Efficient Heat Engines are Powerless
a fundamental tradeoff relation in thermodynamics proved in 2016
Hal Tasakiprerequisitespart 1: some idea about college thermodynamicspart 2: some knowledge about statistical mechanics and stochastic processes
Thermodynamics
quantitatively exact macroscopic phenomenological theory about
formulated entirely within macroscopic description without references to “microscopic world”
possible transitions between equilibrium statesenergy transfer associated with transitions
The second law of thermodynamics is, without a doubt, one of the most perfect laws in physics. Any reproducible violation of it, however small, would bring the discoverer great riches as well as a trip to Stockholm. (Lieb and Yngvason 1997)
a crucial guide in the revolution from classical to
quantum mechanics
What is thermodynamics?
modern textbooks from fully operational points of view
Lieb and Yngvason, “The physics and mathematics of the second law of thermodynamics” (1997)
hold. With the aid or the axiom of choice this can be achieved by considering the formal vectorspace spanned by all systems and choosing a Hamel basis of systems {Γα} in this space such thatevery system can be written uniquely as a scaled product of a finite number of the Γα’s. (See Hardy,Littlewood and Polya, 1934). The choice of an arbitrary state XΓα
in each of these ‘elementary’systems Γα then defines for each Γ a unique XΓ such that (2.17) holds. (If the reader does notwish to invoke the axiom of choice then an alternative is to hypothesize that every system has aunique decomposition into elementary systems; the simple systems considered in the next sectionobviously qualify as the elementary systems.)
For X ∈ Γ we consider the space Γ× Γ0 with its canonical entropy as defined in (2.14), (2.15)relative to the points (XΓ, Z0) and (XΓ, Z1). Using this function we define
S(X) = SΓ×Γ0((X,Z0) | (XΓ, Z0), (XΓ, Z1)). (2.18)
Note: Equation (2.18) fixes the entropy of XΓ to be zero.Let us denote S(X) by λ which, by Lemma 2.3, is characterized by
(X,Z0) ∼A ((1 − λ)(XΓ, Z0),λ(XΓ, Z1)).
By the cancellation law this is equivalent to
(X,λZ0) ∼A (XΓ,λZ1)). (2.19)
By (2.16) and (2.17) this immediately implies the additivity and extensivity of S. Moreover,since X ≺ Y holds if and only if (X,Z0) ≺ (Y,Z0) it is also clear that S is an entropy function onany Γ. Hence S and SΓ are related by an affine transformation, according to Theorem 2.3.
Definition (Consistent entropies). A collection of entropy functions SΓ on state spacesΓ is called consistent if the appropriate linear combination of the functions is an entropy functionon all multiple scaled products of these state spaces. In other words, the set is consistent if themultiplicative constants aΓ, referred to in Theorem 2.5, can all be chosen equal to 1.
Important Remark: From the definition, (2.14), of the canonical entropy and (2.19) it followsthat the entropy (2.18) is given by the formula
S(X) = sup{λ : (XΓ,λZ1) ≺ (X,λZ0)} (2.20)
for X ∈ Γ. The auxiliary system Γ0 can thus be regarded as an ‘entropy meter’ in the spirit of(Lewis and Randall, 1923) and (Giles, 1964). Since we have chosen to define the entropy for eachsystem independently, by equation (2.14), the role of Γ0 in our approach is solely to calibrate theentropy of different systems in order to make them consistent.
Remark about the photon gas: As we discussed in Section II.B the photon gas is special andthere are two ways to view it. One way is to regard the scaled copies Γ(t) as distinct systems andthe other is to say that there is only one Γ and the scaled copies are identical to it and, in particular,must have exactly the same entropy function. We shall now see how the first point of view can bereconciled with the latter requirement. Note, first, that in our construction above we cannot takethe point (U, V ) = (0, 0) to be the fiducial point XΓ because (0, 0) is not in our state space which,according to the discussion in Section III below, has to be an open set and hence cannot containany of its boundary points such as (0, 0). Therefore, we have to make another choice, so let us
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rigorous operational formulationwith a deep physical insight
Hal Tasaki and Glenn Paquette “Thermodynamics: A Novel Approach” (to be published from Oxford UP in 2020?)
田崎晴明『熱力学：現代的な視点から』（培風館）
What is thermodynamics?
Tasaki SasaShimizu
Motivation
Heat engine
non-mechanical transfer of energynon-usable energy usable energy
illustration by Chisato Naruse
a physical system which converts heat into work
heat bath
a central object in thermodynamics
These are perpetual mobile of the 2nd kind,
inhibited by the second law of thermodynamics
Heat engine
W = QH �QL
operates cyclically, interacting with two heat baths
�H
�L
heat bath
heat bath
WQH > 0
QL > 0
�H < �L
in a single cycle absorbs energy from the hot bathQH
expels energy to the cold bathQL
extracted work
a coal-fired power plant� = T�1
(external combustion engine)
Efficiency and power of a heat engine
W = QH �QL
⌘ =W
QH
⌘C := 1� �H
�L
work extracted in a cycle
efficiency of the engine
⌧ period of the cycleW
⌧power of the engine
Carnot’s theorem
thermodynamics:
there is no fundamental
limitation on the power of a
heat engine
�H
�L
heat bath
heat bath Carnot efficiency
a general heat engine (cyclic thermodynamic process)
W
attached to two heat baths
QH > 0
QL > 0
�H < �L
< 1
Carnot cycle�H
�L
�L
�H
the power vanishes
attains the maximum possible efficiency !⌘Cbut only in the quasi-static limit, with period ⌧ " 1
W
⌧# 0
isothermal adiabatic adiabatic
isothermal
Carnot engine is extremely efficient but is totally powerless!!
QUESTION: can there be a heat engine with non-zero power which attains the (maximum) Carnot efficiency?
near Carnot cycle
�L�H +��
�L ����H
J
J
J ' ��
⌘ ' 1� �H +��
�L ���' ⌘C �
⇣ 1
�L+
�H
(�L)2
⌘��
⌧ ' QH +QL
J' QH +QL
��
induce finite current by a temperature-( , heat conductivity)�� > 0
maximum possible efficiency
minimum possible period⌧ ⌘and are related by
⌧ ' (QH +QL)2
�LQH
1
⌘C � ⌘⌧ " 1 ⌘ " ⌘Cas
adiabatic adiabatic
QH, QL > 0
⌘C = 1� �H
�L
-difference
near Carnot cycle
�L�H +��
�L ����H
J
J
⌧ ⌘and are related by⌧ ' (QH +QL)2
�LQH
1
⌘C � ⌘⌧ " 1 ⌘ " ⌘Cas
power W
⌧=
QH �QL
⌧must vanish as the efficiency
approaches the Carnot efficiency⌘
⌘C
what about more general heat engines?
adiabatic adiabatic
General heat engines?QUESTION: can there be a heat engine with non-zero power which attains the (maximum) Carnot efficiency?
?QH
QL
W
thermodynamics has no time scale
thermodynamics alone cannot answer this question
we need some microscopic dynamicalframework
approach based on nonequilibrium statistical mechanics
QUESTION: can there be a heat engine with non-zero power which attains the (maximum) Carnot efficiency?
G. Benenti, K. Saito, and G. Casati, PRL 106, 230602 (2011)
K. Brandner, K. Saito, and U. Seifert, PRL 110, 070603 (2013) V. Balachandran, G. Beneti, and G. Casati, PRB 87, 165419 (2013) J. Stark, et.al. PRL 112, 140601 (2014) B. Sothmann, R. Sanchez, and A. Jordan, EPL 107, 47003 (2014) R. Sanchez, B. Sothmann, and A. Jordan, PRL 114, 146801 (2015) K. Yamamoto, et.al., PRB 94, 121402(R) (2016)
K. Brandner, K. Saito, and U. Seifert, PRX 5, 031019 (2015) K. Proesmans and C. Van den Broeck, PRL 115, 090601 (2015)
M. Mintchev, L. Santoni, and P. Sorba, arXiv:1310.2392 (2013) M. Campisi and R. Fazio, Nature Commun. 7, 11895 (2016) A.E. Allahverdyan, K. V. Hovhannisyan, A. V. Melkikh, and S. G. Gevorkian, Phys. Rev. Lett. 111, 050601 (2013) M. Ponmurugan, arXiv:1604.01912 (2016) M. Polettini and M. Esposito, arXiv:1611.08192 (2016)
concrete models (within linear response)
other approaches
general argument within linear responseyes?
no…
yes???
?QH
QL
W
General heat engines?
Outline of the new resultNaoto Shiraishi, Keiji Saito, and Hal Tasaki Universal Trade-Off Relation between Power and Efficiency for Heat EnginesPhys. Rev. Lett. 117, 190601 (2016)
Naoto Keio U. (now at Gakushuin U.)
KeijiKeio U.
HalGakushuin U.
�H
�L
Settingthe engine is modeled as a classical system of N particles with arbitrary potential and interactions the effect of the heat baths on
the dynamics of the engine is described by random force of the Langevin type
an external agent controls the potential and the interactions with the baths in a periodic manner according to a fixed protocolgeneral and standard framework that can describe any macroscopic engines
QUESTION: can there be a heat engine with non-zero power which attains the (maximum) Carnot efficiency?Our answer: NO, provided that our description is valid
⌧ � (QH +QL)2
⇥̄�LQH
1
⌘C � ⌘
⌧ ' (QH +QL)2
�LQH
1
⌘C � ⌘
?QH
QL
W
our result
⇥̄ < 1 depends on the state and the design of bathsfor the near Carnot engine
⇥̄ ! if the system isclose to equilibrium
W
⌧ ⇥̄�L ⌘(⌘C � ⌘)
General heat engines?
About our main result
?QH
QL
W
W/⌧
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⌘
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⌘C
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0
<latexit sha1_base64="E0H4qpcBLSd+MHc3raGLKiTivB4=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1Fip6Q7KFbfqLkDWiZeTCuRoDMpf/WHM0gilYYJq3fPcxPgZVYYzgbNSP9WYUDahI+xZKmmE2s8Wh87IhVWGJIyVLWnIQv09kdFI62kU2M6ImrFe9ebif14vNeGNn3GZpAYlWy4KU0FMTOZfkyFXyIyYWkKZ4vZWwsZUUWZsNiUbgrf68jppX1W9WrXWrFXqt3kcRTiDc7gED66hDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBe1+Muw==</latexit>
power
1
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not allowed!Carnot 1823
efficiency
not allowed!S.S.T. 2016
allowed
W
⌧ ⇥̄�L ⌘(⌘C � ⌘)
efficient engines are powerless!!
Model Stirling engine. By Richard Wheeler (Zephyris) 2007
Summary and remark We have proved a tradeoff relation between
power and efficiency, which implies that a heat engine with non-zero power can never attain the Carnot efficiency
Inevitable loss in a heat engine with non-zero power is caused by heat current between the engine and the bathsa fundamental limitation on external combustion engines (no such problems for internal combustion engines)
continues to part 2 (which is for experts)
