Efficient Heat Engines
are Powerlessa fundamental tradeoff relation inthermodynamics proved in 2016

Hal Tasaki
prerequisites
part 1: some idea about college thermodynamics
part 2: some knowledge about statisticalmechanics and stochastic processes

Thermodynamics

What is thermodynamics?

quantitatively exact macroscopic
phenomenological theory about
possible transitions bet ween equilibrium states
energy transfer associated with transitions
The second law of thermodynamics is, without a doubt, one of themost perfect laws in physics. Any reproducible violation of it,
however small, would bring the discoverer great riches as well as atrip to Stockholm. (Lieb and Yngvason 1997)

formulated entirely within macroscopic description
without references to “microscopic world”
a crucial guide in therevolution from classical toquantum mechanics

What is thermodynamics?

tent entropies). A collection of entropy functions SΓ on state spa
e appropriate linear combination of the functions is an entropy functducts
theseYngvason,state spaces.“TheIn otherwords,
themathematics
set is consistent if
Liebofandphysics
andreferredtoinTheorem
2.5,
canallbechosenequalto1.
Γ , ofthe second law of thermodynamics” (1997)

rom the definition, (2.14), of the canonical entropy and (2.19) it follo
rigorous
operational
formulation
given by the formulawith a deep physical insight

S(X) = sup{λ : (XΓ , λZ1 ) ≺ (X, λZ0 )}Tasaki(2.

Sasasystem Γ0 can thus
be regarded as an ‘entropy meter’ in the spiritand (Giles, 1964). SinceShimizu
we have chosen to define the entropy for eequation (2.14), the role of Γ0modern
in our approach
is solelyto calibrate
textbooks
fromfullyms in order to make them consistent.
operational points of viewoton gas: As we discussed in Section II.B the photon gas is special aHalTasakiandGlennPaquette“Thermodynamics:A
Novel(t)w it. One way is to regard the scaled copies Γ as distinct systems aApproach”(tobepublishedfromOxford
UPin2020?)
re is only one Γ and the scaled copies are identical to it and, in particu
田崎晴明『熱力学：現代的な視点から』（培風館）

Motivation

Heat engine
a central object in thermodynamics

a physical system which converts heat into work
non-mechanical transfer of energy
non-usable energy

usable energy

heat bath,dnikdn2ehstcifomaelnibyodomlmareuthetpfroepw
ealrade
nsoecehsTehtybdet
ii nhi b
illustration by Chisato Naruse

Heat engine(external combustion engine)
operates cyclically, interacting with t wo heat baths
in a single cycle absorbs energy QH from the hot bath
expels energy QL to the cold bathextracted work W = QH QL
heat bath

H

QH > 0

QL > 0heat bath

LH <WL=T1

a coal-fired power plantEfficiency and power of a heat engine
a general heat engine (cyclic thermodynamic process)attached to t wo heat baths
heat bath H
work extracted in a cycle
W
W = QH QL
QH > 0efficiency of the engine
HW  ⌘ := 1<1C⌘=L
QL > 0
QH
Carnotheat bath L
Carnot’s theorem efficiency
H < L:
scimanydomlathert⌧ period of the cyclenemadnufonsiae
froer
thewopehtnon
Woitatimil
power of the engine
enigne⌧h e atHCarnot cycle
LH

isothermal

adiabatic

isothermalL

adiabatic

attains the maximum possible efficiency ⌘C !
but only in the quasi-static limit, with period ⌧ " 1W
# 0 the power vanishes⌧
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

induce finite current J ' by a temperature> 0,  heat conductivity)-difference(HLJJ
maximum possible efficiency
H+⌘'1

adiabatic

H+' ⌘C⇣ 1adiabatic

L+H⌘⌘C = 1HL

2()
LL
L
QH + QL
QH + QL'minimum possible period ⌧ 'J

⌧ and ⌘ are related by2(QH + QL )1⌧'
 L QH ⌘C ⌘Q H , QL > 0⌧ " 1 as ⌘ " ⌘Cnear Carnot cycle
HLJ

H+adiabaticJ⌧ and ⌘ are related by2(QH + QL )1⌧' L QH ⌘C ⌘Ladiabatic⌧ " 1 as ⌘ " ⌘CW
QH QLpower
must vanish as the efficiency ⌘=⌧⌧approaches the Carnot efficiency ⌘Cwhat about more general heat engines?

General heat engines?

?

QH

QUESTION:
cantherebeaheatengine
Wwith non-zero power which attains the(maximum) Carnot efficiency?

QL

thermodynamics alone cannot answer
this question
thermodynamics has no time scale
we need some microscopic dynamicalframework
approach based on nonequilibrium
statistical mechanics

General heat engines?

?

QHQUESTION: can there be a heat engine
Wwith non-zero power which attains the(maximum) Carnot efficiency?
yes?
general argument within linear response

QL

G. Benenti, K. Saito, and G. Casati, PRL 106, 230602 (2011)

no…
yes???
other approaches

concrete models (within linear response)
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)

Hal
Gakushuin U.

Naoto
Keio U. (now at Gakushuin U.)

Keiji
Keio U.

Outline of the new result
Naoto Shiraishi, Keiji Saito, and Hal Tasaki
Universal Trade-Off Relation between
Power and Efficiency for Heat Engines
Phys. Rev. Lett. 117, 190601 (2016)

HL

Setting
the engine is modeled as aclassical system of N particles
with arbitrary potential andinteractions
the effect of the heat baths on
the dynamics of the engine is describedby random force of the Langevin typean external agent controls the potential and theinteractions with the baths in a periodic manneraccording to a fixed protocol

general and standard framework that candescribe any macroscopic engines

General heat engines?

?

QH

QL

W QUESTION: can there be a heat engine
with non-zero power which attains the(maximum) Carnot efficiency?
Our answer: NO, provided that ourdescription is valid

our result
2(QH + QL )1W
¯ L ⌘(⌘C ⌘)⌧⇥¯ L QH ⌘C ⌘⌧⇥¯ < 1 depends on the state and the design of baths⇥
for the near Carnot engine
2(QH + QL )1⌧'
 L QH ⌘C ⌘¯ !  if the system is⇥close to equilibrium

About our main resultW¯ L ⌘(⌘C⇥⌧W/⌧

power⌘)

not allowed!
S.S.T. 2016

not allowed!
Carnot 1823

allowed
0?

QH
⌘C1

⌘

efficiency

WQL

efficient engines are powerless!!

Summary and remark
We have proved a tradeoff relation between
power and efficiency, which implies that a heatengine with non-zero power can never attain theCarnot efficiency
Inevitable loss in a heat engine with non-zeropower is caused by heat current bet ween the engine
and the baths
a fundamental limitation on external combustion engines(no such problems for internal combustion engines)
continues to part 2 (which is for experts)
Model Stirling engine. By Richard Wheeler (Zephyris) 2007