (61)

In this case, however, the chemical potential di↵erence �µ isfixed, so X1 in Eq. (36) is not an adjustable mechanical force.Instead, we take the convention where the mechanical flux

J2 = �J�µ

T1

�EQD � µ1 � �µ

�, (62)

and the mechanical force

X2 =1

EQD � µ1 � �µ, (63)

which is a valid choice of control parameters here. Note thatX2 is technically not a genuine mechanical force, in the sensethat the condition X2 = 0 (or EQD ! 1) alone cannot achieve

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

P = −J

2X2T

1 w

ith T

1=1,

T2=

1/2

1/X2 = EQD − µ1 − ∆µ

∆µ = 1EQD − µ1 = 2

FIG. 6. The power as a function of 1/X2 = EQD � µ1 � �µ for thefixed-�µ (as �µ = 1) and fixed-q (as EQD � µ1 = 2) cases. The valueEQD � µ1 = 1/⌘C = 2 is set for the same starting point of nonzero Pas a function of 1/X2.

the equilibrium. As a result, the following discussion is nota truly physical linear irreversible thermodynamics approach,but we present its implication for better understanding of thesituation in an ad hoc fashion. Again, the entropy productionrate is given by

S = JX = JtXt + J2X2 . (64)

In this case, in contrast to Eq. (38) in Sec. IV, the linearorder is not the lowest order because of the exponential terms.For computational tractability, first let us suppose that we havea nonlinear leading term given by

J2 = L (X2 + ⇠Xt) Xa2 , (65)

where ⇠ = Jt/J2 = �T1, with the exponent a. When a = 0, werecover the linear irreversible thermodynamics characterizedby the Onsager relation, of course. The optimal power out-put condition with respect to the mechanical force X2 is againEq. (43). With Eqs. (40) and (63), for a given Xt value,

X⇤2 = �a + 1a + 2

⇠Xt , (66)


⌘op =�J⇤2X⇤2T1

T1Jt⇣1 � X⇤2/⇠

⌘ 'a + 1a + 2

Xt =a + 1a + 2

⌘C . (67)

The linear case a = 0 yields Eq. (45) for the tight-couplingheat engine [19] as shown in Sec. IV C. As we already know,the case of fixed �µ > 0 discussed so far involves the expo-nential terms, which would correspond to a! 1, or

⌘op ' ⌘C . (68)

Therefore, the result is again consistent with the previouslyobtained result in Eq. (56).

The drastically di↵erent behaviors for the fixed-�µ andfixed-q cases can be understood by examining the power as

cf) previously used X1 = �µ/T 22

cannot be used here, because �µ is fixed! X1 is not an adjustable force in turn

the thermal flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


the mechanical flux

the mechanical force

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)

which demonstrates that the upper bound of the e�ciency ofany generic heat engine is given by ⌘C , from �S � 0 [27].If we substitute the explicit expressions Eqs. (8) and (9) intoEq. (49), we obtain

T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (51)

while E⇤

QD ' �µ/⌘C , so that

⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)

Therefore, we confirm that ⌘op(�µ) ' ⌘C as ⌘C ! 0. Thisis the consequence of the vanishing scaled entropy S /J or theentropy production during 1/J as S /J / ⌘C and the divergingof scaled heat as Q/J / 1/⌘C .


�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)


⌘C � ⌘op(EQD,T1,T2) =T2S

Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

the power

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (48)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (49)






⌘C � ⌘ =T2S

Q1� 0 , (50)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (51)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (52)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (53)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (54)



Q1


'12⌘C , (55)




X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2. (57)


1/T1 in Eqs. (29) and (30), we obtain

S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (58)

and the power is again given by

W = �T2J2X2 . (59)

Here, we take the limit of small force, Xt ! 0 and X2 ! 0like the case of fixed EQD. Note that in contrast to the casein Sec. IV B, the condition Xt = X2 = 0 does not corresponds

the linear regime where ⌘C ! 0 and 1/EQD ! 0(a perturbative approach from Xt = X2 = 0)

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)





2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


the tight-coupling condition:

7

we have use di↵erent variables for mechanical flux and force,as X1 in Eq. (32) becomes constant in the case of fixed �µ,and thus it cannot be used as a legitimate mechanical forcevariable. Instead, we take the mechanical force defined as

X2 =1

EQD, (56)

which is a valid choice of control parameters in this case, andthe corresponding mechanical flux J2 reads

J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)

and the power is again given by the product of mechanical fluxand force as

W = �T2J2X2 . (58)


1/T1 in Eqs. (29) and (30), we recover the entropy production

S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)

In the case of given �µ, we take the limit of small force,Xt ! 0 and X2 ! 0 similar to the case of fixed EQD inSec. IV B. Note that in contrast to that previous case, the con-dition Xt = X2 = 0 does not corresponds to the equilibriumbecause of the finite value of �µ. As Xt ! 0 and X2 ! 0, orequivalently T1 ! T2 and EQD ! 1, the entropy productionrate in Eq. (9) becomes

S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)

which means that S goes to zero, but it is not exactly zero un-less �µ vanishes. Therefore, although our approach is dealingwith vanishing fluxes in the limit of Xt ! 0 and X2 ! 0, it isnot technically the conventional irreversible thermodynamicsused in Refs. [22, 23], which is the perturbation theory basedon the true equilibrium state. Nevertheless, in the following,we present the same type of irreversible thermodynamics anal-ysis and its implication for better understanding of the situa-tion, with proper justification of using it.

Let us start with the tight-coupling condition between Jt

and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)

implying that the reversible condition for Jt = J2 = J = 0is given by X2 = �⇠0Xt, which is also shown in the conditionS = 0 in Eq. (51) leading to ⌘C = �µ/EQD, or equivalentlyXtT2 = X1�µ. It can also be shown with the direct calcula-tion for the condition of J = 0 in Eq. (7). In other words, thecondition X2 = �⇠0Xt that makes S = J2 (X2 + ⇠0Xt) = 0 alsomakes Jt = J2 = J = 0. Then one can expect that the fluxJ2 contains the factor (X2 + ⇠0Xt) as a zero of J2, just as J1contains (X1 + ⇠Xt) in the case of given EQD value as shownin Eq. (34) in Sec. IV B (see also Fig. 6). This phenomenon ofvanishing fluxes in the reversible limit S = 0 naturally comes

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2 W

FIG. 6. The comparison between the fixed-EQD case (b) and thefixed-�µ case (b), in terms of the mechanical flux (we plot the neg-ative value of the flux for better visualization) and power, where weset T1 = 1 and T2 = 1/2. In panel (a), we fix EQD = 1 and plot themechanical flux J1 and the power W as the functions of the mechan-ical force X1. Both J1 and W vanish at X1 = �⇠Xt = EQD⌘C/T 2

2 = 2,but only W vanishes at X1 = 0. In panel (b), we fix �µ = 1 andplot the mechanical flux J2 and the power W as the functions of themechanical force X2. In this case, both J2 and W vanish at bothX2 = �⇠0Xt = ⌘C/�µ = 1/2 and X2 = 0.

from the Onsager symmetry and the tight-coupling conditionof proportional fluxes to each other, in the conventional irre-versible thermodynamics [22, 23, 28]. Therefore, althoughour derivation for fixed �µ is not technically conventional ir-reversible thermodynamics as we argued in the previous para-graph, there exists a reversible point where J2 = Jt = 0 andX2 = �⇠0Xt at that point, so we take the same formalism asirreversible thermodynamics to proceed further.

In this case, in contrast to Eq. (34), for a given value of Xt

the linear term of X2 is not the lowest order because of theexponential terms involved in the condition of the vanishingJ2 at X2 ! 0 as seen in the expression,

J2 = ��µ

2T2X2e�

1T2X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

1T2X2 , e

T2Xt�1T2X2

◆,

(62)which we obtain by substituting X2 = 1/EQD in Eq. (7).For computational tractability, first let us suppose a nonlinear

heat part

work part

the entropy production rate


7


The fact that ⌘op approaches ⌘C involves an important sin-gular behavior. In the fixed-�µ case, for the optimal valueof E2 = E⇤2, the relationship between the deviation from theCarnot e�ciency and the entropy production in Eq. (26) be-comes

T2SQ1=

E⇤2 � T2⇣E⇤2 + �µ

⌘/T1

E⇤2 + �µ. (57)

By using the condition for E⇤2 as ⌘C ! 0 given by Eq. (54),

E⇤2 'T2�µ

T1⌘C�

T2

⌘Cln (1 � ⌘C) , (58)

as ⌘C ! 0, so Eq. (57) becomes

⌘C � ⌘op =T2SQ1

��µ,E2=E⇤2

'T1⌘2

C(1 � ⌘C)�µ + T1⌘C(1 � ⌘C)

'T1

�µ⌘2

C ! 0 ,

(59)

as ⌘C ! 0 which is consistent with Eq. (56), so ⌘op ' ⌘C as⌘C ! 0 is confirmed. In other words, we achieve the Carnote�ciency at the maximum power by the vanishing �S anddiverging E⇤2 (and thus, diverging Q1) as ⌘C ! 0.


As in Sec. IV, let us consider this problem with irreversiblethermodynamics [21, 22]. The expression for the entropy pro-duction rate is again given by Eq. (32). However, in this case,we use di↵erent variables for mechanical flux and force, tobetter cope with this particular case of fixed �µ. For this, weuse the same thermal flux as in Sec. IV C,


�, (60)

the thermal force

Xt =⌘C

T2. (61)

In this case, however, the chemical potential di↵erence �µ isfixed, so X1 in Eq. (36) is not an adjustable mechanical force.Instead, we take the convention where the mechanical flux

J2 = �J�µ

T1

�EQD � µ1 � �µ

�, (62)

and the mechanical force

X2 =1

EQD � µ1 � �µ, (63)

which is a valid choice of control parameters here. Note thatX2 is technically not a genuine mechanical force, in the sensethat the condition X2 = 0 (or EQD ! 1) alone cannot achieve

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

P = −J

2X2T

1 w

ith T

1=1,

T2=

1/2

1/X2 = EQD − µ1 − ∆µ

∆µ = 1EQD − µ1 = 2

FIG. 6. The power as a function of 1/X2 = EQD � µ1 � �µ for thefixed-�µ (as �µ = 1) and fixed-q (as EQD � µ1 = 2) cases. The valueEQD � µ1 = 1/⌘C = 2 is set for the same starting point of nonzero Pas a function of 1/X2.

the equilibrium. As a result, the following discussion is nota truly physical linear irreversible thermodynamics approach,but we present its implication for better understanding of thesituation in an ad hoc fashion. Again, the entropy productionrate is given by

S = JX = JtXt + J2X2 . (64)

In this case, in contrast to Eq. (38) in Sec. IV, the linearorder is not the lowest order because of the exponential terms.For computational tractability, first let us suppose that we havea nonlinear leading term given by

J2 = L (X2 + ⇠Xt) Xa2 , (65)

where ⇠ = Jt/J2 = �T1, with the exponent a. When a = 0, werecover the linear irreversible thermodynamics characterizedby the Onsager relation, of course. The optimal power out-put condition with respect to the mechanical force X2 is againEq. (43). With Eqs. (40) and (63), for a given Xt value,

X⇤2 = �a + 1a + 2

⇠Xt , (66)


⌘op =�J⇤2X⇤2T1

T1Jt⇣1 � X⇤2/⇠

⌘ 'a + 1a + 2

Xt =a + 1a + 2

⌘C . (67)

The linear case a = 0 yields Eq. (45) for the tight-couplingheat engine [19] as shown in Sec. IV C. As we already know,the case of fixed �µ > 0 discussed so far involves the expo-nential terms, which would correspond to a! 1, or

⌘op ' ⌘C . (68)

Therefore, the result is again consistent with the previouslyobtained result in Eq. (56).

The drastically di↵erent behaviors for the fixed-�µ andfixed-q cases can be understood by examining the power as

cf) previously used X1 = �µ/T 22

cannot be used here, because �µ is fixed! X1 is not an adjustable force in turn

the thermal flux

5


⌘C � ⌘op(q) =T2SQ1

��q,�µ=�µ⇤

'12⌘C , (31)




S = JX = JtXt + J1X1 , (32)



�, (33)


Xt =1T2�

1T1=⌘C

T2, (34)

the mechanical flux

J1 = �JT 2

2

T1, (35)


X1 =�µ

T 22, (36)




⇥

1 �

EQD � µ1

T1⌘C +

�µ

T2

!,

(37)

which leads to

J1 ' q(1 � q)266664X1T 2

2 �T 2

2

T1

�EQD � µ1

�Xt

377775

= q(1 � q)"X1T 2

2 �T2

1 � ⌘C

�EQD � µ1

�Xt

#

' q(1 � q)hX1T 2

2 � T2�EQD � µ1 � �µ

�Xt

i

+ O⇣X2

t , X21 , XtX1

⌘

= q(1 � q)T 22 (X1 � ⇠Xt) + O

⇣X2

t , X21 , XtX1

⌘,

(38)


�µ)/[T2(1 � ⌘C)] ' �(EQD � µ1 � �µ)(1 + ⌘C)/T2 ' �(EQD �


S = J1 (X1 + ⇠Xt) , (39)


P = J�µ = �J1X1T1 . (40)


JtJ1

!=

Ltt Lt1L1t L11

! XtX1

!, (41)


J1 = L (X1 + ⇠Xt) , (42)


dPdX1

��X1=X⇤1

= 0 , (43)


X⇤1 = �12⇠Xt , (44)


⌘op =�J⇤1X⇤1T1

(T1Jt/T2)⇣1 � T2X⇤1/⇠

⌘ '12

T2Xt =12⌘C , (45)



J1 = Lh(X1 + ⇠Xt) + �X2

1 + O⇣X1Xt, X2

t

⌘i, (46)


�⇠

T1= 1 + O (⌘C) , (47)


the mechanical flux

the mechanical force

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in Fig. 3.





equivalently,

⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘.


EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22, (32)


J1X1 = �Q1 � Q2

T2= �

J�µ

T2


6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

the power

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (48)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (49)






⌘C � ⌘ =T2S

Q1� 0 , (50)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (51)




T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (52)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (53)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (54)



Q1


'12⌘C , (55)




X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2. (57)


1/T1 in Eqs. (29) and (30), we obtain

S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (58)

and the power is again given by

W = �T2J2X2 . (59)

Here, we take the limit of small force, Xt ! 0 and X2 ! 0like the case of fixed EQD. Note that in contrast to the casein Sec. IV B, the condition Xt = X2 = 0 does not corresponds

the linear regime where ⌘C ! 0 and 1/EQD ! 0(a perturbative approach from Xt = X2 = 0)

2


EQD

µ1µ2

T1 T2

q

q

�

�



d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)



1 + e�(EQD�µ1)/T1,

✏ =e�(EQD�µ1��µ)/T2

1 + e�(EQD�µ1��µ)/T2,

(4)

or

EQD � µ1 = T1 ln⇥(1 � q) /q

⇤,

EQD � µ1 � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)




�,

Q2 = J�EQD � µ1 � �µ

�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � µ1 � �µ

T2�

EQD � µ1

T1, (10)


P = Q1 � Q2 = J�µ , (11)



⌘ =P

Q1=

�µ

EQD � µ1= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)





2


µ1

µ2

T1 T2

q

q

��

EG

EQD�µ



q/q = e�EQD/T1 ,

✏/✏ = e�(EQD��µ)/T2 ,

(1)


d|Pidt=

�q � ✏ q + ✏q + ✏ �q � ✏

!|Pi . (2)


Po,ss =12

(q + ✏) ,

Pe,ss =12

(2 � q � ✏) ,(3)


q =e�EQD/T1

1 + e�EQD/T1,

✏ =e�(EQD��µ)/T2

1 + e�(EQD��µ)/T2,

(4)

or

EQD = T1 ln⇥(1 � q) /q

⇤,

EQD � �µ = T2 ln [(1 � ✏) /✏] .(5)


I1 = Pe,ssq � Po,ss(1 � q) =12

(q � ✏) ,

I2 = Po,ss(1 � ✏) � Pe,ss✏ =12

(q � ✏) ,(6)


J ⌘12

(q � ✏) , (7)



Q1 = JEQD ,


�.

(8)


S = �Q1

T1+

Q2

T2= JX , (9)


X ⌘EQD � �µ

T2�

EQD

T1, (10)


W = Q1 � Q2 = J�µ , (11)



⌘ =W

Q1=�µ

EQD= 1 �

T2 ln [(1 � ✏) /✏]T1 ln

⇥(1 � q) /q

⇤ , (12)


⌘C = 1 �T2

T1, (13)


7


X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)


W = �T2J2X2 . (58)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)


S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)



and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)

implying that the reversible condition for Jt = J2 = J = 0is given by X2 = �⇠0Xt, which is also shown in the conditionS = 0 in Eq. (51) leading to ⌘C = �µ/EQD, or equivalentlyXtT2 = X1�µ. It can also be shown with the direct calcula-tion for the condition of J = 0 in Eq. (7). In other words, thecondition X2 = �⇠0Xt that makes S = J2 (X2 + ⇠0Xt) = 0 alsomakes Jt = J2 = J = 0. Then one can expect that the flux J2contains the factor (X2 + ⇠0Xt) as a zero of J2 (see also Fig. 6).This phenomenon of vanishing fluxes in the reversible limitS = 0 naturally comes from the Onsager symmetry and thetight-coupling condition of proportional fluxes to each other,

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2W

X2*


2 = 2,but only W vanishes at X1 = 0. In panel (b), we fix �µ = 1 andplot the mechanical flux J2 and the power W as the functions of themechanical force X2. In this case, both J2 and W vanish at bothX2 = �⇠0Xt = ⌘C/�µ = 1/2 and X2 = 0. For each case, we indicatethe optimal values of X

⇤

1 and X⇤

2 that give Wmax.

in the conventional irreversible thermodynamics [22, 23, 28].Therefore, although our derivation for fixed �µ is not techni-cally conventional irreversible thermodynamics as we arguedin the previous paragraph, there exists a reversible point whereJ2 = Jt = 0 and X2 = �⇠0Xt at that point, so we take the sameformalism as irreversible thermodynamics to proceed further.



J2 = ��µ

2T2X2e�

1T2 X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

2T2X2 , e

2(T2Xt�1)T2X2

◆,


In this case, however, X2 is NOT the lowest order:

the tight-coupling condition:

7


X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)


W = �T2J2X2 . (58)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)


S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)



and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)

implying that the reversible condition for Jt = J2 = J = 0is given by X2 = �⇠0Xt, which is also shown in the conditionS = 0 in Eq. (51) leading to ⌘C = �µ/EQD, or equivalentlyXtT2 = X1�µ. It can also be shown with the direct calcula-tion for the condition of J = 0 in Eq. (7). In other words, thecondition X2 = �⇠0Xt that makes S = J2 (X2 + ⇠0Xt) = 0 alsomakes Jt = J2 = J = 0. Then one can expect that the fluxJ2 contains the factor (X2 + ⇠0Xt) as a zero of J2, just as J1contains (X1 + ⇠Xt) in the case of given EQD value as shownin Eq. (34) in Sec. IV B (see also Fig. 6). This phenomenon ofvanishing fluxes in the reversible limit S = 0 naturally comes

(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2 W


2 = 2,but only W vanishes at X1 = 0. In panel (b), we fix �µ = 1 andplot the mechanical flux J2 and the power W as the functions of themechanical force X2. In this case, both J2 and W vanish at bothX2 = �⇠0Xt = ⌘C/�µ = 1/2 and X2 = 0.

from the Onsager symmetry and the tight-coupling conditionof proportional fluxes to each other, in the conventional irre-versible thermodynamics [22, 23, 28]. Therefore, althoughour derivation for fixed �µ is not technically conventional ir-reversible thermodynamics as we argued in the previous para-graph, there exists a reversible point where J2 = Jt = 0 andX2 = �⇠0Xt at that point, so we take the same formalism asirreversible thermodynamics to proceed further.



J2 = ��µ

2T2X2e�

1T2X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

1T2X2 , e

T2Xt�1T2X2

◆,


) ⌘op(�µ) = ⌘C +O�⌘2C

�

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

W = X1 = �µ = 0 W = J = 0(X1 = �⇠Xt)

fixed-EQD fixed-�µ


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble

T1 = 1, T2 = 1/2, EQD = 1 T1 = 1, T2 = 1/2,�µ = 1

7


X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)


W = �T2J2X2 . (58)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)


S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)



and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)


(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2W

X2*



⇤

1 and X⇤

2 that give Wmax.




J2 = ��µ

2T2X2e�

1T2X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

1T2X2 , e

T2Xt�1T2X2

◆,


⌘C = 0.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5m

echa

nica

l flu

x an

d po

wer

X2

−J2W

X2*

W = J = 0(X2 = �⇠0Xt)

W = X2 = J = 0(EQD ! 1)

5


W = �T2J1X1 , (33)




q =e�EQD/T2 e

EQDXt


, ✏ =e�EQD/T2 e

T2X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)



W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)






0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

W = X1 = �µ = 0 W = J = 0(X1 = �⇠Xt)



µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble

T1 = 1, T2 = 1/2, EQD = 1 T1 = 1, T2 = 1/2,�µ = 1

7


X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)


W = �T2J2X2 . (58)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)


S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)



and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)


(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2W

X2*



⇤

1 and X⇤

2 that give Wmax.




J2 = ��µ

2T2X2e�

1T2X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

1T2X2 , e

T2Xt�1T2X2

◆,


⌘C = 0.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5m

echa

nica

l flu

x an

d po

wer

X2

−J2W

X2*

W = J = 0(X2 = �⇠0Xt)

W = X2 = J = 0(EQD ! 1)

5


W = �T2J1X1 , (33)




q =e�EQD/T2 e

EQDXt


, ✏ =e�EQD/T2 e

T2X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)



W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)






J1 ' a linear function of X1

W ' a quadratic function of X1

as ⌘C ! 0

X⇤1 ' �⇠

2Xt as ⌘C ! 0

X⇤1 ' �⇠0Xt as ⌘C ! 0

J2 / e�1/(T2X2)/X2

W / e�1/(T2X2)

near X2 = 0 as ⌘C ! 0

) ⌘op (EQD) '⌘C2 ) ⌘op (�µ) ' ⌘C

0

1x10-47

2x10-47

3x10-47

4x10-47

5x10-47

6x10-47

7x10-47

0 0.002 0.004 0.006 0.008 0.01po

wer

X2

W from the exp form

0

5x10-7

1x10-6

1.5x10-6

2x10-6

2.5x10-6

0 0.002 0.004 0.006 0.008 0.01

pow

er

X1

Wparabola

7


X2 =1

EQD, (56)


J2 = �J�µ

T2EQD = �J

�µ

T2X2, (57)


W = �T2J2X2 . (58)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 . (59)


S 'e�(T2X2)�1

2T

⇣e�µ/T2 � 1

⌘�µ! 0 , (60)



and J2,

Jt/J2 = �T2/�µ ⌘ ⇠0 , (61)


(a)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2

mec

hani

cal f

lux

and

pow

er

X1

−J1 W

X1*

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.1 0.2 0.3 0.4 0.5

mec

hani

cal f

lux

and

pow

er

X2

−J2W

X2*



⇤

1 and X⇤

2 that give Wmax.




J2 = ��µ

2T2X2e�

1T2X2

⇣e

Xt/X2 � e�µ/T2⌘+ O✓e�

1T2X2 , e

T2Xt�1T2X2

◆,


J1 ' a linear function of X1

W ' a quadratic function of X1

as ⌘C ! 0

5


W = �T2J1X1 , (33)




q =e�EQD/T2 e

EQDXt


, ✏ =e�EQD/T2 e

T2X1



J1 = L (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤+ O

⇣X

3t, X3

1

⌘, (34)

where

L =T

22 e�EQD/T2

2�1 + e�EQD/T2

�2 , (35)

⇠ = �EQD/T2 , (36)

� =✓T2

2

◆tanh

EQD

2T2

!. (37)


Jt/J1 = �EQD/T2 = ⇠ , (38)


Jt

J1

!=

L⇠2 L⇠L⇠ L

! Xt

X1

!, (39)



W = �T2J1X1 = �T2LX1 (X1 + ⇠Xt)⇥1 + � (X1 � ⇠Xt)

⇤,


dW

dX1

��X1=X

⇤

1

= 0 ,



X⇤

1 = �⇠

2Xt +

�⇠2

8X

2t. (40)


⌘ = �J1X1T2

Jt

= �X1T2

⇠, (41)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ηop

(∆µ

)

ηc

∆µ = 1ηC


ηC/2


substituting X⇤


⌘op(⇠, �,T, ⌘C) =12⌘C �

⇠�

8T2⌘2

C. (42)


�⇠

T2= �1 + O (⌘C) , (43)






T1 = 1, T2 = 0.99, EQD = 1 T1 = 1, T2 = 0.99,�µ = 1

⌘C = 0.01

W = J = 0(X1 = �⇠Xt)

W = J = 0(X2 = �⇠0Xt)

W = X1 = �µ = 0

X⇤1 ' �⇠

2Xt as ⌘C ! 0 X⇤

1 ' �⇠0Xt as ⌘C ! 0



µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble

W = X2 = J = 0(EQD ! 1)

J2 / e�1/(T2X2)/X2

W / e�1/(T2X2)

near X2 = 0 as ⌘C ! 0

• nonuniversality of the quantum dot heat engine efficiency

• the global optimum:

• the fixed- case:


4

where a0 is the solution of

21 � 2a0

= ln

1 � a0

a0

!, (21)

from the derivation in Appendix A. The expansion form of ⌘opwith respect to ⌘C in Eq. (20) has the same coe�cients up tothe quadratic term to those of the CA e�ciency [3–5] definedas

⌘CA = 1 �p

T2/T1 = 1 �p

1 � ⌘C , (22)


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)

when ⌘C ' 0. As a result, ⌘op and ⌘CA have very similarfunctional forms for ⌘C . 1/2, as shown in Fig. 3. The thirdorder coe�cient (' 0.077 492) in Eq. (20), however, is dif-ferent from 1/16 for the ⌘CA. In other words, the deviationfrom ⌘CA for ⌘op enters from the third order that has not beentheoretically investigated yet. Indeed, ⌘op deviates from ⌘CAfor ⌘C & 1/2, until they coincide at ⌘C = 1. The asymptoticbehavior of ⌘op for ⌘C ! 1 is given by

⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)

as shown in the inset of Fig. 3.



For a given set of temperature values T1 and T2, supposethe quantum dot energy and one of the chemical potentialare given. We take the case of the fixed EQD value (so thefixed q value accordingly) without loss of generality. With thesame procedure as in Appendix A but with the single-valuedfunction optimization with respect to ✏ (or �µ), we obtain⌘op(q,T1,T2) or equivalently,

⌘op(EQD,T2, ⌘C) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘,

(26)with the explicit dependence on T2 [T1 is then determined by⌘C from the relation T1 = T2/(1 � ⌘C)], in contrast to theoptimization with respect to both parameters in Sec. III. Thelinear coe�cient 1/2 is expected from the tight-coupling con-dition [21] that will be detailed later, but the quadratic coef-ficient is in general di↵erent from the value 1/8 for the opti-mized case with respect to both parameters. One can of course

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.

find the condition for the quadratic coe�cient to actually be-come 1/8, which is

EQD

T2tanh

EQD

2T2

!= 2 . (27)

It means that the value of EQD satisfying Eq. (27) with agiven temperature results in the quadratic coe�cient 1/8. Wewill meet this condition again from another standpoint inSec. IV B.



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)

where the entropy production rate is composed of the sum ofthe products of flux and force as follows: the thermal flux

Jt = Q1 = JEQD , (29)

the thermal force representing the temperature gradient

Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22. (32)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6

maximum power takes a completely di↵erent form, in contrastto the case of fixed EQD in Sec. IV. For the sake of conve-nience we rewrite the expression of power in Eq. (11) in termsof energy variables,

W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (44)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (45)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (46)

For the asymptotic behavior at ⌘C ! 0, or equivalentlyT2/T1 ! 1�, the optimal value E

⇤

QD ! 1. If we keep thelowest order terms of e

�E⇤

QD/T2 accordingly, the equation be-comes

⇣e�E⇤

QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (47)


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) , (48)

assuming the given value of T2 as in Sec. IV and T1 = T2/(1�⌘C). Inserting Eq. (48) to the e�ciency, Eq. (12), we finallyarrive at the e�ciency at the maximum power for given �µ,

⌘op(�µ,T2, ⌘C) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O

⇣⌘3

C

⌘. (49)

In contrast to the case where EQD is given (Sec. IV), there-fore, even the linear coe�cient deviates from 1/2 and be-comes unity, along with the negative quadratic coe�cient, asnumerically checked in Fig. 5. This example clearly illustratesthat it is crucial to choose the type of variable constraint im-posed in the two-parameter system which reduces the numberof free parameters to one, and a di↵erent choice can make acompletely di↵erent behavior of e�ciency at the maximumpower. In other words, the path to the global optimizationmatters a lot. In particular, we have demonstrated that the e�-ciency at the optimal power output approximately approachesthe theoretically maximal Carnot e�ciency at the linear orderfor the fixed �µ case, in contrast to the half of Carnot e�-ciency that has been believed to be universal for conventionaltight-coupling engines. It would be a crucial experiment tocheck if the quantum dot engine such as the one introduced inRefs. [13, 19, 20] actually shows ⌘op ' ⌘C + O(⌘2

C) by opti-

mizing the power only with respect to the gate voltage.



⌘C � ⌘ =T2S

Q1� 0 , (50)

which demonstrates that the upper bound of the e�ciency ofany generic heat engine is given by ⌘C , from S � 0 [27].If we substitute the explicit expressions Eqs. (8) and (9) intoEq. (50), we obtain

T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (51)

The above relation indicates that if the energy ratio �µ/EQD isequal to ⌘C , the e�ciency of quantum dot engine can achievethe Carnot e�ciency, which means that the system becomes areversible engine where S = 0.

Another possible way to get the Carnot e�ciency is choos-ing particular energy variables in order to make T2X/EQD van-ish. Indeed, E

⇤

QD in Eq. (48) for a given value of �µmeets thiscondition as ⌘C ! 0 because

T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (52)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (53)

Therefore, the absence of the linear term of ⌘C in the right-hand side of Eq. (53) confirms that ⌘op(�µ) ' ⌘C as ⌘C ! 0.This is the consequence of the vanishing scaled entropy S /J /⌘C and the diverging scaled heat as Q/J / 1/⌘C .

In contrast, for the case of a given value of EQD in Sec. IV,from Eqs. (32) and (40), the series expansion form of �µ⇤ (thatachieves the maximum power) with respect to ⌘C is given by

�µ⇤ =EQD

2⌘C + O

⇣⌘2

C

⌘. (54)



Q1


'12⌘C , (55)

which is consistent with the previous result in Sec. IV. In thiscase, since only the scaled entropy decays S /J / ⌘C in thelimit of ⌘C ! 0, ⌘op(EQD) has the correction in the linearorder of ⌘C .


As in Sec. IV, let us consider this problem with irreversiblethermodynamics [22, 23]. The expression for the entropy pro-duction rate is again given by Eq. (28). However, in this case,


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble

Summary

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)

6=18

EQD

�µ

6=18

✓* ⌘C = 1� T2

T1

◆cf) the Carnot efficiency

where2

1� 2q0= ln

✓1� q0q0

◆ 4

⌘op(⌘C) =12⌘C +

18⌘2

C+

7 � 24q0 + 24q20

96(1 � 2q0)2 ⌘3C+ O

⇣⌘4

C

⌘, (21)


21 � 2a0

= ln

1 � a0

a0

!, (22)


⌘CA = 1 �p

T2/T1 = 1 �p

1 � ⌘C , (23)


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (24)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(25)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (26)





⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘,

(27)with the explicit dependence on T2 [T1 is then determined by⌘C from the relation T1 = T2/(1 � ⌘C)], in contrast to theoptimization with respect to both parameters in Sec. III. The

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.

linear coe�cient 1/2 is expected from the tight-coupling con-dition [21] that will be detailed later, but the quadratic coef-ficient is in general di↵erent from the value 1/8 for the opti-mized case with respect to both parameters. One can of coursefind the condition for the quadratic coe�cient to actually be-come 1/8, which is

EQD

T2tanh

EQD

2T2

!= 2 . (28)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (29)


Jt = Q1 = JEQD , (30)


Xt =1T2�

1T1=⌘C

T2, (31)

the mechanical flux

J1 = �JT2 , (32)

126=

• nonuniversality of the quantum dot heat engine efficiency

• the global optimum:



4


21 � 2a0

= ln

1 � a0

a0

!, (21)


⌘CA = 1 �p

T2/T1 = 1 �p

1 � ⌘C , (22)


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (23)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(24)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (25)





⌘op(EQD,T2, ⌘C) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘,

(26)with the explicit dependence on T2 [T1 is then determined by⌘C from the relation T1 = T2/(1 � ⌘C)], in contrast to theoptimization with respect to both parameters in Sec. III. Thelinear coe�cient 1/2 is expected from the tight-coupling con-dition [21] that will be detailed later, but the quadratic coef-ficient is in general di↵erent from the value 1/8 for the opti-mized case with respect to both parameters. One can of course

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.

find the condition for the quadratic coe�cient to actually be-come 1/8, which is

EQD

T2tanh

EQD

2T2

!= 2 . (27)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (28)


Jt = Q1 = JEQD , (29)


Xt =1T2�

1T1=⌘C

T2, (30)

the mechanical flux

J1 = �JT2 , (31)


X1 =�µ

T22. (32)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6

maximum power takes a completely di↵erent form, in contrastto the case of fixed EQD in Sec. IV. For the sake of conve-nience we rewrite the expression of power in Eq. (11) in termsof energy variables,

W =12

e�EQD/T1

1 + e�EQD/T1�

e�EQD/T2 e

�µ/T2


!�µ . (44)


@W

@EQD

��EQD=E

⇤

QD

= 0 , (45)


QD, which is

e�E⇤

QD/T1

⇣1 + e

�E⇤

QD/T1⌘2

T2

T1=

e�E⇤

QD/T2 e�µ/T2

⇣1 + e

�E⇤

QD/T2e�µ/T2

⌘2 . (46)

For the asymptotic behavior at ⌘C ! 0, or equivalentlyT2/T1 ! 1�, the optimal value E

⇤

QD ! 1. If we keep thelowest order terms of e

�E⇤

QD/T2 accordingly, the equation be-comes

⇣e�E⇤

QD/T2⌘T2/T1 (1 � ⌘C) = e

�E⇤

QD/T2 e�µ/T2 , (47)


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) , (48)

assuming the given value of T2 as in Sec. IV and T1 = T2/(1�⌘C). Inserting Eq. (48) to the e�ciency, Eq. (12), we finallyarrive at the e�ciency at the maximum power for given �µ,

⌘op(�µ,T2, ⌘C) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O

⇣⌘3

C

⌘. (49)

In contrast to the case where EQD is given (Sec. IV), there-fore, even the linear coe�cient deviates from 1/2 and be-comes unity, along with the negative quadratic coe�cient, asnumerically checked in Fig. 5. This example clearly illustratesthat it is crucial to choose the type of variable constraint im-posed in the two-parameter system which reduces the numberof free parameters to one, and a di↵erent choice can make acompletely di↵erent behavior of e�ciency at the maximumpower. In other words, the path to the global optimizationmatters a lot. In particular, we have demonstrated that the e�-ciency at the optimal power output approximately approachesthe theoretically maximal Carnot e�ciency at the linear orderfor the fixed �µ case, in contrast to the half of Carnot e�-ciency that has been believed to be universal for conventionaltight-coupling engines. It would be a crucial experiment tocheck if the quantum dot engine such as the one introduced inRefs. [13, 19, 20] actually shows ⌘op ' ⌘C + O(⌘2

C) by opti-

mizing the power only with respect to the gate voltage.



⌘C � ⌘ =T2S

Q1� 0 , (50)

which demonstrates that the upper bound of the e�ciency ofany generic heat engine is given by ⌘C , from S � 0 [27].If we substitute the explicit expressions Eqs. (8) and (9) intoEq. (50), we obtain

T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (51)

The above relation indicates that if the energy ratio �µ/EQD isequal to ⌘C , the e�ciency of quantum dot engine can achievethe Carnot e�ciency, which means that the system becomes areversible engine where S = 0.

Another possible way to get the Carnot e�ciency is choos-ing particular energy variables in order to make T2X/EQD van-ish. Indeed, E

⇤

QD in Eq. (48) for a given value of �µmeets thiscondition as ⌘C ! 0 because

T2X⇤ = ⌘C E

⇤

QD � �µ ' T2⌘C , (52)

while E⇤


⌘C � ⌘op(�µ,T1,T2) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (53)

Therefore, the absence of the linear term of ⌘C in the right-hand side of Eq. (53) confirms that ⌘op(�µ) ' ⌘C as ⌘C ! 0.This is the consequence of the vanishing scaled entropy S /J /⌘C and the diverging scaled heat as Q/J / 1/⌘C .

In contrast, for the case of a given value of EQD in Sec. IV,from Eqs. (32) and (40), the series expansion form of �µ⇤ (thatachieves the maximum power) with respect to ⌘C is given by

�µ⇤ =EQD

2⌘C + O

⇣⌘2

C

⌘. (54)



Q1


'12⌘C , (55)

which is consistent with the previous result in Sec. IV. In thiscase, since only the scaled entropy decays S /J / ⌘C in thelimit of ⌘C ! 0, ⌘op(EQD) has the correction in the linearorder of ⌘C .


As in Sec. IV, let us consider this problem with irreversiblethermodynamics [22, 23]. The expression for the entropy pro-duction rate is again given by Eq. (28). However, in this case,


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

tunabletuna

ble


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µfixed

tunable


µ1

µ2

T1 T2

q

q

✏✏

EG

EQD�µ

fixedtuna

ble

Summary

cf) ⌘CA = 1�p

1� ⌘C =1

2⌘C +

1

8⌘2C +

1

16⌘3C +

5

128⌘4C +O(⌘5C)

6=18

EQD

�µ

6=18

✓* ⌘C = 1� T2

T1

◆cf) the Carnot efficiency

where2

1� 2q0= ln

✓1� q0q0

◆ 4

⌘op(⌘C) =12⌘C +

18⌘2

C+

7 � 24q0 + 24q20

96(1 � 2q0)2 ⌘3C+ O

⇣⌘4

C

⌘, (21)


21 � 2a0

= ln

1 � a0

a0

!, (22)


⌘CA = 1 �p

T2/T1 = 1 �p

1 � ⌘C , (23)


⌘CA =12⌘C +

18⌘2

C+

116⌘3

C+

5128⌘4

C+ O(⌘5

C) , (24)


⌘op =1 + (1 � q⇤

max)(1 � ⌘C) ln(1 � ⌘C)+ (1 � q

⇤


max)](1 � ⌘C)

+ Oh(1 � ⌘C)2

i,

(25)


11 � q

⇤max= ln

1 � q

⇤max

q⇤max

!, (26)





⌘op(EQD,T1,T2) =12⌘C +

EQD

16T2tanh

EQD

2T2

!⌘2

C+ O

⇣⌘3

C

⌘,

(27)with the explicit dependence on T2 [T1 is then determined by⌘C from the relation T1 = T2/(1 � ⌘C)], in contrast to theoptimization with respect to both parameters in Sec. III. The

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

ηop

(EQ

D)

ηc


2 orderηC/2 + ηC

2/8


C/8 curve.

linear coe�cient 1/2 is expected from the tight-coupling con-dition [21] that will be detailed later, but the quadratic coef-ficient is in general di↵erent from the value 1/8 for the opti-mized case with respect to both parameters. One can of coursefind the condition for the quadratic coe�cient to actually be-come 1/8, which is

EQD

T2tanh

EQD

2T2

!= 2 . (28)




S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2⌘ JtXt + J1X1 , (29)


Jt = Q1 = JEQD , (30)


Xt =1T2�

1T1=⌘C

T2, (31)

the mechanical flux

J1 = �JT2 , (32)

' 2⌘op(⌘C) for the global optimum (as ⌘C ! 0):any practical usage?

126=

8

(a)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ηop

/ [η

op a

t (q*

,ε*)]

ηC

EQD = 2EQD = 2.4

∆µ = 1∆µ = 1.5

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1W

max

/ [W

max

at (

q*,ε*

)]ηC

FIG. 7. The relative e�ciency (a) and maximum power (b) gain forlocal optimum case, compared to the global optimum, where we setT2 = 1. The purple and green curves correspond to the fixed-EQDcase with the indicated values, and the red and black curves corre-spond to the fixed-�µ case with the indicated values.

D. Practical gain of the optimization with the fixed chemicalpotentials

In practice, how much gain can we obtain with this fixed-�µ scheme? In terms of power, the global optimum case hasthat of

W 'T2a0(1 � a0)(1 � 2a0)2 ⌘

2C, (64)

with a0 satisfying Eq. (A8), as ⌘C ! 0 from the conditionof Eqs. (A1a) and (A1b) The fixed-EQD scheme also showsthe same quadratic dependence of the maximum power withrespect to ⌘C as

W 'EQD

4T2

e�EQD/T2

�1 + e�EQD/T2

�2 ⌘2C, (65)

for given EQD value. The fixed-�µ scheme, however, has theasymptotic behavior of

W ' �µe��µ/(T2⌘C )⌘C , (66)

as ⌘C ! 0 from the condition of Eq. (45). Therefore, in the⌘C ! 0 limit, the e�ciency at the maximum power of thefixed-�µ engine becomes twice as much as that of the globaloptimum or the fixed-EQD engine as shown in Fig. 7(a), butthe ratio of the maximum power for the fixed-�µ case to thatfor the global optimum actually vanishes as in Fig. 7(b). How-ever, for a wide range of ⌘C values, we can actually achieve thelarger e�ciency with the fixed �µ value than the e�ciency atthe globally optimized maximum power, while the maximumpower reaches up to a significant fraction of the global opti-mum value, e.g., the case of �µ = 1 at ⌘C ' 0.3 gives ' 30%larger ⌘op than that from the global optimum case and reaches' 70% of the global maximum power (see Fig. 7).

For ⌘C ! 1, as we fix T2, the value of T1 e↵ectively di-verges, so W and Q1 also diverge (but with the finite ratiocorresponding to the e�ciency) in the case of global optimumwithout any variable restriction. However, for the fixed-EQDcase, Q1 in Eq. (8) cannot diverge and thus W also remainsfinite, which leads to the vanishing ratio for fixed EQD inFig. 7(b). For the fixed-�µ case, on the other hand, W = Jµcannot diverge [so the ratio for fixed �µ also vanishes inFig. 7(b)], but that does not prevent Q1 from diverging, re-sulting in ⌘op(�µ)! 0 as ⌘C ! 1 in Fig. 7(a).

VI. CONCLUSIONS AND DISCUSSION

We have demonstrated that a quantum dot heat engine witha nontrivial parameter relation has the nonuniversal form ofe�ciency at the maximum power. In particular, compared tothe global or local optimization involving the controllabilityof chemical potentials of the leads, the single-parameter op-timization by controlling the gate voltage of the quantum dotfor a given chemical potential di↵erence gives the linear coef-ficient 1 instead of 1/2, which has been believed to be univer-sal for tight-coupling engines.

We have investigated the origin of the deviation in terms ofthe vanishing ratio of entropy production to the heat absorbed,and the fact that the lowest order term of mechanical flux isnot linear with respect to the mechanical and thermal forcesin the standpoint of irreversible thermodynamics. We suggestan experimental test to verify that particular single-parameteroptimization case, for instance, by tuning the gate voltage ofthe quantum dot [17] in the quantum dot engines depicted inRefs. [13, 19, 20].

The mathematically identical two-level heat engine modelintroduced in Sec. II B would naturally involves quantum ef-fects in reality when we take atomic-scale systems. A direc-tion for future works would be taking into account the genuinequantum e↵ects [28–37], which are inevitably involved for thesmall-scale systems in which we are interested.

8

(a)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ηop

/ [η

op a

t (q*

,ε*)]

ηC

EQD = 2EQD = 2.4

∆µ = 1∆µ = 1.5

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wm

ax /

[Wm

ax a

t (q*

,ε*)]

ηC




W 'T2a0(1 � a0)(1 � 2a0)2 ⌘

2C, (64)


W 'EQD

4T2

e�EQD/T2

�1 + e�EQD/T2

�2 ⌘2C, (65)


W ' �µe��µ/(T2⌘C )⌘C , (66)







T2 = 1 T2 = 1

global optimum: ⌘op ' ⌘C/2

%

$

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

fixed EQD: ⌘op(EQD) ' ⌘C/2


8

(a)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ηop

/ [η

op a

t (q*

,ε*)]

ηC

EQD = 2EQD = 2.4

∆µ = 1∆µ = 1.5

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1W

max

/ [W

max

at (

q*,ε*

)]ηC




W 'T2a0(1 � a0)(1 � 2a0)2 ⌘

2C, (64)


W 'EQD

4T2

e�EQD/T2

�1 + e�EQD/T2

�2 ⌘2C, (65)


W ' �µe��µ/(T2⌘C )⌘C , (66)







8

(a)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ηop

/ [η

op a

t (q*

,ε*)]

ηC

EQD = 2EQD = 2.4

∆µ = 1∆µ = 1.5

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wm

ax /

[Wm

ax a

t (q*

,ε*)]

ηC




W 'T2a0(1 � a0)(1 � 2a0)2 ⌘

2C, (64)


W 'EQD

4T2

e�EQD/T2

�1 + e�EQD/T2

�2 ⌘2C, (65)


W ' �µe��µ/(T2⌘C )⌘C , (66)







T2 = 1 T2 = 1

global optimum: ⌘op ' ⌘C/2

%

$

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

6


E⇤

QD =�µ

⌘C

�T2

⌘C

ln (1 � ⌘C) . (47)


⌘op(�µ,T1,T2) =�µ

�µ/⌘C � (T2/⌘C) ln (1 � ⌘C),

= ⌘C �T2

�µ⌘2

C+ O⇣⌘3

C

⌘. (48)






⌘C � ⌘ =T2S

Q1� 0 , (49)


T2S

Q1=

T2X

EQD= ⌘C �

�µ

EQD. (50)




T2X⇤ = ⌘C E

⇤

QD � �µ ⇡ T2⌘C , (51)

while E⇤


⌘C � ⌘op(�µ) =T2S

Q1

��µ,EQD=E

⇤

QD

'T2

�µ⌘2

C. (52)



�µ⇤(EQD) =EQD

2⌘C + O

⇣⌘2

C

⌘. (53)



Q1


'12⌘C , (54)




X2 =1

EQD, (55)


J2 = �J�µ

T2EQD = �J

�µ

T2X2(56)



S = �Q1

T1+

Q1

T2�

Q1

T2+

Q2

T2= JtXt + J2X2 , (57)


S 'e�(T X2)�1

2T

⇣e�µ/T� 1⌘�µ! 0 , (58)

fixed EQD: ⌘op(EQD) ' ⌘C/2

&�µ = 1 at ⌘C ' 0.3: ' 20% larger ⌘op thanthat from the global optimum case,while reaching ' 70% of the global maximum power


nonuniversality of heat engine efficiency at maximum power

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