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Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 1/30
Heat and Particle Transport in LowDimensional Mechanical Systems
Carlos Mejía-MonasterioInstitute for Complex Systems, CNR, Florence Italy
http://calvino.polito.it/∼mejia/
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 2/30
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 3/30
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?
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 4/30
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 5/30
Nonequilibrium Statistical Mechanics
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 5/30
Nonequilibrium Statistical Mechanics
〈Ju〉 = −κ∇T ,
κ is the heat conductivity.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 6/30
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 7/30
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 8/30
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 9/30
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
Local thermal equilibrium
0 1 2 3 4ε
10−2
10−1
100
Px(
ε)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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
)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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) .
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 10/30
Rotating-disks Lorentz gas
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 11/30
Rotating-disks Lorentz gas
- 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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 12/30
Rotating-disks Lorentz gas
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 12/30
Rotating-disks Lorentz gas
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 12/30
Rotating-disks Lorentz gas
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 13/30
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 14/30
One dimensional rotating-disks Lorentz gas
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 14/30
One dimensional rotating-disks Lorentz gas
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 15/30
One dimensional rotating-disks Lorentz gas
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
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 ,
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 16/30
One dimensional rotating-disks Lorentz gas
Motion is superdiffusive!
101
102
N
100
101
102
103
κ(N
)κ ~ N
κ ~ N1/3
κ =JU
TN − T1.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 17/30
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 18/30
Enviromental concerns
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 19/30
Thermoelectricity in Billiards
TH
TC
V
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 19/30
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 and
κ = κ′ − TσS2 is the thermal conductivity.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 20/30
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 =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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 21/30
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 21/30
Diluted Polyatomic Ideal Gas
nL , T
Ln
R , T
R
ptγ
Lp
rγ
L
ptγ
R
prγ
R
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 21/30
Diluted Polyatomic Ideal Gas
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
)
)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 21/30
Diluted Polyatomic Ideal Gas
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 21/30
Diluted Polyatomic Ideal Gas
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.
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 22/30
Polyatomic Lorentz gas
TL , µ
LT
R , µ
R
G. Casati, C. M-M and T. Prosen, PRL (2008)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 23/30
Polyatomic Lorentz gas
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).
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 24/30
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
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 25/30
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)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 26/30
Polyatomic Lorentz gas
0 10 20 30d
0
5
10
15
20
ZT 10
110
2
L
0
5
10
15
20
ZT
G. Casati, C. M-M and T. Prosen, PRL (2008)
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 27/30
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
)]
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 28/30
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
)]
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 29/30
Heat Engine
ξT
C
ξT
C
ξT
H
ξT
H
E
Carlos Mejía-Monasterio, November 18, 2008 Heat and Particle Transport - p. 30/30
How future may look like...