Thermodynamic bound on heat to power conversion

Giuliano BenentiCenter for Nonlinear and Complex Systems,

Univ. Insubria, Como, Italy INFN, Milano, Italy

Thanks to collaborators: Wang Jiao, Rongxiang Luo (Xiamen),

Giulio Casati (Como)

Ref.: Phys. Rev. Lett. 121, 080602 (2018)

General motivation

Carnot efficiency can be obtained only for infinitely slow heat engines, so the extracted power vanishesWhat is the maximum allowed efficiency at a given power output?For steady-state (thermoelectric) quantum systems modeled by the Landauer-Büttiker scattering theory a (rather restrictive) upper bound exists

[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]

Is it possible to overcome this bound for interacting systems, thus allowing a better power-efficiency trade-off?

Finite time thermodynamics

In an ideal Carnot engine conversion processes are quasi-static and the extracted power reduces to zero.

How much the efficiency deteriorates when heat to work conversion takes place in a finite time?

Finite time thermodynamics: finite-time steady-state conversion processes or thermodynamic cycles; the efficiency at the maximum output power is an important concept

[Andresen, Angew. Chem. Int. Ed. 50, 2690 (2011)]

Steady-state (thermoelectric) power production

The upper bound to efficiency is given by the Carnot efficiency:

SLeft (L) reservoir

Right (R) reservoir

T ,L L T ,R R

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⌘C = 1� TR

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Charge current

Landauer formalism for thermoelectricity

Heat current from reservoirs:

Jq.� =1h

� �

��dE(E � µ�)�(E)[fL(E)� fR(E)]

Thermoelectric efficiency (power production)

Carnot efficiency

[Mahan and Sofo, PNAS 93, 7436 (1996); Humphrey et al., PRL 89, 116801 (2002)]

Delta-energy filtering and Carnot efficiency

Carnot efficiency obtained in the limit of reversible transport (zero entropy production) and zero output power

If transmission is possible only inside a tiny energy window around E=E✶ then

Heat-to-work conversion through energy filtering

Flow of heat from hot to cold but no flow of charge

[see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)]

Bekenstein-Pendry bound

There is an purely quantum upper bound on the heat current through a single transverse mode

[Bekenstein, PRL 46, 923 (1981); Pendry, JPA 16, 2161 (1983) ]

For a reservoir coupled to another reservoir at T=0 through a -mode constriction which lets particle flow at all energies:

Maximum power of a heat engine

Since the heat flow must be less than the Bekenstein-Pendry bound and the efficiency smaller than Carnot efficiency also the output power must be bounded

Within scattering theory:

[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]

Efficiency optimization (at a given power)Find the transmission function that optimizes the heat-engine efficiency for a given output power

[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]

Trade-off between power and efficiency

Effic

ienc

y

Carnot efficiency

Maximum possible power, P max

gen

forbidden1

2

power generated, Pgen

Result from (nonlinear) scattering theory

[Whitney, PRL 112, 130601 (2014); PRB 91, 115425 (2015)]

increase voltage

Power-efficiency trade-off including phonons

Power output, P

Effi

cien

cy

0

strong phonons

no phonons

weak phonons

[see Whitney, PRB 91, 115425 (2015)]

Boxcar transmission in topological insulators

[Chang et al., Nanolett., 14, 3779 (2014)]

Graphene nanoribbons with heavy adatoms and nanopores

Is it possible to overcome the non-interacting bound?

For P/Pmax<<1,

Bound not favorable for power-efficiency trade-off; due to the fact that delta-energy filtering is the only mechanism to achieve Carnot for noninteracting systems

For interacting systems it is possible to achieve Carnot without delta-energy filtering

Linear response for coupled (particle and heat) flowsStochastic baths: ideal gases at fixed temperature

and electrochemical potential

�µ = µL � µR

�T = TL � TR

(we assume TL > TR, µL < µR)

Onsager relation (for time-reversal symmetric systems):

Positivity of entropy production:

Onsager and transport coefficients

Note that the positivity of entropy production implies that the (isothermal) electric conductance G>0 and the thermal conductance K>0

Local equilibrium

Under the assumption of local equilibrium we can write phenomenological equations with ∇T and ∇µ rather than ΔT and Δµ

In this case we connect Onsager coefficients to electric and thermal conductivity rather than to conductances

charge and heat current densities

� =�

je

�V

�

�T=0

, � =�

jh

�T

�

je=0

Traditional versus quantum thermoelectrics

Relaxation length (tens of n a n o m e t e r s a t r o o m temperature) of the order of the mean free path; inelastic s c a t t e r i n g ( p h o n o n s ) thermalizes the electrons

Structures smaller than the relaxation length (many microns at low temperature); quantum interference effects; Boltzmann transport theory cannot be applied

[see G. B., G. Casati, K. Saito, R. S. Whitney, Phys. Rep. 694, 1 (2017)]

Linear response?

(exhaust gases)

(room temperature)

Linear response for small temperature and electrochemical potential differences (compared to the average temperature) on the scale of the relaxation lengthExhaust pipe: temperature drop over a mm scale: temperature drop of 0.003 K on the relaxation length scale (of 10 nm)

[Vining, Nat. Mater. 8, 83 (2009)]

Maximum efficiency

Find the maximum of η over , for fixed (i.e., over the applied voltage ΔV for fixed temperature difference ΔT)

Within linear response and for steady-state heat to work conversion:

Thermoelectric figure of merit

ZT � L2eh

detL=

GS2

KT

Efficiency at maximum power

Find the maximum of P over , for fixed (over the applied voltage ΔV for fixed ΔT)

Output power

Maximum output power

Power factor

Pmax =T

4L2

eh

LeeF2

h =14

S2G(�T )2

Efficiency at maximum power

�(�max) =�C

2ZT

ZT + 2� �CA �

�C

2

ηCA Curzon-Ahlborn upper bound

P quadratic function of , with maximum at half of the stopping force:

�max

�(�max)Pmax

Efficiency versus power

⇒

Interacting systems, Green-Kubo formulaThe Green-Kubo formula expresses linear response transport coefficients in terms of dynamic correlation functions of the corresponding current operators, cal- culated at thermodynamic equilibrium

Non-zero generalized Drude weights signature of ballistic transport

Conservation laws and thermoelectric efficiencySuzuki’s formula (which generalizes Mazur’s inequality) for finite-size Drude weights

Qm relevant (i.e., non-orthogonal to charge and thermal currents), mutually orthogonal conserved quantities

Assuming commutativity of the two limits,

Momentum-conserving systems

Consider systems with a single relevant constant of motion, notably momentum conservation

Ballistic contribution to vanishes since

ZT =�S2

�T � �1�� �� when ���

(G.B., G. Casati, J. Wang, PRL 110, 070604 (2013))

DeeDhh �D2eh = 0

(� < 1)

For systems with more than a single relevant constant of motion, for instance for integrable systems, due to the Schwarz inequality

Equality arises only in the exceptional case when the two vectors are parallel; in general

detL � L2, � � �, ZT � �0

DeeDhh �D2eh = ||xe||2||xh||2 � �xe,xh� � 0

xi = (xi1, ..., xiM ) =1

2�

��JiQ1��

�Q21�

, ...,�JiQM ��

�Q2M �

�

�xe,xh� =M�

k=1

xekxhk

� �2

Example: 1D interacting classical gas

Consider a one dimensional gas of elastically colliding particles with unequal masses: m, M

ZT depends on the system size

(integrable model)ZT = 1 (at µ = 0)

Quantum mechanics needed: Relation between density and electrochemical potential

Maxwell-Bolzmann distribution of

velocities

Reservoirs modeled as ideal (1D) gases

injection rates

grand partition function

density

de Broglie thermal wave length

Non-decaying correlation functions

Anomalous thermal transport

Carnot efficiency at the thermodynamic limit

ZT =�S2

kT

[R. Luo, G. B., G. Casati, J. Wang, PRL 121, 080602 (2018)]

Delta-energy filtering mechanism?

A mechanism for achieving Carnot different from delta-energy filtering is needed

Validity of linear response

The agreement with linear response improves with N

Non-interacting classical bound (but quantum mechanics needed)

charge current

heat current

Maxwell-Boltzmann distribution

(in 1D)de Broglie thermal

wave length

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

PêPmax

hêhC

Overcoming the non-interacting bound

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Multiparticle collision dynamics (Kapral model) in 2D

S t r e a m i n g s t e p : f r e e propagation during a time τ

Collision step: random rotations of the velocities of the particles in cells of linear size a with respect to the center of mass velocity:

Momentum is conserved

Overcoming the (2D) non-interacting bound

Results can be extended to cooling

linear response

numerical data

Applications for cold atoms?

ConclusionsThanks to interactions, for a given power it is possible to overcome the bound of efficiency which applies for classical non-interacting systems

Non-integrable momentum-conserving systems exhibit a power-efficiency trade-off which is optimal within linear response

Our results are based on the fact that such systems can achieve the Carnot efficiency at the thermodynamic limit without delta-energy filtering

Extension of our results to purely quantum systems?