Efficiency bounds for nonequilibrium heat engines

0.31 MB·English

by  Pankaj Mehta

2 Efficiency bounds for nonequilibrium heat engines
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2 PankajMehta,1∗ AnatoliPolkovnikov,1∗

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J 1DepartmentofPhysics,BostonUniversity,

4 DepartmentofPhysics,BostonUniversity, 590CommonwealthAve.,Boston,MA02215,USA
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] ∗Towhomcorrespondence shouldbeaddressed; E-mail: [email protected]; [email protected]
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Increasing theefficiencyofengines isalong-standing technological problem. Thesec-
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a ond law of thermodynamics ensures that the efficiency of any engine must be below
t
s
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t the Carnot bound. Theidealized engines traditionally considered in thermodynamics
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operateusingtworeservoirs, ahotreservoir thatprovidesenergyandacoldreservoir
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n thatservesasanentropysink. Hereweconsideradifferentbutmorecommonclassof
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nonequilibriumengines with a single reservoir. Byextending fundamental thermody-
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1 namicrelationstononequilibriumprocesses,wefindarigorousthermodynamicbound
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for the efficiency of single-reservoir engines which is below the Carnot efficiency. We
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5 showthattheefficiencyoftheseenginescanbeincreasedbyusingnon-ergodicengines
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0 toperformworkandillustratetheseideasusingsimpleexamples. Theseresultssuggest
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1 anewgeneralstrategyfordesigningmoreefficientengines.
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Introduction Heat engines are systems that convert heat or thermal energy into macroscopic work.
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a
Heatengines playamajorroleinmodern technology andarecrucial toourunderstanding ofthermody-
namics(1,2). Examplesofheatengines include conventional combustion engine suchasthose foundin
cars and airplanes, various light emitting devices, aswellasnaturally occurring engines such asmolec-
ular motors (3). A conventional heat engine consists of two heat reservoirs, a hot reservoir that serves
asasource ofenergy andacoldreservoir thatserves asanentropy sink. Theefficiency ofsuch engines
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is fundamentally limited by the second law of thermodynamics providing an upper bound given by the
efficiencyofaCarnotengineoperating atthesametemperatures (1),
T
c
η = 1 , (1)
c
− T
h
withT andT thetemperature ofthecoldandhotreservoirs respectively.
c h
Realenginesoftendiffersignificantlyfromtheidealized,two-reservoir enginesconsideredinclassi-
cal thermodynamics. Theyoperate withasingle bath, such asthe atmosphere, that serves asanentropy
sink. Instead ofahigh temperature bath, energy issuddenly deposited inthesystem atthe beginning of
each cycle and is converted into mechanical work. The most common example of this are combustion
engines such as those found in cars where energy is deposit in the system through the combustion of a
fuel. Currently,themostrealisticmodelsdescribingcombustionenginesarebasedontheOttocycle(1),
withacorresponding efficiency. whichislessthanη withappropriately chosentemperaturesT andT .
c c h
One can ask some natural questions: is the Carnot efficiency a good bound for the efficiencies of such
single-reservoir engines or are these engines better described by a different bound? Are there realistic
processes that allow you to realize these bounds? Can we overcome the thermodynamic bounds if we
useengines whicharenotcompletely ergodic?
To address these questions, wegeneralize the fundamental relations of thermodynamics to describe
large, nonequilibrium quenches in systems coupled to a thermal bath. We use these relations to derive
newboundsfortheefficiencyofnonequilibrium enginesthatoperatewithasinglebath. Weanalyzeour
bounds in two different regimes, a local equilibrium regime where the system quickly thermalizes with
itself(butnotthebath),andanon-erdgodic regimewherethethermalization timesaremuchlongerthan
timescalesonwhichworkisperformed. Wedemonstrateourresultsusingasimpleexamplessuchasan
idealgasthatdrivesapistonandamagneticgasengine.
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Generalized Thermodynamic Identities Most applications of thermodynamics are connected to the
fundamental thermodynamic relation(4)
dE = TdS dλ, (2)
−F
where E is the energy if the system, T is the temperature, S is the entropy, λ is some external macro-
scopic parameter, and is the generalized force. When λ is the volume stands for the pressure and
F F
the fundamental relation takes the most familiar form dE = TdS PdV. The fundamental relation
−
mathematically encodes the fact that the energy of a system in equilibrium is a unique function of the
entropy and external parameters. For quasistatic processes, one can associate the first term with the
supplied heat and the second term with the work done on the system by changing the coupling λ. The
fundamental relation canalsobeintegrated forquasistatic processes andonecanexplicitly compute the
totalwork,heatetc. However,howtogeneralizethesecalculations tostronglynonequilibrium processes
where changes in energy, entropy, etc. can be large, is still largely an open question. Using the second
law ofthermodynamics, onecan prove various inequalities. Inparticular, ifweprepare asystem Aina
thermalstatewithtemperatureT andletitequilibrate withabathattemperatureT thenthesecondlaw
A
ofthermodynamics impliestworelatedinequalties: (seeAppendixA)
T ∆S ∆E 0, T∆S ∆E 0. (3)
A A A A A
− ≤ − ≥
The first inequality is applicable to the case where an energy ∆E is deposited in the system in a
A
nonequilibrium fashion, for example, by an external energy pulse, and the second inequality describes
the relaxation ofasystem back toequilibrium. Itimplies that thefree energy ofthesystem can only go
downduringtherelaxation (4).
In this work we establish two main closely related results, which refine the inequalities (3) to arbi-
trarynon-equilibriumprotocolsusingtheconceptofrelativeentropy. Relativeentropy,ortheKullback-
Leibler Divergence, is well known in information theory (5,6) and appears naturally in statistical me-
chanics within the context of large deviation theory (7). In deriving our results, wewill use “quantum”
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notations and restrict ourselves to discrete probability distributions. Our results also equally apply to
classical systems with continuous probability distributions and can be derived from the corresponding
“quantum” results by multiplying all distributions by an appropriately chosen density of states (see (8)
andAppendixC).Thesegeneral resultsvalidforbothquantum andclassical systemsarecloselyrelated
to those recently obtained by S.Deffner and E.Lutz(9,10) forquantum systems butdeviate in asubtle
but important way that is crucial to our discussion (see Sec A for details). Consider a system A with
external parameter λ = λ(1) and a λ-dependent energy spectrum (1) (λ ) which is coupled to
n n 1
E ≡ E
a thermal bath at temperature T. We assume that the bath is insensitive to the coupling λ (Figure 1).
Initially, the system is prepared in equilibrium with the bath and is described by a Boltzmann distribu-
tion of the form p(1) = exp[ (1)/T]/Z with Z = e−βǫ(n1). In stage I, the system undergoes
n −En 1 1 n
an arbitrary process where λ is changed from λ(1) to λ(2P) , resulting in a new non-equilibrium state, q
n
(see Appendix A for discussion about the meaning of q ). We do not assume that during this process
n
thesystem isthermally isolated. Duringprocess II,thesystem re-equilibrates withthebath, eventually
reaching anewBoltzmanndistribution withλ =λ(2),p(2) = exp[ (2)/T]/Z .
n n 2
−E
DuringstageI,thetotalchangeinenergyinthesystem∆EI canbedividedintotwoparts,adiabatic
work,WI ,andheat,QI,
ad
∆EI = WI +QI. (4)
ad
Adiabatic workisdefined asthechange inenergy thatwould result from adiabatically changing thepa-
rametersfromλ(1) toλ(2). Physically, itmeasureschangesintotalenergystemmingformtheparameter
dependence oftheenergyspectrum (potential energy). Bydefinition, theheatistheremainingcontribu-
tion to the change in energy (2). Thus in our language heat includes both the non-adiabatic part of the
work and the conventional thermodynamic heat. The heat generated during process I can be explicitly
calculated (seeAppendixA):
QI = T∆SI +TS (q p(2)) TS (p(1) p(2)), ∆SI = S(q) S(p(1)) (5)
r r
|| − || −
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Equilibrium (1) Nonequilibrium Equilibrium (2)
λ(1) I λ(2) II λ(2)
Energy
Relaxation
injection
p(1,)ε(1) q ,ε(2) p(2,)ε(2)
n n T n n T n n T
Figure1: Generalized nonequilibrium quenches. Asystem parameterized byλiscoupled toanexternal
bath at temperature T. Initially, λ = λ(1) and the system is in equilibrium and is described by the
(1)
Boltzmann distribution p . I, energy is suddenly injected into the system while changing λ from
n
λ(1) to λ(2). The system is now described by possibly nonthermal distribution, q . During stage II of
n
the process, the system relaxes and equilibrates with the external bath after which it is described by a
Boltzmanndistribution p(2) withλ = λ(2).
n
where the S(p) p logp is the diagonal entropy of a probability distribution p (11) and
≡ − n n n
S (q p) q log(qP/p )istherelative entropy between thedistributions q and p. Wehave shown
r || ≡ n n n n
previously thPatforlargeergodicsystems,thediagonalentropy isequivalenttotheusualthermodynamic
entropy (11). Note that for a cyclic process where λ(1) = λ(2) the last term in Eq. (5) vanishes since
p(1) = p(2). During process II, the system re-equilibrates withthe bath by exchanging heat, QII, with
thereservoir.
Onecanshowthat(seeAppendixA)
QII = T∆SII TS (q p(2)), ∆SII = S(p(2)) S(q). (6)
r
− || −
The importance of relative entropy for describing relaxation of nonequilibrium distributions has been
discussed in previous for different setups both in quantum and classical systems (10,12–15). Taken
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together, (4),(5),and (6)constitute thenonequilibrium identities thatwillbeexploited nexttocalculate
bounds fortheefficiencyofengines thatoperatewithasingleheatbath.
A B
la
m
re
hto Energy
sI
Q Thermalization
H
cita cita Equilibrate
b b
aid aid with bath
A A Tq
la
m
re Perform
hto Work
sI
Q
L
C D
0.8
Energy
0.7 η
Thermalization η 0.6 c η
y0.5 mt η
Equilibrate cn0.4 3
with bath Tq eic0.3 η5/3
iffE0.2
Perform 0.1
Work 00 0.5 1 1.5 2 2.5 3
τ
Figure 2: Comparison of Carnot engines and single-heat bath engines (A) Carnot engines function by
using two heat reservoirs, a hot reservoir that serves as a source of energy and a cold reservoir that
serves as an entropy sink. (B) In the thermalization regime, energy is injected into the engine. The
gaswithin the engine quickly equilibrates withitself. Thegasthen performs mechanical work andthen
relaxes to back to its initial state. (C). In the nonequilibrium regime, the system thermalizes on time
scales much slower than time scales on which work is performed. (D). (blue) Maximum efficiency as
a function of excess energy (ratio of injected energy to initial energy), τ, for Carnot engine, η , (red)
c
true thermodynamic bound, η , (magenta) actual efficiency of nonequilibrium engine which acts as
mt
an effective one-dimensional gas, η (see main text), and (green) actual efficiency of three-dimensional
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idealgasLenoirengine, η .
5/3
Maximum Efficiency of Engines Figure 2 summarizes the single-reservoir engines analyzed in this
workandcomparesthemwithCarnotengines(1). Theengineisinitiallyinequilibriumwiththeenviron-
ment (bath) at a temperature T and the system is described by the equilibrium probability distribution
0
p . In the first stage, excess energy, ∆Q, is suddenly deposited into the system. This can be a pulse
eq
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electromagnetic wave, burst of gasoline, current discharge etc. In second stage, the engine converts the
excess energy into workand reaches mechanical equilibrium withthe bath . Finally, the system relaxes
backtotheinitialequilibrium state. Ofcoursesplittingthecycleintothreestagesisratherschematicbut
itisconvenientfortheanalysisoftheworkoftheengine. Suchanenginewillonlyworkiftherelaxation
time of the system and environment is slow compared to the time required to perform the work. Other-
wise the energy will be simply dissipated to the environment and no work will be done (see discussion
inRef.(16)).
The initial injection of energy, ∆Q results in the corresponding entropy increase ∆SI = S(q)
−
S(p )ofthesystem, whereS is thediagonal entropy andq describes the system immediately after the
eq
addition of energy. Because by assumption the environment is not affected during this initial stage, the
totalentropychangeofthesystemandenvironmentisalsojust∆SI. Bytheendofthecycle,theentropy
of the system returns to its initial value. Thus, from the second law of thermodynamics, the increase in
entropy of the environment must be greater than equal to ∆SI. This implies that the minimal amount
of heat that must be dissipated into the environment during the cycle is T ∆SI. An engine will work
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optimally if no extra entropy beyond ∆SI is produced during the system-bath relaxation since then all
oftheremaining energy injected intothesystem isconverted towork. Thus,themaximalworkthatcan
beperformedbytheengineduringacycleisW = ∆Q T ∆SI. Foracyclicprocesssuchastheone
m 0
−
considered here, substituting (5) into the expression for W implies that the maximum efficiency of a
m
nonequilibrium engine,η ,isgivenby
nme
W T ∆SI T S (q p)
m O 0 r
η = = 1 = || . (7)
mne
∆Q − ∆Q ∆Q
Equation (7) is the main result of this paper. It relates the maximum efficiency of an engine to the
relative entropy of the intermediate nonequilibrium distribution and the equilibrium distribution. We
nextconsidervariouslimitsandapplications ofthisresult. WepointthatEq.(5)alsoallowsustoextend
the maximum efficiency bound to a more general class of engines, like Otto engines, where during the
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firststageofthecycleonesimultaneously changestheexternalparameterλfromλ toλ . Inthiscase,
1 2
T S (q p(2)) S (p(1) p(2))
0 r r
η = || − || , (8)
mne
∆Q
(cid:2) (cid:3)
where p(1) and p(2) stand for equilibrium Gibbs distributions corresponding to the couplings λ and λ
1 2
at the beginning and the end of the process I respectively. Since the second term is negative, changing
the external parameter during the first stage can only reduce the engine efficiency, though this may be
desirable forotherpractical reasonsunrelated tothermodynamics.
Efficiency of Thermal Engines An important special case of our bound is the limit where the the
relaxation ofparticleswithintheengineisfastcomparedtothetimescaleonwhichtheenginepreforms
work(seeFigure2). Thisisthenormalsituationinmechanicalenginesbasedoncompressing gasesand
liquids. In this case, after the injection of energy the particles in the engine quickly thermalize and can
bedescribed byagasataneffectivetemperature T(E) (dS/dE)−1 thatdependsontheenergyofthe
≡
gas. ItisshowninAppendixC,thatinthiscase,(7)reduces to
T ∆SI 1 E+∆Q T
η =1 0 = dE′ 1 0 . (9)
mt − ∆Q ∆Q − T(E′)
ZE (cid:18) (cid:19)
Bydefinitionη isthetrueupperboundforthermalefficiencyofasinglereservoir engine.
mt
It is easy to see that η is the integrated Carnot efficiency and thus it is always smaller that the
mt
Carnot efficiency corresponding to the same heating Q = ∆Q (see Fig. 2). This efficiency bound
h
becomes verysimple foridealgases whereT(E) E. Assuming thatinthebeginning ofthecycle the
∝
systemisinequilibriumwithenvironment,onehasthatthemaximalefficiencyofanequilibriumengines
thatthermalizes is
1
η = 1 log(1+τ), (10)
mt
− τ
whereτ = ∆Q/E = ∆T/T . Forcomparison theequivalent Carnotefficiencyis
0
τ
η = . (11)
c
τ +1
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Itisinterestingthattheresultforη isvalidforarbitraryidealgasesanddoesnotdependondimension-
mt
ality or thetype of dispersion (linear, quadratic etc.) or thenumber of internal degrees offreedom. Itis
alsovalidformixturesofidealgaseswithdifferentmassesanddispersionrelations. Theexpression(10)
canbeextended tothesituations wheretheinitial temperature oftheengine isdifferent from thatofthe
environment (seeAppendixC).
Higher efficiency bound for non-thermal distributions. Another interesting limit is when the full
thermalization time in the system is long compared to the time required to perform the work. We call
engines that work in these parameter regime nonequilibrium engines. This situation can be realized in
small systems, integrable or nearly integrable systems with additional conservation laws or the systems
where different degrees offreedom are weakly coupled like e.g. kinetic and spin degrees of freedom of
molecules, electrons and phonons inmetalsand semiconductors andsoon. Insuch systems theprocess
of relaxation typically occurs in two stages. The system first undergoes a fast relaxation to a quasi
steady-state, prethermalized distribution. Subsequently, the system then very slowly relaxes to the true
equilibrium distribution. The notion of prethermalization mechanism was first suggested in the context
of cosmology (17). Since ithas been confirmed tooccur both experimentally and theoretically inmany
physicalsituationsincludingoneandtwodimensionalturbulence(18),weaklyinteractingfermions(19),
quenches inlowdimensional dimensional superfluids (seeRef.(20)forreview).
Prethermalization iswellknownfromstandardthermodynamics wheretwoormoreweaklycoupled
systems first quickly relax to local equilibrium states and then slowly relax with each other. From a
microscopicpointofview,prethermalizationisequivalenttodephasingwithrespecttoafastHamiltonian
H where the density matrix effectively becomes diagonal withrespect to the eigenstates ofH . Itwas
0 0
also recently realized that thermalization can be also understood as dephasing with respect to the full
Hamiltonian ofthesystem through theeigenstate thermalization hypothesis (21–23). Inthelanguage of
kinetictheoryofweaklyinteractingparticles,prethermalization impliesafastlossofcoherencebetween
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particles governed by the noninteracting Hamiltonian H = mv2/2 followed by a much slower
0 j j
relaxation ofthenon-equilibrium distribution functiontotheBoPltzmannformduetosmallinteractions.
Theefficiencyofanonequilibrium engineisgivenby(7)withqnowrepresentingtheprethermalized
distribution. A simple minimization shows that the numerator of (7) for a fixed energy increase ∆Q
has a minimum precisely for the Gibbs distribution (see Appendix D). Thus, any non-equilibrium state
can only increase the maximum possible efficiency of the engine. Alternatively this statement can be
understood from the fact that the Gibbs distribution maximizes the entropy for a given energy (24) .
Finally, notice that the first equality in (7) implies that the maximum value of η is achieved for a
mne
process whentheprethermalized non-equilibrium statehasthesamediagonal entropy astheinitialstate
i.ewheretheprobability q arepermutations oftheprobabilities p .
n n
Maximumefficiencyofcoherentnon-equilibriumengines. Finallywebrieflydiscuss theefficiency
bound for coherent engines which preserve coherence between particles while performing macroscopic
work. Such engines are sensitive not only to conserved or approximately conserved quantities (like
energyandvelocitydistributionforweaklyinteractinggas)butalsotonon-conserveddegreesoffreedom
(like precise positions ofparticles atagiven momentoftime). Inpractice, such engines canberealized
only for very small, non-interacting systems with long coherence times. Such engines have the highest
efficiency bound still given by Eq. (7) but with S standing for the full relative entropy of the non-
r
equilibrium distribution q withrespect totheequilibrium distribution p(seeAppendixAfordetails):
Svn(q p) = Tr[ρ(log(ρ) log(p))]. (12)
r || −
Wewillnotfurtherdiscusssuchenginessincetheyarenotthermal.
Some simple examples Let us start from the simplest ideal gas single reservoir engine which pushes
thepiston. Theengine undergoes theLenoircycleasillustrated inFigure2. Firstapulseofenergy ∆Q
is deposited to the gas via e.g. a gasoline burst. The gas immediately thermalizes at a new temperature
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