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Microscopic Hamiltonian dynamics perturbed by a conservative noise - - PowerPoint PPT Presentation

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Microscopic Hamiltonian dynamics perturbed by a conservative noise C edric Bernardin (with G. Basile and S. Olla) CNRS, Ens Lyon April 2008 C edric Bernardin (with G. Basile and S. Olla) Fourier law Introduction C edric Bernardin

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Microscopic Hamiltonian dynamics perturbed by a conservative noise

C´ edric Bernardin (with G. Basile and S. Olla)

CNRS, Ens Lyon

April 2008

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Introduction

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Introduction

Fourier’s law : Consider a macroscopic system in contact with two heat baths with different temperatures Tℓ = Tr. When the system reaches its steady state < · >ss, one expects Fourier’s law holds: < J(q) >ss= −κ(T(q))∇(T(q)), q macroscopic point

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Introduction

Fourier’s law : Consider a macroscopic system in contact with two heat baths with different temperatures Tℓ = Tr. When the system reaches its steady state < · >ss, one expects Fourier’s law holds: < J(q) >ss= −κ(T(q))∇(T(q)), q macroscopic point J(q) is the energy current; T(q) the local temperature; κ(T) the conductivity.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Introduction

Fourier’s law : Consider a macroscopic system in contact with two heat baths with different temperatures Tℓ = Tr. When the system reaches its steady state < · >ss, one expects Fourier’s law holds: < J(q) >ss= −κ(T(q))∇(T(q)), q macroscopic point J(q) is the energy current; T(q) the local temperature; κ(T) the conductivity. If system has (microscopic) size N, finite conductivity means < J >ss∼ N−1.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

H =

  • x∈Λ
  • p2

x

2mx + W (qx) 2 +

  • y∼x

V (qx − qy) 4

  • ,

Λ ⊂ Zd

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

H =

  • x∈Λ
  • p2

x

2mx + W (qx) 2 +

  • y∼x

V (qx − qy) 4

  • ,

Λ ⊂ Zd W : pinning potential; V : interaction potential

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

H =

  • x∈Λ
  • p2

x

2mx + W (qx) 2 +

  • y∼x

V (qx − qy) 4

  • ,

Λ ⊂ Zd W : pinning potential; V : interaction potential If V (r) = α|r|2, W (q) = ν|q|2 (harmonic chain), < · >ss is an explicit Gaussian measure and < J >ss∼ 1 : Fourier’s law is false (Lebowitz, Lieb, Rieder ’67)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

H =

  • x∈Λ
  • p2

x

2mx + W (qx) 2 +

  • y∼x

V (qx − qy) 4

  • ,

Λ ⊂ Zd W : pinning potential; V : interaction potential If V (r) = α|r|2, W (q) = ν|q|2 (harmonic chain), < · >ss is an explicit Gaussian measure and < J >ss∼ 1 : Fourier’s law is false (Lebowitz, Lieb, Rieder ’67) Non linearity is extremely important to have normal heat conduction.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Hamiltonian microscopic models : Fermi-Pasta-Ulam chains

H =

  • x∈Λ
  • p2

x

2mx + W (qx) 2 +

  • y∼x

V (qx − qy) 4

  • ,

Λ ⊂ Zd W : pinning potential; V : interaction potential If V (r) = α|r|2, W (q) = ν|q|2 (harmonic chain), < · >ss is an explicit Gaussian measure and < J >ss∼ 1 : Fourier’s law is false (Lebowitz, Lieb, Rieder ’67) Non linearity is extremely important to have normal heat conduction. But it is not sufficient : It has been observed experimentally and numerically for nonlinear chains that if d ≤ 2 and momentum is conserved (⇔ W = 0, unpinned) then conductivity is still infinite (finite otherwise).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Motivations/Goal

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Motivations/Goal

Give a rigorous derivation of Fourier’s law from the microscopic model.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Motivations/Goal

Give a rigorous derivation of Fourier’s law from the microscopic model. If Fourier’s law does not hold, κN ∼ Nδ, universality of the diverging order δ of the conductivity?

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Motivations/Goal

Give a rigorous derivation of Fourier’s law from the microscopic model. If Fourier’s law does not hold, κN ∼ Nδ, universality of the diverging order δ of the conductivity? Numerical simulations are not conclusive (δ ∈ [0.25; 0.47] for the same models) and subject of intense debate.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

FPU chains are mathematically very difficult to study. We perturb the Hamiltonian dynamics by a stochastic noise. These stochastic perturbations simulate (qualitatively) the long time (chaotic) effect of the deterministic nonlinear model.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

FPU chains are mathematically very difficult to study. We perturb the Hamiltonian dynamics by a stochastic noise. These stochastic perturbations simulate (qualitatively) the long time (chaotic) effect of the deterministic nonlinear model. FPU chains conserve total energy H. If the system is unpinned (W = 0), it conserves also total momentum

x px.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

FPU chains are mathematically very difficult to study. We perturb the Hamiltonian dynamics by a stochastic noise. These stochastic perturbations simulate (qualitatively) the long time (chaotic) effect of the deterministic nonlinear model. FPU chains conserve total energy H. If the system is unpinned (W = 0), it conserves also total momentum

x px.

Two different noises:

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

FPU chains are mathematically very difficult to study. We perturb the Hamiltonian dynamics by a stochastic noise. These stochastic perturbations simulate (qualitatively) the long time (chaotic) effect of the deterministic nonlinear model. FPU chains conserve total energy H. If the system is unpinned (W = 0), it conserves also total momentum

x px.

Two different noises: Noise 1 = only energy conservative

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

FPU chains are mathematically very difficult to study. We perturb the Hamiltonian dynamics by a stochastic noise. These stochastic perturbations simulate (qualitatively) the long time (chaotic) effect of the deterministic nonlinear model. FPU chains conserve total energy H. If the system is unpinned (W = 0), it conserves also total momentum

x px.

Two different noises: Noise 1 = only energy conservative Noise 2 = energy and momentum conservative

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

The generator L (adjoint of the Fokker-Planck operator) has two terms L = A + γS

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

The generator L (adjoint of the Fokker-Planck operator) has two terms L = A + γS A is the Liouville operator A =

  • x
  • ∂H

∂px ∂qx − ∂H ∂qx ∂px

  • C´

edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

The generator L (adjoint of the Fokker-Planck operator) has two terms L = A + γS A is the Liouville operator A =

  • x
  • ∂H

∂px ∂qx − ∂H ∂qx ∂px

  • S is a diffusion on the shell of constant kinetic energy (noise

1) or of constant kinetic energy and constant momentum (noise 2).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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The Models

The generator L (adjoint of the Fokker-Planck operator) has two terms L = A + γS A is the Liouville operator A =

  • x
  • ∂H

∂px ∂qx − ∂H ∂qx ∂px

  • S is a diffusion on the shell of constant kinetic energy (noise

1) or of constant kinetic energy and constant momentum (noise 2). γ > 0 regulates the strength of the noise.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Construction of the noise

Example : Noise 1, energy conserving, mx = 1, d=1

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Construction of the noise

Example : Noise 1, energy conserving, mx = 1, d=1 For every nearest neigbor atoms x and x + 1, surface of constant kinetic energy e S1

e = {(px, px+1) ∈ R2; p2 x + p2 x+1 = e}

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Construction of the noise

Example : Noise 1, energy conserving, mx = 1, d=1 For every nearest neigbor atoms x and x + 1, surface of constant kinetic energy e S1

e = {(px, px+1) ∈ R2; p2 x + p2 x+1 = e}

The following vector field Xx,x+1 is tangent to S1

e

Xx,x+1 = px+1∂px − px∂px+1 so X 2

x,x+1 generates a diffusion on S1 e (Brownion motion on

the circle).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Construction of the noise

Example : Noise 1, energy conserving, mx = 1, d=1 For every nearest neigbor atoms x and x + 1, surface of constant kinetic energy e S1

e = {(px, px+1) ∈ R2; p2 x + p2 x+1 = e}

The following vector field Xx,x+1 is tangent to S1

e

Xx,x+1 = px+1∂px − px∂px+1 so X 2

x,x+1 generates a diffusion on S1 e (Brownion motion on

the circle). We define S = 1 2

N−2

  • x=1

X 2

x,x+1

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Construction of the noise

Example : Noise 1, energy conserving, mx = 1, d=1 For every nearest neigbor atoms x and x + 1, surface of constant kinetic energy e S1

e = {(px, px+1) ∈ R2; p2 x + p2 x+1 = e}

The following vector field Xx,x+1 is tangent to S1

e

Xx,x+1 = px+1∂px − px∂px+1 so X 2

x,x+1 generates a diffusion on S1 e (Brownion motion on

the circle). We define S = 1 2

N−2

  • x=1

X 2

x,x+1

Noise 2, d ≥ 1 ... are of the same type.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?. Problem : If Tℓ = Tr, we don’t know < · >ss

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?. Problem : If Tℓ = Tr, we don’t know < · >ss LINEAR RESPONSE THEORY (GREEN-KUBO):

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?. Problem : If Tℓ = Tr, we don’t know < · >ss LINEAR RESPONSE THEORY (GREEN-KUBO): Non-rigorous perturbative arguments predict: lim

Tℓ,Tr→T lim N→∞ κN

is given by the Green-Kubo formula κGK(T) (space-time variance of the current at equilibrium)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?. Problem : If Tℓ = Tr, we don’t know < · >ss LINEAR RESPONSE THEORY (GREEN-KUBO): Non-rigorous perturbative arguments predict: lim

Tℓ,Tr→T lim N→∞ κN

is given by the Green-Kubo formula κGK(T) (space-time variance of the current at equilibrium) Advantage : κGK requires to consider the dynamics at

  • equilibrium. We know the equilibrium measure (Gibbs

measure with temperature T).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Evaluation of the conductivity

NON EQUILIBRIUM SETTING : system in contact with two heat baths (Tℓ = Tr). κN is of same order as N < J >ss. κN ∼ Nδ, δ =?. Problem : If Tℓ = Tr, we don’t know < · >ss LINEAR RESPONSE THEORY (GREEN-KUBO): Non-rigorous perturbative arguments predict: lim

Tℓ,Tr→T lim N→∞ κN

is given by the Green-Kubo formula κGK(T) (space-time variance of the current at equilibrium) Advantage : κGK requires to consider the dynamics at

  • equilibrium. We know the equilibrium measure (Gibbs

measure with temperature T). Inconvenient : The quantity to evaluate is a dynamical quantity.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Green Kubo formula

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Green Kubo formula

Energy ex of atom x is (mx = 1 to simplify) ex = p2

x

2 + W (qx) 2 +

  • y∼x

V (qx − qy) 2

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Green Kubo formula

Energy ex of atom x is (mx = 1 to simplify) ex = p2

x

2 + W (qx) 2 +

  • y∼x

V (qx − qy) 2 Local conservation of energy: ex(t) − ex(0) = −

d

  • k=1

(Jx−ek,x(t) − Jx,x+ek(t))

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Green Kubo formula

Energy ex of atom x is (mx = 1 to simplify) ex = p2

x

2 + W (qx) 2 +

  • y∼x

V (qx − qy) 2 Local conservation of energy: ex(t) − ex(0) = −

d

  • k=1

(Jx−ek,x(t) − Jx,x+ek(t)) The current of energy J has the decomposition: Jx,x+ek(t) = t jx,x+ek(s)ds + Mx,x+ek(t)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Green Kubo formula

Energy ex of atom x is (mx = 1 to simplify) ex = p2

x

2 + W (qx) 2 +

  • y∼x

V (qx − qy) 2 Local conservation of energy: ex(t) − ex(0) = −

d

  • k=1

(Jx−ek,x(t) − Jx,x+ek(t)) The current of energy J has the decomposition: Jx,x+ek(t) = t jx,x+ek(s)ds + Mx,x+ek(t) with M martingale (mean 0 w.r.t. any initial condition) jx,x+ek = −1 2(∇V )(qx+ek − qx) · (px+ek + px) − γ∇ekp2

x

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Consider the system of length N. Linear response theory predicts the following expression for the conductivity:

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Consider the system of length N. Linear response theory predicts the following expression for the conductivity: [κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2 Eeq.     1 √ tNd

  • x∈Td

N

Jx,x+e1(t)  

2

  • Eeq. is the expectation w.r.t. EQUILIBRIUM, meaning the uniform

measure on the shell {H =

x∈Td

N ex = NdT}. C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Consider the system of length N. Linear response theory predicts the following expression for the conductivity: [κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2 Eeq.     1 √ tNd

  • x∈Td

N

Jx,x+e1(t)  

2

  • Eeq. is the expectation w.r.t. EQUILIBRIUM, meaning the uniform

measure on the shell {H =

x∈Td

N ex = NdT}.

If the conductivity is infinite, how to obtain the diverging order of the conductivity κN of the system of length N ?

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Consider the system of length N. Linear response theory predicts the following expression for the conductivity: [κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2 Eeq.     1 √ tNd

  • x∈Td

N

Jx,x+e1(t)  

2

  • Eeq. is the expectation w.r.t. EQUILIBRIUM, meaning the uniform

measure on the shell {H =

x∈Td

N ex = NdT}.

If the conductivity is infinite, how to obtain the diverging order of the conductivity κN of the system of length N ? κN ∼ κGK with truncation of the time up to time tN = N (phononic picture).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Consider the system of length N. Linear response theory predicts the following expression for the conductivity: [κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2 Eeq.     1 √ tNd

  • x∈Td

N

Jx,x+e1(t)  

2

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 48

Consider the system of length N. Linear response theory predicts the following expression for the conductivity: [κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2 Eeq.     1 √ tNd

  • x∈Td

N

Jx,x+e1(t)  

2

 The current of energy J has the decomposition: Jx,x+ek(t) = t jx,x+ek(s)ds + Mx,x+ek(t) with M martingale and jx,x+ek = −1 2(∇V )(qx+ek − qx) · (px+ek + px) − γ∇ekp2

x

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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[κGK]1,1 (T) = lim

t→∞ lim N→∞

1 2T 2Ndt Eeq  

  • x

t jx,x+e1(s)ds 2  +γ d (Martingale contribution)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 1: energy conserving

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 1: energy conserving Momentum is NOT conserved.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 1: energy conserving Momentum is NOT conserved. Homogenous chain : mx = 1

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 1: energy conserving Momentum is NOT conserved. Homogenous chain : mx = 1 Theorem (B., Olla, JSP’05) κGK is finite (pinned or unpinned) in any dimension. Fourier’s law holds and linear response theory is correct: System of length N in contact with two Langevin baths at temperature Tℓ and Tr in its steady state lim

N→∞ N < jx,x+1 >ss= κGK(Tr − Tℓ)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

Numerical simulations and rigorous results (Dahr, Lebowitz, O’Connor, Rubin-Greer...), non-equilibrium setting, disordered chain (without noise)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

Numerical simulations and rigorous results (Dahr, Lebowitz, O’Connor, Rubin-Greer...), non-equilibrium setting, disordered chain (without noise) Harmonic chain without pinning: κN ∼ Nδ with δ ∈ [−1/2, 1/2] depending on the spectral properties of the baths !!!

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

Numerical simulations and rigorous results (Dahr, Lebowitz, O’Connor, Rubin-Greer...), non-equilibrium setting, disordered chain (without noise) Harmonic chain without pinning: κN ∼ Nδ with δ ∈ [−1/2, 1/2] depending on the spectral properties of the baths !!! Harmonic chain with harmonic pinning: κN ∼ e−cN.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

Numerical simulations and rigorous results (Dahr, Lebowitz, O’Connor, Rubin-Greer...), non-equilibrium setting, disordered chain (without noise) Harmonic chain without pinning: κN ∼ Nδ with δ ∈ [−1/2, 1/2] depending on the spectral properties of the baths !!! Harmonic chain with harmonic pinning: κN ∼ e−cN. Anharmonic chain (with or without pinning): κN = O(1).

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case, random masses

Numerical simulations and rigorous results (Dahr, Lebowitz, O’Connor, Rubin-Greer...), non-equilibrium setting, disordered chain (without noise) Harmonic chain without pinning: κN ∼ Nδ with δ ∈ [−1/2, 1/2] depending on the spectral properties of the baths !!! Harmonic chain with harmonic pinning: κN ∼ e−cN. Anharmonic chain (with or without pinning): κN = O(1). Theorem (B., ’08) Harmonic system with random masses and energy conservative noise 1.The conductivity defined by Green-Kubo formula is strictly positive and bounded above: 0 < c− ≤ κKG ≤ C+ < +∞

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 2: energy/momentum conserving

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 2: energy/momentum conserving Theorem (Basile, B., Olla, PRL’06) C1,1(t) = lim

N→∞

  • x

jx,x+e1(t)

  • , j0,e1(0)
  • eq.

C1,1(t) = T 2 4π2d

  • [0,1]d(∂k1ω(k))2e−tγψ(k)dk

where ω is the dispertion relation of the harmonic chain ω(k) = (ν + 4α

d

  • j=1

sin2(πkj))1/2, ψ(k) ∼ k2

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

Noise 2: energy/momentum conserving Corollary C1,1(t) ∼ t−d/2 in the unpinned case (ν = 0) C1,1(t) ∼ t−d/2−1 in the pinned case (ν > 0)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

κN = 1 2T 2NdtN Eeq  

  • x

tN jx,x+e1(s)ds 2  + γ d, tN = N

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Harmonic case : V (r) = αr 2, W (q) = νq2

κN = 1 2T 2NdtN Eeq  

  • x

tN jx,x+e1(s)ds 2  + γ d, tN = N Corollary If the system is unpinned (ν = 0) then ”truncated” Green-Kubo formula for κN gives:

  • κN ∼ N1/2 if d = 1

κN ∼ log N if d = 2 In all other cases κN is bounded in N and converges to κGK.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Anharmonic case, Canonical version of GK

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Results : Anharmonic case, Canonical version of GK

Theorem (Basile,B.,Olla’08) For d ≥ 3, if W > 0 is ”general” or if W = 0 and 0 < c− ≤ V ” ≤ C+ < +∞ then κN ≤ C. For d = 2, if W = 0 and 0 < c− ≤ V ′′ ≤ C+ < ∞ κN ≤ C(log N)2. For d = 1, if W = 0 and 0 < c− ≤ V ′′ ≤ C+ < ∞, then κN ≤ C √ N. In any dimension, if V are quadratic and W > 0 is ”general” then κN ≤ C.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 69

Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ. FPU (α) : δ = 0.35 for γ small to δ = 0.48 for γ larger.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 70

Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ. FPU (α) : δ = 0.35 for γ small to δ = 0.48 for γ larger. FPU (β): δ = 0.41 for γ small to δ = 0.47 for γ larger.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 71

Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ. FPU (α) : δ = 0.35 for γ small to δ = 0.48 for γ larger. FPU (β): δ = 0.41 for γ small to δ = 0.47 for γ larger. Thermal conductivity increases with strength of the noise !!!

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 72

Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ. FPU (α) : δ = 0.35 for γ small to δ = 0.48 for γ larger. FPU (β): δ = 0.41 for γ small to δ = 0.47 for γ larger. Thermal conductivity increases with strength of the noise !!! Theoretical arguments are controversial. For FPU (β):

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 73

Anharmonic case : Numerical simulations, (EPJ’08) , Basile, Delfini, Lepri, Livi, Olla, Politi

Simulations (d = 1) for unpinned systems with energy/momentum conservative noise 2. The strength of the noise is regulate by γ. Then κN ∼ Nδ. FPU (α) : δ = 0.35 for γ small to δ = 0.48 for γ larger. FPU (β): δ = 0.41 for γ small to δ = 0.47 for γ larger. Thermal conductivity increases with strength of the noise !!! Theoretical arguments are controversial. For FPU (β): Kinetic approach : δ = 2/5 (Perverzev, Lukkarinen and Spohn) MCT approach : δ = 1/2 (Delfini, Lepri,Livi, Politi)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

Eeq. ∞ dte−λt

  • x

jx,x+e1(t)

  • j0,e1(0)
  • C´

edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

Eeq. ∞ dte−λt

  • x

jx,x+e1(t)

  • j0,e1(0)
  • =

∞ dte−λt

  • x

etLjx,x+e1

  • , j0,e1
  • eq.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

Eeq. ∞ dte−λt

  • x

jx,x+e1(t)

  • j0,e1(0)
  • =

∞ dte−λt

  • x

etLjx,x+e1

  • , j0,e1
  • eq.

=

  • (λ − L)−1
  • x

jx,x+e1

  • , j0,e1
  • eq.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

Eeq. ∞ dte−λt

  • x

jx,x+e1(t)

  • j0,e1(0)
  • =

∞ dte−λt

  • x

etLjx,x+e1

  • , j0,e1
  • eq.

=

  • (λ − L)−1
  • x

jx,x+e1

  • , j0,e1
  • eq.

Solve the resolvent equation (λ − L)hλ =

x jx,x+e1 and

compute limN→∞ < j0,e1 hλ >eq..

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Proof of the harmonic case

Eeq. ∞ dte−λt

  • x

jx,x+e1(t)

  • j0,e1(0)
  • =

∞ dte−λt

  • x

etLjx,x+e1

  • , j0,e1
  • eq.

=

  • (λ − L)−1
  • x

jx,x+e1

  • , j0,e1
  • eq.

Solve the resolvent equation (λ − L)hλ =

x jx,x+e1 and

compute limN→∞ < j0,e1 hλ >eq.. Function hλ is local in the energy conservative case and non-local in the energy/momentum case. For d ≥ 3 or ν > 0, the decay of hλ is sufficient to assure a finite conductivity.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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First order correction to local equilibrium

What is h0 = limλ→0 hλ ?

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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First order correction to local equilibrium

What is h0 = limλ→0 hλ ? Consider the system in contact with two reservoirs with temperature Tℓ = β−1

ℓ

and Tr = β−1

r

.

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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First order correction to local equilibrium

What is h0 = limλ→0 hλ ? Consider the system in contact with two reservoirs with temperature Tℓ = β−1

ℓ

and Tr = β−1

r

. If Tℓ = Tr = β−1, NESS < · >ss is the Gibbs mesure Z −1 exp (−βH)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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First order correction to local equilibrium

What is h0 = limλ→0 hλ ? Consider the system in contact with two reservoirs with temperature Tℓ = β−1

ℓ

and Tr = β−1

r

. If Tℓ = Tr = β−1, NESS < · >ss is the Gibbs mesure Z −1 exp (−βH) Let fss the density of < · >ss w.r.t. the local equilibrium state < · >ℓe= Z −1 exp (−β(x/N)H) , β(q) = βℓ + (βr − βℓ)q

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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SLIDE 84

First order correction to local equilibrium

What is h0 = limλ→0 hλ ? Consider the system in contact with two reservoirs with temperature Tℓ = β−1

ℓ

and Tr = β−1

r

. If Tℓ = Tr = β−1, NESS < · >ss is the Gibbs mesure Z −1 exp (−βH) Let fss the density of < · >ss w.r.t. the local equilibrium state < · >ℓe= Z −1 exp (−β(x/N)H) , β(q) = βℓ + (βr − βℓ)q If δT = Tr − Tℓ is small fss = 1 + δTh0 + o((δT)2)

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : upper bounds

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : upper bounds

Eeq   N

  • x

jx,x+e1(s)ds 2 

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : upper bounds

Eeq   N

  • x

jx,x+e1(s)ds 2  ≤ 10N

  • x

jx,x+e1

  • , (N−1 − L)−1
  • x

jx,x+e1

  • eq

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : upper bounds

Eeq   N

  • x

jx,x+e1(s)ds 2  ≤ 10N

  • x

jx,x+e1

  • , (N−1 − γS)−1
  • x

jx,x+e1

  • eq

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law

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Anharmonic case : upper bounds

Eeq   N

  • x

jx,x+e1(s)ds 2  ≤ 10N

  • x

jx,x+e1

  • , (N−1 − γS)−1
  • x

jx,x+e1

  • eq

(N−1 − γS)−1

  • x

jx,x+e1

  • =

d

  • j=1
  • x,y

GN(x − y)pj

xV ′(qj y+e1 − qj y)

where GN(z) is the solution of the resolvent equation N−1GN(z) − 2γ(∆GN)(z) = −1 2 [δ0(z) + δe1(z)]

C´ edric Bernardin (with G. Basile and S. Olla) Fourier law