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Colloquium: Fundamental aspects of steady state heat to work conversion

Giuliano Benenti∗ and Giulio Casati†

CNISM & Center for Nonlinear and Complex Systems, Universita degli Studi dell’Insubria, Via Valleggio 11, 22100 Como,

Italy and

Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy

Tomaz Prosen‡

Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia.

Keiji Saito§

Department of Physics, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

(Dated: July 27, 2018)

We review theoretical approaches to analyzing efficiency of steady state heat to work conversion

which is crucial in the timely problem of optimizing efficiency of small-scale heat engines and

refrigerators. A rather abstract perspective of non-equilibrium statistical mechanics and dynamical

system’s theory is taken to view at this very practical problem. Several recently discovered

general mechanisms of optimizing the figure of merit of thermoelectric efficiency are discussed,

also in connection to breaking time-reversal symmetry of the microscopic equations of motion.

Applications of these theoretical and mathematical ideas to practically relevant models are pointed

out.

Contents

I. INTRODUCTION 1

II. OVERVIEW OF BASIC CONCEPTS OF

NON-EQUILIBRIUM THERMODYNAMICS 2

A. Linear response, Onsager reciprocal relations 3

III. THERMODYNAMIC EFFICIENCIES 4

A. Finite-time thermodynamics 4

B. Figure of merit for thermodynamic efficiency 6

C. Systems with broken time-reversal symmetry 7

IV. NON-INTERACTING SYSTEMS,

LANDAUER-BUTTIKER FORMALISM 9

A. Energy filtering 9

B. Noise and probe reservoirs 10

V. INTERACTING SYSTEMS 12

A. Green-Kubo formula 12

B. Conservation laws and thermoelectric transport 12

C. Thermalization 13

D. Local equilibrium and non-equilibrium steady states 14

VI. NUMERICAL APPROACHES 14

A. Classical 15

B. Quantum 15

VII. MODELS OF THERMODYNAMIC ENGINES 16

A. Stochastic thermodynamics 16

B. Heat engine with blowtorch effect 17

C. Driven quantum dot 18

D. Quantum heat engines 18

∗Electronic address: giuliano.benenti@uninsubria.it†Electronic address: giulio.casati@uninsubria.it‡Electronic address: tomaz.prosen@fmf.uni-lj.si§Electronic address: saitoh@rk.phys.keio.ac.jp

VIII. PHENOMENOLOGICAL LAWS AND

THERMOELECTRIC TRANSPORT 20

A. Wiedemann-Franz law 20

B. Mott’s formula 20

C. Thermoelectricity 20

IX. CONCLUSIONS 21

Acknowledgments 22

References 22

I. INTRODUCTION

The need of providing a sustainable energy to the worldpopulation is becoming increasingly important. It islikely that in the following decades the efforts of the sci-entific community will be increasingly addressed to thisdirection and in particular to the heat to work trans-formation. An important possibility under investiga-tion is the thermoelectric power generation and refriger-ation. In spite of relevant progress made in the last yearsthe efficiency of thermoelectric technology remains toolow (Dresselhaus et al., 2007; Dubi and Di Ventra, 2011;Goldsmid, 2010; Shakouri, 2011; Snyder and Toberer,2008; Sootsman et al., 2009; Vineis et al., 2010). Indeedsuch efficiency depends on physical properties of a givenmaterial namely the electrical conductivity σ, the ther-mal conductivity κ, and the Seebeck coefficient S, andis expressed by a non-dimensional quantity, often calledfigure of merit, ZT = (σS2/κ)T (Ioffe, 1957). High ef-ficiency requires high ZT values which seem difficult toachieve. After more than 50 years from Ioffe’s discoverythat doped semiconductors exhibit a relatively large ther-moelectric effect (Ioffe, 1957; Ioffe and Stil’bans, 1959),and in spite of recent achievements, the most efficient

2

actual devices still operate at ZT around 1. Certainly,in consideration of the importance of the problem, evena small improvement would be most welcome. However,it is generally accepted that ZT ≈ 3 is a target value forefficient competing thermoelectric technology and, so far,no clear paths exist which may lead to reach that target.

In such a situation it is probably useful to investigate adifferent approach namely an approach which starts fromfirst principles i.e. from the fundamental microscopic dy-namical mechanisms which determine the phenomenolog-ical laws of heat and particles transport. In this connec-tion, as it is well known, the enormous achievements innonlinear dynamical systems and the new tools devel-oped have led to a much better understanding of thestatistical behavior of dynamical systems. For exam-ple, the question of the derivation of the phenomeno-logical Fourier law of heat conduction from the dynami-cal equations of motion has been studied in great detail(Dhar, 2008; Lepri, Livi, and Politi, 2003). Theoreticalwork in this direction even led to the possibility to con-trol the heat current and devise heat diodes, transistors,and thermal logic gates (Li et al., 2012). Preliminary ex-perimental results have also been obtained (Chang et al.,2006; Kobayashi, Teraoka, and Terasaki, 2009). We areconfident that this theoretical approach, combined withthe present sophisticated numerical techniques, may leadto substantial progress on the way of improving the longstanding problem of thermoelectric efficiency. An addi-tional motivation in favor of this approach is that ther-moelectric technology, at small sizes (e.g. at micro ornano-scale), is expected to be more efficient than tra-ditional conversion systems. Indeed the efficiency of me-chanical engines decrease very rapidly at low power level.The recent progress in engineering nanostructured mate-rials opens now new possibilities. The study of dynamicalcomplexity of these structures may lead to the design ofnew strategies for developing materials with high thermo-electric efficiency. Nanostructures may allow to controlthe thermal and electrical conductivity e.g. with appro-priate scattering mechanisms (Dresselhaus et al., 2007;Shakouri, 2011; Vineis et al., 2010).

In summary, what is required is a better understandingof the fundamental dynamical mechanisms which con-trol heat and particles transport. The combined effortsof physicists and mathematicians working in nonlineardynamical systems and statistical mechanics, condensedmatter physicists, and material scientists may prove use-ful to contribute substantially to the progress in this fieldof great importance for both energy supply and the en-vironmental concern.

The purpose of the present Colloquium is to intro-duce the basic tools and fundamental results on steadystate heat to work conversion, mainly from a rather ab-stract, statistical physics and dynamical system’s per-spective, yet with a clear focus toward potential applica-tions. We hope our paper might help bridging the gapamong rather diverse communities and research fields,such as non-equilibrium statistical mechanics, mathemat-

ical physics and dynamical systems, mesoscopic physics,and strongly correlated many-body systems of condensedmatter. Our line of presentation is going from more ab-stract to more phenomenological. We start with a shortoverview of non-equilibrium thermodynamics in sectionII, where fundamental results on linear response theoryand Onsager reciprocity relations are discussed. In sec-tion III we then explain basic abstract definitions of ther-moelectric heat to work conversion efficiency. In sec-tion IV we review the microscopic Landauer-Buttikerframework for non-interacting systems and discuss themain non-interacting concepts for controlling thermo-dynamic efficiency: energy filtering, external noise andprobe reservoirs. We believe that most of exciting fu-ture investigations on heat to work conversion will bedevoted to strongly interacting systems, therefore we setthe stage in section V by reviewing state of the art onunderstanding thermalization in equilibrium and local-thermalization near-equilibrium in closed and open inter-acting systems. As the analysis of strongly interactingsystems is mainly relying on numerical simulations, weoutline in section VI some of the key ideas and methodsfor efficient simulation of non-equilibrium steady states ofclassical and quantum open many-body systems. In sec-tion VII we then discuss some simple models of thermo-electric engines and stress their importance from eitherexact-solvability or practical-relevance perspective. Insection VIII we outline some phenomenological and em-pirical laws governing thermoelectric phenomena, withthe emphasis on open theoretical problems. We concludein Sec. IX with some remarks on future prospects of thefield.

II. OVERVIEW OF BASIC CONCEPTS OF

NON-EQUILIBRIUM THERMODYNAMICS

Non-equilibrium thermodynamics (Callen, 1985;de Groot and Mazur, 1984) describes processes on thebasis of two types of parameters: thermodynamic forces

Xi (also known as generalized forces or affinities) drivingirreversible processes, and the fluxes Ji characterizingthe response of the system to the applied forces. Morespecifically, we will consider a generic setup for theextraction of work from a heat flow. The systemperforms work W = −Fx against an external force F ,with thermodynamically conjugate variable x. The forcecan be of mechanical, chemical, or electrical nature. Thethermodynamic force is X1 = F/T , with T being thetemperature of the system. The thermodynamic fluxis J1 = x, where the dot denotes the time derivative.The output power reads P = W = −J1X1T . We areconsidering heat to work conversion, that is, the workis performed by converting a part of the amount ofheat Q1 flowing from the hot reservoir at temperatureT1 (we assume T1 > T2). The thermodynamic force is

X2 = 1/T2 − 1/T1, and the heat current reads J2 = Q1.

For instance, in thermoelectric power generation (see

3

FIG. 1 Schematic drawing of steady-state thermoelectric heatto work conversion. A system S is in touch with two reservoirsat temperatures T1, T2 and electrochemical potentials µ1, µ2.We assume T1 > T2 and µ1 < µ2, so that the generalizedforces X1 < 0 and X2 > 0. In thermoelectric power gen-eration, both charge and heat flow from the hot to the coldreservoir, i.e., J1 > 0 and J2 > 0. In refrigeration, J1 < 0 andJ2 < 0. Note that, while fluxes are one dimensional (alongthe direction connecting the two reservoirs), the motion ofparticles or other degrees of freedom inside the system can betwo or three dimensional.

Fig. 1) F = ∆V = ∆µ/e, where e is the electron chargeand ∆V = V1 − V2 (V1 < V2) the voltage difference be-tween the two reservoirs at electrochemical potentials µ1

and µ2, x is the total charge transferred from reservoir1 to reservoir 2, X1 = ∆V/T , and J1 is the steady-stateelectric current. We can also write J1 = eJρ, with Jρbeing the particle current.

A. Linear response, Onsager reciprocal relations

Assuming that the generalized forces are small, therelationship between fluxes and forces is linear:

J1 = L11X1 + L12X2,

J2 = L21X1 + L22X2.(1)

These relations are referred to as phenomenological (cou-pled) transport equations or linear response equationsor kinetic equations and the coefficients La,b are knownas Onsager coefficients. Since we are assuming smallthermodynamic forces, the temperature difference ∆T =T1 − T2 is small compared to T1 ≈ T2 ≈ T , so thatX2 = ∆T/T 2.Perhaps we should also stress that we focus on the sta-

tionary and steady state situations, where all the forcesand responsive currents are independent of time, on av-erage, apart from fluctuations.The positivity of the entropy production rate,

S = J1X1 + J2X2 ≥ 0, (2)

implies for the Onsager coefficients that

L11 ≥ 0,L22 ≥ 0,

L11L22 −1

4(L12 + L21)

2 ≥ 0.(3)

Assuming the property of time-reversal invariance ofthe equations of motion, Onsager (1931) derived fun-damental relations, known as Onsager reciprocal rela-

tions for the cross coefficients of the Onsager matrix:La,b = Lb,a. When an external magnetic field B is ap-plied to the system, the laws of physics remain unchangedif time t is replaced by −t, provided that simultaneouslythe magnetic field B is replaced by −B. In this casethe Onsager-Casimir relations (Casimir, 1945; Onsager,1931) read

La,b(B) = Lb,a(−B). (4)

At zero magnetic field, we recover the Onsager reciprocalrelations La,b = Lb,a. Note that only the diagonal coef-ficients are bound to be even functions of the magneticfield: La,a(B) = La,a(−B), while in general, for a 6= b,La,b(B) 6= La,b(−B).The Onsager coefficients are related to the familiar

transport coefficients. In the case of thermoelectricitywe have

G =

(

J1∆V

)

∆T=0

=L11

T, (5)

Ξ =

(

J2∆T

)

J1=0

=1

T 2

detL

L11, (6)

S = −(

∆V

∆T

)

J1=0

=1

T

L12

L11, (7)

where G is the (isothermal) electric conductance, Ξ thethermal conductance, S the thermopower (or Seebeck co-efficient), and L denotes the Onsager matrix with matrixelements La,b (a, b = 1, 2). The Peltier coefficient

Π =

(

J2J1

)

∆T=0

=L21

L11(8)

is related to the thermopower via the Onsager recipro-cal relation: Π(B) = TS(−B). Note that the Onsager-Casimir relations imply G(−B) = G(B) and Ξ(−B) =Ξ(B), but in general do not impose the symmetry of theSeebeck coefficient under the exchange B → −B.We can eliminate in the phenomenological equations

(1) the Onsager matrix elements in favor of the transportcoefficients G,Ξ, S,Π, thus obtaining

J1 = G∆V +GS∆T,

J2 = GΠ∆V + (Ξ +GSΠ)∆T.(9)

By eliminating ∆V from these two equations we obtainan interesting interpretation of the Peltier coefficient. In-deed, the entropy current reads

JS =J2T

=Π

TJ1 +

Ξ

T∆T. (10)

4

Hence, Π/T can be understood as the entropy trans-ported by the electron flow J1. Since J1 = eJρ, eachelectron carries an entropy of eΠ/T . This contributionto the entropy current adds to the last term in (10), whichis independent of the electric current. For time-reversalsymmetric systems, the same interpretation applies tothe Seebeck coefficient, since in this case S = Π/T . Theheat flow J2 = TJS is the sum of two terms, ΠJ1 andΞ∆T . While the last term is irreversible, the first oneis reversible, that is, it changes sign when reversing thedirection of the current. It can be intuitively understoodthat efficient energy conversion requires to minimize irre-versible, dissipative processes with respect to reversibleprocesses. Hence, it is desirable to have a large Peltiercoefficient and a small heat conductance.The heat dissipation rate Q can be computed from the

entropy production rate (2):

Q = T S =J21

G+

Ξ

T(∆T )2 + J1(Π− TS)

∆T

T, (11)

where the first term is the Joule heating, the second termis the heat lost by thermal resistance and the last term,which disappears for time-reversal symmetric systems,can be negative when J1(Π−TS) < 0, thus reducing thedissipated heat. It is clear from (11) that to minimizedissipative effects for a given electric current and thermalgradient, we need a large electric conductance and lowthermal conductance.Under the assumption of local equilibrium, we can write

coupled equations like (1), connecting local fluxes to lo-cal forces, expressed in terms of gradients ∇µ, ∇T ratherthan ∆µ, ∆T (see, for instance, Callen (1985)). In thiscase, Eqs. (5) and (6) can be written with on the left-handside the electric conductivity σ and the thermal conduc-

tivity κ rather than the conductances G and Ξ.

III. THERMODYNAMIC EFFICIENCIES

A. Finite-time thermodynamics

A cornerstone result goes back to Carnot (1824). Ina cycle between two reservoirs at temperatures T1 andT2 (T1 > T2), the efficiency η, defined as the ratio ofthe performed work W over the heat Q1 extracted fromthe high temperature reservoir, is bounded by the Carnotefficiency ηC :

η =W

Q1≤ ηC = 1− T2

T1. (12)

The Carnot efficiency is obtained for a quasi-static trans-formation which requires infinite time and therefore theextracted power, in this limit, reduces to zero. Animportant question is how much the efficiency dete-riorates when the cycle is operated in a finite time.This is the central question in the field of finite-time

thermodynamics (for a recent review, see Andresen(2011)). In particular, endoreversible thermodynamics

(Hoffmann, Burzler, and Schubert, 1997; Rubin, 1979)views a thermodynamic system as a collection of re-versible subsystems which interact in an irreversible man-ner.A very important concept is that of efficiency at max-

imum power. An upper bound for the output power Wof a heat engine can be deduced for the endoreversibleCurzon-Ahlborn (CA) engine depicted in Fig. 2. TheCA engine consists of two heat baths at temperatures T1

and T2 and a reversible Carnot engine operating betweeninternal temperatures T1i and T2i (T1 > T1i > T2i > T2).The two processes of heat transfer, from the hot reservoirto the system and from the system to the cold reservoir,are the only irreversible processes in the CA engine. Theoutput work W is the difference between the heat Q1 ab-sorbed from the hot reservoir and the heat −Q2 (Q1 > 0,Q2 < 0) evacuated to the cold reservoir (W = Q1 +Q2).Heat transfers take place during the isothermal strokesof the Carnot cycle, with the working fluid (the system)at internal temperatures T1i and T2i. We further assumethat the rate of heat flow Q1 (Q2) is proportional to thetemperature difference T1−T1i (T2−T2i) between the hot(cold) reservoir and the working fluid, the proportional-ity factor being the heat conductance Ξ1 (Ξ2). Therefore,we need a time t1 (t2) to transfer an amount Q1 (Q2) ofheat, so that

Q1 = Ξ1t1(T1 − T1i), (13)

−Q2 = Ξ2t2(T2i − T2). (14)

Finally, we assume that the time spent in the adiabaticstrokes of the Carnot cycle is negligible compared to thetimes of the isothermal strokes, so that the total time ofcycle is approximately given by t = t1+t2. Such assump-tion is justified if the relaxation time for the working fluidis fast enough to allow to operate quickly the adiabatictrasformations. The output power reads

P =W

t=

Q1 +Q2

t=

k1t1(T1 − T1i) + k2t2(T2 − T2i)

t1 + t2.

(15)Taking into account that the internal Carnot engine op-erating between temperatures T1i and T2i has efficiencyηCi = 1 − T2i/T1i = 1 + Q2/Q1 and using the relationsQ1 +Q2 = W and tj = Qj/[Ξj(Tj − Tji)], (j = 1, 2), wecan express the power as

P =Ξ1Ξ2αβ(T1 − T2 − α− β)

Ξ1αT2 + Ξ2βT1 + αβ(Ξ1 − Ξ2), (16)

where we have defined α = T1 − T1i and β = T2i − T2.By maximizing the power with respect to the internaltemperatures T1i and T2i we obtain

Pmax = Ξ1Ξ2

(√T1 −

√T2√

Ξ1 +√Ξ2

)2

. (17)

The efficiency at the maximum power Pmax, commonlyreferred to as Curzon-Ahlborn upper bound (Chambadal,

5

FIG. 2 Schematic drawing of the endoreversible engine for theCurzon-Ahlborn cycle. The two heat baths at temperaturesT1 and T2 are coupled for times t1 and t2 to the system S(the working fluid, with output work per cycle equal to W ) byheat conductances Ξ1 and Ξ2. The system S is considered asa Carnot engine operating between the internal temperaturesT1i and T2i (T1 > T1i > T2i > T2).

1957; Curzon and Ahlborn, 1975; Novikov, 1958; Yvon,1955), is given by

ηCA = 1−√

T2

T1= 1−

√

1− ηC . (18)

Remarkably, the CA efficiency is independent of the con-ductances Ξ1 and Ξ2.It is interesting to remark that the Curzon-

Ahlborn efficiency is invariant under concatenation(Van den Broeck, 2005). We consider two thermal ma-chines working in a tandem, the first one between thehot source at temperature T1 and a second heat bathat intermediate temperature Ti, the second one betweenthis latter bath and the cold source at temperature T2.The first machine absorbs heat Q1, delivers work W ′

and evacuates heat |Qi| = Q1 −W ′, the second machinereuses heat |Qi| and outputs work W ′′. If both machinesfunction at the CA efficiency, also the overall efficiency(W ′+W ′′)/Q1 is given by ηCA = 1−

√

T2/T1. This self-concatenation property also holds for machines workingat Carnot efficiency.The CA efficiency is not a universal upper

bound. Efficiencies at maximum power notonly below, but also above ηCA have been re-ported (Allahverdyan, Johal, and Mahler, 2008;Esposito, Lindenberg, and Van den Broeck, 2009a;Izumida and Okuda, 2008; Schmiedl and Seifert, 2008;Sothmann and Buttiker, 2012). Yet ηCA describesthe efficiency of actual thermal plants reasonably well(Curzon and Ahlborn, 1975; Esposito et. al., 2010a),and therefore the range of validity of ηCA as upper boundfor the efficiency at maximum power has been widelydiscussed in several papers (Apertet et. al., 2012a,b;Esposito, Lindenberg, and Van den Broeck, 2009b;Esposito et. al., 2010a; Gaveau, Moreau, and Schulman,2010; Nakpathomkun, Xu, and Linke, 2010; Seifert,2011; Van den Broeck, 2005; Yakazawa and Shakouri,

2012); for a recent review, see Tu (2012). The CA bound(18) is an exact and universal bound only for systemswith (i) time-reversal symmetry and (ii) within a regimeof linear response (see Sec. III.B below). In the presenceof left-right symmetry in the system, the CA bound isexact up to quadratic order in the deviation from equi-librium (Esposito, Lindenberg, and Van den Broeck,2009b). That is, to second order in ηC ,

ηCA =ηC2

+η2C8. (19)

The CA efficiency was derived for the Carnot cy-cle in the limit of low and symmetric dissipation byEsposito et. al. (2010a). They considered a Carnot en-gine which operates under reversible conditions at theCarnot efficiency when the cycle duration becomes in-finitely long. In that limit, the system entropy increase∆S = Q1/T1 during the isothermal transformation atthe hot temperature T1 is equal to the system entropydecrease −∆S = Q2/T2 during the isothermal trans-formation at the cold temperature T2. Hence, there isno overall entropy production and the Carnot efficiencyηC = 1+Q2/Q1 = 1−T2/T1 is achieved. Esposito et. al.

(2010a) consider the weak dissipation regime and assumethat the system relaxation is much faster than the timest1 and t2 spent in the isothermal strokes, so that theoverall cycle duration is to a good approximation givenby t1 + t2. In the low dissipation regime the entropyproduction is proportional to 1/t1 and 1/t2, so that itvanishes in the limit of infinite-time cycle where it is sup-posed that the Carnot efficiency is recovered. Thereforethe amount of heat entering the system from the hot(cold) reservoir is, to first order in 1/t1 and 1/t2,

Q1 = T1

(

∆S − Σ1

t1

)

, Q2 = T2

(

−∆S − Σ2

t2

)

,

(20)with Σ1 and Σ2 coefficients depending on the specificimplementation. The maximum of the output power

P =Q1 +Q2

t1 + t2=

(T1 − T2)∆S − T1Σ1/t1 − T2Σ2/t2t1 + t2

(21)is obtained when ∂P/∂t1 = ∂P/∂t2 = 0. This leads tothe efficiency at the maximum output power

η(Pmax) =ηC

(

1 +√

T2Σ2

T1Σ1

)

(

1 +√

T2Σ2

T1Σ1

)2

+ T2

T1

(

1− Σ2

Σ1

)

. (22)

Note that this result was also obtained in the contextof stochastic thermodynamics by Schmiedl and Seifert(2008). Within linear response, the Curzon-Ahlborn ef-ficiency is recovered for symmetric dissipation, Σ1 = Σ2.From (22) we obtain

η− =ηC2

≤ η(Pmax) ≤ η+ =ηC

2− ηC. (23)

6

with the lower and upper bound reached in the limitsof completely asymmetric dissipation, for Σ2/Σ1 → ∞and Σ2/Σ1 → 0, respectively. The lower and up-per bound coincide in the linear response regime whereη− = η+ = ηCA = ηC/2. The same upper boundas in (23) was obtained with a different approach byGaveau, Moreau, and Schulman (2010).

B. Figure of merit for thermodynamic efficiency

Within the linear response, the efficiency of steadystate heat to work conversion reads

η =W

Q1

=−TX1J1

J2=

−TX1(L11X1 + L12X2)

L21X1 + L22X2, (24)

where J2 = Q1 > 0 and the power P = W > 0. Themaximum of η over X1, for fixed X2, is achieved for

X1 =L22

L21

(

−1 +

√

detL

L11L22

)

X2. (25)

For systems with time-reversal symmetry (so that L12 =L21), the maximum efficiency is given by

ηmax = ηC

√ZT + 1− 1√ZT + 1 + 1

, (26)

where the figure of merit

ZT =L212

detL(27)

is a dimensionless parameter. In the case of thermoelec-tricity, ZT , expressed in terms of the electric conductanceG, the thermal conductance Ξ and the thermopower S,reads

ZT =GS2

ΞT. (28)

For systems with local equilibrium, the figure of meritcan be expressed in therms of the material constants,the electric conductivity σ and the thermal conductivityκ, rather than G and Ξ: ZT = (σS2/κ)T . The onlyrestriction imposed by thermodynamics (more precisely,by the positivity of the entropy production rate) is ZT ≥0 and ηmax is a monotonous growing function of ZT , withηmax = 0 when ZT = 0 and ηmax → ηC when ZT → ∞(full curve in Fig. 3).Note that ZT diverges (thus leading to Carnot effi-

ciency) if and only if the the Onsager matrix L is ill-conditioned, namely the condition number

cond(L) =[Tr(L)]2

detL(29)

diverges and therefore the system (1) becomes singular.That is, J2 = cJ1, the proportionality factor c being in-dependent of the values of the applied thermodynamic

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ZT0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η/η

C

FIG. 3 Linear response efficiency for heat to work conversion,in units of Carnot efficiency ηC , as a function of the figure ofmerit ZT . The top and the bottom curve correspond to themaximum efficiency ηmax and to the efficiency at the maxi-mum power η(Pmax), respectively.

forces. In short, within linear response (and withoutexternal magnetic fields or other effects breaking time-reversal symmetry) the Carnot efficiency is obtained ifand only if charge and energy flows are proportional (tightcoupling condition, also known as strong coupling in theliterature).The output power

P = −TX1J1 = −TX1(L11X1 + L12X2) (30)

is maximal when

X1 = − L12

2L11X2 (31)

and is given by

Pmax =T

4

L212

L11X2

2 =ηC4

L212

L11X2. (32)

Using Eqs. (5) and (7) we can also write

Pmax =1

4S2G(∆T )2. (33)

We can see from this last equation that the maximumpower is directly set by the combination S2G, known forthis reason as power factor. Note that P is a quadraticfunction ofX1 and the maximum is obtained for the value(31) corresponding to half of the stopping force,

Xstop1 = −L12

L11X2, (34)

that is, to the value for which the motion halts, J1 = 0.For systems with time reversal symmetry, the efficiencyat maximum power reads (Van den Broeck, 2005)

η(Pmax) =ηC2

ZT

ZT + 2. (35)

7

This quantity also is a monotonous growing function ofZT , with η(Pmax) = 0 when ZT = 0 and η(Pmax) →ηC/2 when ZT → ∞ (dashed curve in Fig. 3). Note thatfor small ZT we have η(Pmax) ≈ ηmax ≈ (ηC/4)ZT . Thedifference between η(Pmax) and ηmax becomes relevantonly for ZT > 1.Note that we can establish an efficiency versus power

plot. We can express the ratio between the power at agiven value of X1 and the maximum power as a functionof the force ratio r = X1/X

stop1 :

P

Pmax= 4r(1− r). (36)

This relation can be inverted:

r =1

2

[

1±√

1− P

Pmax

]

, (37)

with the plus sign for r ≥ 1/2 and the minus sign forr ≤ 1/2. Inserting this latter relation into Eq. (24) wecan express the efficiency (normalized to the Carnot effi-ciency) as

η

ηC=

P

Pmax

2

(

1 +2

ZT∓√

1− P

Pmax

) , (38)

where the minus sign corresponds to r ≥ 1/2, the plussign to r ≤ 1/2. Plots of the normalized efficiency versusthe normalized power are shown in Fig. 4, for severalvalues of the figure of merit ZT . Note that, while forlow values of ZT the maximum efficiency is close to theefficiency at maximum power, for large ZT the differencebecomes relevant (see also Fig. 3). For ZT = ∞ theCarnot efficiency is achieved at the stopping power X1 =Xstop

1 (r = 1).When the force ratio exceeds one, r > 1, the thermo-

electric device works as a refrigerator. In this case themost important benchmark is the coefficient of perfor-

mance η(r) = J2/P (J2 < 0, P < 0), given by the ratioof the heat current extracted from the cold system overthe absorbed power. By optimizing this quantity withinlinear response, we obtain

η(r)max = η(r)C

√ZT + 1− 1√ZT + 1 + 1

, (39)

where η(r)C = T2/(T1 − T2) ≈ 1/(TX2) is the efficiency

of an ideal, dissipationless refrigerator. Since the ratio

η(r)max/η

(r)C for refrigeration is equal to the ratio ηmax/ηC

for thermoelectric power generation, ZT is the figure ofmerit for both regimes.

C. Systems with broken time-reversal symmetry

The same analysis as above can be repeated when time-reversal symmetry is broken, say by a magnetic field

0 0.2 0.4 0.6 0.8 1

P/Pmax

0

0.2

0.4

0.6

0.8

1

η/η C

FIG. 4 Relative efficiency η/ηC versus normalized powerP/Pmax. From bottom to top: ZT = 1, 5, 100, and∞. In eachcurve the lower branch corresponds to a force ratio r ≤ 1/2,the upper branch to r ≥ 1/2. Maximum efficiency is alwaysachieved on the upper branch.

B (or by other effects such as the Coriolis force). Inthis case the maximum efficiency and the efficiency atmaximum power are both determined by two parame-ters (Benenti, Saito, and Casati, 2011): the asymmetryparameter

x =L12

L21=

S(B)

S(−B)(40)

and the “figure of merit”

y =L12L21

detL=

G(B)S(B)S(−B)

Ξ(B)T. (41)

The maximum efficiency reads

ηmax = ηC x

√y + 1− 1√y + 1 + 1

, (42)

while the efficiency at maximum power is

η(Pmax) =ηC2

xy

2 + y. (43)

In the particular case x = 1, y reduces to the ZT fig-ure of merit of the time-symmetric case, Eq. (42) re-duces to Eq. (26), and Eq. (43) to Eq. (35). While ther-modynamics does not impose any restriction on the at-tainable values of the asymmetry parameter x, the pos-itivity of entropy production implies h(x) ≤ y ≤ 0 ifx ≤ 0 and 0 ≤ y ≤ h(x) if x ≥ 0, where the functionh(x) = 4x/(x − 1)2. Note that limx→1 h(x) = ∞ andtherefore there is no upper bound on y(x = 1) = ZT .For a given value of the asymmetry x, the maximum(over y) η(Pmax) of η(Pmax) and the maximum ηmax ofηmax are obtained for y = h(x):

η(Pmax) = ηCx2

x2 + 1, (44)

8

ηmax =

ηC x2 if |x| ≤ 1,

ηC if |x| ≥ 1.(45)

The functions η(Pmax)(x) and ηmax(x) are drawn inFig. 5. In the case |x| > 1, it is in principle possibleto overcome the CA limit within linear response andto reach the Carnot efficiency, for increasingly smallerand smaller figure of merit y as the asymmetry param-eter x increases. The Carnot efficiency is obtained fordetL = (L12 − L21)

2/4 > 0 when |x| > 1, that is, thetight coupling condition is not fulfilled.The output power at maximum efficiency reads

P (ηmax) =ηmax

4

|L212 − L2

21|L11

X2. (46)

Therefore, always within linear response, it is allowedfrom thermodynamics to have Carnot efficiency andnonzero power simultaneously when |x| > 1. Sucha possibility can be understood on the basis of thefollowing argument (Brandner, Saito, and Seifert, 2013;Brandner and Seifert, 2013). We first split each currentJi (i = 1, 2) into a reversible and an irreversible part,defined by

J revi =

2∑

j=1

Lij − Lji

2Xj , J irr

i =

2∑

j=1

Lij + Lji

2Xj . (47)

It is readily seen from Eq. (2) and (47) that only theirreversible part of the currents contributes to the entropyproduction:

S = J irr1 X1 + J irr

2 X2. (48)

The reversible currents J revi vanish for B = 0. On

the other hand, for broken time-reversal symmetry thereversible currents can in principle become arbitrarilylarge, giving rise to the possibility of dissipationlesstransport.While in the time-reversal case the linear response nor-

malized maximum efficiency ηmax/ηC and coefficient of

performance η(r)max/η

(r)C for power generation and refrig-

eration coincide, this is no longer the case with brokentime-reversal symmetry. For refrigeration the maximumvalue of the coefficient of performance reads

η(r)max = η(r)C

1

x

√y + 1− 1√y + 1 + 1

. (49)

For small fields, x is in general a linear function of themagnetic field, while y is by construction an even func-tion of the field. As a consequence, a small external mag-netic field either improves power generation and worsensrefrigeration or vice-versa, while the average efficiency

1

2

[

ηmax(B)

ηC+

η(r)max(B)

η(r)C

]

=ηmax(0)

ηC=

η(r)max(0)

η(r)C

, (50)

-5 -4 -3 -2 -1 0 1 2 3 4 5x

0

0.2

0.4

0.6

0.8

1

η/η C

FIG. 5 Ratio η/ηC as a function of the asymmetry parameterx, with η = η(Pmax) (dashed curve) and η = ηmax (full curve).For x = 1, η(Pmax) = ηC/2 and ηmax = ηC are obtained fory(x = 1) = ZT = ∞.

up to second order corrections. Due to the Onsager-Casimir relations, x(−B) = 1/x(B) and therefore byinverting the direction of the magnetic field one can im-prove either power generation or refrigeration.

Onsager relations do not impose the symmetryx = 1, i.e., we can have S(B) 6= S(−B). However,as discussed in Sec. IV.B below, in the non-interactingcase S(−B) = S(B) as a consequence of the sym-metry properties of the scattering matrix (Datta,1995). On the other hand, this symmetry may beviolated when electron-phonon and electron-electroninteractions are taken into account. While the See-beck coefficient has always been found to be an evenfunction of the magnetic field in two-terminal purelymetallic mesoscopic systems (Godijn et al., 1999;van Langen, Silvestrov, and Beenakker, 1998), mea-surements for certain orientations of a bismuth crystal(Wolfe, Smith, and Haszko, 1963), Andreev interferome-ter experiments (Eom, Chien, and Chandrasekhar, 1998)and recent theoretical studies (Jacquod and Whitney,2010; Saito et al., 2011; Sanchez and Serra, 2011) haveshown that systems in contact with a superconductor orsubject to inelastic scattering can exhibit non-symmetricthermopower, i.e., S(−B) 6= S(B). So far, investigationsof various classical (Horvat et al., 2012) and quantum(Saito et al., 2011) dynamical models have shownarbitrarily large values of the asymmetry x, but corre-spondingly with low efficiency. However, efficiency atmaximum power beyond the CA limit for x > 1 has beenrecently shown in Balachandran, Benenti, and Casati(2013); Brandner, Saito, and Seifert (2013);Brandner and Seifert (2013) (see Sec. IV.B below).

9

IV. NON-INTERACTING SYSTEMS,

LANDAUER-BUTTIKER FORMALISM

Exact calculation of thermodynamic efficiencies ispossible for non-interacting models by means of theLandauer-Buttiker approach. This approach describesthe coherent flow of electrons through a channel. All dis-sipative and phase-breaking processes are limited to thecontacts (reservoirs). The electric and thermal currentsare expressed in terms of the scattering (transmission)properties of the system (Datta, 1995; Imry, 1997):

J1 =e

h

∫ ∞

−∞

dEτ(E)[f1(E)− f2(E)], (51)

J2 =1

h

∫ ∞

−∞

dE(E − µ1)τ(E)[f1(E)− f2(E)]. (52)

Here, e is the electron charge, h the Planck’s constant,τ(E) the transmission probability for a particle with en-ergy E to transit from terminal (reservoir) 1 to terminal 2(0 ≤ τ(E) ≤ 1), and fi(E) = exp[(E−µi)/kBTi]+1−1is the Fermi distribution of the particles injected fromreservoir i. Note that J2 = Q1 is the heat current fromthe hot reservoir (T1 > T2).The Onsager coefficients La,b can be derived from the

linear expansion of the currents (51) and (52). We obtain

L11 = e2TI0, L12 = L21 = eT I1, L22 = TI2. (53)

Here, the integrals In have been defined as

In =1

h

∫ ∞

−∞

dE(E − µ)nτ(E)

(

− ∂f

∂E

)

, (54)

where the derivative of the Fermi function, −∂f/∂E =1/4kBT cosh2[(E − µ)/kBT ], is a bell-shaped functioncentered at µ and has a width of the order of kBT .It immediately follows that conductances and the ther-mopower can be expressed in terms of the integrals In:

G = e2I0, Ξ =1

T

(

I2 −I21I0

)

, S =1

eT

I1I0

. (55)

While Landauer-Buttiker approach describes coher-ent quantum transport, semiclassical transport canbe described by means of the Boltzmann equation.Here we consider transport processes that occur muchslower than the relaxation to local equilibrium andtreat collisions within the relaxation-time approximation(Ashcroft and Mermin, 1976). That is, collisions drivethe electronic system to local thermodynamic equilib-rium under the assumption that the distribution of elec-trons emerging from collisions does not depend on thestructure of their non-equilibrium distribution prior tothe collision and that collisions do not alter local equilib-rium. We can then express conductivities and the ther-mopower in terms of the integrals

Kn =

∫ ∞

−∞

dE(E − µ)nΣ(E)

(

− ∂f

∂E

)

. (56)

Here Σ(E) ≈ D(E)tR(E)ν(E)2 is the transport distribu-tion function, where D(E) is the density of states, tR(E)the electron relaxation time, and ν(E) the electron groupvelocity. We obtain (Mahan and Sofo, 1996)

σ = e2K0, κ =1

T

(

K2 −K2

1

K0

)

, S =1

eT

K1

K0. (57)

A. Energy filtering

An interesting question is what transmission func-tion τ(E) (or transport distribution function Σ(E) inthe Boltzmann approach) provides the largest thermody-namic efficiencies. Mahan and Sofo (1996) showed that,within linear response, a delta-shaped transmission func-tion leads to Carnot efficiency. As discussed in Sec. III.B,ZT diverges if and only if the Onsager matrix L is ill-conditioned, that is, in the tight coupling limit J2 = cJ1,with c independent of the applied forces. The tight cou-pling condition is achieved when the transmission is pos-sible only within a tiny energy window around E = E⋆

(energy filtering). In this case from Eq.(54) we obtainIn ≈ (E⋆ − µ)nI0, and therefore

ZT =GS2

ΞT =

I21I0I2 − I21

→ ∞. (58)

The energy filtering mechanism allows us to achievethe Carnot efficiency also beyond linear response(Humphrey et al., 2002; Humphrey and Linke, 2005). Inthis case, assuming T1 > T2, µ1 < µ2, J1 > 0 and J2 > 0,the efficiency for power generation is given by

η =[(µ2 − µ1)/e]J1

J2

=(µ2 − µ1)

∫∞

−∞ dEτ(E)[f1(E)− f2(E)]∫∞

−∞ dE(E − µ1)τ(E)[f1(E)− f2(E)].

(59)

When the transmission is possible only within a tiny en-ergy window around E = E⋆, the efficiency reads

η =µ2 − µ1

E⋆ − µ1. (60)

We have f1(E⋆) = f2(E⋆), namely the occupation ofstates is the same in the two reservoirs at different tem-peratures and electrochemical potentials, when

E⋆ − µ1

T1=

E⋆ − µ2

T2⇒ E⋆ =

µ2T 1− µ1T2

T1 − T2. (61)

Substituting such E⋆ into Eq. (60), we obtain the Carnotefficiency η = ηC = 1 − T2/T1. Note that Carnot effi-ciency is obtained in the limit J1 → 0, corresponding toreversible transport (zero entropy production) and zero

output power.High values of ZT can still be achieved if rather

than delta-shaped transmission function one considers

10

sharply rising transmission functions (O’Dwyer et al.,2005; Vashaee and Shakouri, 2004). The advantage ofstep over narrow transmission functions is that good effi-ciencies can be obtained without greatly reducing power.

B. Noise and probe reservoirs

The Landauer-Buttiker approach provides a rigorousframework for the description of coherent quantumtransport. The transport can be only partially coherentwhen inelastic scattering, due to the interactions of theelectrons with phonons, photons, and other electrons, istaken into account. A very convenient way to introduceinelastic scattering is by means of a third terminal (orconceptual probe), whose parameters (temperature andchemical potential) are chosen self-consistently so thatthere is no net average flux of particles and heat betweenthis terminal and the system (see Fig. 6). In mesoscopicphysics, probe reservoirs are commonly used to simulatephase-breaking processes in partially coherent quantumtransport, since they introduce phase-relaxation withoutenergy damping (Buttiker, 1988). The advantage ofsuch approach lies in its simplicity and independencefrom microscopic details of inelastic processes. Probeterminals have been widely used in the literature andproved to be useful to unveil nontrivial aspects ofphase-breaking processes (Datta, 1995), heat trans-port and rectification (Bandyopadhyay and Segal,2011; Bolsterli, Rich, and Visscher, 1970;Bonetto, Lebowitz, and Lukkarinen, 2004;Bonetto et al., 2009; Dhar, 2008; Pereira,2010; Roy and Dhar, 2007; Saito, 2006;Segal and Nitzan, 2005), and thermoelectric trans-port (Bedkihal, Bandyopadhyay, and Meister,2013; Entin-Wohlman and Aharony, 2012;Entin-Wohlman, Imry, and Aharony, 2010;Hershfield, Muttalib, and Pekola, 2013; Horvat et al.,2012; Jacquet, 2009; Jiang, Entin-Wohlman, and Imry,2012, 2013,b; Jordan et al., 2013; Ruokola and Ojanen,2012; Saito et al., 2011; Sanchez and Serra, 2011;Sanchez and Buttiker, 2011; Sothmann and Buttiker,2012; Sothmann et al., 2012, 2013). Note that someof the above models consider the third terminal as abosonic (phonons, photons, or magnons) rather thana fermionic bath and cannot be treated within theLandauer-Buttiker approach for non-interacting parti-cles. In that case one has to use other methods such asthe Keldysh technique (Entin-Wohlman and Aharony,2012; Entin-Wohlman, Imry, and Aharony, 2010).

The approach can be generalized to any number np ofprobe reservoirs. We call J1,k and J2,k the charge andheat currents from the kth terminal (at temperature Tk

and electrochemical potential µk), with k = 3, ..., n de-noting the np = n − 2 probes. Due to the steady-stateconstraints of charge and energy conservation,

∑

k J1,k =0 and

∑

k(J2,k + µkJ1,k) = 0, we can express, for in-stance, the currents from the second reservoir as a func-

FIG. 6 Schematic drawing of partially-coherent thermoelec-tric transport, with the third terminal acting as a probe reser-voir mimicking inelastic scattering. The temperature T3 andthe chemical potential µ3 of the third reservoir are such thatthe net average electric and heat currents through this reser-voir vanish: J1,3 = J2,3 = 0. This setup can be generalizedto any number of probe reservoirs, k = 3, ..., n, by settingJ1,k = J2,k = 0 for all probes.

tion of the remaining 2(n− 1) currents. The correspond-ing generalized forces are given by X1,k = ∆µk/(eT ) andX2,k = ∆Tk/T

2, with ∆µk = µk − µ, ∆Tk = Tk − T ,µ = µ2, and T = T2. The linear response relations be-tween currents and thermodynamic forces,

Ji,k =

2∑

j=1

n∑

l=1(l 6=2)

Lik;jlXj,l, (62)

are expressed in terms of an Onsager matrix L of size2(n− 1). We then impose the condition of zero averagecurrents through the probes, J1,k = J2,k = 0 for k =3, ..., n to reduce the Onsager matrix to a 2 × 2 matrixL′ connecting the fluxes Ji,1 through the first reservoir

and the conjugated forces Xi,1:

(

J1,1J2,1

)

=

(

L′11 L′12L′21 L′22

)(

X1,1

X2,1

)

. (63)

The reduced matrix L′ fulfills the Onsager-Casimir rela-

tions and represents the Onsager matrix for two-terminalinelastic transport modeled by means of self-consistentreservoirs. Details of the reduction from L to L

′ forn = 3 reservoirs are provided in Saito et al. (2011). Notethat, if the average flow of particles and heat through theprobe reservoirs is zero, then thermodynamic efficienciescan be computed by means of the standard two-terminalformulas (42) and (43), with the parameters x = L′12/L

′21

and y = L′12L′21/detL

′. However, the transport betweenthese two terminals is no longer fully coherent, since theelectrons can be absorbed and emitted by the probe ter-minals.Electric and heat current can be conveniently com-

puted, for any number of probes, by means of the multi-terminal Landauer-Buttiker formula (Datta, 1995):

J1,k =e

h

∫ ∞

−∞

dE∑

l

[τl←k(E)fk(E)− τk←l(E)fl(E)],

(64)

11

J2,k =1

h

∫ ∞

−∞

dE(E − µk)∑

l

[τl←k(E)fk(E)

−τk←l(E)fl(E)],

(65)

where τl←k(E) is the transmission probability from ter-minal k to terminal l at the energy E. Charge conser-vation and the requirement of zero current at zero biasimpose

∑

k

τk←l =∑

k

τl←k = Ml, (66)

with Ml being the number of modes in the lead l. More-over, in the presence of a magnetic field B we have

τk←l(B) = τl←k(−B). (67)

The last relation is a consequence of the unitarity of thescattering matrix S(B) that relates the outgoing waveamplitudes to the incoming wave amplitudes at the dif-ferent leads. The time reversal invariance of unitary dy-namics leads to S(B) = t

S(−B), which in turn implies(67) (Datta, 1995). In the two-terminal case, Eq. (66)means τ1←2 = τ2←1. Hence, we can conclude from thisrelation and Eq. (67) that τ2←1(B) = τ2←1(−B), thusimplying that the Seebeck coefficient is a symmetric func-tion of the magnetic field.The third (probe) terminal can break the symme-

try of the Seebeck coefficient. We can have S(−B) 6=S(B), that is, L′12 6= L′21 in the reduced Onsagermatrix L

′. Arbitrarily large values of the asymme-try parameter x = S(B)/S(−B) = L′12/L

′21 were ob-

tained in Saito et al. (2011) by means of a three-dotAharonov-Bohm interferometer model. The asymme-try was found also for chaotic cavities, ballistic mi-crojunctions (Sanchez and Serra, 2011), and randomHamiltonians drawn from the Gaussian unitary en-semble (Balachandran, Benenti, and Casati, 2013). InSanchez and Serra (2011) it was shown tha the asymme-try is a higher-order effect in the Sommerfeld expansionand therefore disappears in the low temperature limit.The asymmetry was demonstrated also in the frameworkof classical physics, for a three-terminal deterministicrailway switch transport model (Horvat et al., 2012). Insuch model, only the values zero and one are allowed forthe transmission functions τj←i(E), i.e., τj←i(E) = 1 ifparticles injected from terminal i with energy E go to ter-minal j and τj←i(E) = 0 is such particles go to a terminalother than j. The transmissions τj←i(E) are piecewiseconstant in the intervals [Ei, Ei+1], (i = 1, 2, ...), withswitching τj←i = 1 → 0 or viceversa possible at thethreshold energies Ei, with the constraints (66) alwaysfulfilled.In all the above instances, it was not possible to find

at the same time large values of asymmetry parameter(40) and high thermoelectric efficiency. Such failure wasexplained by Brandner, Saito, and Seifert (2013) and isgeneric for non-interacting three-terminal systems. In

that case, when the magnetic field B 6= 0, current con-servation, which is mathematically expressed by unitarityof the scattering matrix S, imposes bounds on the On-sager matrix stronger than those derived from positivityof entropy production. We have

L11L22 −1

4(L12 + L21)

2 ≥ 3

4(L12 − L21)

2. (68)

Such constraint reduces to the third inequality of Eq. (3)only in the time-symmetric case L12 = L21, while itis in general a stronger inequality, since the right-handside of Eq. (68) is strictly positive when L12 6= L21.As a consequence, Carnot efficiency can be achieved inthe three-terminal setup only in the time-symmetric caseB = 0. On the other hand, the Curzon-Ahlborn linearresponse bound ηCA = ηC/2 for the efficiency at maxi-mum power can be overcome for moderate asymmetries,1 < x < 2, with a maximum of 4ηC/7 at x = 4/3.The bounds obtained by Brandner, Saito, and Seifert(2013) are in practice saturated in a quantum trans-mission model reminiscent of the above described rail-way switch model (Balachandran, Benenti, and Casati,2013). Multi-terminal cases with more than three ter-minals were also discussed for noninteracting electronictransport (Brandner and Seifert, 2013). By increasingthe number np of probe terminals, the constraint fromcurrent conservation on the maximum efficiency and theefficiency at maximum power becomes weaker than thatimposed by (68). However, the bounds (44) and (45)from the second law of thermodynamics are saturatedonly in the limit np → ∞. It is an interesting open ques-tion whether similar bounds on efficiency, tighter thatthose imposed by the positivity of entropy production,exist in more general transport models for interactingsystems.Probe-reservoir models unveil several other nontriv-

ial aspects of inelastic processes. For instance, thethird reservoir may be a phonon bath connected to ananostructure (e.g., a molecule), and it has been shownthat such setup can be very favorable for thermoelectricenergy conversion (Jiang, Entin-Wohlman, and Imry,2012), notably the setup can act as a refriger-ator for the local phonon system. The cooling

by heating phenomenon can also be interpreted interms of a third, photonic terminal powering re-frigeration. Pekola and Hekking (2007) (see alsoMuhonen, Meschke, and Pekola (2012); Peltonen et al.

(2011); Van den Broeck and Kawai (2006)) consideredthe case in which the photons emitted by a hot resis-tor can extract heat from a cold metal, providing theenergy needed to electrons to tunnel to a superconductor(separated from the metal by a thin insulating junction;no voltage is applied over the junction). If the tempera-ture of the resistor is suitably set, only the high energyelectrons are removed from the metal, thus cooling it.Such Brownian refrigerator is still to be experimentallydemonstrated. Similar mechanisms have been discussedfor cooling a metallic lead, connected to another, higher

12

temperature lead by means of two adjoining quantumdots (Cleuren, Rutten, and Van den Broeck, 2012) orfor cooling an optomechanical system (Mari and Eisert,2012). In both cases, refrigeration is powered by absorp-tion of photons.

V. INTERACTING SYSTEMS

A. Green-Kubo formula

The Green-Kubo formula expresses linear responsetransport coefficients in terms of dynamic correlation

functions of the corresponding current operators, cal-culated at thermodynamic equilibrium (see for instanceKubo, Toda, and Hashitsume (1985); Mahan (1990)):

La,b = limω→0

ReLa,b(ω), (69)

La,b(ω) = limǫ→0

∫ ∞

0

dte−i(ω−iǫ)t limΩ→∞

1

Ω

∫ β

0

dτ〈JaJb(t+ iτ)〉,

where β = 1/kBT (kB it the Boltzmann constant),〈 · 〉 =

tr( · ) exp−βH

/tr exp(−βH) denotes the ther-modynamic expectation value, Ω is the system’s volume,and the currents are Ja = 〈Ja〉, with Ja the currentoperator. Note that in extended systems, the operatorJa =

∫

Ω d~rj(~r) is an extensive quantity, ja(~r) is the cur-rent density operator, satisfying the continuity equation

dρa(~r, t)

dt=

i

~[H, ρa] = −∇ · j(~r, t), (70)

where ρa is the density of the corresponding conservedquantity, say, energy, electric charge, magnetization etc.Eq. (70) can be equally well written in classical mechan-ics, provided the commutator is substituted by the Pois-son bracket multiplied by the factor i~. The real partof La,b(ω) can be decomposed into a δ-function at zerofrequency defining a generalized Drude weight Da,b (fora = b this is the conventional Drude weight) and a regularpart Lreg

a,b(ω):

ReLa,b(ω) = 2πDa,bδ(ω) + Lrega,b(ω). (71)

The matrix of Drude weights can be within linear re-sponse also expressed in terms of time-averaged current-current correlations directly:

Da,b = limt→∞

1

t

∫ t

0

dt limΩ→∞

1

Ω

∫ β

0

dτ〈JaJb(t+ iτ)〉. (72)

Note that for finite systems, i.e. disregarding the ther-modynamic limit Ω → ∞, it is possible to give aspectral representation of both Da,b and Lreg

a,b in termsof the eigenenergies and eigenstates of the system andof the corresponding Boltzmann weights (Kohn, 1964;Zotos, Naef, and Prelovsek, 1997; Zotos and Prelovsek,2004).

The linear response Kubo formalism has beenrecently used to investigate the thermoelectric proper-ties of one-dimensional integrable and non-integrablestrongly correlated lattice models (Arsenault et al.,2013; Chaikin and Beni, 1976; Deng et al., 2012;Furukawa, Ikeda, and Sakai, 2005; Peterson et al.,2007; Peterson and Shastry, 2010; Shastry, 2009;Zemljic and Prelovsek, 2007). Thermoelectrics ofstrongly correlated materials are of fundamental interest.Moreover, experimental results have revealed that somematerials, such as sodium cobalt oxide, posses unusuallylarge thermopower (Terasaki, Sasago, and Uchinokura,1997; Wang et al., 2003), due in part to strong electroninteractions (Peterson, Shastry, and Haerter, 2007).

B. Conservation laws and thermoelectric transport

In interacting (quantum) many-body systems onehas to often resort to more abstract mathematicalframeworks to describe linear response theory, Green-Kubo formulae and Onsager reciprocity relations(Jaksic, Ogata, and Pillet, 2006). For describing corre-lated transport in extended interacting quantum systemswith local interaction, one can use the formalism of C∗

algebraic dynamical systems (Bratelli and Robinson,1997). There we can take advantage of an emerging

causality (Bravyi, Hastings, and Verstraete, 2006) as theconsequence of the Lieb-Robinson bounds stating thatcorrelations, even in nonrelativistic systems with localinteractions, propagate with a finite maximum velocitywhich is essentially given by the strength of the inter-action. In addition, one can use the Mermin-Wagnertheorem to conclude that in one- and two-dimensionalsystems the static correlations always decay exponen-tially in non-zero temperature (Gibbsian) equilibrium.Consequently, one can prove finite-temperature ballistictransport for systems with relevant local conserva-tion laws (Ilievski and Prosen, 2013; Prosen, 2011).Similarly, one can prove (Prosen, 2013) nonvanish-ing lower bounds on Green-Kubo diagonal diffusivetransport coefficients La,a (69) in terms of quadratically-extensive conserved quantities (in non-ballistic caseswhen linearly-extensive – local conserved quantitiesdo not exist, or are irrelevant, such as in the caseof one-dimensional half-filled fermi Hubbard chain ornon-magnetized Heisenberg spin 1/2 chain). A constant

of motion Km is by definition relevant if 〈J Km〉 6= 0,where J is the current under consideration. Ballisticfinite-temperature transport as a consequence of the ex-istence of relevant conservation laws is a typical featureof completely integrable strongly interacting systems, assuggested some time ago by Zotos, Naef, and Prelovsek(1997) (see also Garst and Rosch (2001);Heidrich-Meisner, Honecker, and Brenig (2005)). Ina similar way, the theory of quantum integrable systemsimplies ballistic coupled transport, i.e. thermoelectricand thermomagnetic, properties of such systems, e.g.

13

in the anisotropic Heisenberg XXZ spin 1/2 chain(Furukawa, Ikeda, and Sakai, 2005; Sakai, 2005).More specifically, let us consider a strongly interacting

system with a set of M constants of motion Km, m =1, . . . ,M , i.e. Hermitian operators Km which commutewith the Hamiltonian and among themselves, [H, Km] =

0, [Km, Kl] = 0, and which can always be chosen to be or-

thogonal, i.e. 〈KmKl〉 = δk,l〈K2m〉, via a Gram-Schmidt

procedure. Furthermore, let us assume that Km are lin-early extensive, i.e. 〈K2

m〉 ∝ Ω. Then, provided the set

Km exhausts all extensive conserved quantities (in thethermodynamic limit Ω → ∞), the matrix of general-ized Drude weights (72) can be expressed explicitly bymeans of Suzuki’s formula (Benenti, Casati, and Wang,2013; Suzuki, 1971):

Da,b = limΩ→∞

1

2Ω

M∑

m=1

〈JaKm〉〈JbKm〉〈K2

m〉. (73)

At zero frequency ω = 0, the elements of Onsager ma-trix La,b can be replaced by Da,b in the expression forthe figure of merit of thermoelectric, thermomagnetic orthermochemical efficiency ZT .Conservation laws have an interesting consequence for

thermoelectric efficiency, when there is a single relevantconserved quantity, M = 1 in Eq. (73). In that caseall Onsager matrix elements La,b are size independentand therefore also the electric conductance G ∝ L11

and the thermopower S ∝ L12/L11 are size indepen-dent. On the other hand, the thermal conductanceΞ ∝ det(L)/L11 drops with the system size. Indeed theDrude weight contribution to det(L) vanishes, since it isgiven by D11D22−D2

12, which vanishes as a consequenceof Eq. (73) which for M = 1 states that the matrix Da,b

has rank 1. Since the electric conductance is ballistic, thethermopower size-independent and the thermal conduc-tance subbalistic (i.e., it drops with the system size), wecan conclude that the figure of merit ZT = (GS2/Ξ)T di-verges with the system size (Benenti, Casati, and Wang,2013). Note that these conclusions for the thermal con-ductance and the figure of merit do not hold whenM > 1,as is typical for completely integrable systems. In thatcase we have, in general, D11D22 − D2

12 6= 0, so thatthermal conductance is ballistic and therefore ZT issize-independent. This result is not limited to quan-tum systems and has no dimensional restrictions; it hasbeen illustrated by means of a diatomic chain of hard-point colliding particles (Benenti, Casati, and Wang,2013), where the divergence of the figure of meritwith the system size (Casati, Wang, and Prosen, 2009)cannot be explained in terms of the energy filter-ing mechanism (Saito, Benenti, and Casati, 2010), andin a two-dimensional system connected to reservoirs(Benenti, Casati, and Mejıa-Monasterio, 2013), with thedynamics simulated by the multiparticle collision dy-namics method (Malevanets and Kapral, 1999). In both(classical) models collisions are elastic and the compo-nent of momentum along the direction of the charge and

heat flows is the only relevant constant of motion. Diver-gence of ZT has been also predicted, by different theoret-ical arguments, for a quantum wire with weak electron-electron interactions, in the limit of infinite wire length(Micklitz, Rech, and Matveev, 2010).

When the underlying many-body system is stronglynon-integrable, or when all the extensive local conserva-tion laws are irrelevant for the transporting quantities,i.e. when 〈JaKm〉 = 0 for all m, then the transport istypically diffusive. In the latter case one often finds diffu-sive transport even if the system is completely integrable(Prosen and Znidaric, 2009; Steinigeweg, 2011). Then,within the linear response approach, one has to applythe regularized Onsager matrix Lreg

a,b(ω) of Eq. (71) forestimation of ZT .

C. Thermalization

One of the essential prerequisites for using themethods of canonical statistical mechanics is estab-lishing precise conditions under which the system dis-cussed can be claimed to be in thermal equilibrium.The understanding of thermalization in closed (iso-lated) but complex quantum systems is one of themost intriguing problems in quantum physics, withdeep connections with the field of quantum chaos(Aberg, 1990; Benenti, Casati, and Shepelyansky, 2001;Deutsch, 1991; Flambaum, Izrailev, and Casati, 1996;Jacquod and Shepelyansky, 1997; Srednicki, 1994). The-oretical interest in these issues resurfaced periodi-cally, until very recently (Polkovnikov et al., 2011;Rigol, Dunjko, and Olshanii, 2008). Recent progress hasbeen made in understanding the conditions under whichclosedbut complex systems undergo relaxation to equi-librium. The conditions for thermalization are essen-tially related to the systems’ integrability and localiza-tion properties (e.g. due to disorder). Non-ergodic sys-tems, such as those possessing some number of exact lo-cal conservation laws Km (for example, completely in-tegrable systems) undergo relaxation to a generalizedGibbs state (Barthel and Schollwock, 2011), which canagain facilitate application of standard statistical me-chanics methods.

Thermalization of simple or complex quantum sys-tems immersed into large (many-body), complex, orchaotic environment can be described very conve-niently by the methods of open quantum systems(Breuer and Petruccione, 2002). One can treat alsoextended many-body systems, say particle (or spin)chains, in this way by using a convenient setup inwhich only boundary (local, on-site) degrees of free-dom are directly coupled with the environment. Withinthe Markovian approximation, the system’s many-bodydensity matrix undergoes a time-evolution dictatedby the Lindblad-Gorini-Kossakowski-Sudarshan equation(Gorini, Kossakowski, and Sudarshan, 1976; Lindblad,

14

1976)

d

dtρ(t) = Lρ(t), (74)

where the Liouvillian superoperator is defined as

Lρ := − i

~[H, ρ] +

∑

µ

(

LµρL†µ − 1

2L†µLµ, ρ

)

. (75)

The Hamiltonian H is here considered to be a sum oflocally interacting terms, H =

∑N−1n=1 h[n,n+1] and Lµ

are the Lindblad (or so-called quantum jump) opera-tors, which are in the simplest case supported only atthe boundary sites, n = 1 or n = N , of the system.This setup can be justified (Benenti et al., 2009) in theregime of weak tunneling interaction between differentconstituents (particles, spins) of the system, but providesalso a more general paradigm of open many-body quan-tum systems with fully coherent bulk dynamics and inco-herent boundary conditions, which is particularly suitedfor studying non-equilibrium steady state phenomena,such as quantum transport (Wichterich et al., 2007). Ithas been demonstrated by numerical simulations thatcoupling a many-body locally interacting quantum sys-tem to a pair of equal Markovian baths through the sys-tem’s ends in this way, results in thermalization if, andonly if, the system’s bulk Hamiltonian is not integrable(via Bethe ansatz) (Znidaric et al., 2010). Further theo-retical work is needed to deeper understand these results.

D. Local equilibrium and non-equilibrium steady states

Using the just described boundary driven open sys-tem’s setup, but putting small biases on the rates withwhich Lindblad jump operators Lµ target certain locallycanonical states near the boundaries, one can addressalso the problem of bulk transport properties of the sys-tem. For example, measuring expectation values of thecurrent observables in the steady states of the Lindbladequation reveals, in the thermodynamic limit N → ∞,bulk conductivities (Prosen and Znidaric, 2009), and inprinciple also the off-diagonal elements of the Onsagermatrix. Nevertheless, the problem of establishing localequilibrium in such situations seems to be quite nontriv-ial (Prosen and Znidaric, 2010). Namely, fixed points ofLiouvillean dynamics can describe a variety of qualita-tively distinct non-equilibrium quantum phases, rangingfrom equilibrium-like states where spatial correlations de-cay exponentially and where local equilibrium can be welldefined, to states where spatial correlations only dependon the bias (voltage, temperature drop, etc.) betweenthe reservoirs, but not on the system size. In the lat-ter case one lacks any sensible definition of local equi-librium. Nevertheless, the approach to non-equilibriumsteady states in terms of the methods of open quantumsystems appears to be very promising, in particular sincestrong interactions in the system can easily be treated.

It is, however, difficult to control the dependence of theresults on the type of reservoirs used, as the reservoirs’degrees of freedom are traced out in the very setup.Markovian master equation models can also be used

to treat heat and particle/spin transport in models withconservative noise in the bulk (e.g. dephasing noisewhich conserves the number of quasi-particle excitations)(Manzano et al., 2012; Znidaric, 2010). Such models ofnoise, close in spirit to the probe terminals discussedin Sec. IV.B, can be interesting for applications to cou-pled transport. They offer elegant ways of treating un-wanted degrees of freedom in the bulk, such as lat-tice vibrations, and they often lead to surprising re-sults, such as noise-induced enhancement of transport(Mendoza-Arenas et al., 2013).

Complementarily, instead of using quantum mas-ter equations, one may use another, perhaps morestandard approach to non-equilibrium steady statesvia a Keldysh formalism of non-equlibrium Green’sfunctions (Haug and Jauho, 2008), where one essen-tially discusses the scattering of elementary quasi-particle excitations between two or more infinitenon-interacting Hamiltonian reservoirs. This approachwas used, among other things, to study heat transportin driven nanoscale engines (Arrachea et al., 2012;Arrachea, Moskalets, and Martin-Moreno, 2007) andspin heterostructures (Arrachea, Lozano, and Aligia,2009). Clearly, for this method to be feasible the fullself-energy of the central system has to be somehowtractable. Thus the Keldysh technique is usuallyused when the bulk dynamics is simple and detailsof coupling to the reservoirs (i.e. physical leads) areimportant. Open quantum system’s approach, on theother hand, has the advantage of allowing numericaland non-perturbative treatments for the many-bodycentral (bulk) Hamiltonians. Another setup in which theKeldysh treatment of infinite Hamiltonian reservoirs canbe explicitly implemented is when the bulk dynamicscan be treated with integrable effective quantum fieldtheory. For example, if the bulk Hamiltonian is critical,having massless (gapless) low energy excitations andif the temperatures of the reservoirs are small, onemay use conformal field theory to describe effectivenon-equilibrium steady states (Bernard and Doyon,2013).

VI. NUMERICAL APPROACHES

Numerical computations in various classical andquantum dynamical transport problems in low dimen-sions, in particular in one-dimensional particle chains,have a rich and long history (see e.g. the reviews ofLepri, Livi, and Politi (2003) and Dhar (2008), andreferences therein). In this section we only mentionclassical and quantum molecular dynamics approacheswhich are suitable for efficient computer simulations.In the context of simulating coupled heat and matter

15

transport from deterministic classical dynamical sys-tem’s perspective one has to mention the referencesMejıa-Monasterio, Larralde and Leyvraz (2001) andMejıa-Monasterio, Larralde and Leyvraz (2003) whichprovided the first numerical measurements of theOnsager matrix for interacting chaotic classical gasses.

A. Classical

The aforementioned and related studies simulate trans-port in an extended classical particle chain, or a parti-cle container, by using a hybrid deterministic/stochasticmethod. The bulk dynamics is simulated deterministi-cally using the standard techniques of molecular dynam-ics, i.e. solving Hamilton’s equations for all particles’coordinates and momenta, while at the extreme (left andright) ends of the system, particles are exchanged withthe left/right reservoirs (baths), or their energy is ex-changed, in a stochastic fashion, so that after the eventof stochastic interaction the particle’s density and energyis distributed according to the grand-canonical distribu-tion, determined by the temperatures and the chemicalpotentials of the baths. This can be viewed as a classicalanalog of the boundary driven open quantum many-bodysetup described in the previous section, and can be im-plemented efficiently to estimate numerically the figure ofmerit ZT (Casati, Mejıa-Monasterio, and Prosen, 2008).There is also an alternative deterministic approach of

the Nose-Hoover thermostats (Evans and Holian, 1985),in which also the dynamics of end (left or right) particlesis purely deterministic, but it is dissipative and typicallychaotic, so that these end particles are correctly thermal-ized (only for the average values, not for correlations).Nose-Hoover baths are nevertheless somewhat less usedthan the stochastic ones, as one can never be sure whendeterministic dynamics of the baths may bring in someunwanted (spurious) correlation effects into the system’sdynamics.

B. Quantum

For quantum systems, boundary driven locallyinteracting Lindblad equation (74) is particularlysuitable since it allows for efficient numerical sim-ulation of the steady state in terms of the time-dependent-density-matrix-renormalization-groupmethod (tDMRG) (Daley et al., 2004; Schollwock,2011; White and Feiguin, 2004) in the Liouvillespace of linear operators acting on wave functions(Prosen and Znidaric, 2009). The reason for efficiencyof this method in the long time limit as comparedto the usual tDMRG algorithm lies in typically ef-fective suppression of entanglement (correlations) inthe operator space due to the coupling to Markovianbaths. In the Liouvillean tDMRG approach, the stateof the system, say of N quantum spins 1/2 or qubits

(abstract two-level systems) described by Pauli matricesσ0 ≡ 11, σ1,2,3 ≡ σx,y,z, encoded in a many-body densityoperator is at all times t represented in the form of amatrix product ansatz,

ρ(t) =∑

sj∈0,1,2,3

〈L|A(s1)1 (t) · · ·A(sN )

N (t)|R〉σs1 ⊗ · · · ⊗ σsN ,

(76)

by means of a set of 4n time-dependent matrices A(s)j (t),

and an appropriate pair of boundary vectors 〈L|, |R〉, ofsome finite dimensionD. The simplest strategy for calcu-lating the density matrix of non-equilibrium steady stateρ∞ = limt→∞ ρ(t) is then simply to simulate time evo-

lution of the master equation (74), ρ(t) = exp(Lt)ρ(0).Namely, exp(Lt) can be decomposed into a product oflocal operators for systems with local interactions andlocal Lindblad dissipators Lµ which can be handled fullywithin the ansatz (76). The crucial approximation of themethod lies in the fact that application of local two-siteinteraction operators h[j,j+1] on (76) results in ampli-fication of dimension D to 4D, which in turn need tobe truncated to D by means of the singular value de-composition. The overall error of such truncations isrelated to the growth of entanglement entropy (in theHilbert-Schmidt space of density operators), which canbe intuitively understood to be smaller for systems withdissipation as compared to fully coherent (Hamiltonian)evolution. Having matrix product representation ρ∞ ofthe steady state one can then compute easily the localobservables such as energy density, particle density ormagnetization profiles and currents from which the fullOnsager matrix can be calculated.One can also apply linear response approach and cal-

culate Onsager coefficients from Green-Kubo expressions(69) based on tDMRG calculations of current-currenttime-correlation functions of pure Hamiltonian (coher-ent) dynamics. Here accessible time scales t∗, due toentanglement entropy growth, are much smaller than inLiouville space approach with dissipative boundaries, butt∗ does not need to be longer than the time up to whichall current-current correlations essentially vanish. State-of-the-art algorithm for such calculations is described inKarrasch, Bardarson, and Moore (2013).In cases where the coupling among the particles is non-

local, say we have residual long-range (e.g. Coulomb)interaction, or some other complications arise which pro-hibit efficient use of tDMRG, one can simulate quantummaster equation using the method of quantum trajecto-ries (see, for example, Mejıa-Monasterio and Wichterich(2007)). The idea there is to represent the density oper-ator as an expectation of |Ψ〉〈Ψ| where the many-bodywavefunction Ψ is a solution of a stochastic Schrodingerequation dΨ(t) = −(i/~)HΨdt + dξ, with dξ being anappropriate stochastic process simulating the action ofthe baths. The advantage of this technique is that non-Markovian effects can be treated easily and intuitively.For even more general problems, one can always

solve the many-body quantum master equation numer-

16

ically exactly, but the computational complexity then,of course, grows exponentially with the systems size.For modeling the proper canonical heat baths in quan-tum transport problems one often uses the Redfieldequation (Redfield, 1957). Redfield master equationis derived physically via projection operator technique(Kubo, Toda, and Hashitsume, 1985). It is a crucialproperty of the Redfield master equation that the steadystate is guaranteed to have the Gibbs distribution.Such approach has a wide applicabilities in many-bodysystems and proved also useful to investigate thermaltransport (Saito, 2003; Saito, Takesue, and Miyashita,2003; Segal and Nitzan, 2005; Wu and Segal, 2008) andelectron transport (Galperin et al., 2009). A quan-tum Langevin equation approach was used to com-pute thermal conductance through molecular wires(Segal, Nitzan, and Hanggi, 2003).

The time-dependent density functional theory was ex-tended to include open quantum systems evolving un-der a master equation (Burke, Car, and Gebauer, 2005;Tempel et al., 2011) or in interaction with external baths(Di Ventra and D’Agosta, 2007). Applications of thedensity functional theory to thermoelectricity are dis-cussed in D’Agosta (2013); Vignale, Eich, and Di Ventra(2013).

VII. MODELS OF THERMODYNAMIC ENGINES

Since the pioneering work by Carnot (1824), a hugenumber of physical phenomena have been recognized asheat engines, including thermoelectric phenomena. Inoriginal Carnot’s idea, the maximum efficiency was stud-ied for an ideal gas confined by a piston and is boundedby the Carnot efficiency. The Carnot efficiency is achiev-able for the celebrated Carnot cycle, that consists ofisothermal and adiabatic processes. Otto invented amore practical heat engine (Otto engine) (Callen, 1985)which consists of adiabatic compression, heat addition atconstant volume, adiabatic expansion, and rejection ofheat at constant volume. In all heat engines the max-imum efficiency is obtained for a quasi-static processwhere asymptotically vanishing power output is gener-ated. From the practical viewpoint, finite power withhigh efficiency is desirable. This consideration opens theway to the concept of finite-time thermodynamics, dis-cussed in Sec. III.A. Paradigms of finite-time thermo-dynamic engine are the Carnot or Otto cycles with afinite time period. Recent technological developmentsenable us to consider and realize finite-time thermody-namic devices with high controllability. Main issues offinite-time thermodynamics are universal nature of effi-ciency at maximum power and new ideas for making suchdevices. In this section, we discuss several paradigms ofthermodynamic heat engines.

A. Stochastic thermodynamics

Stochastic thermodynamics is a framework to studynon-equilibrium thermodynamics in small systems likecolloids or biomolecules driven out of equilibrium(Seifert, 2012; Sekimoto, 2010). In most cases, thisfield treats classical systems. The system is always at-tached to a thermal environment like water. The timescales of environment and system are sufficiently sepa-rated and the system’s dynamics is well described by theLangevin equation. Suppose that one colloidal particleis trapped by an external potential and the dynamics isoverdamped; the Langevin equation of motion is givenby

x = µF (x, λ(t)) + η(t) , (77)

where x(t) is the particle’s coordinate, η(t) a Langevinthermal noise satisfying 〈η(t)η(t′)〉 = 2Dδ(t− t′) with Dbeing the diffusion constant. The function F (x, λ(t)) isa time-dependent force field, which is given by

F (x, λ(t)) = −∂xV (x, λ(t)) + f , (78)

where V (x, λ(t)) is a potential and f is a nonconservativeforce which can not be expressed as the gradient of a po-tential. The parameter λ(t) is a time-dependent externalcontrol parameter. In equilibrium, the diffusion constantD and the mobility µ are related by the Einstein relationD = Tµ.The Langevin dynamics is endowed with a thermody-

namic interpretation by applying the energy balance toany individual stochastic trajectory:

δU = δQ− δW, (79)

where δQ is the heat absorbed from the environment,δU the variation of internal energy, and δW the workperformed by the particle. Note that −δW is the workapplied to the particle due to a time-dependent potential(i.e., λ changes in time) or to a nonconservative forcef . Since the dynamics is overdamped, the kinetic energydoes not change in time. Hence, the variation δU ofthe internal energy is equivalent to the change δV of thepotential. The work applied to the particle reads

− δW = (∂V/∂λ)λdt+ fdx, (80)

and therefore the absorbed heat is

δQ = δV + δW = −Fdx . (81)

The work and heat defined above are the basis to considerthermodynamic efficiency in stochastic thermodynamics.An intriguing and solvable heat engine in the frame-

work of stochastic thermodynamics was introduced bySchmiedl and Seifert (2008). Suppose that one parti-cle is trapped by a time-dependent harmonic potentialV (x, λ(t)) = λ(t)x2/2 without nonconservative force, i.e.f = 0. We consider a cycle, depicted in Fig. 7, composedof the following four steps.

17

1.

2.

T1

T2

3.

4.

V

V

V

V

FIG. 7 (color online). Schematic picture of a stochastic ther-modynamic engine. In each plot the curve shows the potentialV versus x, the filled region is limited by the curve p(x), rep-resenting the (time-dependent) probability density to find thetrapped particle at x.

1. Isothermal transition at the hot temperature T1

during 0 ≤ t < t1. The potential V (x, λ(t)) changesin time and work is extracted (W > 0).

2. Adiabatic transition (instantaneously) from the hottemperature T1 to the cold temperature T2.

3. Isothermal transition at the cold temperature T2

during the time interval t1 ≤ t < t1+ t3; V (x, λ(t))changes in time and work is applied to the particle(W < 0).

4. Adiabatic instantaneous transition from the coldtemperature T2 to the hot temperature T1.

This heat engine captures important aspects of finite-time thermodynamics. Exact analysis of this modelshows that the next leading order η2C/8 in the CAformula is correct (Schmiedl and Seifert, 2008). Later,this result was proved in a more general context(Esposito, Lindenberg, and Van den Broeck, 2009b).The above described engine is quite realistic, sinceit models the trapping of a colloidal particle in atime-dependent harmonic potential. The stochas-tic heat engine was experimentally demonstrated(Blickle and Bechinger, 2012).The CA efficiency was also studied in gas systems

where the Carnot cycle is performed within a finite timecycle. The validity of CA efficiency was numericallydiscussed by means of molecular dynamics simulations(Izumida and Okuda, 2008, 2009a,b).

B. Heat engine with blowtorch effect

Buttiker and Landauer’s (BL) model is one exam-ple of heat engine (Buttiker, 1987; Landauer, 1988;

van Kampen, 1988). In BL model a particle is trapped ina periodic potential V (x) and subject to a nonuniform,spatially periodic temperature profile. Although thereis no time-dependent driving this system satisfies Curie’sprinciple (Curie, 1894), namely, rectification effect occursas it is not ruled out by symmetries. This situation is an-alyzed using the Langevin dynamics where the particleis alternately attached along the spatial coordinate, tothermal baths at different temperatures. The equationof motion is given by

mx = −γ(x)x− V ′(x)− f +√

2γ(x)T (x) ξ(t),(82)

where γ(x) is the coefficient of a viscous friction, f is theexternal force, and ξ is a white Gaussian noise satisfying〈ξ(t)ξ(t′)〉 = δ(t − t′). We assume that the potentialand temperature depend on the position and are periodicwith period L, and take (T (x), γ(x)) = (T1(2), γ1(2)) forx ≤ (>)L/2, with T1 > T2. A schematic picture for thepotential and temperature is presented in Fig. 8.The energetics and transport properties of the BL

motor have been studied by many authors. Landauer(1988) proposed the idea of the BL motor and showedthe physical significance of nonuniform temperature inchanging the relative stability of otherwise locally stablestates, a phenomenon he termed as the blowtorch effect.Periodic temperature with a periodic potential inducesa net transport of Brownian particles (Buttiker, 1987;Landauer, 1988; Parrondo and de Cisneros, 2002). In thehot region the Brownian particle can move more easilythan in the cold region. Hence, a finite net current is gen-erated. Efficiency η is calculated by η = W/Q1, where W

is the work done by the particle per unit time: W = f〈x〉.The denominator Q1 is the heat supply (per unit time)from the hot region, evaluated as (see Sekimoto (2010))

Q1 = 〈(−γ1x +√2γ1T1ξ(t))x〉, where the statistical av-

erage is taken only over the hot region, 0 < x ≤ L/2.In the overdamped limit (x → 0) the net cur-

rent can be derived using the Fokker-Planck equa-tion. Then one can show that, under suitableconditions, the efficiency can reach the Carnot effi-ciency (Asfaw and Bekele, 2004, 2007; Matsuo and Sasa,2000). However, it was pointed out that reachingthe Carnot efficiency is problematic due to the abruptchange of temperature at the boundaries (Ai et al.,2005; Ai, Wang, and Liu, 2006; Derenyi and Astumian,1999; Hondou and Sekimoto, 2000). Carnot efficiency isunattainable due to the irreversible heat flow via kineticenergy. Recent first principle calculations using molec-ular dynamics simulation support the unattainability ofCarnot efficiency (Benjamin and Kawai, 2008).Brownian motors driven by temporal rather than

spatial temperature oscillations are discussed inReimann et al. (1996), where the potential has brokenspatial symmetry (“ratchet” potential). The constructiverole of Brownian motion for various physical and tech-nological setups is reviewed in Hanggi and Marchesoni(2009).

18

V

L

T1 T2

0 L/2

f

FIG. 8 (color online). Schematic picture of Buttiker-Landauer heat engine.

C. Driven quantum dot

Among electric devices, the quantum dot (QD) hasthe potential to provide many kinds of thermodynamicengines (Esposito, Lindenberg, and Van den Broeck,2009a; Esposito et al., 2010b; Humphrey et al., 2002;Nakpathomkun, Xu, and Linke, 2010). Finite-time heatengines can be illustrated by means of quantum-dotsystems, where one controls the gate voltage and in timeto change the on-site energy of the dot. We assume thata single-level quantum dot with time-dependent energyǫ(t) is attached to one lead which has temperatureT (= 1/β) and chemical potential µ(t). Although theengine should work at the nanoscale, the dynamics isassumed to be classical, in that the off-diagonal elementsof the density matrix are neglected in the time-evolution.Let p(t) be the occupation probability for the state ofthe quantum dot. Then the time-evolution is governedby the following master equation:

p(t) = −W1(t)p(t) +W2(t) [1− p(t)] , (83)

where the transition rates read

W1(t) = C[

e−β[ǫ(t)−µ(t)] + 1]−1

,

W2(t) = C[

eβ[ǫ(t)−µ(t)] + 1]−1

,(84)

with a constant C. The internal energy U(t) of thequantum-dot system at time t is defined as U(t) =ε(t)p(t); the output work δW (t) and the absorbed heatδQ(t) are given by

δW (t) = − [dǫ(t)− dµ(t)] p(t) , (85)

δQ(t) = [ǫ(t)− µ(t)] dp(t) . (86)

Esposito et al. (2010b) proposed a solvable engine us-ing the following cycle (see Fig. 9):

1. Isothermal process: The quantum dot is in contactwith a cold lead at temperature T2 and chemicalpotential µ2. The energy is raised during a finitetime τa as ǫ(t) : ǫ0 → ǫ1 (ǫ1 > ǫ0).

T1

T20

1

2

1.

2.

3.

4.

T2

T1

3

FIG. 9 (color online). Schematic picture of a heat engine thatconsists of a driven quantum dot.

2. Adiabatic process: The quantum dot is discon-nected from the lead, and the energy is abruptlylowered as ǫ(t) : ǫ1 → ǫ2 (ǫ1 > ǫ2).

3. Isothermal process: The quantum dot is connectedto a hot lead with temperature T1 and chemicalpotential µ1. The energy is lowered during a finitetime τb, ǫ(t) : ǫ2 → ǫ3 (ǫ2 > ǫ3).

4. Adiabatic process: The dot is disconnected, andthe energy abruptly returns to the original value:ǫ(t) : ǫ3 → ǫ0.

The period of one cycle is τ = τa + τb, the output poweris given by P = W/τ = Q/τ =

∫ τ

0 dt p(t) [ε(t)− µ(t)].Finding the set of parameters that maximize the powermay be done with a variational equation. In particular,the CA efficiency is recovered in the limit of weak dissi-pation (Esposito et al., 2010b).

D. Quantum heat engines

Quantum mechanics and thermodynamics have a deepconnection, whose investigation started from thermody-namic studies by Planck (1901) and Einstein (1917). Dueto recent progress of micro-fabrication technology, quan-tum effects in small heat engines have become an impor-tant subject.The expectation value of the measured energy of a

whole quantum system is U =∑

i piEi, where Ei arethe energy levels and pi are the corresponding occupa-tion probabilities. This implies that

dU =∑

i

(Eidpi + pidEi) , (87)

19

from which the absorbed heat is recognized to be δQ =∑

iEidpi and the output work δW = −∑i pidEi. Then,the first law dU = δQ− δW is satisfied.The quantum-mechanical analogue of Carnot cyclic en-

gine is introduced by identifying isothermal processeswith quantum processes at constant expectation valueof the energy (Bender, Brody, and Meister, 2000). For asingle quantum mechanical particle confined to a poten-tial well, the Carnot efficiency in this framework takes theform ηC = 1 − E2/E1, where E1 and E2 are the energyexpectation values during the isothermal transformationsat temperatures T1 and T2, respectively (T1 > T2). En-tropy and temperature were discussed within this frame-work (Bender, Brody, and Meister, 2002). Extensionsof this approach include ideal Fermi gas with an arbi-trary number of particles (Wang et al., 2012), Otto cycle(Wu et al., 2010),and the investigation of the efficiencyat maximum power for the engine introduced by (Abe,2011).Quantum version of the Carnot heat engine and Otto

heat engine for finite temperature was defined with-out ambiguities in (Quan et al., 2007). Effect of multi-level systems was invesitigated (Quan, Zhang, and Sun,2005). A class of quantum heat engines consisting oftwo subsystems interacting with a work source was stud-ied to maximize the extracted work under various con-straints (Allahverdyan, Johal, and Mahler, 2008). Con-cepts from quantum information theory also providenew insights into the working of quantum heat engines(Kieu, 2004, 2006; Maruyama, Nori, and Vedral, 2008;Quan et al., 2006; Zhou and Segal, 2010).Open systems attached to a thermal environment can

be analyzed by means of the quantum master equation inthe form (Kubo, Toda, and Hashitsume, 1985; Lindblad,1976; Redfield, 1957)

ρ =i

~

[

ρ,H(λ(t))]

−∑

α

Γα(t) ρ(t) , (88)

where ρ is the density matrix of the system and Γα(t)denotes a superoperator representing dissipative effectsarising from a reservoir labeled by index α. The param-eter λ is a control parameter for the system. Then theoutput work δW and the heat δQα absorbed from the αreservoir are given by

δW = −Tr [ ρ(t)∂λH(λ) ] λdt , (89)

δQα = −Tr[

H(λ)Γα(t)ρ(t)]

dt . (90)

The master equation approach reproduces the Carnotefficiency for the Carnot cycle (Alicki, 1979). Theperformance of a quantum heat engine or heat pumpwith the working fluid composed of noninteractingtwo-level systems was investigated by using a mas-ter equation (Feldmann and Kosloff, 2000). Kosloff(1984) showed that two coupled oscillators interact-ing with hot and cold quantum reservoir exhibitCurzon-Ahlborn efficiency in the limit of weak cou-pling. The quantum master equation was applied

to analyze the performance of heat engines workingwith spins (Chen, Lin, and Hua, 2002; Geva and Kosloff,1992; He, Lin, and Hua, 2002), harmonic oscillators(Arnaud, Chusseau, and Philippe, 2002; Lin and Chen,2003a,b; Lin, Chen, and Hua, 2003), and multi-level sys-tems (Geva and Kosloff, 1994). Characteristics of thesteady state achieved by the iteration of cyclic processand monotonic approach to the limit cycle was dis-cussed making use of the quantum conditional entropy(Feldmann and Kosloff, 2004). The unavoidable irre-versible loss of power in a heat engine was considered forharmonic systems in the framework of quantum masterequation approach (Rezek and Kosloff, 2006).

Fifty years after the pioneering works on quantummechanics and thermodynamics by Planck (1901) andEinstein (1917), models of lasers and masers were realizedas quantum heat engines (Scovil and Schultz-DuBois,1959) and the relation between the quantum efficiencyof the maser and the Carnot cycle was investigated. De-tailed balance imposed by thermodynamics limits the ef-ficiency of quantum heat engines, including solar cells(Shockley and Queisser, 1961). Scully et al. analyzedthe performance of a quantum heat engine operating bymeans of the radiation pressure from a single mode ra-diation field which drives a piston engine or a photon-Carnot engine. They pointed out that the phase asso-ciated with the atomic coherence provides a new con-trol parameter, which can be varied to increase thetemperature of the radiation field and to extract workfrom a single heat bath, while the real physics be-hind the second law of thermodynamics is not violated(Scully, 2002; Scully et al., 2003). This photon-Carnotengine was discussed in a quantum information context(Dillenschneider and Lutz, 2009) and decoherence mech-anisms were studied (Quan, Zhang, and Sun, 2006).

Breakdown of detailed balance due to quantum co-herence has been discussed for a quantum-dot pho-tocell (Scully, 2010) and a photon-Carnot quantumheat engine (Scully et al., 2003). Noise-induced co-herence was proposed as a potential ingredient to en-hance quantum power (Scully et al., 2011). QuantumOtto engine in two-atom cavity quantum electrodynam-ics (Wang, Liu, and He, 2009) was studied to under-stand the role of thermal entanglement, and realiza-tion of Otto engine in single-modes fields was discussed(Wang, Wu and He, 2012).

Photosynthetic reaction center is a quantum enginewhere energy is supplied from the light and hence it hassimilarities to heat engine (Dorfman et al., 2013). Thisalso provides an important intersection between physicsand biology. The rich history and ongoing studies suggestthat the deep underlying connection between thermody-namics and quantum mechanics may be useful for im-proving the design and boosting the efficiencies of light-harvesting devices.

The concept of ideal quantum heat engine was intro-duced in cold bosonic atoms confined to a double well po-tential where thermalization occurs, and operation of a

20

heat engine with a finite quantum heat bath was theoreti-cally demonstrated (Fialko and Hallwood, 2012). A ther-moelectric heat engine with ultracold fermionic atomswas demonstrated, both theoretically and experimentally(Brantut et al., 2013).The fundamental limits to the dimensions of

(steady-state) self-contained quantum machines, i.e.machines working without the supply of externalwork but only due to interactions with thermalbaths at various temperatures, were investigated byBrunner et al. (2012); Linden, Popescu, and Skrzypczyk(2010); Skrzypczyk et al. (2011). In particular, itwas shown that also a small self-contained refrig-erator consisting of three qubits, each one in con-tact with a thermal reservoir, can achieve theCarnot efficiency (Skrzypczyk et al., 2011). InVenturelli, Fazio, and Giovannetti (2013) an electronicquantum refrigerator based on four quantum dots in con-tact with as many thermal reservoirs was theoreticallyinvestigated.

VIII. PHENOMENOLOGICAL LAWS AND

THERMOELECTRIC TRANSPORT

A. Wiedemann-Franz law

In a wide range of macroscopic electronic conduc-tion, the electron conductivity and the electronic con-tribution to the thermal conductivity are not indepen-dent. They are connected to each other by an em-pirical relation called the Wiedemann-Franz (WF) law(Wiedemann and Franz, 1853). The WF law states thatthe ratio of the thermal conductivity κ to the electricconductivity σ of a metal is proportional to the temper-ature:

κ

σ= LT , (91)

where the constant value L is known as the Lorenz num-ber. In non-interacting electronic systems at low temper-atures, the Lorenz number is given by

L =π2

3

(

kBe

)2

. (92)

This law is derived by using either kinetic theory or Lan-dauer formula. In the kinetic theory approach (Kittel,2004), one uses the expression of electronic conductivityσ ∼ ne2τ/m, where n is the electronic density and τ isthe mean-free time. Thermal conductivity κ is given byκ ∼ Cv2F τ/3, with C the specific heat and vF the Fermivelocity. The specific heat for a free electronic system atlow temperatures is given by C ∼ π2nk2BT/mv2F , whichimmediately yields the constant value (92) for the Lorenznumber. Another approach is based on the Landauer’sexpressions of electric and thermal conductance, Eq. (55).In these expressions, we assume that the transmission isweakly energy-dependent, hence

τ(E) ≈ τ(µ) . (93)

Under this assumption, one can derive the WF law for theratio of thermal and electric conductances: Ξ/G = LT .Both derivations of the WF law are substantiated by theSommerfeld expansion (Ashcroft and Mermin, 1976) ofintegrals (54) or (56) to the lowest order in kBT/EF ,with EF being the Fermi energy. Such expansion is validfor a smooth function τ(E) or Σ(E). Note that to derivethe WF law the off-diagonal Onsager coefficient L12 hasto be neglected, i.e., one needs L11L22 ≫ L2

12 in orderto approximate the thermal conductance Ξ with L22/T

2

(van Houten et al., 1992).

B. Mott’s formula

The Mott’s formula states that the Seebeck co-efficient S in a metal is approximately given bythe formula (Cutler and Mott, 1969; Macdonald, 2006;Mott and Davis, 1971)

S ∼ π2k2BT

3e∂E lnG(E)

∣

∣

∣

E=µ, (94)

where G(E) is the electric conductance at chemical po-tential E in the leads. This relation is obtained underthe assumption that the system is non-interacting andthat conduction mainly occurs around the Fermi energy.Thus, the transmission probability is approximated asfollows:

τ(E) ≈ τ(µ) + ∂Eτ(µ) (E − µ) . (95)

The Mott’s relation (94) is derived after inserting thisexpansion into (55). An analogous derivation is obtainedby considering the conductivity σ(E) rather than con-ductance G(E) and (57) rather than (55).

C. Thermoelectricity

The Mott’s formula tells us that a sharp energy-dependence of electric conductivity is crucial forincreasing the Seebeck effect. This is consistent withthe importance of energy filtering structure of trans-mission discussed in Sec. IV.A. Both the WF lawand the Mott’s relation follow from the single particleFermi-liquid (FL) theory, so that the Sommerfeldexpansion can be applied. If the FL theory holds innon-interacting systems, the WF law is valid in thepresence of arbitrary disorder (Chester and Thellung,1961; Jonson and Mahan, 1980). When the WF law isvalid, it is not possible to obtain large thermoelectricefficiency. Indeed, the WF law is obtained under thecondition L11L22 ≫ L2

12, hence the figure of meritZT = L2

12/detL ≈ L212/L11L22 ≪ 1. This implies

that to get large ZT one should search for physicalsituations where the WF law is violated. The WF law islargely violated in low-dimensional interacting systemsthat exhibit non-FL behavior (Dora, 2006; Garg et al.,

21

2009; Kane and Fisher, 1996; Li and Orignac, 2002;Wakeham et al., 2011) and in small systems wheretransmission can show a significant energy de-pendence (Bergfield, Jacquod, and Stafford, 2010;Bergfield, Solis, and Stafford, 2010; Boese and Fazio,2001; Vavilov and Stone, 2005). Properties of reservoirsfor such small systems may affect the validity regimesof the WF law (Balachandran, Bosisio, and Benenti,2012). Departures of the WF law in the regime of non-linear transport were investigated in Lopez and Sanchez(2013). The possibility of engineering the Lorenz numberin superlattices was discussed in Bian et al. (2007).

Hicks and Dresselhaus theoretically studied quantum-well structures in low dimensions and showed thatlayering has the potential to increase the thermoelec-tric figure of merit (Hicks and Dresselhaus, 1993a,b).Indeed the layering may reduce the phonon thermalconductivity as phonons can be scattered by the inter-faces between layers. Moreover, sharp features in theelectronic density of states, favorable for thermoelectricconversion (see the discussion of Sec. IV.A), are inprinciple possible due to quantum confinement. Thisproposal was experimentally demonstrated (Hicks et al.,1996). Moreover, it motivated interest in the thermo-electric properties of low-dimensional nanostructures.For experimental evidence of energy filtering, seeZide et al. (2006), where barriers in a superlattice wereused to limit the transport to those electrons withsufficiently high energy. As a result, a dramatic increaseof the Seebeck coefficient was shown together with arelatively modest decrease of the electrical conduc-tivity. Significant energy-dependence of transmissionwas found in graphene (Zuev, Chang, and Kim, 2009).Thermopower measurements have been used to studysemiconductor nanostructures such as quantum pointcontacts (Molenkamp et al., 1990) and quantum dots inthe Coulomb blockade regime (Molenkamp et al., 1994;Staring et al., 1993). The thermoelectric properties of aquantum dot can be used to obtain thermal rectification,as reported in the experiments by Scheibner et al.(2008). First principle calculation for silicon nanowiresshows that in nanowires small diameter is desiredto increase ZT and that isotopic doping can in-crease the value of ZT significantly (Shi et al., 2009).Sharp electronic resonances can be found also inmolecules weakly coupled to electrodes and for thisreasons molecular junctions might be efficient forthermoelectric conversion (Balachandran et al., 2012;Bergfield, Solis, and Stafford, 2010; Dubi and Di Ventra,2011; Finch, Garcıa-Suarez, and Lambert, 2009;Ke et al., 2009; Leijnse, Wegewijs, and Flensberg,2010; Pauly, Viljias, and Cuevas, 2008; Reddy et al.,2007; Widawsky et al., 2013). Photo-Seebeck effect,i.e. the contribution of photo-induced carriers to thethermopower, was experimentally demonstrated for awide-gap oxide semiconductor (Okazaki et al., 2012).

Small systems such as quantum dots have an advantagefor making narrow energy window in transmission and

breaking the WF law. On the other hand, small systemsexhibit large fluctuations in physical quantities. In openchaotic quantum dots, thermopower shows significantfluctuations. The distribution of parametric conductancederivatives was analyzed using random matrix theory(Brouwer et al., 1997) and fluctuations of thermopowerwere discussed in van Langen, Silvestrov, and Beenakker(1998). Thermopower fluctuations exhibit a non-Gaussian distribution, experimentally demonstrated inGodijn et al. (1999). Effects of the spectrum edges of achaotic scatterer on thermopower distributions and itsuniversal aspects were studied in Abbout et al. (2013).Fluctuations and nonlinear response of thermoelectric-ity were studied in terms of time-reversal symmetry(Iyoda, Utsumi, and Kato, 2010) in analogy to the ex-pansion using the fluctuation theorem for electric trans-port (Andrieux et al., 2009; Saito and Utsumi, 2008).Nernst effect was theoretically studied to get strictbounds on efficiency (Stark et al., 2013).

IX. CONCLUSIONS

In this Colloquium we have presented a simpleand self-contained account of few main theoreticalapproaches to discuss the problem of thermoelectricefficiency and the efficiency of steady-state heat to workconversion in general. Even though the problem has along history, we believe that many recent theoreticaladvances described here, being in turn stimulated bynew generations of experiments with nano-scale systems,should be somewhat taken from a new, more abstractperspective. Namely, we believe that the powerfulmachinery of non-equilibrium statistical mechanicsand dynamical system’s theory has not yet been fullyexplored in connection to coupled heat and electric, mag-netic or particle transport and in particular in analyzingthe figure of merit of thermoelectric, thermomagneticor thermochemical efficiency. This being particularlyso in view of some very recent new fundamental resultson the behavior of thermoelectric efficiency in thepresence of time-reversal symmetry breaking of theunderlying equations of motion, such as by meansof the magnetic field (Benenti, Saito, and Casati,2011; Brandner, Saito, and Seifert, 2013;Brandner and Seifert, 2013).

The central question which identifies this paper is:What limits, if any, the microscopic dynamical laws – fora particular model, or for a particular non-equilibriumsteady-state setup – impose on the thermodynamic heat-to-work efficiency? While the theory for non-interactingsystems discussed here seems to be well understood,the understanding of general mechanisms connected tostrong interactions are only beginning to emerge. Thisjustifies the importance of efficient numerical simula-tions of interacting systems at this point. While thisColloquium is focused on linear transport, thermoelec-tric devices often operate in the nonlinear regime of

22

transport. In that regime, reciprocity relations breakdown, as experimentally observed in a four-terminalmesoscopic device (Matthews et al., 2013), and the fig-ure of merit ZT fails to describe thermoelectric per-formance (Meair and Jacquod, 2013; Whitney, 2013b;Zebarjadi, Esfarjani, and Shakouri, 2007). The observedbreakdown of the Onsager-Casimir relations increasingthermal bias could in principle allow for improved ther-moelectric efficiencies. In the nonlinear regime, rectifica-tion effects occur and their impact on thermoelectricity isstill not well understood. In this regard, the recently de-veloped (Meair and Jacquod, 2013; Sanchez and Lopez,2013; Whitney, 2013) scattering theory of nonlinear ther-moelectricity could pave the way to deeper investigationsof quantum coherent conductors working beyond the lin-ear response regime.

We expect to see in near future a burst of applicationsof fundamental ideas on abstract dynamical mechanismsfor analyzing, or engineering particular practically rele-vant models, or for designing experiments and technolog-ical applications.

Acknowledgments

G.B. and G.C. acknowledge the support by MIUR-PRIN. T.P. acknowledges support by the grants P1-0044and J1-5439 of Slovenian Research Agency (ARRS). K.S.was supported by MEXT (23740289) .

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