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Heat and Particle Transport in LowDimensional Mechanical Systems

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

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.




• 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?



from the Microscopic to the Macroscopic

Reversible Microscopic Dynamics.

qj =∂H

∂pj; pj = −∂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.




〈Ju〉 = −κ∇T ,

κ is the heat conductivity.



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)

Chains of oscillators with geometric constraintsG. Casati, J. Ford, F. Vivaldi, and W. M. Visscher, PRL 52, 1861 (1984).T. Prosen and M. Robnik, J. Phys. A 25, 3449 (1992).



Fourier’s Law

What are the ingredients of the microscopic dynamics that lead to the observedmacroscopic transport?

TL

TR

TL

TRφ θ

R

D. Alonso, R. Artuso, G. Casati, and I. Guarneri, PRL 82, 1859 (1999).B. Li, G. Casati, and J. Wang, Phys. Rev. E 67, 021204 (2003).

M. Cencini, F. Cecconi, M. Falcioni, and A. Vulpiani, arXiv:0804.0776(2008).



Fourier’s Law

For noninteracting particle systems with no globally conserved quantities(globally ergodic), positive Lyapunov exponents (chaos) is, “in general”, asufficient condition to ensure macroscopic transport.

However, without interactions the very definition of temperature is at bestproblematic, due to the lack of local thermal equilibrium.

LTE: the intensive thermodynamic variables are well defined at each point ofthe system, and the relations amongst these variables are the same as inequilibrium thermodynamics.



Rotating-disks Lorentz gas

v′n = −vn

v′t = vt − 2η1+η (vt − Rω)

Rω′ = Rω + 21+η (vt − Rω)

η =Θ

mR2

ωα

α’

ξC-

ξC+

ξH+

ξH-

C. M-M, H. Larralde and F. Leyvraz, PRL 86, 5417 (2001)




Genuine many-body interactingparticle system.

ωα

α’

ξC-

ξC+

ξH+

ξH-

• 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)




The bath is an ideal gas at equilibrium densityρ and temperature T which exchanges particleswith the system through a physical wall.

Particles are absorbed with probability

Pabs = 1 − e−α/|vn| ,

where α determines the wall’s resistance.

Particles are emitted at rate γ with a velocity dis-tributed according to

Pn(vn) = 1kT |vn| exp

(

−mv2

n

2kT

)

Pt(vt) =√

m2πkT exp

(

−mv2

t

2kT

)

SystemBath




At equilibrium

−50.00 0.00 50.00ω

0.00

0.01

0.02

0.03

0.04

P(ω

)

0.00 20.00 40.00v

0.000

0.020

0.040

0.060

P(v

)

In a “microcanonical” situation there is energy equipartition among all thed.o.f.




Local thermal equilibrium

0 1 2 3 4ε

10−2

10−1

100

Px(

ε)




Local thermal equilibrium

0 4 8 12 16 20 24 281.00

1.05

1.10

1.15

0 4 8 12 16 20 24 28x

1.00

1.05

1.10

1.15

T(x

)

0 4 8 12 16 20 24 28x

5.3

5.32

5.34

5.36

5.38

5.4

5.42

5.44

−µ/

T

The gas of particles behaves locally as an ideal gas.

µ = T ln( ρ

T

)




Linear transport

Ju = Luu ∂x

(

1T

)

+ Luρ ∂x

(

− µT

)

,

Jρ = Lρu ∂x

(

1T

)

+ Lρρ ∂x

(

− µT

)

,

Onsager reciprocity relations: L is symmetric, Luρ = Lρu.

Green-Kubo relations

Lab =1

2LCkJaJb

(ω) ,

whereCkJaJb

(t) = 〈Jka(t)J∗kb(0)〉t ,

andJkρ(t) =

∑Ln=1 ei2πkn/LJnρ(t) ,

Jku(t) =∑L

n=1 ei2πkn/LJnu(t) .




Linear transport

Green-Kubo Gradients

Lρρ 0.1050 ± 0.003 0.1030 ± 0.002

Lρu 0.1276 ± 0.0016 0.1271 ± 0.0017

Luρ 0.1276 ± 0.0016 0.1272 ± 0.0048

Luu 0.7920 ± 0.012 0.7710 ± 0.005

H. Larralde, F. Leyvraz and C. M-M, JSP 113, 197 (2003)




- NESS assumed to exist. Uniqueness can beproven.

- In the zero coupling limit (RW approximation) thedensity profiles are

ρ(ξ) = |Γ|√

π

2T (ξ)j(ξ) ,

and

T (ξ) =1

3

Q(ξ)

j(ξ),

wherej(ξ) = 2(jL + (jR − jL)ξ) ,

and

Q(ξ) = 2(qL + (qR − qL)ξ) , q =3

2jT .

j>

q>

j<

q<

γR

Rs

JRjRjLJL

J.-P. Eckmann and L. Young, Commun. Math. Phys. 262 237 (2006).




At finite coupling the transition rates j and q depend on the densities ρ and T ,as well as on the local gradients, yielding “dynamical memory” terms.

JL,k = αJL,k jL,k + (1 − αJ

R,k) jR,k ,

JR,k = (1 − αJL,k) jL,k + αJ

R,k jR,k ,

QL,k = αQL,k qL,k + (1 − αQ

R,k) qR,k ,

QR,k = (1 − αQL,k) qL,k + αQ

R,k qR,k .

The reflection probabilities α satisfying

α(jL, jR, qL, qR) = αG

(

j3/2

q1/2

)

+ ǫ

(

j3/2

q1/2,jR − jLjR + jL

,qR − qL

qR + qL

)

.

J-P Eckmann, C. M-M, and E. Zabey, JSP 123, 1339 (2006).




10-2

10-1

100

∆T/T

10-5

10-4

10-3

10-2

10-1

∆α

0.55

0.56

0.57

αJ

10-2

10-1

100

∆T/T

0.54

0.56

0.58

αQ

α(jL, jR, qL, qR) = αG

(

j3/2

q1/2

)

+ ǫ

(

j3/2

q1/2,jR − jLjR + jL

,qR − qL

qR + qL

)

.

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




When interaction can be neglected

J(ξ) = 2(jL + ξ∆j) +(1 − 2αJ)(1 − 2ξ)

1 + (N − 1)αJ∆j ,

Q(ξ) = 2(qL + ξ∆q) +(1 − 2αQ)(1 − 2ξ)

1 + (N − 1)αQ∆q ,

with ∆j = jR − jL and ∆q = qR − qL.

The macroscopic currents are

Jρ = − 1 − αJ

1 + (N − 1)αJ∆j , Ju = − 1 − αQ

1 + (N − 1)αQ∆q .

Taking interaction into account, in the infinite volume limit

Jρ =∆j

N∫ 1

0W J(ξ)dξ

, with W J(ξ) =αJ

G

(

n(ξ))

−AJ

1 − αJG

(

n(ξ))

+ AJ.



One dimensional rotating-disks Lorentz gas

Discrete space: a varying number of particles of mass m and N equidistantfixed scatterers of mass M . x ∈ (0, 1).

At the boundaries, particles are injected from and absorbed to stochasticparticle reservoirs.

The particles collide with the scatterers with probability γ/N (γ ∈ [0, N ]).For γ = N the bulk dynamics is deterministic.

Elastic collision rules for the momenta(

P

p

)

= S

(

P

p

)

, S =

(

−σ 1 − σ

1 + σ σ

)

,

and σ = (M − m)/(M + m) is −1 ≤ σ ≤ 1.

P Collet and J-P Eckmann, arXiv:0804.3025 (2008).L A Bunimovich and M A Khlabystova, J. Stat. Phys., 112 1207, (2003).




Under the given 1D dynamics, the reservoir distribution

f(p) dp =Θ(±p)

(2πmkT )1/2

e−p2/2mkT

|p| dp ,

is the only distribution that admits stationary solutions that preserve thedistribution of the scatterer’s momenta




Under the given 1D dynamics, the reservoir distribution

f(p) dp =Θ(±p)

(2πmkT )1/2

e−p2/2mkT

|p| dp ,

is the only distribution that admits stationary solutions that preserve thedistribution of the scatterer’s momenta

In 1D f(p) has an infrared singularity.The particle’s density diverges.However, the number of particles withmomentum p > p0 has a limit as t → ∞.Thus the stationary state is well defined.

0 5 10 15 20 25 30|p|/dp

0

1

2

3

4

5

p n t(p

)

102

103

104

105

t

50

60

70

80

90

n(t)

t

P Collet, J-P Eckmann, and C.M-M, arXiv:0810.4464 (2008).




Kinetic Grad-limit: N → ∞, i/N = x ∈ [0, 1], and γ = 1 the NESS exists and isunique.

Furthermore, the particle distribution (F (p) = |p|f(p))

F (p) dp = (2πmkT )−1/2

e−p2/2mkT dp ,

and scatterer’s distribution

g(P ) dP = (2πMkT )−1/2

e−P 2/2MkT dP ,

satisfy a Boltzmann equation:

p∂xF (p, x) = γ|p|∫

dP(

F (p, x)g(P , x) − F (p, x)g(P , x))

,

0 =

∫

dp(

F (p, x)g(P , x) − F (p, x)g(P , x))

,

P Collet and J-P Eckmann, arXiv:0804.3025 (2008).




The solution of the Boltzmann equation is a bona fide limit of the discreteparticle models.

-100 -50 0 50p

0

0.01

0.02

0.03

0.04

0.05

F(p

,x)

-60 -40 -20 0 20 40 60p

-0.1

-0.05

0

0.05

0.1

[F(p

) -

FN(p

)] x

102

P Collet, J-P Eckmann, and C.M-M, arXiv:0810.4464 (2008).




Density profiles depend on the scattering probability γ.

80

90

100

110

120

ρ E(x

)

0 0.2 0.4 0.6 0.8 1x

12

14

16

18

ρ(x)




The particle’s motion is persistent

-1 -0.5 0 0.5 1σ

0

0.2

0.4

0.6

0.8

1

µ(σ)

µ(σ) ≡ P (v′ > 0|v > 0) =1

2−( m

2πkT

)1/2∫ ∞

0

dv erf

(

(M − m)v

(8MkT )1/2

)

e−mv

2

2kT ,




Furthermore, the particle’s move in “a-sort-of” Levy walk

10-2

100

102

104

τ10

-9

10-7

10-5

10-3

10-1

101

ψ(τ

)

σ = 0.5

σ = 0




Motion is superdiffusive!

10-2

100

102

104

t

10-1

101

103

105

107

<x2 (t

)>

0 0.2 0.4 0.6 0.8 1σ

1.85

1.90

1.95

2.00

α(σ)




Motion is superdiffusive!

101

102

N

100

101

102

103

κ(N

)κ ~ N

κ ~ N1/3

κ =JU

TN − T1.



Thermoelectric effect

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.

Thomas J. Seebeck (1821)



Enviromental concerns



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 and

κ = 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 =σS2

κ=

L2un

det 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 = pt (γ> − γ<)

Ju = pt (ε> − ε<)

For noninteracting particles:- pt is a property of the geometry of the billiard only.- does not depend on the density or the temperature of the particles.- has no spatial dependence: pt;LR = pt;RL.

- If particles interact then none of these properties is true




nL , T

Ln

R , T

R

ptγ

Lp

rγ

L

ptγ

R

prγ

R




Jρ = pt(γL − γR)

=λpt

(2πm)1/2

(

ρLT1/2L − ρRT

1/2R

)

=−λpt

(2πm)1/2

L∇(

ρT 1/2)

=−λptL

cd (2πm)1/2

∇(

T (d+1)/2e(µ/T ))

=−λptL

cd (2πm)1/2

(

d + 1

2T (d−1)/2∇(T ) + T (d+1)/2∇

(µ

T

)

)

e(µ/T )

=−λptL

(2πm)1/2

(

d + 1

2ρT−1/2∇(T ) + T 1/2∇

(µ

T

)

)

=λptL

(2πm)1/2

(

d + 1

2ρT 3/2∇

(

1

T

)

+ ρT 1/2∇(

−µ

T

)

)




Jn =λptL

(2πm)1/2

(

d + 1

2ρT 3/2∇

(

1

T

)

+ ρT 1/2∇(

−µ

T

)

)

Ju =d + 1

2

λptL

(2πm)1/2

(

d + 3

2ρT 5/2∇

(

1

T

)

+ ρT 3/2∇(

−µ

T

)

)

The transport coefficients are

Lρρ =λptL

(2πm)1/2

ρT 1/2

Lρu = Luρ =d + 1

2

λptL

(2πm)1/2

ρT 3/2

Luu =(d + 1)(d + 3)

4

λptL

(2πm)1/2

ρT 5/2




Taking ∇(µ/T ) = e∇φ/T one identifies:

σ =e2Lρρ

T=

e2λptLρ

(2πmT )1/2,

κ =det L

T 2Lρρ=

λptLρ

(2πm)1/2

(

d + 1

2

)

T 1/2 ,

S =Luρ

eTLρρ=

d + 1

(2e).

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





ω2

ω1

ω3

...Dint

ω

Compound molecule: Each “particle" of mass m can be imagined as a stack ofDint small identical disks of mass m/Dint and radius r ≪ R, rotating freely andindependently at a constant angular velocity ωi, i = 1, . . .Dint. The center ofmass of the particle moves with velocity ~v = (vx, vy).



Energy mixing

v′n

v′t

ω′1

ω′2

...

ω′D

=

−1 0 0 0 · · · 0

0 1−Dη1+Dη

2η1+Dη

2η1+Dη · · · 2η

1+Dη

0 21+Dη 1 − 2

D(1+Dη) − 2D(1+Dη) · · · − 2

D(1+Dη)

0 21+Dη − 2

D(1+Dη) 1 − 2D(1+Dη) · · · − 2

D(1+Dη)

.... . .

...

0 21+Dη − 2

D(1+Dη) − 2D(1+Dη) · · · 1 − 2

D(1+Dη)

vn

vt

ω1

ω2

...

ωD

Here η = Θ/mR2




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 10

110

2

L

0

5

10

15

20

ZT




Without interactions

0 5 10 15 20D

0

0.1

0.2

0.3

0.4

0.5

η max

(D)

0 0.005 0.01 0.015 0.02 0.025∇ (µ/T)

0.02

0.04

0.06

0.08

0.1

η[∇

(µ/T

)]



With interactions

0 5 10 15 20D

0

0.1

0.2

0.3

0.4

0.5

η max

(D)

0 0.005 0.01 0.015 0.02 0.025∇ (µ/T)

0

0.02

0.04

0.06

0.08

0.1

η[∇

(µ/T

)]



Heat Engine

ξT

C

ξT

C

ξT

H

ξT

H

E



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