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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 1/32

Control of Heat Flow in Classical andQuantum Complex Systems

Carlos Mejía-MonasterioInstitute for Complex Systems, CNR, Florence Italy

http://calvino.polito.it/∼mejia/

http://calvino.polito.it/~mejia/http://calvino.polito.it/~mejia/


Nonequilibrium Statistical Mechanics

Thermodynamics of systems at equilibrium , relies on firmly establishedprinciples and phenomenological laws.

These laws are of empirical nature and rest on some statistical assumptions.

Nonequilibrium Thermodynamics , is far from being understood.

Given a particular classical, many-body Hamiltonian system, neither pheno-menological nor fundamental transport theory can predict whether or not thisspecific Hamiltonian system leads to realistic macroscopic transport.

http://calvino.polito.it/~mejia/



• What are the ingredients of the microscopic dynamics that lead to theobserved macroscopic transport?

• Given a microscopic mechanical model, is it possible to control themacroscopic transport in terms of a small set of parameters of the mi-croscopic dynamics?




• What are the ingredients of the microscopic dynamics that lead to theobserved macroscopic transport? NESS

• Given a microscopic mechanical model, is it possible to control themacroscopic transport in terms of a small set of parameters of the mi-croscopic dynamics? small systems

+ Mathematical modelling and numerical simulation of simple modelsystems.



from the Microscopic to the Macroscopic

Reversible Microscopic Dynamics.

q̇j =∂H

∂pj; ṗj = −

∂H

∂qj

Irreversible Macroscopic Transport.

Jn = Lnn∇ (µ/T ) + Lnu∇ (1/T )

Ju = Lun∇ (µ/T ) + Luu∇ (1/T )

Onsager reciprocity relations : microscopic reversibility ! macroscopicsymmetry of conjugated nonequilibrium processes.

Fluctuation-Dissipation Theorem : reversible fluctuations at equilibrium !irreversible dissipation occurring out of equilibrium.

Nonequilibrium Fluctuation Theorems : microscopic foundation for thesecond law of thermodynamics.



Small Systems

Molecular fluctuations play a fundamental role for transport processes(phase transitions, nucleation, chemical reactions, DNA mutations).

FT implies that dynamical fluctuations contrary to the thermodynamic forcesare likely to occur in small systems.

This has strong implications for the behaviour of nano and molecular machinesand even for living organisms

Nano-scales: size of the fluctuations are of the same order of the magnitude ofthe observables

L Rondoni and C. M-M, Nonlinearity 20, R1 (2007)



Small Systems

Chemically tunable nano-propellers

(nano-propeller)

Carbon nanotubes with attached aromatic blades.B Wang and P Král, PRL 98, 266102 (2007)


surf-pho.aviMedia File (video/avi)


Small Systems

Jarzynski equality〈e−βW

〉

A→B= eβ[F (B)−F (A)]

Reconstruction of free-energy landscapes of protein folding.

F. Ritort, Ad. Chem. Phys. 137, (2007, Ed. Wiley & Sons)



Fourier’s Law

��

��

Ju = −κ∇T ,

κ is the heat conductivity.Joseph Fourier (1822)



Fourier’s Law

Classical systems

H =N∑

i=1

(p2i

2mi+ U(qi) + V (qi+1 − qi)

)

+ bath’s coupling

The harmonic chain does not satisfies Fourier’s law.Z. Rieder, J. L. Lebowitz and E. Lieb, J. Math. Phys. 8, 1073 (1967).

FPU chain shows anomalous transport.S. Lepri, R. Livi and A. Politi, Phys. Rep. 377, 1 (2003)



Fourier’s Law

Rotating Lorentz gas

ωα

α’

ξC-

ξC+

ξH+

ξH-

Genuine many-body interacting particle system.• Local Thermal Equilibrium• Normal transport of heat and matter• Onsager reciprocity relations• Green-Kubo formulas

C. M-M, H. Larralde and F. Leyvraz, PRL 86, 5417 (2001)H. Larralde, F. Leyvraz and C. M-M, JSP 113, 197 (2003)J-P Eckmann, C. M-M, and E. Zabey, JSP 123, 1339 (2006)J-P Eckmann and C. M-M, PRL 97, 094301 (2006)C. M-M and L. Rondoni, JSP (2008)



Fourier’s Law

Classical systems

For systems with no globally conserved quantities (globally ergodic),positive Lyapunov exponents (chaos) is, “in general”, a sufficient con-dition to ensure macroscopic transport.

C. Casati and C. M-M, AIP Conf. Proc. 965, 221 (2007)



Fourier’s Law

Quantum systems TL TR

H =N−2∑

n=0

Hn +h

2(σL + σR)

︸︷︷︸

coupling

.

Diffusive vs ballistic behaviour and thermal conductivity in low dimen-sional magnetic systems is a long standing a controversial issue.

σ(ω) = 2πDδ(ω) + σreg(ω) ,

σreg(ω > 0) =1 − e−βω

ωLRe

∫∞

0

dteiωt〈j(t)j(0)〉 .

D is the so-called thermal Drude coefficient



Fourier’s Law

Quantum systems TL TR

H =N−2∑

n=0

Hn +h

2(σL + σR)

︸︷︷︸

coupling

.

• Stochastic quantum bathsC. M-M, T. Prosen and G. Casati, EPL 72, 520 (2005)

• Quantum Master Equation in Lindblad form (Monte Carlo Wave Function)C. M-M and H. Wichterich, Eur. Phys. J. ST, 151, 113 (2007)

Fourier’s law sets in at the transition to quantum chaos



Thermal Rectification




A thermal rectifier is the analogue of an electric diode: Is a device with theability to carry the energy flow in one preferred direction.





��

��





Dynamical control of the transmission probability (pt;LR 6= pt;RL).

• Completely new technological devices that take advantage of heat flow.• Heat pumps (highly efficient cooling micro-devices).• Nano-technology engineering.• Micro-devices to control heat in chemical reactions.• Control of energy flow in bio-molecules.




First theoretical mechanism M. Terraneo, M. Peyrard and G. Casati, PRL 88,094302 (2002).

H =N∑

i=1

p2i2mi

+ Ui(qi) +K

2(qi − qi−1)2 ,

whereUi(qi) = Di

(e−αiqi − 1

)2.

Model for DNA denaturationM. Peyrard and A. R. Bishop, PRL 62, 2755 (1989)




weaklyanharmonic

weaklyanharmonic

nonlinear

Tuning of the nonlinearity switches between overlap and no overlap of theeffective “phonon bands”.

∆ =max{|J+|, |J−|}min{|J+|, |J−|} ≈ 2.




Solid State Thermal Rectifier : C W Chang, D Okawa, A Majumdar and A Zett;Science 314, 1121 (2006)



Thermal Rectification in billiards

Interaction induced thermal rectifier

j<q<

j>

Rs

jL,qLq>

jR,qRδ

(-) (+)

Jn =tnLγ> − tnRγ<1 − α+Lα−R

; Ju =tuLε> − tuRε<1 − β+L β−R

,

tn and tu are the particle and energy transmission probabilities.γ and ε are the particle and energy injection rates.

J.-P. Eckmann and C. M-M, PRL 97, 094301 (2006)




10-2

10-1

100

101

102

103

interaction

68.5

69

69.5

70

α G ;

β G

-1 -0.5 0 0.5 1(γ

> - γ

<)/(γ

> + γ

<)

-3

-2

-1

0

1

2

3

103

ε ; ν

a b





For certain temperature gradients, the system becomes insulating:

(1 − β−L )(1 − β+L )(1 − β−R )(1 − β+R)

=ε<ε>

.





Magnetically induced thermal rectifier

TL TRR

λ

- fast particles of velocity v > vc, always enter theright cell, and thus contribute to the left to rightenergy flow.

- Instead, slow particles of velocity v < vc, suchthat the gyro-magnetic radius ρ(v) = mv/(eB) isless than λ/2 will be mostly reflected.

- Critical temperature:

Tc =(eBcλ)

2

8mkB,

G. Casati, C. Mejia-Monasterio, T. Prosen; PRL 98, 104302 (2007)




Magnetically induced thermal rectifier

Let J+ be the heat current if TL < TR and J− if TL > TR.

TL < TR

For a closed billiard J+ ∝ p+t ∼ 2ρ(v)λ .A particle with velocity v is transmitted if it crosses the interface at adistance from the upper boundary shorter than 2ρ(v).Thus, J+ ∝ 2

√2mkBτmin/eBλ

TL > TRp−t = 1. Thus J

− ∝ 1.

∆ = p−t /p+t ∝

1√Tmin

.





10-7

10-5

10-3

10-1

101

τmin

10-1

100

101

102

103

104

∆(τ

min)

Arbitrarily large rectifications!





Two coupled QD with different magnetic properties.

Diluted 2DEG:

• λ = 100nm• B = 1T• Tc ∼ 0.5K• Setting Tmin ∼ 10−3K and Tmax ∼ 10K one obtains ∆ ∼ 10.




Quantum Dot as a Thermal Rectifier : R Scheiber, M König, D Reuter, A.D.Wieck, H Buhmann and L.W. Molenkamp; arXiv:cond-mat/0703514v1



Thermoelectricity

Thermoelectricity concerns the conversion of temperature differences intoelectric potential or vice-versa.

It can be used to perform useful electrical work or to pump heat from cold to hotplace, thus performing refrigeration.



Thermoelectricity in Billiards

TH

TC

V




Ju = −κ′∇T − TσS∇φ ,Je = −σS∇T − σ∇φ ,

Je = eJρ is the electric current,

E ≡ −∇φ is the electric field,σ is the electric conductivity,

S = E/∇T when Je = 0 is the Seebeck coefficient andκ = κ′ − TσS2 is the thermal conductivity.




Jn = Lnn∇ (µ/T ) + Lnu∇ (1/T )Ju = Lun∇ (µ/T ) + Luu∇ (1/T )

For ergodic gases of noninteracting particles the so-called TE figure-of-meritZT is

ZT =L2undet L

,

where

η = ηcarnot ·√

ZT + 1 − 1√ZT + 1 + 1

,

Therefore, the Carnot’s limit ZT = ∞ is reached if the Onsager matrix issingular det L = 0.

G. Casati, C. M-M and T. Prosen, PRL (2008)



Diluted Polyatomic Ideal Gas

In the context of classical physics this happens for instance in the limit oflarge number of internal degrees of freedom, provided the dynamics isergodic.

Consider an ergodic gas of non-interacting particles with Dint internal

degrees of freedom enclosed in a D dimensional container, d = D + Dint.Then

Jn = t (γ> − γ − ε






Jn =λptL

(2πm)1/2

(d + 1

2ρT 3/2∇

(1

T

)

+ ρT 1/2∇(

−µT

))

Ju =d + 1

2

λptL

(2πm)1/2

(d + 3

2ρT 5/2∇

(1

T

)

+ ρT 3/2∇(

−µT

))







ZT =d + 1

2, d = D + Dint

e.g. ZT = 2 for dilute mono-atomic gas in 3 dimensions.

ZT is independent of the sample size L.



Polyatomic Lorentz gas

TL , µ

LT

R , µ

R





0 10 20 30 40 50d

10-2

100

102

104

106

108

Lab

Luu

Luρ

Lρρ





0 10 20 30d

0

5

10

15

20

ZT 101 102L

0

5

10

15

20

ZT




Final Remarks

We have discussed the problem of heat conduction in classical and quantumsystems.

We have shown, different microscopic mechanisms by which, the heat flow canbe controlled.

High thermal rectification and high thermoelectric efficiency can be observed insimple mechanical systems.

Such phenomena are also observed in quantum magnetic systems.

Experimental prototypes at nano scales are possible.

Whether biomolecules exploit rectification of heat is still a completely openproblem.


Nonequilibrium Statistical MechanicsNonequilibrium Statistical MechanicsNonequilibrium Statistical Mechanics

from the Microscopic to the MacroscopicSmall SystemsSmall SystemsSmall SystemsFourier's LawFourier's LawFourier's LawFourier's LawFourier's LawFourier's Law

Thermal RectificationThermal RectificationThermal RectificationThermal RectificationThermal Rectification

Thermal RectificationThermal RectificationThermal RectificationThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermal Rectification in billiardsThermoelectricityThermoelectricity in BilliardsThermoelectricity in Billiards

Thermoelectricity in BilliardsDiluted Polyatomic Ideal GasDiluted Polyatomic Ideal GasDiluted Polyatomic Ideal Gas

Polyatomic Lorentz gasPolyatomic Lorentz gasPolyatomic Lorentz gasFinal Remarks

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