heat transport in low-dimensional systems with asymmetric...
TRANSCRIPT
-
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
Results: (3) Spectrum analysis
Slop~Slop~--0.30.3
Slop~Slop~--0.170.17
Slop~Slop~--0.080.080~
modelβFPUfor~)(
modelβFPUfor~)(
0
0.3
S
S
-
FPU model:
Results: (3) Spectrum analysis
)(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
Discussion: mechanism
-
Discussion: mechanism
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
Discussion: mechanism
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)
Discussion: mechanism
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
-
CrossCross--correlation between jq(0,0) and jm(x,t) correlation between jq(0,0) and jm(x,t)
Discussion: mechanism
-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
Local energy current Local energy current
Local mass currentLocal mass current
-
CrossCross--correlation between jq(0,0) and jm(x,t) correlation between jq(0,0) and jm(x,t)
Discussion: mechanism
-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
Local energy current Local energy current
Local mass currentLocal mass current
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
1d gas FPU 1d gas FPU 4 LatticeLattice
HeatHeat
EnergyEnergy
MomentumMomentum
Mass densityMass density
(t=300)(t=300)
-
DiffusionDiffusion distribution functions
1d gas FPU 1d gas FPU 4 LatticeLattice
HeatHeat
EnergyEnergy
MomentumMomentum
Mass densityMass density
(t=300)(t=300)
-
DiffusionDiffusion distribution functions
1d gas FPU 1d gas FPU 4 LatticeLattice
HeatHeat
EnergyEnergy
MomentumMomentum
Mass densityMass density
(t=300)(t=300)
S. Chen, Y. Zhang, J. Wang and H. Zhao, arXiv:1106.2896v2
-
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
S. Chen, Y. Zhang, J. Wang and H. Zhao, arXiv:1106.2896v2
-
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)
Discussion: connection to heat conduction
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
Discussion: connection to heat conduction
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
S. Chen, Y. Zhang, J. Wang and H. Zhao, arXiv:1106.2896v2
-
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
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o d e m o
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