Thermal transport in classical nonlinear lattices · 2018. 9. 17. · I Spatial constraints can signi cantly alter transport properties: long{time tails and breakdown of hydrodynamics

Thermal transport in classical nonlinear lattices

Stefano Lepri

Istituto dei Sistemi Complessi ISC-CNR Firenze

[email protected]

Stefano Lepri (ISC-CNR) 1 / 48

Introduction

Nonequilibrium statistical mechanics of open, many-particles systems.

Steady states, derivation of phenomenological laws (Fick, Fourier)

Our favourite example: energy transport in low-dimensional lattices:

I Minimal, nonperturbative model for nonequilibrium states[Rieder, Lebowitz, Lieb, 1967; Casati, Ford, Vivaldi and Wisscher, 1984 ...]

I Spatial constraints can significantly alter transport properties:long–time tails and breakdown of hydrodynamics

I Beyond phonons: the role of nonlinear excitationsI Low-dimensional materials (nanotubes, polymers ...) modeling

nanoscale heat transfer


Introduction









Introduction









Introduction









Introduction









Introduction






I Beyond phonons: the role of nonlinear excitations

I Low-dimensional materials (nanotubes, polymers ...) modelingnanoscale heat transfer


Introduction









Bibliography


Experimental situations

Nanotube/Nanowire

heater

thermal contact

substrate

Cantilever

heater

molecular chains


Classical nonlinear oscillators on a lattice

Chain of N coupled oscillators with n.n. coupling: (pl = mlxl)

H =

N∑l=1

[p2l

2ml+ V (xl+1 − xl)

].

Equations of motion :

mnxn = −Fn + Fn−1 ; Fn = −V ′(xn+1 − xn) ,

Boundary conditions e.g. periodic xN+1 = x0 + L.

Displacement from eq. position xn = na+ un, L = Na chain length

Symmetry un → un + cst: momentum conservation,∑

l pl const.

”Acoustic dispersion” in the harmonic limit


Conservation laws

L =

N∑n=1

(xn+1 − xn) ≡N∑n=1

rn

P =∑N=1

mxn ≡N∑n=1

pn

E =

N∑n=1

[p2n

2mn+ V (rn)

]≡

N∑n=1

en

Microcanonical equilibrium: (L,P,E) (usually P = 0)

Local conservaton laws of densities (rn, pn, en)

Local currents


Example 1: the Fermi-Pasta-Ulam (FPU) model

V (z) =g2

2(z − a)2 +

g3

3(z − a)3 +

g4

4(z − a)4

Historical namesg3 = 0: ”FPU-β modelg3 6= 0: ”FPU-α+ β model


Example 2: The Toda chain

V (z) = exp(−z),

N integrals of motion, rl = xl+1 − xl

Q1 =

N∑l=1

pl

Q2 =

N∑l=1

p2l

2+ e−rl

Q3 =

N∑l=1

p3l

3+ (pl + pl+1)e−rl

Q4 =N∑l=1

p4l

4+ (p2

l + plpl+1 + p2l+1)e−rl +

1

2e−2rl + e−rle−rl+1


Example 3: The rotor model

Also Hamiltonian XY model:

θn = pn , pn = sin(θn+1 − θn)− sin(θn − θn−1) .

Important difference: θn are phases

Two conserved quantities: momentum and energy

Phase slips


Models with substrate (pinning) potential

H =

N∑l=1

[p2l

2ml+ U(xl) + V (xl+1 − xl)

]. (1)

The substrate potential U breaks the invariance xl → xl + const. thetotal momentum is no longer a constant of the motion. Accordingly,all branches of the dispersion relation have a gap at zerowavenumber. (optical dispersion).

Only one conserved quantity: energy


Example 1: discrete Klein-Gordon

H =

N∑l=1

[pl

2m

2+ U(ul) +

1

2C(ul+1 − ul)2

];

Frenkel-Kontorova potential

U(z) = −U0 cos

(2πz

a

).


Example 2: the Discrete Nonlinear Schrodinger equation

iψn = −ψn+1 − ψn−1 + ν|ψn|2ψn

”Generic model” in different fields: biomolecules, BEC, optics

Nonintegrable many-body problem

Nonlinear localized solution (discrete breathers)

Negative absolute temperatures

Finite-temperature coupled transport

[S. Iubini’s talk ....]


Simulating lattice heat transport

Equilibrium MD: correlation of the heat flux J

κGK =1

kBT 2dlimt→∞

∫ t

0dt lim

L→∞L−d〈J(t) · J(0)〉

Microscopic exp. for 1D lattice: J = a2

∑n(un+1 + un)V ′(un+1 − un)

Non-equilibrium MD: boundary thermostats

T+ T−

κ(L, T±) =J

(T+ − T−)/L

Deterministic (e.g Nose-Hoover) or stochastic (e.g Langevin) thermostats ...


Simulating lattice heat transport

Equilibrium MD: correlation of the heat flux J

κGK =1

kBT 2dlimt→∞

∫ t

0dt lim

L→∞L−d〈J(t) · J(0)〉

Microscopic exp. for 1D lattice: J = a2

∑n(un+1 + un)V ′(un+1 − un)

Non-equilibrium MD: boundary thermostats

T+ T−

κ(L, T±) =J

(T+ − T−)/L

Deterministic (e.g Nose-Hoover) or stochastic (e.g Langevin) thermostats ...


Anomalous transport in one dimension

RLL 1967

... For a physical system (anharmonic coupling) we wouldexpect j to reach a limiting value proportional to ∇T .

... The thermal conductivity diverges ..... indicating thatchaotic behavior is not enough to ensure the Fourier law.


Anomalous transport in one dimension

RLL 1967

... For a physical system (anharmonic coupling) we wouldexpect j to reach a limiting value proportional to ∇T .

... The thermal conductivity diverges ..... indicating thatchaotic behavior is not enough to ensure the Fourier law.


Evidences

In nonlinear, nonintegrable, momentum-conserving chains:

I Thermal conductivity diverges

II Current correlation have long-time tails

III Energy superdiffusion

IV Anomalous relaxation of fluctuations

V Temperature profiles are ”strange“


Anomalous transport in 1D (I)

Breakdown of Fourier’s law: diverging finite-size conductivity

κ(L) ∝ Lα

FPU g3 = 1, NH thermostats T+ = 0.21, T− = 0.19, free boundaries

101

102

103

L

101

102

κ(L)

L0.35


Anomalous transport in 1D (II)

Nonintegrable power-law decay at large t:

〈J(t)J(0)〉 ∝ t−(1+δ) (−1 < δ < 0)

Microcanonical FPU g3 = 1, N = 1024, periodic boundaries

10-4

10-2

100

ω10

-2

100

102

S(ω)

ω−0.33


Anomalous transport in 1D (III)

Superdiffusion of energy perturbations δe(x, t):

σ2(t) ∝ tβ

Microcanonical, diatomic hard-points gas, t = 40, 80, 160, 320, 640, 1280, 2560, 3840 γ = 3/5

Levy walk model: δe(x, t) = t−γf(x/tγ), β = 3− 1/γ [Cipriani et al. 2005]


Detour: Brownian versus anomalous diffusion


Levy Walks

The simplest model of superdiffusion with finite speeds:

Ballistic motion with speed ±v in between “collisions”

Path length distribution

ψ(l) ∝ l−µ−1

Superdiffusion:〈x2〉 ∼ tγ

γ =

2 0 < µ ≤ 1

3− µ 1 < µ < 2

1 µ ≥ 2


Propagator

PDF to find in x at time t, a particle initially at x = 0

P (x, t) = PL(x, t) + t1−µ[δ(x− vt) + δ(x+ vt)]

PL(x, t) ∝

t−1/µ exp

[−(ηx/t1/µ)2

]|x| . t1/µ

t x−µ−1 t1/µ . |x| < vt

0 |x| > vt


Propagator

x/t1/µ

log

P(x

,t)


Anomalous transport in 1D (IV)

“Fast” (i.e. superexponential) relaxation of spontaneous fluctuations

Microcanonical FPU model, dynamical structure factor S(q, ω)

-5 0 5

10-2

10-1

100

10-2

10-1

100

101

102

(ω − ωmax

)/γ(q)

10-6

10-5

10-4

10-3

10-2

10-1

100

S(q,ω

)q=0.049

x-7/3

Linewidths: γ(q) ∝ qz with z < 2 (e.g. FPU: z ≈ 1.5− 1.6).


Anomalous transport in 1D (V)

The temperature field is nonlinear also for small ∆T

0 0.2 0.4 0.6 0.8 1x

0

0.5

1T(x)

Fractional diffusion equation with sources:

DαxT = −σ(x)

σ ∝ x−32

Boundary conditions are very important!Stefano Lepri (ISC-CNR) 24 / 48

“Hyperscaling” among exponents?

How to compare the two equilibrium with nonequilibrium methodsCut–off in the Green-Kubo formula:

κ(L) ∝∫ L/vs

0〈J(t)J(0)〉 dt ∝ L−δ

vs propagation velocity of excitations (sound waves).

Consistency with linear response implies α = −δ, from continuity eq.β = α+ 1 etc.


How generic is the anomaly?

Observed for all momentum-conserving models (acoustic dispersion).Exception: the 1D XY potential V (x) = − cosx [Giardina et al., 2000]

0 500 1000

N

0

2

4

6

8

κ

0 2 4 61/T

100

101

102

103

104

105

κ , τ


What if un → un + cst is broken?

Normal conduction: κ <∞, fast correlation decayExample: the discrete φ4−model

U(x) =a

2x2 +

b

4x4 .

0 100 200 300

N

0.0

0.1

0.2

κ

0 2000 4000

N

0

200

400

600

800

κa b


Is it model-dependent?

Model Reference α −δ

FPU-β Lepri et al (1998) 0.37 0.37FPU-α Lepri (2000) . 0.44 -Diatomic FPU r=2 Vassalli (1999) 0.43 compatibleDiatomic Toda r=2 Hatano (1999) 0.35-0.37 0.35

Vassalli (1999) 0.39 compatibleDiatomic Toda r=8 Vassalli (1999) 0.44 compatibleDiatomic hard points Hatano (1999) 0.35 -

Grassberger et al. (2002) 0.32 0.34Pencase Deutsch et al. (2003) 0.34 -Random collision Deutsch et al. (2003) 0.33(1) compatible

SWNT (Tersoff) Maruyama (2002) 0.32 -


Two-dimensional lattices

Less developed theory! one may expect

Long tail t−1 (with log corrections?)

κ ∼ logL



0.0 0.2 0.4 0.6 0.8 1.00.5

1.0

1.5

purely quartic modelFPU model

FPU model

NY

(a)

1024 32 1024 64 2048 32 2048 64

T i

i/NX

(b)

XY

NX

[L. Wang, N.Li, P. Hanggi, 2016]Stefano Lepri (ISC-CNR) 30 / 48


100 101 102 103 104

10-4

10-3

10-2

10-1

512 512 1024 512 512 1024 1024 1024 -1

cX()

FPU latttice

100 101 102 103 104

10-3

10-2

10-1

512 512 1024 512 512 1024 1024 1024 -1

FPU latttice

cX()

100 101 102 103 10410-4

10-3

10-2

10-1

512 512 1024 512 512 1024 1024 1024 -1

(c)

(b)

cX()

purely quartic latttice

(a)

101 102 103 1042

4

6

101 103

10

L0.25

GK

L

512 512 1024 512 512 1024 1024 1024

(f)

FPU- lattice

GK(L

)

L

101 102 103 1040

10

20

30

40

512 512 1024 512 512 1024 1024 1024

FPU- lattice

GK(L

)

L

101 102 103 104

2

4

6

8

512 512 1024 512 512 1024 1024 1024

(e)

purely quartic lattice

GK(L

)

L

(d)

[L. Wang, N.Li, P. Hanggi, 2016]


The 2D hamiltonian XY model

H =∑r

p2r

2+∑〈r,r′〉

[1− cos(θr′ − θr)] ,

[Delfini et al., 2006]Stefano Lepri (ISC-CNR) 32 / 48

Universality and Theoretical approaches

1 Fluctuating hydrodynamics: random fields of deviations of theconserved quantities with respect to their stationary values. The roleof fluctuations is taken into account by renormalization group or somekind of self-consistent theory.

2 Mode-coupling theory: this is closely related to the above, as itamounts to solving (self-consistently) some approximate equations forthe correlation functions of the fluctuating random fields.

3 Kinetic theory : phonon transport by means of the Boltzmann(Peierls) equation.

4 Exact solution of specific models: the original microscopicHamiltonian dynamics is replaced by some suitable stochastic onewhich can be treated by probabilistic methods (e.g. Fokker-Planck).


Theoretical approaches

Dynamical RG [Narayan and Ramaswamy, 2002]

Fluctuating hydrodynamics of a 1D fluid α = 1/3

Mode–coupling equationsNonlinear coupling of sound modes

[Delfini et al. 2007, VanBeijeren 2012] α = 1/3

Fluctuating hydrodynamics [Spohn 2014]Connection with Kardar-Parisi-Zhang growth

α = 1/3

Universality of dynamical scaling!


Nonlinear fluctuating hydrodynamics

[Spohn,2014]

Effective dynamics of the 3 conserved quantities

rn = un+1 − un; un; en

Fluctuations around the equilibrium values

rn = `+ U1; un = U2; en = e+ U3

Write hydrodynamic equations up to second order forU = (U1, U2, U3)

U = −∂x [AU + UGU + ∂xCU +Bξ]

Coupled, noisy Burgers equation (or Kardar-Parisi-Zhang equations)


Nonlinear fluctuating hydrodynamics:predictions

Linear limit: two propagating sound modes and one diffusive heatmode

To leading order, the oppositely moving sound modes are decoupledfrom the heat mode and satisfy noisy Burgers equations. For the heatmode, the leading nonlinear correction is from the sound modes.

Universal dynamical exponents (again!)

Predictions for the scaling functions too e.g. compute the function hsuch that

S(q, ω) ∼ fKPZ((ω − ωmax)/q3/2)

Correlation of the heat mode is Levy-stable distribution function

Estimate of sub-leading terms

Confirmation of two universality classes


Modes

Hydrodynamic fluctuations of fields should be a combination of these.


Modes

sound modeheat mode

+ct-ct

~ t5/3

~ t2/3

Hydrodynamic parts of correlators should be a combination of these.


Numerical check: FPU displacement correlator

10-4

10-3

10-2

10-1

ω

10-4

10-2

100

102

104

106

108

S(k

,ω)

arb

. u

nit

s

-40 -20 0 20 40

(ω −ω max

) / λnum

k3/2

0

0.5

1

S(k

,ω)

/Sm

ax

k=0.0122k=0.0245k=0.0490KPZ


Numerical check: FPU energy correlator

0 10 20 30 40 50 60

ω/q5/3

0

0.2

0.4

0.6

0.8

1S

e(q,ω

) /S

e(q,0

)

e=0.1 α=0.1

Levy peak: Lorenzian with width q5/3


Numerical test: FPU αβ vs β

Effective exponent: δeff = d lnSd lnω

-12 -10 -8 -6

ln ν

-0.5

-0.4

-0.3

-0.2

0

δeff

g3=1


Numerical test: FPU αβ vs β

Effective exponent: δeff = d lnSd lnω

-12 -10 -8 -6

ln ν

-0.5

-0.4

-0.3

-0.2

0

δeff

g3=1

g3=0


Numerical check:energy flux correlator

δeff(ω) =d lnS

d lnω.

-5 -4 -3 -2 -1 0log

10ω

-2

-1.5

-1

-0.5

0

δeff

α=0.1

α=0.25

α=0.50

α=1.0

e=0.5

-5 -4 -3 -2 -1 0log

10ω

-2

-1.5

-1

-0.5

0

δeff

α=0.1

α=0.25

α=0.50

α=1.0

e=0.1

Why this difference despite the convincing KPZ scaling in the otherobservables? Yet an open issue ....


A solvable stochastic model

Deterministic dynamics and interaction with the baths

qn = pn

pn = ω2(qn+1 − 2qn + qn−1) + δn,1(ξ+ − λq1) + δn,N (ξ− − λqN )

ξ± independent Wiener processes, 〈ξ±〉 = 0, 〈ξ2±〉 = 2λkBT±

Fixed boundary conditions: q0 = qN+1

Random collisions: exchange momenta (with rate γ)

pi ←→ pi+1


A solvable stochastic model

Deterministic dynamics and interaction with the baths

qn = pn

pn = ω2(qn+1 − 2qn + qn−1) + δn,1(ξ+ − λq1) + δn,N (ξ− − λqN )

ξ± independent Wiener processes, 〈ξ±〉 = 0, 〈ξ2±〉 = 2λkBT±

Fixed boundary conditions: q0 = qN+1

Random collisions: exchange momenta (with rate γ)

pi ←→ pi+1


Master equation

Phase-space probability density P (x, t), xµ in (q1, . . . , qN , p1, . . . , pN )

∂P

∂t= (L + Lcol)P .


Master equation


∂P

∂t= (L + Lcol)P .

Deterministic forces and the coupling with the heat baths:

LP =∑µ,ν

[aµν

∂

∂xµ(xνP ) +

dµν2

∂2P

∂xµ∂xν

]

a =

(0 −I

ω2g + kI λr

), d =

(0 00 2λkBT (r + ηs)

)

rij = δij(δi1 + δiN ), sij = δij(δi1 − δiN ), gij = 2δij − δi+1,j − δi,j+1

T = (T+ + T−)/2, η = (T+ − T−)/T


Master equation


∂P

∂t= (L + Lcol)P .

Collisional term

LcolP = γ∑i

[P (. . . pi+1, pi . . .)− P (. . . pi, pi+1 . . .)]


Covariance matrix

In terms of N×N blocks:

uij = 〈qiqj〉, vij = 〈pipj〉, zij = 〈qipj〉

c ≡(u zz† v

)Evolution equation:

c = d− ac− ca† − γ(

0 zggz† w

)

wij ≡

vi+1j + vi−1j + vij−1 + vij+1 − 4vij |i− j| > 1

vi±1j + vij∓1 − 2vij i− j = ±1

vi−1j−1 + vi+1j+1 − 2vij i = j

We can study the steady state and the relaxation to it!


Covariance matrix

In terms of N×N blocks:

uij = 〈qiqj〉, vij = 〈pipj〉, zij = 〈qipj〉

c ≡(u zz† v

)Evolution equation:

c = d− ac− ca† − γ(

0 zggz† w

)

wij ≡

vi+1j + vi−1j + vij−1 + vij+1 − 4vij |i− j| > 1

vi±1j + vij∓1 − 2vij i− j = ±1

vi−1j−1 + vi+1j+1 − 2vij i = j

We can study the steady state and the relaxation to it!


Steady state: main results

Temperature profile

For y ≡ 2i/N − 1

T (y) = T + ∆T

√2

(√

8− 1)ζ(3/2)

∑odd n

n−3/2 cos(nπ

2(y + 1)

),

Heat flux

J =j√N

=∆T

8(√

8− 1)ζ(3/2)

√π3ω3

γN

Thermal conductivity

κ ≡ J

∆T/N=

1

8(√

8− 1)ζ(3/2)

√π3ω3N

γ



Temperature profile

For y ≡ 2i/N − 1

T (y) = T + ∆T

√2

(√

8− 1)ζ(3/2)

∑odd n

n−3/2 cos(nπ

2(y + 1)

),

Heat flux

J =j√N

=∆T

8(√

8− 1)ζ(3/2)

√π3ω3

γN


κ ≡ J

∆T/N=

1

8(√

8− 1)ζ(3/2)

√π3ω3N

γ



Temperature profile

For y ≡ 2i/N − 1

T (y) = T + ∆T

√2

(√

8− 1)ζ(3/2)

∑odd n

n−3/2 cos(nπ

2(y + 1)

),

Heat flux

J =j√N

=∆T

8(√

8− 1)ζ(3/2)

√π3ω3

γN


κ ≡ J

∆T/N=

1

8(√

8− 1)ζ(3/2)

√π3ω3N

γ


Temperature profiles and comparison with nonlinear model

-1 -0.5 0 0.5 1y

0.9

0.95

1

1.05

1.1T

(y)

FPU Analytic


Conclusions

Correlations in d = 1, 2 leads to anomalous energy transport anddiffusion

Universality (toy vs. realistic models)

Interpretation as a Levy process

Fluctuating hydrodynamics: connection with Kardar-Parisi-Zhang

Solvable models

Open problem: the quasi-integrable limit


Documents

Thermal transport in classical nonlinear lattices · 2018. 9. 17. · I Spatial constraints can signi cantly alter transport properties: long{time tails and breakdown of hydrodynamics