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Efficiency at the maximum power output
for simple two-level heat engine
Sang Hoon Lee 

School of Physics, Korea Institute...
Carnot engine
source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg
S. Carnot, ...
Carnot engine
source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg
S. Carnot, ...
the Carnot e ciency ⌘C =
Weng
|Qh|
=
|Qh| |Qc|
|Qh|
= 1
Tc
Th
quasi-static
(the 1st law of thermodynamics)
the Carnot e ciency ⌘C =
Weng
|Qh|
=
|Qh| |Qc|
|Qh|
= 1
Tc
Th
quasi-static
(the 1st law of thermodynamics)
0: cyclic proce...
the Carnot e ciency ⌘C =
Weng
|Qh|
=
|Qh| |Qc|
|Qh|
= 1
Tc
Th
quasi-static
(the 1st law of thermodynamics)
0: cyclic proce...
Th
Tc
hot reservoir
cold reservoir
Thw
Tcw
Endoreversible engine
• P. Chambadal, Les Centrales Nuclaires (Armand Colin, Pa...
“endoreversibility”
Th
Tc
hot reservoir
cold reservoir
Thw
Tcw
during t1
irreversible heat conduction
the input energy (li...
“endoreversibility”
the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1
r
Tc
Th
Th
Tc
hot reservoir
cold reservoir
Thw...
the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1
r
Tc
Th
the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1
r
Tc
Th
Q. Is this a universal formula for power-
maximizing effici...
the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1
r
Tc
Th
Q. Is this a universal formula for power-
maximizing effici...
our simple two-level heat engine model
R1
R2
relaxation with
relaxation with
Q1
Q2
E1
E2
T1
T2
q
✏
t1
t2
during t1
during ...
our simple two-level heat engine model
R1
R2
relaxation with
relaxation with
Q1
Q2
E1
E2
T1
T2
q
✏
t1
t2
during t1
during ...
our simple two-level heat engine model
R1
R2
relaxation with
relaxation with
Q1
Q2
E1
E2
T1
T2
q
✏
t1
t2
during t1
during ...
our simple two-level heat engine model
R1
R2
relaxation with
relaxation with
Q1
Q2
E1
E2
T1
T2
q
✏
t1
t2
during t1
during ...
our simple two-level heat engine model
R1
R2
relaxation with
relaxation with
Q1
Q2
E1
E2
T1
T2
q
✏
t1
t2
during t1
during ...
our simple two-level heat engine model
R1
E1
T1
q
R2
E2
T2
✏
0
˜q
0
˜✏
during during
stochastic Markov processes
⌧1 ⌧2
|P1...
hWi = (E1 E2)P1 and hW0
i = (E1 E2)P2
where P1 (P2) is the population in E1 (E2) at R1 (R2), respectively
independent of t...
hWneti = hWi hW0
i = (P1 P2)(E1 E2)
= (P1 P2){T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]}
e ciency ⌘ =
hWneti
hQ1i
=
hWi hW0
i
hQ1i
= 1...
Let t1 = t2 = ⌧/2, then
in terms of ⌧, the maximum power is achieved for ⌧ ! 0, as
Power ! hWneti/4 and the power is monot...
Let t1 = t2 = ⌧/2, then
in terms of ⌧, the maximum power is achieved for ⌧ ! 0, as
Power ! hWneti/4 and the power is monot...
!
T2q⇤
(1 q⇤
)
T1✏⇤(1 ✏⇤)
= 1at (q⇤
, ✏⇤
) (global optimum)
(8)
d.
system
✏
✏
!#
,
(9)
1 e ⌧
)
ormula
t of net
. (10)
of n...
We verify that the (q⇤
, ✏⇤
) is indeed the maximum point by
using the relations of second derivatives
0
BBBBB@
@2
hPi
@q2...
!
T2q⇤
(1 q⇤
)
T1✏⇤(1 ✏⇤)
= 1at (q⇤
, ✏⇤
) (global optimum)
) ✏⇤
=
1
2
1
r
1
4T2
T1
q⇤(1 q⇤)
!
substituting ✏⇤
(q⇤
) to
1
...
! f(T2/T1, q⇤
) = 0 ! q⇤
(T1, T2) ! ✏⇤
(T1, T2) ! (hWnetimax, ⌘op)
f(T2/T1, q⇤
) = ln
✓
1 q⇤
q⇤
◆
T2
T1
ln
0
B
B
@
1 +
r
1...
schematically . . .
✏
q
✏ = q
net power < 0
schematically . . .
✏
q
✏ = q
net power < 0
⌘C from 0 to 1
(T2/T1 from 1 to 0)
<Wnet>(τ → ∞), T1 = 1, T2 = 1/100
0.1 0.2 0...
) ✏⇤
=
1
2
1
r
1
4T2
T1
q⇤(1 q⇤)
!
) ⌘op = 1
T2
T1
✓
ln[(1 ✏⇤
)/✏⇤
]
ln[(1 q⇤)/q⇤]
◆
=
q⇤ 1
2
+
1
2
r
1
4T2
T1
q⇤(1 q⇤)
q⇤...
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
q*(ηc→0) = ε*(ηc→0)
q*(ηc=1)
optimaltransitionrates
ηc
q*
ε*
ηc→0 and 1 asymptotes...
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
q*(ηc→0) = ε*(ηc→0)
q*(ηc=1)
optimaltransitionrates
ηc
q*
ε*
ηc→0 and 1 asymptotes...
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
q*(ηc→0) = ε*(ηc→0)
q*(ηc=1)
optimaltransitionrates
ηc
q*
ε*
ηc→0 and 1 asymptotes...
q⇤
= q0 + a1⌘C + a2⌘2
C + a3⌘3
C + O
⇣
⌘4
C
⌘
. (22) ⌘op =
1
2
⌘C
= ln
✓
1 q⇤
q⇤
◆
(1 ⌘C) ln
1 +
p
(1 2q⇤)2 + 4⌘Cq⇤(1 q⇤)
1
p
(1 2q⇤)2 + 4⌘Cq⇤(1 q⇤)
!
q⇤ 1
2
+
1
2
p
(1 2q⇤)2 + 4⌘Cq⇤(1 q⇤)...
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
q*(ηc→0) = ε*(ηc→0)
q*(ηc=1)
optimaltransitionrates
ηc
q*
ε*
ηc→0 and 1 asymptotes...
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
q*(ηc→0) = ε*(ηc→0)
q*(ηc=1)
optimaltransitionrates
ηc
q*
ε*
ηc→0 and 1 asymptotes...
(22). Then,
⌘op =
1
(1 2q0) ln[(1 q0)/q0]
⌘C
+
a1
q0 3q2
0+2q3
0
+
[q2
0+2a1 q0(1+4a1)] ln[(1 q0)/q0]
(1 2q0)3
ln2
[(1 q0)...
(22). Then,
⌘op =
1
(1 2q0) ln[(1 q0)/q0]
⌘C
+
a1
q0 3q2
0+2q3
0
+
[q2
0+2a1 q0(1+4a1)] ln[(1 q0)/q0]
(1 2q0)3
ln2
[(1 q0)...
a1)] ln[(1 q0)/q0]+2[ 2q0 +a1 4a1 2a2 +4q0(4a1
q3
0(1+a1+4a2) 2q2
0(1+3a1+8a2
1+12a2)] ln2
[(1 q0)/q
q0)4
{ 2a2
1+[(1 2q0)...
a1)] ln[(1 q0)/q0]+2[ 2q0 +a1 4a1 2a2 +4q0(4a1
q3
0(1+a1+4a2) 2q2
0(1+3a1+8a2
1+12a2)] ln2
[(1 q0)/q
q0)4
{ 2a2
1+[(1 2q0)...
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
ηop
ηc
at (q*, ε*)
ηCA = 1−√1−ηc
ηc/(2−ηc)
ηc/2
ηc→1 asymptote
0.88
0.92
0.96
1
0....
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
ηop
ηc
at (q*, ε*)
ηCA = 1−√1−ηc
ηc/(2−ηc)
ηc/2
ηc→1 asymptote
0.88
0.92
0.96
1
0....
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
ηop
ηc
at (q*, ε*)
ηCA = 1−√1−ηc
ηc/(2−ηc)
ηc/2
ηc→1 asymptote
0.88
0.92
0.96
1
0....
For ⌘C ' 1, we need to consider the logarithmic correc-
n due to the functional form. The series expansion of the
nctional...
For ⌘C ' 1, we need to consider the logarithmic correc-
n due to the functional form. The series expansion of the
nctional...
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
Efficiency at the maximum power output for simple two-level heat engine
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Efficiency at the maximum power output for simple two-level heat engine

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the slides for 2016 Workshop on Special Topics in Statistical Physics & Complex Systems (SPnCS) @ Chosun Univ., Gwangju, 22 December, 2016.

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Efficiency at the maximum power output for simple two-level heat engine

  1. 1. Efficiency at the maximum power output for simple two-level heat engine Sang Hoon Lee 
 School of Physics, Korea Institute for Advanced Study 
 http://newton.kias.re.kr/~lshlj82 in collaboration with Jaegon Um (CCSS, CTP and Department of Physics and Astronomy, SNU) and Hyunggyu Park (School of Physics & Quantum Universe Center, KIAS) 2016 Workshop on Special Topics in Statistical Physics & Complex Systems (SPnCS) @ Chosun Univ., Gwangju, 22 December, 2016. SHL, J. Um, and H. Park, e-print arXiv:1612.00518.
  2. 2. Carnot engine source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg S. Carnot, Réflexions Sur La Puissance Motrice Du Feu Et Sur Les Machines Propres À Développer Cette Puissance (Bachelier Libraire, Paris, 1824). the Sadi Carnot 
 (1796-1832)
  3. 3. Carnot engine source: http://web2.uwindsor.ca/courses/physics/high_schools/2013/SteamEngine/images/PVgraph.jpg S. Carnot, Réflexions Sur La Puissance Motrice Du Feu Et Sur Les Machines Propres À Développer Cette Puissance (Bachelier Libraire, Paris, 1824). the Sadi Carnot 
 (1796-1832) the Carnot e ciency ⌘C = Weng |Qh| = |Qh| |Qc| |Qh| = 1 Tc Th quasi-static (the 1st law of thermodynamics)
  4. 4. the Carnot e ciency ⌘C = Weng |Qh| = |Qh| |Qc| |Qh| = 1 Tc Th quasi-static (the 1st law of thermodynamics)
  5. 5. the Carnot e ciency ⌘C = Weng |Qh| = |Qh| |Qc| |Qh| = 1 Tc Th quasi-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. quasi-static
  6. 6. the Carnot e ciency ⌘C = Weng |Qh| = |Qh| |Qc| |Qh| = 1 Tc Th quasi-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. 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
  7. 7. Th Tc hot reservoir cold reservoir Thw Tcw Endoreversible engine • 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).
  8. 8. “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 engine operated at Thw and Tcw ! Qh Thw = Qc Tcw during t2 irreversible heat conduction the heat rejected (linear heat conduction) Qc = t2(Tcw Tc) Endoreversible engine • 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).
  9. 9. “endoreversibility” the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1 r Tc Th 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 engine operated 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 with respect to t1 and t2 Endoreversible engine • 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 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 0.30 Larderello (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 an engine violates Carnot's principle which states that work can be done any time there is a difference in temperature. The fact that the hot and cold reservoirs are not at the same temperature as the working fluid they are in contact with means that work can and is done at the hot and cold reservoirs. The result is tantamount to coupling the high and low temperature parts of the cycle, so that the cycle collapses.[7] In the Carnot cycle there is strict necessity that the working fluid be at the same temperatures as the heat reservoirs they are in contact with and that they are separated by adiabatic transformations which prevent thermal contact. The efficiency was first derived by William Thomson [8] in his study of an unevenly heated body in which the adiabatic partitions between bodies at different temperatures are removed and maximum work is performed. 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 the Chambadal-Novikov-Curzon-Ahlborn efficiency. See also Heat engine An introduction to endoreversible thermodynamics is given in the thesis by Katharina Wagner.[4] It is also introduced by Hoffman et al.[9][10] A thorough discussion of the concept, together with many applications in engineering, is given in the book by Hans Ulrich Fuchs.[11] References 1. 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 1-58 3. 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.html 5. A Bejan, J. Appl. Phys., vol. 79, pp. 1191–1218, 1 Feb. 1996 http://dx.doi.org/10.1016/S0035-3159(96)80059-6 6. Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed. ed.). John Wiley &
  10. 10. the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1 r Tc Th
  11. 11. the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1 r Tc Th Q. Is this a universal formula for power- maximizing efficiency, or does endoreversibility guarantee it?
  12. 12. the (Chambadal-Novikov-)Curzon-Ahlborn e ciency ⌘CA = 1 r Tc Th Q. Is this a universal formula for power- maximizing efficiency, or does endoreversibility guarantee it? A. No.The linear heat conduction 
 is essential. ˙Q = ↵(Th Tc) We introduce a different type of engine with 
 non-(CN)CA optimal efficiency. • L. Chen and Z. Yan, J. Chem. Phys. 90, 3740 (1988): • F. Angulo-Brown and R. Páez-Hernández, J. Appl. Phys. 74, 2216 (1993):
 (Dulong-Petit law of cooling)˙Q = ↵(Th Tc)n ˙Q = ↵ (Tn h Tn c )
  13. 13. our simple two-level heat engine model R1 R2 relaxation with relaxation with Q1 Q2 E1 E2 T1 T2 q ✏ t1 t2 during t1 during t2 W = E1 E2W0 = E1 E2 q/(1 q) = exp( E1/T1) ✏/(1 ✏) = exp( E2/T2) 0 E1 q(setting the Boltzmann constant kB ⌘ 1 for notational convenience)
  14. 14. our simple two-level heat engine model R1 R2 relaxation with relaxation with Q1 Q2 E1 E2 T1 T2 q ✏ t1 t2 during t1 during t2 W = E1 E2W0 = E1 E2 q/(1 q) = exp( E1/T1) ✏/(1 ✏) = exp( E2/T2) 0 E2 ✏ (setting the Boltzmann constant kB ⌘ 1 for notational convenience)
  15. 15. our simple two-level heat engine model R1 R2 relaxation with relaxation with Q1 Q2 E1 E2 T1 T2 q ✏ t1 t2 during t1 during t2 W = E1 E2W0 = E1 E2 q/(1 q) = exp( E1/T1) ✏/(1 ✏) = exp( E2/T2) 0 E2 ✏ (setting the Boltzmann constant kB ⌘ 1 for notational convenience)
  16. 16. our simple two-level heat engine model R1 R2 relaxation with relaxation with Q1 Q2 E1 E2 T1 T2 q ✏ t1 t2 during t1 during t2 W = E1 E2W0 = E1 E2 q/(1 q) = exp( E1/T1) ✏/(1 ✏) = exp( E2/T2) 0 q E1 (setting the Boltzmann constant kB ⌘ 1 for notational convenience)
  17. 17. our simple two-level heat engine model R1 R2 relaxation with relaxation with Q1 Q2 E1 E2 T1 T2 q ✏ t1 t2 during t1 during t2 W = E1 E2W0 = E1 E2 q/(1 q) = exp( E1/T1) ✏/(1 ✏) = exp( E2/T2) 0 q E1 (setting the Boltzmann constant kB ⌘ 1 for notational convenience)
  18. 18. our simple two-level heat engine model R1 E1 T1 q R2 E2 T2 ✏ 0 ˜q 0 ˜✏ during during stochastic Markov processes ⌧1 ⌧2 |P1i(t1 = 0) |P1i(t1 = ⌧1) = |P2i(t2 = 0) |P2i(t2 = ⌧2) |P2i(t2 = ⌧2) = |P1i(t1 = 0) W = E1 E2 W0 = E1 E2 Q1 Q2 sh = T2 T1 sc + hWneti T1 sh
  19. 19. hWi = (E1 E2)P1 and hW0 i = (E1 E2)P2 where P1 (P2) is the population in E1 (E2) at R1 (R2), respectively independent of t1 and t2 (no P1 and P2 dependency) and ⌘ ! ⌘Carnot = 1 T2/T1 when ✏ ' q total entropy change S = Q1 T1 + Q2 T2 hQ1i = (P1 P2)T1 ln[(1 q)/q], hQ2i = (P1 P2)T2 ln[(1 ✏)/✏] hWneti = hWi hW0 i = (P1 P2)(E1 E2) = (P1 P2){T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} e ciency ⌘ = hWneti hQ1i = hWi hW0 i hQ1i = 1 T2 T1 ⇢ ln[(1 ✏)/✏] ln[(1 q)/q] 0 = (P1 P2)E1 = (P1 P2)E2 (from the Schnakenberg entropy formula, or equivalently, the 1st law: hEi = hQi hWi for each half of the cycle)
  20. 20. hWneti = hWi hW0 i = (P1 P2)(E1 E2) = (P1 P2){T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} e ciency ⌘ = hWneti hQ1i = hWi hW0 i hQ1i = 1 T2 T1 ⇢ ln[(1 ✏)/✏] ln[(1 q)/q] P1(t1 ! 1, t2 ! 1) = q and P2(t1 ! 1, t2 ! 1) = ✏, as expected meaningful only for q > ✏, or hWneti > 0 ˙P1 = (1 q)P1 + q(1 P1) ˙P2 = (1 ✏)P2 + ✏(1 P2) ! P1 = q A1e t0 1 , P2 = ✏ A2e t0 2 (0  t0 1  t1 and 0  t0 2  t2) P1(t0 1 = 0, t2) = P2(t1, t2) and P2(t1, t0 2 = 0) = P1(t1, t2) ! A1 = q P2, A2 = ✏ P1 and let t0 1 = t1, t0 2 = t2 ! P1 = q(1 e t1 ) + ✏(1 e t2 )e t1 1 e (t1+t2) P2 = ✏(1 e t2 ) + q(1 e t1 )e t2 1 e (t1+t2)
  21. 21. Let t1 = t2 = ⌧/2, then in terms of ⌧, the maximum power is achieved for ⌧ ! 0, as Power ! hWneti/4 and the power is monotonically decreased as ⌧ is increased. hWneti = (q ✏)  (1 e ⌧/2 )2 1 e ⌧ {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} time: decoupled overall factor hWneti(⌧ ! 1) = (q ✏) {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} Power hPi = hWneti ⌧ = q ✏ ⌧  (1 e ⌧/2 )2 1 e ⌧ {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} (still decoupled even when t1 6= t2)
  22. 22. Let t1 = t2 = ⌧/2, then in terms of ⌧, the maximum power is achieved for ⌧ ! 0, as Power ! hWneti/4 and the power is monotonically decreased as ⌧ is increased. hWneti = (q ✏)  (1 e ⌧/2 )2 1 e ⌧ {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} time: decoupled overall factor hWneti(⌧ ! 1) = (q ✏) {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} Power hPi = hWneti ⌧ = q ✏ ⌧  (1 e ⌧/2 )2 1 e ⌧ {T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} our goal: to find (q, ✏) = (q⇤ , ✏⇤ ) maximizing hPi @hPi @q q=q⇤,✏=✏⇤ = @hPi @✏ q=q⇤,✏=✏⇤ = 0 (still decoupled even when t1 6= t2) hWneti = hWi hW0 i = (P1 P2)(E1 E2) = (P1 P2){T1 ln[(1 q)/q] T2 ln[(1 ✏)/✏]} e ciency ⌘ = hWneti hQ1i = hWi hW0 i hQ1i = 1 T2 T1 ⇢ ln[(1 ✏)/✏] ln[(1 q)/q] substitute (q, ✏) = (q⇤ , ✏⇤ ) here, then ⌘op ⌘ ⌘(q⇤ , ✏⇤ ) is the e ciency at the maximum power output
  23. 23. ! T2q⇤ (1 q⇤ ) T1✏⇤(1 ✏⇤) = 1at (q⇤ , ✏⇤ ) (global optimum) (8) d. system ✏ ✏ !# , (9) 1 e ⌧ ) ormula t of net . (10) of net 1. The condition for the maximum power output For a given T2/T1 value, the maximum power output con- dition for the two-variable function is @hPi @q q=q⇤,✏=✏⇤ = @hPi @✏ q=q⇤,✏=✏⇤ = 0 , (12) which leads to 1 T2 T1 ln[(1 ✏⇤ )/✏⇤ ] ln[(1 q⇤)/q⇤] = q⇤ ✏⇤ q⇤(1 q⇤) ln[(1 q⇤)/q⇤] , (13a) and 1 T2 T1 ln[(1 ✏⇤ )/✏⇤ ] ln[(1 q⇤)/q⇤] = (T2/T1)(q⇤ ✏⇤ ) ✏⇤(1 ✏⇤) ln[(1 q⇤)/q⇤] , (13b) from Eq. (11). By eliminating the left-hand side of Eqs. (13a) and (13b), we obtain the following simple relation T2q⇤ (1 q⇤ ) T1✏⇤(1 ✏⇤) = 1 , (14a)
  24. 24. We verify that the (q⇤ , ✏⇤ ) is indeed the maximum point by using the relations of second derivatives 0 BBBBB@ @2 hPi @q2 q=q⇤,✏=✏⇤ 1 CCCCCA 0 BBBBB@ @2 hPi @✏2 q=q⇤,✏=✏⇤ 1 CCCCCA 0 BBBBB@ @2 hPi @q@✏ q=q⇤,✏=✏⇤ 1 CCCCCA 2 = T2 1 ✏⇤ (1 ✏⇤ ) q⇤ (1 q⇤ ) (2q⇤ ✏⇤ q⇤ ✏⇤ )2 4T2 1 ✏⇤2 (1 ✏⇤ )2 ✏⇤2 (1 ✏⇤ )2 q⇤2 (1 q⇤ )2 = T1 (q⇤ ✏⇤ )2 ✏⇤ (1 ✏⇤ )q⇤3 (1 q⇤ )3 > 0 , (17) where we use the relation in Eq. (14a), and @2 hPi @q2 q=q⇤,✏=✏⇤ = T1[q⇤ + (1 2q⇤ )✏⇤ ] q⇤2(1 q⇤)2 < 0 , @2 hPi @✏2 q=q⇤,✏=✏⇤ = T2[q⇤ + (1 2q⇤ )✏⇤ ] ✏⇤2(1 ✏⇤)2 < 0 . (18) Therefore, the procedure to calculate the e ciency for given T2/T1 at the maximum power output seems straight- forward now. First, find the q⇤ value satisfying Eq. (15) and 0 0.2 0.4 0.6 0.8 1 0 ηop FIG. 5. The of the Carnot timal q⇤ valu Ahlborn e ci the lower bou Figure 3 sho ! T2q⇤ (1 q⇤ ) T1✏⇤(1 ✏⇤) = 1at (q⇤ , ✏⇤ ) (global optimum) ! T2q⇤ (1 q⇤ ) T1✏⇤(1 ✏⇤) = 1
  25. 25. ! T2q⇤ (1 q⇤ ) T1✏⇤(1 ✏⇤) = 1at (q⇤ , ✏⇤ ) (global optimum) ) ✏⇤ = 1 2 1 r 1 4T2 T1 q⇤(1 q⇤) ! substituting ✏⇤ (q⇤ ) to 1 T2 T1 ✓ ln[(1 ✏⇤ )/✏⇤ ] ln[(1 q⇤)/q⇤] ◆ = T2 T1 q⇤ ✏⇤ ✏⇤(1 ✏⇤) ln[(1 q⇤)/q⇤] Finding the global maximum theoretically 1 T2 T1 ✓ ln[(1 ✏⇤ )/✏⇤ ] ln[(1 q⇤)/q⇤] ◆ = q⇤ ✏⇤ q⇤(1 q⇤) ln[(1 q⇤)/q⇤] or ! f(T2/T1, q⇤ ) = 0 ! q⇤ (T1, T2) ! ✏⇤ (T1, T2) ! (hWnetimax, ⌘op)
  26. 26. ! f(T2/T1, q⇤ ) = 0 ! q⇤ (T1, T2) ! ✏⇤ (T1, T2) ! (hWnetimax, ⌘op) f(T2/T1, q⇤ ) = ln ✓ 1 q⇤ q⇤ ◆ T2 T1 ln 0 B B @ 1 + r 1 4T2 T1 q⇤(1 q⇤) 1 r 1 4T2 T1 q⇤(1 q⇤) 1 C C A q⇤ 1 2 + 1 2 r 1 4T2 T1 q⇤(1 q⇤) q⇤(1 q⇤) numerically found q⇤ (⌘C) and ✏⇤ (⌘C) 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes
  27. 27. schematically . . . ✏ q ✏ = q net power < 0
  28. 28. schematically . . . ✏ q ✏ = q net power < 0 ⌘C from 0 to 1 (T2/T1 from 1 to 0) <Wnet>(τ → ∞), T1 = 1, T2 = 1/100 0.1 0.2 0.3 0.4 0.5 q 0.1 0.2 0.3 0.4 0.5 ε 0 0.05 0.1 0.15 0.2 0.25 0.3 <Wnet>(τ → ∞), T1 = 1, T2 = 1/10 0.1 0.2 0.3 0.4 0.5 q 0.1 0.2 0.3 0.4 0.5 ε 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 <Wnet>(τ → ∞), T1 = 1, T2 = 1/2 0.1 0.2 0.3 0.4 0.5 q 0.1 0.2 0.3 0.4 0.5 ε 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 <Wnet>(τ → ∞), T1 = 1, T2 = 9/10 0.1 0.2 0.3 0.4 0.5 q 0.1 0.2 0.3 0.4 0.5 ε 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 P(T2/T1 = 9/10) P(T2/T1 = 1/2) P(T2/T1 = 1/10) P(T2/T1 = 1/100) q q q ✏ ✏ ✏ ✏ q hWneti(⌧ ! 1) = (q ✏) {T1 ln[(1 q) q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 ⌘ q0 the root of 2 1 2q⇤ = ln ✓ 1 q⇤ q⇤ ◆ q⇤ (⌘C = 1) ' 0.217 811 705 719 800 ✏⇤ (⌘C = 1) = 0 the root of 1 1 q⇤ = ln ✓ 1 q⇤ q⇤ ◆
  29. 29. ) ✏⇤ = 1 2 1 r 1 4T2 T1 q⇤(1 q⇤) ! ) ⌘op = 1 T2 T1 ✓ ln[(1 ✏⇤ )/✏⇤ ] ln[(1 q⇤)/q⇤] ◆ = q⇤ 1 2 + 1 2 r 1 4T2 T1 q⇤(1 q⇤) q⇤(1 q⇤) ln[(1 q⇤)/q⇤] = q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) ln[(1 q⇤ )/q⇤ ] = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) f(T2/T1, q⇤ ) = ln ✓ 1 q⇤ q⇤ ◆ T2 T1 ln 0 B B @ 1 + r 1 4T2 T1 q⇤(1 q⇤) 1 r 1 4T2 T1 q⇤(1 q⇤) 1 C C A q⇤ 1 2 + 1 2 r 1 4T2 T1 q⇤(1 q⇤) q⇤(1 q⇤) ! f(T2/T1, q⇤ ) = 0 ! q⇤ (T1, T2) ! ✏⇤ (T1, T2) ! (hWnetimax, ⌘op) the series expansion at ⌘C ! 0
  30. 30. 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes the numerically found functional form of stant for a moment (which is clearly not q⇤ is also a function of T2/T1 to satisfy q⇤ (⌘C) ' q⇤ min + 0.0457524⌘C +
  31. 31. 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes the numerically found functional form of stant for a moment (which is clearly not q⇤ is also a function of T2/T1 to satisfy q⇤ (⌘C) ' q⇤ min + 0.0457524⌘C + f(T2/T1 = 0, q⇤ ) = ln ✓ 1 q⇤ q⇤ ◆ 1 1 q⇤ = 0 ! q⇤ ' 0.217 811 705 719 800
  32. 32. 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes the numerically found functional form of stant for a moment (which is clearly not q⇤ is also a function of T2/T1 to satisfy q⇤ (⌘C) ' q⇤ min + 0.0457524⌘C + f(T2/T1 = 0, q⇤ ) = ln ✓ 1 q⇤ q⇤ ◆ 1 1 q⇤ = 0 ! q⇤ ' 0.217 811 705 719 800 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) = 0 exactly 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 engine is shrunk to the line q = ✏ and there cannot be any positive work). Therefore, let us examine the case ⌘C ' 0 using the series expansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) asym respe ⌘C in wher 4a1)] 4q3 0(1 2q0)4 q2 0)2 q ply ⌘
  33. 33. q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C
  34. 34. = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) 0 = f(⌘C, q⇤ ) q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C ues, along with various asymptotic cases: the Curzon-Ahlborn ncy ⌘CA in Eq. (28), the upper bound ⌘C/(2 ⌘C) and the lower ⌘C/2 in Ref. [19], and the function in Eq. (32) for ⌘C 0.65. ituting Eq. (22) into Eq. (18) and expanding the left-hand 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 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) ng the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) which the lower bound for q⇤ (⌘C ! 0) = q0 = ! 0) ' 0.083 221 720 199 517 7 found numerically !0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = q⇤ (⌘C ! 0) q. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) unction of ⌘C, where the asymptotic behaviors derived hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is tonically increased and ✏⇤ is monotonically decreased, is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = q⇤ (⌘C = 1), = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustrates the situ- on the (q, ✏) plane. The linear coe cient a1 in Eq. (22) e written in terms of q0 when we let the coe cient of the atic term in Eq. (23) to be zero, as 4 0.8 1 satisfying Eq. (18), as a (⌘C ! 0) = ✏⇤ (⌘C ! 0) B 2. ✏⇤ (⌘C = 1) = 0 (the cates Eq. (34). q 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as 0.2 0.4 0.6 0.8 1 ηc = 1 − T2 / T1 q* ε* (ηc→0) = ε*(ηc→0) q*(ηc=1) ηc→1 asymptote y found q⇤ and ✏⇤ values satisfying Eq. (18), as a T2/T1, along with the q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) es presented in Sec. III B 2. ✏⇤ (⌘C = 1) = 0 (the e ⌘C ! 1 asymptote indicates Eq. (34). . . . = q rk q ( C 0) = ( C 0) 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*andε* 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 a function 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 (the horizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34). schematically . . . = q no net work q ( C 0) = ( C 0) 0.083 221 720 199 517 7 side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , from which the lower bound for q⇤ (⌘C ! 0 ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 found [lim⌘C!0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = by Eq. (16b)]. Figure 3 shows the numerical solu as a function of ⌘C, where the asymptotic behav above hold when ⌘C ' 0 and ⌘C ' 1. It seem monotonically increased and ✏⇤ is monotonically as ⌘C is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustra ation on the (q, ✏) plane. The linear coe cient a1 can be written in terms of q0 when we let the coe quadratic term in Eq. (23) to be zero, as
  35. 35. 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) 0 = f(⌘C, q⇤ ) C 2 (1 2q0) ln[(1 q0)/q0] 2q0 1 ⌘C + q0(1 q0) 2a1(1 2q0) 2(1 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) rom 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = a0(1 a0) 2(1 2a0) . (25) 0 q⇤ (⌘C ! 0) = q0 = ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C ues, along with various asymptotic cases: the Curzon-Ahlborn ncy ⌘CA in Eq. (28), the upper bound ⌘C/(2 ⌘C) and the lower ⌘C/2 in Ref. [19], and the function in Eq. (32) for ⌘C 0.65. ituting Eq. (22) into Eq. (18) and expanding the left-hand 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 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) ng the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) which the lower bound for q⇤ (⌘C ! 0) = q0 = ! 0) ' 0.083 221 720 199 517 7 found numerically !0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = q⇤ (⌘C ! 0) q. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) unction of ⌘C, where the asymptotic behaviors derived hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is tonically increased and ✏⇤ is monotonically decreased, is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = q⇤ (⌘C = 1), = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustrates the situ- on the (q, ✏) plane. The linear coe cient a1 in Eq. (22) e written in terms of q0 when we let the coe cient of the atic term in Eq. (23) to be zero, as 4 0.8 1 satisfying Eq. (18), as a (⌘C ! 0) = ✏⇤ (⌘C ! 0) B 2. ✏⇤ (⌘C = 1) = 0 (the cates Eq. (34). q 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as 0.2 0.4 0.6 0.8 1 ηc = 1 − T2 / T1 q* ε* (ηc→0) = ε*(ηc→0) q*(ηc=1) ηc→1 asymptote y found q⇤ and ✏⇤ values satisfying Eq. (18), as a T2/T1, along with the q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) es presented in Sec. III B 2. ✏⇤ (⌘C = 1) = 0 (the e ⌘C ! 1 asymptote indicates Eq. (34). . . . = q rk q ( C 0) = ( C 0) 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*andε* 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 a function 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 (the horizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34). schematically . . . = q no net work q ( C 0) = ( C 0) 0.083 221 720 199 517 7 side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , from which the lower bound for q⇤ (⌘C ! 0 ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 found [lim⌘C!0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = by Eq. (16b)]. Figure 3 shows the numerical solu as a function of ⌘C, where the asymptotic behav above hold when ⌘C ' 0 and ⌘C ' 1. It seem monotonically increased and ✏⇤ is monotonically as ⌘C is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustra ation on the (q, ✏) plane. The linear coe cient a1 can be written in terms of q0 when we let the coe quadratic term in Eq. (23) to be zero, as
  36. 36. 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) 0 = f(⌘C, q⇤ ) C 2 (1 2q0) ln[(1 q0)/q0] 2q0 1 ⌘C + q0(1 q0) 2a1(1 2q0) 2(1 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) rom 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = a0(1 a0) 2(1 2a0) . (25) 0 0 ned from series expansion en by the condition ⌘C = 1, 1 q⇤ ) and q⇤ (⌘C = 1) ' erically and ✏⇤ (⌘C = 1) = 0 always satisfies Eq. (18) re- he optimal q⇤ is meaningless ating regime for the engine here cannot be any positive e the case ⌘C ' 0 using the t to ⌘C, as ⌘2 C + O ⇣ ⌘3 C ⌘ . (22) ⌘C!0 C C C by Eq. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) as a function of ⌘C, where the asymptotic behaviors derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = q0(1 q0) 2(1 2q0) . (25) With the relations of coe cients in hand, we find the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of q⇤ (⌘C ! 0) = q0 = ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 0 q q ( C = 1) 0.217 811 705 719 800 ( C = 1) = 0 llustration of the optimal transition rates (q⇤ , ✏⇤ ) for the max- ower output as the T2/T1 value varies. Asymptotic behaviors obtained from series expansion upper bound for q⇤ is given by the condition ⌘C = 1, ng ln[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 11 705 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 from Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- s of q⇤ values, so finding the optimal q⇤ is meaningless , when ⌘C = 0, the operating regime for the engine nk to the line q = ✏ and there cannot be any positive Therefore, let us examine the case ⌘C ' 0 using the xpansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) a1 = q0(1 q0) 2(1 2q0) . (25) Similarly, the coe cient a2 in Eq. (22) can also be written in terms of q0 alone, by letting c3(q0, a1, a2) = 0 in Eq. (23) and using 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 the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(q0, a1, a2) = {2(1 2q0)2 a1[q2 0 + 2a1 q0(1 + 4a1)] ln[(1 q0)/q0]+2[ 2q4 0 +a1 4a2 1 2a2 +4q0(4a2 1 +3a2)+ 4q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q0]+(1 q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C ues, along with various asymptotic cases: the Curzon-Ahlborn ncy ⌘CA in Eq. (28), the upper bound ⌘C/(2 ⌘C) and the lower ⌘C/2 in Ref. [19], and the function in Eq. (32) for ⌘C 0.65. ituting Eq. (22) into Eq. (18) and expanding the left-hand 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 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) ng the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) which the lower bound for q⇤ (⌘C ! 0) = q0 = ! 0) ' 0.083 221 720 199 517 7 found numerically !0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = q⇤ (⌘C ! 0) q. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) unction of ⌘C, where the asymptotic behaviors derived hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is tonically increased and ✏⇤ is monotonically decreased, is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = q⇤ (⌘C = 1), = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustrates the situ- on the (q, ✏) plane. The linear coe cient a1 in Eq. (22) e written in terms of q0 when we let the coe cient of the atic term in Eq. (23) to be zero, as 4 0.8 1 satisfying Eq. (18), as a (⌘C ! 0) = ✏⇤ (⌘C ! 0) B 2. ✏⇤ (⌘C = 1) = 0 (the cates Eq. (34). q 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as 0.2 0.4 0.6 0.8 1 ηc = 1 − T2 / T1 q* ε* (ηc→0) = ε*(ηc→0) q*(ηc=1) ηc→1 asymptote y found q⇤ and ✏⇤ values satisfying Eq. (18), as a T2/T1, along with the q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) es presented in Sec. III B 2. ✏⇤ (⌘C = 1) = 0 (the e ⌘C ! 1 asymptote indicates Eq. (34). . . . = q rk q ( C 0) = ( C 0) 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*andε* 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 a function 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 (the horizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34). schematically . . . = q no net work q ( C 0) = ( C 0) 0.083 221 720 199 517 7 side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , from which the lower bound for q⇤ (⌘C ! 0 ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 found [lim⌘C!0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = by Eq. (16b)]. Figure 3 shows the numerical solu as a function of ⌘C, where the asymptotic behav above hold when ⌘C ' 0 and ⌘C ' 1. It seem monotonically increased and ✏⇤ is monotonically as ⌘C is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustra ation on the (q, ✏) plane. The linear coe cient a1 can be written in terms of q0 when we let the coe quadratic term in Eq. (23) to be zero, as
  37. 37. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C ⇣ ⌘ (26) 3. The asymptotic form when q⇤ ' Another type of asymptotic behavior can be we assume q⇤ ' ✏⇤ . If we write q = ✏ + ↵ 0 < ↵/✏ ⌧ 1, then ln[(1 q)/q] ' ln[(1 ✏)/✏] ↵/[✏(1 ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C +O ⇣ ⌘3 C ⌘ . (26) Using Eqs. (24) and (25), Eq. (26) becomes simply we 0 < and q t 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 tration of the optimal transition rates (q⇤ , ✏⇤ ) for the max- output as the T2/T1 value varies. ymptotic behaviors obtained from series expansion er bound for q⇤ is given by the condition ⌘C = 1, n[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 05 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 m Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- can be written in terms of q0 when we quadratic term in Eq. (23) to be zero, a1 = q0(1 q0 2(1 2q0 Similarly, the coe cient a2 in Eq. (2 terms of q0 alone, by letting c3(q0, a1 using the relations in Eqs. (24) and (2 a2 = 7q0(1 q0 24(1 2q0 With the relations of coe cients asymptotic behavior of ⌘op in Eq. (1 respect to ⌘C after substituting q⇤ as ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘ + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1 (1 2q ln2 [(1 q0)/q0 + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , where d3(q0, a1, a2) = {2(1 2q0) 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) 0 = f(⌘C, q⇤ ) C 2 (1 2q0) ln[(1 q0)/q0] 2q0 1 ⌘C + q0(1 q0) 2a1(1 2q0) 2(1 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) rom 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = a0(1 a0) 2(1 2a0) . (25) 0 0 ned from series expansion en by the condition ⌘C = 1, 1 q⇤ ) and q⇤ (⌘C = 1) ' erically and ✏⇤ (⌘C = 1) = 0 always satisfies Eq. (18) re- he optimal q⇤ is meaningless ating regime for the engine here cannot be any positive e the case ⌘C ' 0 using the t to ⌘C, as ⌘2 C + O ⇣ ⌘3 C ⌘ . (22) ⌘C!0 C C C by Eq. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) as a function of ⌘C, where the asymptotic behaviors derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = q0(1 q0) 2(1 2q0) . (25) With the relations of coe cients in hand, we find the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of q⇤ (⌘C ! 0) = q0 = ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 because 0 < ✏ < q < 1/2. By substituting ✏⇤ as a function of q⇤ in Eq. (15b) to Eq. (13) or Eq. (14), we get the optimum condition f(T2/T1, q⇤ ) = 0, which is explicitly q⇤ 1 2 + 1 2 r 1 4T2 T1 q⇤(1 q⇤) q⇤ (1 q⇤ ) q⇤(1 q⇤) q⇤(1 q⇤) 3 77777777775 q⇤ 1 2 + 1 2 q (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) = 0 . (16) or in terms of ⌘C = 1 T2/T1, ⌘op = q⇤ 1 2 + 1 2 q (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) ln[(1 q⇤ )/q⇤ ] . (17b) pper bound for q⇤ is given by the condition ⌘C = 1, g ln[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 1 705 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 rom Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- of q⇤ values, so finding the optimal q⇤ is meaningless when ⌘C = 0, the operating regime for the engine to the line q = ✏ and there cannot be any positive Therefore, let us examine the case ⌘C ' 0 using the pansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + O ⇣ ⌘3 C ⌘ . (22) monotonically increased and ✏⇤ as ⌘C is increased, i.e., q⇤ min = q⇤ ( ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). ation on the (q, ✏) plane. The lin can be written in terms of q0 whe quadratic term in Eq. (23) to be z a1 = a0(1 2(1 With the relations of coe c asymptotic behavior of ⌘op in Eq respect to ⌘C after substituting q 0 q q ( C = 1) 0.217 811 705 719 800 ( C = 1) = 0 llustration of the optimal transition rates (q⇤ , ✏⇤ ) for the max- ower output as the T2/T1 value varies. Asymptotic behaviors obtained from series expansion upper bound for q⇤ is given by the condition ⌘C = 1, ng ln[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 11 705 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 from Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- s of q⇤ values, so finding the optimal q⇤ is meaningless , when ⌘C = 0, the operating regime for the engine nk to the line q = ✏ and there cannot be any positive Therefore, let us examine the case ⌘C ' 0 using the xpansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) a1 = q0(1 q0) 2(1 2q0) . (25) Similarly, the coe cient a2 in Eq. (22) can also be written in terms of q0 alone, by letting c3(q0, a1, a2) = 0 in Eq. (23) and using 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 the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(q0, a1, a2) = {2(1 2q0)2 a1[q2 0 + 2a1 q0(1 + 4a1)] ln[(1 q0)/q0]+2[ 2q4 0 +a1 4a2 1 2a2 +4q0(4a2 1 +3a2)+ 4q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q0]+(1 q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C ues, along with various asymptotic cases: the Curzon-Ahlborn ncy ⌘CA in Eq. (28), the upper bound ⌘C/(2 ⌘C) and the lower ⌘C/2 in Ref. [19], and the function in Eq. (32) for ⌘C 0.65. ituting Eq. (22) into Eq. (18) and expanding the left-hand 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 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) ng the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) which the lower bound for q⇤ (⌘C ! 0) = q0 = ! 0) ' 0.083 221 720 199 517 7 found numerically !0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = q⇤ (⌘C ! 0) q. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) unction of ⌘C, where the asymptotic behaviors derived hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is tonically increased and ✏⇤ is monotonically decreased, is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = q⇤ (⌘C = 1), = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustrates the situ- on the (q, ✏) plane. The linear coe cient a1 in Eq. (22) e written in terms of q0 when we let the coe cient of the atic term in Eq. (23) to be zero, as 4 0.8 1 satisfying Eq. (18), as a (⌘C ! 0) = ✏⇤ (⌘C ! 0) B 2. ✏⇤ (⌘C = 1) = 0 (the cates Eq. (34). q 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as 0.2 0.4 0.6 0.8 1 ηc = 1 − T2 / T1 q* ε* (ηc→0) = ε*(ηc→0) q*(ηc=1) ηc→1 asymptote y found q⇤ and ✏⇤ values satisfying Eq. (18), as a T2/T1, along with the q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) es presented in Sec. III B 2. ✏⇤ (⌘C = 1) = 0 (the e ⌘C ! 1 asymptote indicates Eq. (34). . . . = q rk q ( C 0) = ( C 0) 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*andε* 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 a function 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 (the horizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34). schematically . . . = q no net work q ( C 0) = ( C 0) 0.083 221 720 199 517 7 side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , from which the lower bound for q⇤ (⌘C ! 0 ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 found [lim⌘C!0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = by Eq. (16b)]. Figure 3 shows the numerical solu as a function of ⌘C, where the asymptotic behav above hold when ⌘C ' 0 and ⌘C ' 1. It seem monotonically increased and ✏⇤ is monotonically as ⌘C is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustra ation on the (q, ✏) plane. The linear coe cient a1 can be written in terms of q0 when we let the coe quadratic term in Eq. (23) to be zero, as
  38. 38. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C ⇣ ⌘ (26) 3. The asymptotic form when q⇤ ' Another type of asymptotic behavior can be we assume q⇤ ' ✏⇤ . If we write q = ✏ + ↵ 0 < ↵/✏ ⌧ 1, then ln[(1 q)/q] ' ln[(1 ✏)/✏] ↵/[✏(1 ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C +O ⇣ ⌘3 C ⌘ . (26) Using Eqs. (24) and (25), Eq. (26) becomes simply we 0 < and q t 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 tration of the optimal transition rates (q⇤ , ✏⇤ ) for the max- output as the T2/T1 value varies. ymptotic behaviors obtained from series expansion er bound for q⇤ is given by the condition ⌘C = 1, n[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 05 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 m Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- can be written in terms of q0 when we quadratic term in Eq. (23) to be zero, a1 = q0(1 q0 2(1 2q0 Similarly, the coe cient a2 in Eq. (2 terms of q0 alone, by letting c3(q0, a1 using the relations in Eqs. (24) and (2 a2 = 7q0(1 q0 24(1 2q0 With the relations of coe cients asymptotic behavior of ⌘op in Eq. (1 respect to ⌘C after substituting q⇤ as ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘ + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1 (1 2q ln2 [(1 q0)/q0 + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , where d3(q0, a1, a2) = {2(1 2q0) 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc q* ε* ηc→0 and 1 asymptotes = ln ✓ 1 q⇤ q⇤ ◆ (1 ⌘C) ln 1 + p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) 1 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) ! q⇤ 1 2 + 1 2 p (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) 0 = f(⌘C, q⇤ ) C 2 (1 2q0) ln[(1 q0)/q0] 2q0 1 ⌘C + q0(1 q0) 2a1(1 2q0) 2(1 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) rom 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = a0(1 a0) 2(1 2a0) . (25) 0 0 ned from series expansion en by the condition ⌘C = 1, 1 q⇤ ) and q⇤ (⌘C = 1) ' erically and ✏⇤ (⌘C = 1) = 0 always satisfies Eq. (18) re- he optimal q⇤ is meaningless ating regime for the engine here cannot be any positive e the case ⌘C ' 0 using the t to ⌘C, as ⌘2 C + O ⇣ ⌘3 C ⌘ . (22) ⌘C!0 C C C by Eq. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) as a function of ⌘C, where the asymptotic behaviors derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as a1 = q0(1 q0) 2(1 2q0) . (25) With the relations of coe cients in hand, we find the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of q⇤ (⌘C ! 0) = q0 = ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 because 0 < ✏ < q < 1/2. By substituting ✏⇤ as a function of q⇤ in Eq. (15b) to Eq. (13) or Eq. (14), we get the optimum condition f(T2/T1, q⇤ ) = 0, which is explicitly q⇤ 1 2 + 1 2 r 1 4T2 T1 q⇤(1 q⇤) q⇤ (1 q⇤ ) q⇤(1 q⇤) q⇤(1 q⇤) 3 77777777775 q⇤ 1 2 + 1 2 q (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) = 0 . (16) or in terms of ⌘C = 1 T2/T1, ⌘op = q⇤ 1 2 + 1 2 q (1 2q⇤)2 + 4⌘Cq⇤(1 q⇤) q⇤ (1 q⇤ ) ln[(1 q⇤ )/q⇤ ] . (17b) pper bound for q⇤ is given by the condition ⌘C = 1, g ln[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 1 705 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 rom Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- of q⇤ values, so finding the optimal q⇤ is meaningless when ⌘C = 0, the operating regime for the engine to the line q = ✏ and there cannot be any positive Therefore, let us examine the case ⌘C ' 0 using the pansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + O ⇣ ⌘3 C ⌘ . (22) monotonically increased and ✏⇤ as ⌘C is increased, i.e., q⇤ min = q⇤ ( ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). ation on the (q, ✏) plane. The lin can be written in terms of q0 whe quadratic term in Eq. (23) to be z a1 = a0(1 2(1 With the relations of coe c asymptotic behavior of ⌘op in Eq respect to ⌘C after substituting q 0 q q ( C = 1) 0.217 811 705 719 800 ( C = 1) = 0 llustration of the optimal transition rates (q⇤ , ✏⇤ ) for the max- ower output as the T2/T1 value varies. Asymptotic behaviors obtained from series expansion upper bound for q⇤ is given by the condition ⌘C = 1, ng ln[(1 q⇤ )/q⇤ ] = 1/(1 q⇤ ) and q⇤ (⌘C = 1) ' 11 705 719 800 found numerically and ✏⇤ (⌘C = 1) = 0 from Eq. (16b). ⌘C = 0 always satisfies Eq. (18) re- s of q⇤ values, so finding the optimal q⇤ is meaningless , when ⌘C = 0, the operating regime for the engine nk to the line q = ✏ and there cannot be any positive Therefore, let us examine the case ⌘C ' 0 using the xpansion of q⇤ with respect to ⌘C, as q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) a1 = q0(1 q0) 2(1 2q0) . (25) Similarly, the coe cient a2 in Eq. (22) can also be written in terms of q0 alone, by letting c3(q0, a1, a2) = 0 in Eq. (23) and using 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 the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(q0, a1, a2) = {2(1 2q0)2 a1[q2 0 + 2a1 q0(1 + 4a1)] ln[(1 q0)/q0]+2[ 2q4 0 +a1 4a2 1 2a2 +4q0(4a2 1 +3a2)+ 4q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q0]+(1 x- 1, ' 0 e- ss ne ve he 2) 0 3 0 1 2 using 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 the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2q0) ln[(1 q0)/q0] ⌘C + a1 q0 3q2 0+2q3 0 + [q2 0+2a1 q0(1+4a1)] ln[(1 q0)/q0] (1 2q0)3 ln2 [(1 q0)/q0] ⌘2 C + d3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(q0, a1, a2) = {2(1 2q0)2 a1[q2 0 + 2a1 q0(1 + 4a1)] ln[(1 q0)/q0]+2[ 2q4 0 +a1 4a2 1 2a2 +4q0(4a2 1 +3a2)+ 4q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q0]+(1 2q0)4 { 2a2 1+[(1 2q0)a2 1 2(1 q0)q0a2] ln[(1 q0)/q0]}}/[(1 q2 0)2 q2 0]. Using Eqs. (24), (25) and (26), Eq. (27) becomes sim- ply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q2 0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) q⇤ = q0 + a1⌘C + a2⌘2 C + a3⌘3 C + O ⇣ ⌘4 C ⌘ . (22) ⌘op = 1 2 ⌘C ues, along with various asymptotic cases: the Curzon-Ahlborn ncy ⌘CA in Eq. (28), the upper bound ⌘C/(2 ⌘C) and the lower ⌘C/2 in Ref. [19], and the function in Eq. (32) for ⌘C 0.65. ituting Eq. (22) into Eq. (18) and expanding the left-hand 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 a0)a0(1 2a0)3 ⌘2 C +O ⇣ ⌘3 C ⌘ = 0 . (23) ng the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , (24) which the lower bound for q⇤ (⌘C ! 0) = q0 = ! 0) ' 0.083 221 720 199 517 7 found numerically !0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = q⇤ (⌘C ! 0) q. (16b)]. Figure 3 shows the numerical solution (q⇤ , ✏⇤ ) unction of ⌘C, where the asymptotic behaviors derived hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is tonically increased and ✏⇤ is monotonically decreased, is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = q⇤ (⌘C = 1), = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustrates the situ- on the (q, ✏) plane. The linear coe cient a1 in Eq. (22) e written in terms of q0 when we let the coe cient of the atic term in Eq. (23) to be zero, as 4 0.8 1 satisfying Eq. (18), as a (⌘C ! 0) = ✏⇤ (⌘C ! 0) B 2. ✏⇤ (⌘C = 1) = 0 (the cates Eq. (34). q 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the quadratic term in Eq. (23) to be zero, as 0.2 0.4 0.6 0.8 1 ηc = 1 − T2 / T1 q* ε* (ηc→0) = ε*(ηc→0) q*(ηc=1) ηc→1 asymptote y found q⇤ and ✏⇤ values satisfying Eq. (18), as a T2/T1, along with the q⇤ (⌘C ! 0) = ✏⇤ (⌘C ! 0) es presented in Sec. III B 2. ✏⇤ (⌘C = 1) = 0 (the e ⌘C ! 1 asymptote indicates Eq. (34). . . . = q rk q ( C 0) = ( C 0) 0.083 221 720 199 517 7 Substituting Eq. (22) into Eq. (18) and expanding the left-hand side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , (23) where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2) 6q5 0(5 + 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a1 +132a2 1 + 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 (1 q0)2 q2 0]. Letting the linear coe cient to be zero yields 2 1 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 derived above hold when ⌘C ' 0 and ⌘C ' 1. It seems that q⇤ is monotonically 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 the 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*andε* 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 a function 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 (the horizontal axis). The ⌘C ! 1 asymptote indicates Eq. (34). schematically . . . = q no net work q ( C 0) = ( C 0) 0.083 221 720 199 517 7 side 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 ⌘2 C + c3(q0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ = 0 , where c3(q0, a1, a2) = [10q6 0 + 3a2 1 6q0(a2 1 + a2 6a1 +8a2) 12q3 0(1+6a1 +16a2 1 +9a2)+q2 0(1+18a 42a2)+q4 0(31+90a1 +96a2 1 +120a2)]/[6(1 2q0)5 Letting the linear coe cient to be zero yields 2 1 2q0 = ln 1 q0 q0 ! , from which the lower bound for q⇤ (⌘C ! 0 ✏⇤ (⌘C ! 0) ' 0.083 221 720 199 517 7 found [lim⌘C!0 U(⌘C, q⇤ ) = 1 2q⇤ , thus ✏⇤ (⌘C ! 0) = by Eq. (16b)]. Figure 3 shows the numerical solu as a function of ⌘C, where the asymptotic behav above hold when ⌘C ' 0 and ⌘C ' 1. It seem monotonically increased and ✏⇤ is monotonically as ⌘C is increased, i.e., q⇤ min = q⇤ (⌘C ! 0), q⇤ max = ✏⇤ min = 0, and ✏⇤ max = ✏⇤ (⌘C ! 0). Figure 4 illustra ation on the (q, ✏) plane. The linear coe cient a1 can be written in terms of q0 when we let the coe quadratic term in Eq. (23) to be zero, as 2 1 8
  39. 39. a1)] ln[(1 q0)/q0]+2[ 2q0 +a1 4a1 2a2 +4q0(4a1 q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q q0)4 { 2a2 1+[(1 2q0)a2 1 2(1 q0)q0a2] ln[(1 q0)/q0] 2 0)2 q2 0]. Using Eqs. (24), (25) and (26), Eq. (27) becom ly ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q2 0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . No closed-form solution, but we get the series expansion at cf) ⌘CA = 1 p 1 ⌘C = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) ✓ * ⌘C = 1 T2 T1 ◆ ⌘C ! 0
  40. 40. a1)] ln[(1 q0)/q0]+2[ 2q0 +a1 4a1 2a2 +4q0(4a1 q3 0(1+a1+4a2) 2q2 0(1+3a1+8a2 1+12a2)] ln2 [(1 q0)/q q0)4 { 2a2 1+[(1 2q0)a2 1 2(1 q0)q0a2] ln[(1 q0)/q0] 2 0)2 q2 0]. Using Eqs. (24), (25) and (26), Eq. (27) becom ly ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q2 0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . different! ' 0.077 492 = 0.0625 strong 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). strong coupling + symmetry between the reservoirs (“left-right” symmetry)
 ref) M. Esposito, K. Lindenberg, and C. Van den Broeck, PRL 102, 130602 (2009). The deviation from ⌘CA for ⌘op enters from the third order. No closed-form solution, but we get the series expansion at cf) ⌘CA = 1 p 1 ⌘C = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) ✓ * ⌘C = 1 T2 T1 ◆ ⌘C ! 0
  41. 41. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ηop ηc at (q*, ε*) ηCA = 1−√1−ηc ηc/(2−ηc) ηc/2 ηc→1 asymptote 0.88 0.92 0.96 1 0.97 0.98 0.99 1 ηop ηc : very similar to up to a certain point, but clearly different!⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28)cf) ⌘CA = 1 p 1 ⌘C = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C +
  42. 42. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ηop ηc at (q*, ε*) ηCA = 1−√1−ηc ηc/(2−ηc) ηc/2 ηc→1 asymptote 0.88 0.92 0.96 1 0.97 0.98 0.99 1 ηop ηc M. Esposito et al., PRL 105, 150603 (2010)’s upper and lower bounds, respectively deviation very similar : very similar to up to a certain point, but clearly different!⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28)cf) ⌘CA = 1 p 1 ⌘C = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C +
  43. 43. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ηop ηc at (q*, ε*) ηCA = 1−√1−ηc ηc/(2−ηc) ηc/2 ηc→1 asymptote 0.88 0.92 0.96 1 0.97 0.98 0.99 1 ηop ηc M. Esposito et al., PRL 105, 150603 (2010)’s upper and lower bounds, respectively deviation very similar : very similar to up to a certain point, but clearly different! “Curzon-Ahlborn regime” “log correction regime” ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24q0 + 24q0 96(1 2q0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28)cf) ⌘CA = 1 p 1 ⌘C = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C +
  44. 44. For ⌘C ' 1, we need to consider the logarithmic correc- n due to the functional form. The series expansion of the nctional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) we take only the zeroth order term, we obtain the identity (1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is eady mentioned in the first part of this subsection. Similar the ⌘C ' 0 case, by letting each coe cient to be zero, we d the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) d b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) ich are well consistent with the numerical solution as own in Fig. 3. the ⌘C ! 1 asymptote in Eq. (36) for with the numerical solution as oe cients in hand, we find the in Eq. (19) by expanding it with ting q⇤ as the series expansion of a0)/a0] ⌘C +2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 (1 a0)/a0] ⌘2 C C + O ⇣ ⌘4 C ⌘ , (27) 2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 (1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 (25), and (26), ⌘op in Eq. (27) be- 24a0 + 24a2 0 6(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) ble to find the coe cients in terms rder in principle. We would like nsion form of ⌘op in Eq. (28) has ts up to the quadratic term to those functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 0 = f(⌘C, q⇤ ) mptote in Eq. (36) for merical solution as hand, we find the y expanding it with series expansion of [(1 a0)/a0] ⌘2 C (27) a2 0 + 2a1 a0(1 + a4 0 + a1 4a2 1 2a2 + a2 0(1 + 3a1 + 8a2 1 + +[(1 2a0)a2 1 2(1 2(2a0 1)5 ln3 [(1 , ⌘op in Eq. (27) be- 3 C + O ⇣ ⌘4 C ⌘ . (28) coe cients in terms ple. We would like ⌘op in Eq. (28) has adratic term to those efined as 1 ⌘C , (29) + O(⌘5 C) , (30) functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) well consistent with the numerical solution as g. 3. relations of coe cients in hand, we find the behavior of ⌘op in Eq. (19) by expanding it with C after substituting q⇤ as the series expansion of 2). Then, 1 2a0) ln[(1 a0)/a0] ⌘C a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) , a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 sing Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- ly ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) ethod, we are able to find the coe cients in terms an arbitrary order in principle. We would like e that the expansion form of ⌘op in Eq. (28) has ame coe cients up to the quadratic term to those n-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) ansion form 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) q =qmax + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) shown in Fig. 3. With the relations of coe cients in hand, we find the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2a0) ln[(1 a0)/a0] ⌘C + a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C + d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(a0, a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 4a1)] ln[(1 a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + 4a0(4a2 1 + 3a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 12a2)] ln2 [(1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 a0)a0a2] ln[(1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 a0)/a0]}. Using Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- comes simply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) With this method, we are able to find the coe cients in terms of a0 up to an arbitrary order in principle. We would like to emphasize that the expansion form of ⌘op in Eq. (28) has exactly the same coe cients up to the quadratic term to those of the Curzon-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) with the expansion form ⌘CA = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C) + ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] ) (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2a0) ln[(1 a0)/a0] ⌘C + a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C + d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(a0, a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 4a1)] ln[(1 a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + 4a0(4a2 1 + 3a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 12a2)] ln2 [(1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 a0)a0a2] ln[(1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 a0)/a0]}. Using Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- comes simply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) With this method, we are able to find the coe cients in terms of a0 up to an arbitrary order in principle. We would like to emphasize that the expansion form of ⌘op in Eq. (28) has exactly the same coe cients up to the quadratic term to those of the Curzon-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) with the expansion form ⌘CA = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) 0 = ln 1 q⇤ max q⇤ max + " 1 bln q⇤ max(1 + ( 1 b1 q⇤ max(1 + O h (1 ⌘C)2 i . If we take only the ze ln[(1 q⇤ max)/q⇤ max] = already mentioned in t to the ⌘C ' 0 case, by find the relations amon bln and b1 = q⇤ max(1 q⇤ m which are well consi shown in Fig. 3. Again, the asymptot can be deduced from ⌘C) > 0, using Eq. (32 ⌘op =1 + (1 q⇤ m + (1 q⇤ ma + O h (1 based on the relations Fig. 5, the asymptotic C/(2 ⌘C) and the lower symptote in Eq. (36) for numerical solution as n hand, we find the by expanding it with he series expansion of ] ln[(1 a0)/a0] )3 ⌘2 C (27) a1[a2 0 + 2a1 a0(1 + 2a4 0 + a1 4a2 1 2a2 + 2a2 0(1 + 3a1 + 8a2 1 + 2 1 +[(1 2a0)a2 1 2(1 /{2(2a0 1)5 ln3 [(1 6), ⌘op in Eq. (27) be- 2 0 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) e coe cients in terms ciple. We would like of ⌘op in Eq. (28) has uadratic term to those defined as p 1 ⌘C , (29) ⌘4 C + O(⌘5 C) , (30) tion due to the functional form. The series expansion of the functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C) + ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] ) (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather the (singular) series expansion at ⌘C ! 1
  45. 45. For ⌘C ' 1, we need to consider the logarithmic correc- n due to the functional form. The series expansion of the nctional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) we take only the zeroth order term, we obtain the identity (1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is eady mentioned in the first part of this subsection. Similar the ⌘C ' 0 case, by letting each coe cient to be zero, we d the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) d b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) ich are well consistent with the numerical solution as own in Fig. 3. the ⌘C ! 1 asymptote in Eq. (36) for with the numerical solution as oe cients in hand, we find the in Eq. (19) by expanding it with ting q⇤ as the series expansion of a0)/a0] ⌘C +2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 (1 a0)/a0] ⌘2 C C + O ⇣ ⌘4 C ⌘ , (27) 2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 (1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 (25), and (26), ⌘op in Eq. (27) be- 24a0 + 24a2 0 6(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) ble to find the coe cients in terms rder in principle. We would like nsion form of ⌘op in Eq. (28) has ts up to the quadratic term to those functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 0 = f(⌘C, q⇤ ) mptote in Eq. (36) for merical solution as hand, we find the y expanding it with series expansion of [(1 a0)/a0] ⌘2 C (27) a2 0 + 2a1 a0(1 + a4 0 + a1 4a2 1 2a2 + a2 0(1 + 3a1 + 8a2 1 + +[(1 2a0)a2 1 2(1 2(2a0 1)5 ln3 [(1 , ⌘op in Eq. (27) be- 3 C + O ⇣ ⌘4 C ⌘ . (28) coe cients in terms ple. We would like ⌘op in Eq. (28) has adratic term to those efined as 1 ⌘C , (29) + O(⌘5 C) , (30) functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) well consistent with the numerical solution as g. 3. relations of coe cients in hand, we find the behavior of ⌘op in Eq. (19) by expanding it with C after substituting q⇤ as the series expansion of 2). Then, 1 2a0) ln[(1 a0)/a0] ⌘C a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) , a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 sing Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- ly ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) ethod, we are able to find the coe cients in terms an arbitrary order in principle. We would like e that the expansion form of ⌘op in Eq. (28) has ame coe cients up to the quadratic term to those n-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) ansion form 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) q =qmax + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C)+ ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) shown in Fig. 3. With the relations of coe cients in hand, we find the asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2a0) ln[(1 a0)/a0] ⌘C + a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C + d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(a0, a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 4a1)] ln[(1 a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + 4a0(4a2 1 + 3a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 12a2)] ln2 [(1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 a0)a0a2] ln[(1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 a0)/a0]}. Using Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- comes simply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) With this method, we are able to find the coe cients in terms of a0 up to an arbitrary order in principle. We would like to emphasize that the expansion form of ⌘op in Eq. (28) has exactly the same coe cients up to the quadratic term to those of the Curzon-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) with the expansion form ⌘CA = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C) + ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] ) (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather asymptotic behavior of ⌘op in Eq. (19) by expanding it with respect to ⌘C after substituting q⇤ as the series expansion of ⌘C in Eq. (22). Then, ⌘op = 1 (1 2a0) ln[(1 a0)/a0] ⌘C + a1 a0 3a2 0+2a3 0 + [a2 0+2a1 a0(1+4a1)] ln[(1 a0)/a0] (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C + d3(a0, a1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) where d3(a0, a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 4a1)] ln[(1 a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + 4a0(4a2 1 + 3a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 12a2)] ln2 [(1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 a0)a0a2] ln[(1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 a0)/a0]}. Using Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- comes simply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) With this method, we are able to find the coe cients in terms of a0 up to an arbitrary order in principle. We would like to emphasize that the expansion form of ⌘op in Eq. (28) has exactly the same coe cients up to the quadratic term to those of the Curzon-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) with the expansion form ⌘CA = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) 0 = ln 1 q⇤ max q⇤ max + " 1 bln q⇤ max(1 + ( 1 b1 q⇤ max(1 + O h (1 ⌘C)2 i . If we take only the ze ln[(1 q⇤ max)/q⇤ max] = already mentioned in t to the ⌘C ' 0 case, by find the relations amon bln and b1 = q⇤ max(1 q⇤ m which are well consi shown in Fig. 3. Again, the asymptot can be deduced from ⌘C) > 0, using Eq. (32 ⌘op =1 + (1 q⇤ m + (1 q⇤ ma + O h (1 based on the relations Fig. 5, the asymptotic C/(2 ⌘C) and the lower symptote in Eq. (36) for numerical solution as n hand, we find the by expanding it with he series expansion of ] ln[(1 a0)/a0] )3 ⌘2 C (27) a1[a2 0 + 2a1 a0(1 + 2a4 0 + a1 4a2 1 2a2 + 2a2 0(1 + 3a1 + 8a2 1 + 2 1 +[(1 2a0)a2 1 2(1 /{2(2a0 1)5 ln3 [(1 6), ⌘op in Eq. (27) be- 2 0 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) e coe cients in terms ciple. We would like of ⌘op in Eq. (28) has uadratic term to those defined as p 1 ⌘C , (29) ⌘4 C + O(⌘5 C) , (30) tion due to the functional form. The series expansion of the functional form in Eq. (18) near ⌘C = 1 in terms of q⇤ =q⇤ max + bln(1 ⌘C) ln(1 ⌘C) + b1(1 ⌘C) + O h (1 ⌘C)2 i . (32) is written as 0 = " ln 1 q⇤ max q⇤ max ! 1 1 q⇤ max # + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C) + ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] ) (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather 0 a0)/a0] ⌘2 C (27) + 2a1 a0(1 + a1 4a2 1 2a2 + + 3a1 + 8a2 1 + 2a0)a2 1 2(1 a0 1)5 ln3 [(1 in Eq. (27) be- O ⇣ ⌘4 C ⌘ . (28) cients in terms We would like in Eq. (28) has tic term to those ed as ⌘C , (29) O(⌘5 C) , (30) max max + " 1 bln q⇤ max(1 q⇤ max)2 # (1 ⌘C) ln(1 ⌘C) + ( 1 b1 q⇤ max(1 q⇤ max)2 + ln[q⇤ max(1 q⇤ max)] ) (1 ⌘C) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather q⇤ max ' 0.217 811 705 719 800 0 2a3 0 + 0 1 0 1 0 0 (1 2a0)3 ln2 [(1 a0)/a0] ⌘2 C 1, a2)⌘3 C + O ⇣ ⌘4 C ⌘ , (27) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + }/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 /a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 s. (24), (25), and (26), ⌘op in Eq. (27) be- 2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) we are able to find the coe cients in terms trary order in principle. We would like he expansion form of ⌘op in Eq. (28) has e cients up to the quadratic term to those orn e ciency [3–5] defined as 1 p T2/T1 = 1 p 1 ⌘C , (29) form 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) qmax(1 qmax) + O h (1 ⌘C)2 i . (33) If we take only the zeroth order term, we obtain the identity ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that is already mentioned in the first part of this subsection. Similar to the ⌘C ' 0 case, by letting each coe cient to be zero, we find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34) and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35) which are well consistent with the numerical solution as shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' 1 can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36) based on the relations in Eqs. (34) and (35). As shown in Fig. 5, the asymptotic form in Eq. (36) only holds in a rather 0 where d3(a0, a1, a2) = {{2(1 2a0)2 a1[a2 0 + 2a1 a0(1 + 4a1)] ln[(1 a0)/a0]}/[a0(a0 1)] + 2[ 2a4 0 + a1 4a2 1 2a2 + 4a0(4a2 1 + 3a2) + 4a3 0(1 + a1 + 4a2) 2a2 0(1 + 3a1 + 8a2 1 + 12a2)] ln2 [(1 a0)/a0]+{(1 2a0)4 { 2a2 1 +[(1 2a0)a2 1 2(1 a0)a0a2] ln[(1 a0)/a0]}}/[(1 a0)2 a2 0]}/{2(2a0 1)5 ln3 [(1 a0)/a0]}. Using Eqs. (24), (25), and (26), ⌘op in Eq. (27) be- comes simply ⌘op = 1 2 ⌘C + 1 8 ⌘2 C + 7 24a0 + 24a2 0 96(1 2a0)2 ⌘3 C + O ⇣ ⌘4 C ⌘ . (28) With this method, we are able to find the coe cients in terms of a0 up to an arbitrary order in principle. We would like to emphasize that the expansion form of ⌘op in Eq. (28) has exactly the same coe cients up to the quadratic term to those of the Curzon-Ahlborn e ciency [3–5] defined as ⌘CA = 1 p T2/T1 = 1 p 1 ⌘C , (29) with the expansion form ⌘CA = 1 2 ⌘C + 1 8 ⌘2 C + 1 16 ⌘3 C + 5 128 ⌘4 C + O(⌘5 C) , (30) ln[(1 q⇤ max)/q⇤ max] = 1/(1 q⇤ max) exactly at ⌘C = 1 that i already mentioned in the first part of this subsection. Simila to the ⌘C ' 0 case, by letting each coe cient to be zero, w find the relations among the coe cients as bln = q⇤ max(1 q⇤ max)2 , (34 and b1 = q⇤ max(1 q⇤ max)2 1 + ln[q⇤ max(1 q⇤ max)] , (35 which are well consistent with the numerical solution a shown in Fig. 3. Again, the asymptotic behavior of ⌘op in Eq. (19) for ⌘C ' can be deduced from the series expansion in terms of (1 ⌘C) > 0, using Eq. (32), which is ⌘op =1 + (1 q⇤ max)(1 ⌘C) ln(1 ⌘C) + (1 q⇤ max) ln[q⇤ max(1 q⇤ max)](1 ⌘C) + O h (1 ⌘C)2 i , (36 based on the relations in Eqs. (34) and (35). As shown i Fig. 5, the asymptotic form in Eq. (36) only holds in a rathe the (singular) series expansion at ⌘C ! 1 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 q*(ηc→0) = ε*(ηc→0) q*(ηc=1) optimaltransitionrates ηc = 1 − T2 / T1 q* ε* ηc→0 and 1 asymptotes

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