heat transport in low-dimensional systems with asymmetric...

2371-14

Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries

Hong ZHAO

15 - 24 October 2012

Dept.of Physics, Inst.of Theoretical Physics & Astrophysics, Xiamen University People's Republic of China

Heat Transport in Low-dimensional Systems with Asymmetric Inter-particle Interaction (Asymmetric Heat Conduction and Negative Differential Thermal

Resistance in Nonlinear Systems)

Transport and relaxation in one-dimensional models

Hong Zhao Department of Physics,Institute of Theoretical Physics and Astrophysics，Xiamen University.

Advanced Workshop on Energy Transport—Trieste - Italy, 15 - 24 October 2012

Hong ZhaoJiao WangYong ZhangDahai He

Shunda ChenYi Zhong Daxing Xiong

Complex Systems Group, Xiamen University:

Recent works of our group

1. Breakdown of the power-law decay of current-current correlation in 1D lattices (Phys. Rev. E 85, 060102(R) (2012); S. Chen et.al., arXiv:1204.5933)

2. Diffusion of heat, energy, momentum and mass in 1D systems (S. Chen et.al., arXiv:1106.2896))

3. Non-universal heat conduction of 1D momentum conserving lattices ( D. Xiong et.al., PRE 85, 020102(R) (2012))

4. How to correct finite-size effects in calculating the current-current correlation ( S. Chen, et. al., arXiv:1208.0888)

Background & Problem ModelsResultsPossible mechanism

Outline

FluctuationFluctuation--DissipationDissipation--TheoryTheory

b

baba FLJ

0)(

21limlim dttCL abLab

Onsager coefficients can be calculated Onsager coefficients can be calculated by the Greenby the Green--Kubo formula: Kubo formula:

Where, Where,

Background

)0()()( baab JtJtC

MvJxxVxJ

m

i

iiq

)(

mmmqmqm

mqmqqqq

FLFLJ

FLFLJ

Background

Onsager reciprocal relation:Onsager reciprocal relation:

But in what conditions we haveBut in what conditions we have

Problem I:

baab LL

？0 baab LL

The conventional hydrodynamic approach assumes that the current correlation decay rapidly (i.e., exponentially) as

which guarantees the convergence of the G-K formula

However, after Alder (1970) numerically evidenced the ‘long time tail’

the G-K formula encounter the problem of divergence:

~ .abL const

Background

( ) ~ tabC t e

~abL L

-( ) ~ ,abC t t

0

1lim lim ( )2ab L ab

L C t dt

In recent decades, the hydrodynamic analysis has been extended to lattice systems. At present, for momentum-conserving 1D fluids and lattices, it is actually believed that the current correlation should decay in the power-law manner, and resulting in a divergent thermal conductivity as

Background

~ ,L

though some counterexamples with size-independent thermal conductivities have been found

Counterexamples: the rotator model [PRL. 84, 2144 (2000); PRL, 84, 2381 (2000)], a 1D lattice in effective magnetic fields [J. Stat. Mech. P05009 (2005)], the variant ding-a-ling model [PRE 82, 061118 (2010)],lattice models with asymmetric inter-particle interactions [PRE 85, 060102(R) (2012)]

Does really the power-law decay so common inmomentum-conserving 1D systems?

Problem II:

ttCoretC abt

ab ~)(~)(

Which one should be more general in real materials?

Models: Lattices with asymmetric inter-particle interactions

with

(c) ( ) [( ) 2( ) 1] L-J modelm nc cc c

x xV xx x x x

2 3 41 1(b) ( ) FPU- model2 3 4

V x x x x

21(a) ( ) ( )2

rxV x x r e

2

1( )2i

i ii

pH V q q am

Models

21(a) ( ) ( )2

rxV x x r e

Thermal expansion null thermal expansion negative thermal expansion

Models2 3 41 1( ) ( ) FPU model

2 3 4b V x x x x

-0.5 0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20V(

x)

x

=0 =1 =1.5 =1.75 =2

Temperature~0.1Temperature~0.1

Models

(c) ( ) [( ) 2( ) 1] L-J modelm nc cc c

x xV xx x x x

0 2 4 6

0.0

0.5

1.0

(12,6) (6,3) (4,2) (2,1)

Vx

x Temperature~0.5Temperature~0.5

Methods:

( ) ~ cost. ( ) ~ cos .( 0)

(

exponential decay:

power-law decay) ( ) ~

:~

tJJ

JJ

C t e S t

C t t N S

Note: the heat flux is equal to the energy flux in lattices

q e ii i

HJ J xx

Livi Prof. ：

[S. Lepri, R. Livi, A. Politi, Physics Reports 377, 1 (2003)]

)(;);( StcJJ

Results: (1) Equilibrium simulation

An overview for all of the three modelsAn overview for all of the three models

100 101 102 103

10-2

10-1

100C

JJ(t)

/ C

JJ (

0)

t

FPU- LJ FPU- - Exp+Harmonic

N=4096N=4096

Results: (1) Equilibrium simulation

for model (a) with r=1.5:

0 1000 2000 3000

10-2

10-1

100

100 1000

0.01

0.1

1

(b)

CJJ

(t) /

CJJ

(0)

L=512 L=1024 L=2048

t

L=512 L=1024 L=2048

t

(a)

Results: (1) Equilibrium simulationfor model (b) and (c)

N=4096

Reliable time range Reliable time range is t

T JLJx T

Results: (2) Nonequilibrium simulation

Y. Zhong et al, PRE 85, 060102(R) (2012)

Results: (2) Nonequilibrium simulation

Model (b)

Model (c)

Model (a) α=2

(m,n)=(12,6)

S. Chen et.al., arXiv:1204.5933

Pow

er s

pect

rum

Po

wer

spe

ctru

m

FrequencyFrequency

Uniform noiseUniform noise

Random walk Random walk processprocess

Results: (3) Spectrum analysis

0~)( 2

S

0~)(

cS

)(log 10

Prof. Livi: ,)0()( LS

FPU model


Slop~Slop~--0.30.3

Slop~Slop~--0.170.17

Slop~Slop~--0.080.080~

modelβFPUfor~)(

modelβFPUfor~)(

0

0.3

S

S

FPU model:


)(log10 )(log10

Correlation lengthCorrelation lengthRandom noiseRandom noise

int, 0, no thermal expensionp vc c P

Symmetry of inter-particle interactions is essential in determining physical properties of systems

Asymmetric potential:

Symmetric potential:int, 0, thermal expensivep vc c P

2

34

0

Bgx k Tc

x

2 3 4U( x) =cx - gx - f x

g 0：

g=0：

Discussion: mechanism

C. Kittel, Introduction to Solid State Physics (7ed., Wiley, 1996), p130.C. Kittel, Introduction to Solid State Physics (7ed., Wiley, 1996), p130.

Nonequilibrium case: Thermal expansion can induce Mass gradient, which may provide additional scattering to the flux

thermal expansion.

negative thermal expansion.

symmetry potential: no thermal expansion.

Phonon scattering + massPhonon scattering + mass--gradient scatteringgradient scattering



Equilibrium case: Thermal expansion can induce thecoupling between heat currentheat current and mass currentmass current

i

qiq jJ

)()()(

)0()()(

)0()()(

tvtMtJ

JtJtC

JtJtC

Mm

qmmq

mqqm

1. Onsager reciprocal relations

2. Symmetry interaction

3. Asymmetry interaction 0

0

mqqm

mqqm

mqqm

LL

LL

LL


Coupling between the energy and mass currents Coupling between the energy and mass currents

The asymmetry induce the coupling between the local currents of heat and massThe asymmetry induce the coupling between the local currents of heat and massS. Chen et.al., arXiv:1204.5933 (2012)

0 50 100

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

(b)

corr

elat

ion

t

(a)

Chen-fig.3FPU model:

2

CrossCross--correlation between jq(0,0) and jm(x,t) correlation between jq(0,0) and jm(x,t)


t=0t=0

t=tt=t

Local energy current Local energy current

Local mass currentLocal mass current

( , ) (0,0) ( , )q mc x t j j x t

--spatiotemporal cross correlation betweenthe fluctuations of local energy currents and mass currents



-2 0 0 -1 0 0 0 1 0 0 2 0 0

0 .0 0

0 .0 1

0 .0 2

0 .0 3

t= 8 0G a s m o d e l

x

t=0t=0

t=tt=t





-2 0 0 -1 0 0 0 1 0 0 2 0 0

0 .0 0

0 .0 1

0 .0 2

0 .0 3

G a s

F P U - -

x

t=0t=0

t=tt=t



Additional scattering is induced by the lattice featureAdditional scattering is induced by the lattice feature

Summary: resultsFor momentum conserving 1d lattice with proper degree ofasymmetry:

(1) while in the symmetric case

(2) for the heat conductivity:

(a) Equilibrium simulation: The current-current correlation decays faster than the power law;

(b) Nonequilibrium simulation: The thermal conductivity converges in the thermodynamic limit.

(c) Power spectrum analysis: A size-independent heat conductivity

0qm mqL L 0 mqqm LL

Summary: Mechanism

fluids: mass-current couplingLattice with asymmetric interactionsMass-current coupling + lattice scattering + nonlinear

interactions Lattice with symmetric interactionslattice scattering+ nonlinear interactionsAlso guesses! We are trying to find more details….

fluidsfordecaylawpowerlatticesfordecayrapid

asymmetry

decaylawpowersymmetry

a guess

The transport theories of hydrodynamics andkinetics may apply to

momentum conserving fluids & momentum conserving lattice with symmetric interactions

but may not apply tomomentum conserving lattice with asymmetricinteractions

Three Puzzles

(1) Why it deviates the hydrodynamic prediction?

(2) Does there exist a transition from the power-law decay to the rapid decay with the increase of the asymmetry? — open question

FPU model, 1 1, 1 power-law decay1, 2 exponential decay0.1, 1 exponential decay0.1, 2 exponential decay

TTTT

-0.5 0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20

V(x

)

x

=0 =1 =1.5 =1.75 =2

0 2 4 6

0.0

0.5

1.0

(12,6) (6,3) (4,2) (2,1)

Vx

x

-2 0 20

1

2

3

V(x)

x

(3) The convergence length ?

If, say for graphene , the potential is nearIf, say for graphene , the potential is near--symmetry, the symmetry, the convergence length should be quit longconvergence length should be quit long

A big guess

Breather physics

Soliton physics

Phonon physics

KAM regimeKAM regime

Hydrodynamic Hydrodynamic regime?regime?

TemperatureTemperature

Thanks for your attention!

Finally, I would like to thanks profs. S. Lepri, R. Livi, A. Politi for helpful discuss. They pushed us keeping studying the mechanism of the observed phenomena

Part II. Diffusion of heat, energy, momentum and mass in 1D systems

1.Problem & Background

2.Methods

3.Models

4.Results & Discussions

In equilibrium systems, a perturbation will induce fluctuations of energy, mass, momentum, heat , local flux and etc.

How does a local fluctuation spread or diffuse into other parts of the system?

Question:

Hydrodynamic prediction for the problemHydrodynamic prediction for the problem

Heat modeHeat mode

Sound modeSound modeSound modeSound mode

JP Hansen, IR McDonald, Theory of Simple Liquids, 3rd edition, 2006.

Our aim

Temperature fluctuationsTemperature fluctuations

Explore the diffusion or relax behavior of local deviations by direct simulationby direct simulation

Background： Distribution function and diffusion classification

Probability distribution function (PDF) ρ(r,t)

ttrdrtrrtr ~)(),()( 222

motionballisticnerdiffusiodiffusionnormalonsubdiffusi

,2,sup,1,1,,1

Diffusion can represent not only the diffusive motion but also regular motion

Methods

Methods:Nonequilibrium extraction

)0,0()0,0(~ AAA Add a kick

B. Li and J. Wang, PRL 91 (2003);P. Cipriani, S. Denisov, and A. Politi PRL 94 (2005)

)0,0()0,0(~)0,0(),(~),(

AAAtrAtr

Temperature TTemperature T

Methods:Equilibrium extraction

drrAA

trAAtrsystemCanonical a )0,()0,0(),()0,0(),(:

TT

11

)0,()0,0(),()0,0(),(:

NdrrAA

trAAtrsystemicalMicrocanon a

H. Zhao, PRL 96, (2006);P.Huang and H. Zhao, arXiv:1106.2866v1;S. Chen, Y. Zhang, J. Wang and H. Zhao, arXiv:1106.2896v2

∆A(0,0)

∆A(0,0) Temperature TTemperature T

drrAAtrAA

AAtrAtrA

)0,()0,0(),()0,0(

)0,0()0,0(~),(),(~

if the Fluctuation–Dissipation Relation (FDR) remains correct:

Equilibrium and nonequilibrium methods are equivalent

2 FPU lattice

3 Lattice model4

41

21

1

2

)(4

)(21

2axxaxx

mpH iiii

N

i i

i

421

1

2

)(4

)(21

2ixaxx

mpH iii

N

i i

i

1 1D gas with alternating masses

Models

Results

An example: the distribution function of massAn example: the distribution function of mass--density fluctuations (The dynamic structure factor) density fluctuations (The dynamic structure factor) of the 1d gas at t=300of the 1d gas at t=300

(0,0)m

2a rea o f cen tr t p ea kL a n d a u P la czek ra tioa rea o f s id e p ea ks

S. Chen, Y. Zhang, J. Wang and H. Zhao, arXiv:1106.2896v2

11

)0,()0,0(),()0,0(),(

Ndrrmm

trmmtrm

Temperature T=2Temperature T=2

Diffusion distribution functions

1d gas FPU 1d gas FPU 4 LatticeLattice

HeatHeat

EnergyEnergy

MomentumMomentum

Mass densityMass density

(t=300)(t=300)

DiffusionDiffusion distribution functions


HeatHeat

EnergyEnergy

MomentumMomentum


(t=300)(t=300)

Summary for the gas model

(1) Sound mode carries the total momentum, energy, and 1/3 mass-density fluctuations and spreads outwards at the sound speed

(2) Heat mode diffuses superdiffusively

(3) Energy and heat are separated completely

Summary for the FPU model

(1) Sound mode carries the total momentum, total mass-density, partial energy fluctuations

(2) Heat mode diffuses superdiffusively, its PDF shows a three-peak structure in a transient process, but asymptotically it will evolve into single-peak structure one

(3) The PDF of the energy always keep the three-peak structure.

Summary for the

(1) Sound mode disappears;

(2) The diffusion behavior of energy and heat are identical, they diffuse normally.

4 Lattice

Discussion: the heat and energy diffusion in Discussion: the heat and energy diffusion in the FPU latticethe FPU lattice

33.1~ th 0.5~h t


Cross correlation between Q-P and E-P

Area conservedArea conserved

Discussion: Discussion: the heat and energy diffusion in the heat and energy diffusion in the FPU latticethe FPU lattice

Discussion: What the sideDiscussion: What the side--peaks on are?peaks on are?

Move at a supersonic speed and decay in power-law;

They may be the long lived heat waves.

),( txQ

Discussion: Heat wave in shortDiscussion: Heat wave in short--time scaletime scale

4 Lattice

Moves at a 2/3 sound speed, decays exponentially.

Can we establish the universal connection between heat conduct and energy diffusion?

No!No!

(1) (1) By definition, they are different quantities

Discussion: connection to heat conduction

(2) Practically, we have shown that energy diffusion mayhave any connection to that of heat in the case of the gas model

Tow formula connecting the exponents are presented

L~

It has been found that the heat conductivity of 1D momentum conserved systems goes as:

It has been declared that the energy in these systems diffuse as: 2( ) ~x t t

/221

P. Cipriani, S. Denisov and A. Politi, Phys. Rev. Lett. 94, (2005); B. Li, J. Wang, L. Wang and G. Zhang, Chaos 15, (2005); H. Zhao, Phys. Rev. Lett. 96, (2006); L. Delfini, at.al, Eur. Phys. J. Special Topics 146, (2007)


B. Li and J. Wang, Phys. Rev. Lett. 91, (2003)

S. Denisov et. al. Phys. Rev. Lett. 91, (2003

4

2( ) ~x t t

1d gas FPU1d gas FPU


Connection between the heat diffusion and heat conductivity may exist

LatticeLattice4

( , )Q x t x xtThe function is invariant upon rescaling

1d gas FPU1d gas FPU LatticeLattice4

ββ =2=2λλ ==1.176, 1.201,1.000

/221

√

Discussion: connection to heat conduct


T T

T

T

Discussion: TwoDiscussion: Two--dimensional systems: dimensional systems:

2D Gas with L-J potential

Diffusion of a particle

Diffusion of the energy and momentum

Discussion: TwoDiscussion: Two--dimensional systems: dimensional systems:

J.H. Yang et al, PRE 83, 052104 (2011)

Check of the FDR

A: equilibrium method

B &C: nonequilibrium method, B: δH=5E C: δH=2E

P Hwang and H Zhao, arXiv:1106.2866v1;

100 200 300 400 500

0.025

0.030

0.035he

at &

ene

rgy

curre

nt

t

d e m o d e m o d e m o d e m o d e m o






The heat and energy flux in 1D gas modelThe heat and energy flux in 1D gas model

Discussion: diffusion of the local-flux fluctuations and how to approach the autocorrelation function of the global flux

1d gas FPU1d gas FPU LatticeLattice4

The area decay with time

Diffusion:

The diffusion of local-flux fluctuations in systems with periodic boundary conditions

1d gas FPU1d gas FPU

The area decays as a function of time

Diffusion:

Periodic behavior of the autocorrelation function of the global flux.

Diffusion:

The autocorrelation function of the global flux is reliable only for

Diffusion:

vlLt

2

L:system size, l: the width of the peaks, v: the speed of the peaks

heat transport in low-dimensional systems with asymmetric...

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