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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/
Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 2/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 3/32
Nonequilibrium Statistical Mechanics
• 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?
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 3/32
Nonequilibrium Statistical Mechanics
• 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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 4/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 5/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 6/32
Small Systems
Chemically tunable nano-propellers
(nano-propeller)
Carbon nanotubes with attached aromatic blades.B Wang and P Král, PRL 98, 266102 (2007)
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surf-pho.aviMedia File (video/avi)
Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 7/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 8/32
Fourier’s Law
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Ju = −κ∇T ,
κ is the heat conductivity.Joseph Fourier (1822)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 9/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 10/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 11/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 12/32
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
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 12/32
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
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 13/32
Thermal Rectification
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 13/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 13/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 13/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 13/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 14/32
Thermal Rectification
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 15/32
Thermal Rectification
weaklyanharmonic
weaklyanharmonic
nonlinear
Tuning of the nonlinearity switches between overlap and no overlap of theeffective “phonon bands”.
∆ =max{|J+|, |J−|}min{|J+|, |J−|} ≈ 2.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 16/32
Thermal Rectification
Solid State Thermal Rectifier : C W Chang, D Okawa, A Majumdar and A Zett;Science 314, 1121 (2006)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 17/32
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 18/32
Thermal Rectification in billiards
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
J.-P. Eckmann and C. M-M, PRL 97, 094301 (2006)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 19/32
Thermal Rectification in billiards
For certain temperature gradients, the system becomes insulating:
(1 − β−L )(1 − β+L )(1 − β−R )(1 − β+R)
=ε<ε>
.
J.-P. Eckmann and C. M-M, PRL 97, 094301 (2006)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 20/32
Thermal Rectification in billiards
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 21/32
Thermal Rectification in billiards
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
.
G. Casati, C. Mejia-Monasterio, T. Prosen; PRL 98, 104302 (2007)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 22/32
Thermal Rectification in billiards
10-7
10-5
10-3
10-1
101
τmin
10-1
100
101
102
103
104
∆(τ
min)
Arbitrarily large rectifications!
G. Casati, C. Mejia-Monasterio, T. Prosen; PRL 98, 104302 (2007)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 23/32
Thermal Rectification in billiards
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 24/32
Thermal Rectification in billiards
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
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 25/32
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 26/32
Thermoelectricity in Billiards
TH
TC
V
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 26/32
Thermoelectricity in Billiards
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 27/32
Thermoelectricity in Billiards
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)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 28/32
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
Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 28/32
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 =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
))
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 28/32
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
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.
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 29/32
Polyatomic Lorentz gas
TL , µ
LT
R , µ
R
G. Casati, C. M-M and T. Prosen, PRL (2008)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 30/32
Polyatomic Lorentz gas
0 10 20 30 40 50d
10-2
100
102
104
106
108
Lab
Luu
Luρ
Lρρ
G. Casati, C. M-M and T. Prosen, PRL (2008)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 31/32
Polyatomic Lorentz gas
0 10 20 30d
0
5
10
15
20
ZT 101 102L
0
5
10
15
20
ZT
G. Casati, C. M-M and T. Prosen, PRL (2008)
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Carlos Mejía-Monasterio, May 19, 2008 Control of Heat Flow - p. 32/32
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.
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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