Macroscopic scattering by a Langevin heat bath in contact with a - - PowerPoint PPT Presentation

Macroscopic scattering by a Langevin heat bath in contact with a harmonic chain.

Stefano Olla CEREMADE, Universit´ e Paris-Dauphine, PSL

Supported by ANR LSD

Nice, November 27, 2017

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thermal-scattering

thermal boundaries

Thermal boundaries appear in macroscopic equations for the evolution of energy or temperatures profiles.

T+ T−

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thermal-scattering

thermal boundaries

Thermal boundaries appear in macroscopic equations for the evolution of energy or temperatures profiles.

T+ T−

Basic example: heat equation for a material in contact with heat bath at the boundaries: ∂tu(t,y) = ∂y (D(u)∂yu(t,y)), y ∈ [0,L] u(t,0) = T+, u(t,L) = T−.

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Microscopic modeling: Langevin stochastic thermostats

q(t),p(t) positions and velocities of the system at time t: dq(t) = p(t)dt dp(t) = −∂qV (q(t)) − γp(t)dt + √ 2γTdw(t) The equilibrium measure is the canonical Gibbs at temperature T: e−H(p,q)/T ZT , H(p,q) = p2 2 + V (q).

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Microscopic modeling: Langevin stochastic thermostats

q(t),p(t) positions and velocities of the system at time t: dq(t) = p(t)dt dp(t) = −∂qV (q(t)) − γp(t)dt + √ 2γTdw(t) The equilibrium measure is the canonical Gibbs at temperature T: e−H(p,q)/T ZT , H(p,q) = p2 2 + V (q). What is the effect of the thermostat in the macroscopic evolution, after a rescaling of space and time?

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A kinetic limit

Consider an infinite one dimensional chain of coupled harmonic

• scillators, with a Langevin thermostat at site 0.

p = {px, x ∈ Z}, q = {qx, x ∈ Z} H(p,q) ∶= 1 2 ∑

x

p2

x + 1

2 ∑

x,x′ αx−x′qxqx′

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A kinetic limit

Consider an infinite one dimensional chain of coupled harmonic

• scillators, with a Langevin thermostat at site 0.

p = {px, x ∈ Z}, q = {qx, x ∈ Z} H(p,q) ∶= 1 2 ∑

x

p2

x + 1

2 ∑

x,x′ αx−x′qxqx′

dqx(t) = px(t)dt, x ∈ Z dpx(t) = −(α ∗ q(t))x +(−γp0(t)dt + √ 2γTdw(t))δ0,x, where {w(t), t ≥ 0} is a standard Wiener process.

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Coupled Harmonic Oscillators

ˆ f (k) = ∑

x

fxe−i2πkx k ∈ Π ∼ [0,1]

▸ ˆ

α(k) ∈ C∞(Π).

▸ ω(k) =

√ α(k): dispersion relation.

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Coupled Harmonic Oscillators

ˆ f (k) = ∑

x

fxe−i2πkx k ∈ Π ∼ [0,1]

▸ ˆ

α(k) ∈ C∞(Π).

▸ ω(k) =

√ α(k): dispersion relation. In the n.n. unpinned chain (acoustic chain) : ω(k) = ∣sin(πk)∣.

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Coupled Harmonic Oscillators

ˆ f (k) = ∑

x

fxe−i2πkx k ∈ Π ∼ [0,1]

▸ ˆ

α(k) ∈ C∞(Π).

▸ ω(k) =

√ α(k): dispersion relation. In the n.n. unpinned chain (acoustic chain) : ω(k) = ∣sin(πk)∣. ˆ ψ(t,k) ∶= ω(k)ˆ q(t,k) + iˆ p(t,k).

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Coupled Harmonic Oscillators

ˆ f (k) = ∑

x

fxe−i2πkx k ∈ Π ∼ [0,1]

▸ ˆ

α(k) ∈ C∞(Π).

▸ ω(k) =

√ α(k): dispersion relation. In the n.n. unpinned chain (acoustic chain) : ω(k) = ∣sin(πk)∣. ˆ ψ(t,k) ∶= ω(k)ˆ q(t,k) + iˆ p(t,k). d ˆ ψ(t,k) = −iω(k) ˆ ψ(t,k) dt − iγp0(t)dt + i √ 2γTdw(t)

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Coupled Harmonic Oscillators

ˆ f (k) = ∑

x

fxe−i2πkx k ∈ Π ∼ [0,1]

▸ ˆ

α(k) ∈ C∞(Π).

▸ ω(k) =

√ α(k): dispersion relation. In the n.n. unpinned chain (acoustic chain) : ω(k) = ∣sin(πk)∣. ˆ ψ(t,k) ∶= ω(k)ˆ q(t,k) + iˆ p(t,k). d ˆ ψ(t,k) = −iω(k) ˆ ψ(t,k) dt − iγp0(t)dt + i √ 2γTdw(t) In integral form ˆ ψ(t,k) = e−iω(k)t ˆ ψ(0,k) − iγ ∫

t 0 e−iω(k)(t−s)p0(s)ds

+i √ 2γT ∫

t 0 e−iω(k)(t−s)dw(t)

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Wigner distribution

η ∈ R, ̂ Wε(t,η,k) ∶= ε 2E[ ˆ ψ∗ (ε−1t,k − εη 2 ) ˆ ψ (ε−1t,k + εη 2 )] ̂ Wε(t,η,k) ∶= ̂ Wε(t,−η,k)∗ and the inverse Fourier transform in η Wε(t,y,k) = ∫ ̂ Wε(t,η,k)ei2πηy dη ∈ R, y ∈ R,

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Wigner distribution

η ∈ R, ̂ Wε(t,η,k) ∶= ε 2E[ ˆ ψ∗ (ε−1t,k − εη 2 ) ˆ ψ (ε−1t,k + εη 2 )] ̂ Wε(t,η,k) ∶= ̂ Wε(t,−η,k)∗ and the inverse Fourier transform in η Wε(t,y,k) = ∫ ̂ Wε(t,η,k)ei2πηy dη ∈ R, y ∈ R, Wε(t,y,k) ⇀

ε→0 W (t,y,k) ≥ 0,

as distribution When γ = 0 it is easy to prove that ∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = 0

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Easy case: γ = 0, no thermostat

From waves to particle: ˆ ψ(t,k) = ˆ ψ(0,k)e−iω(k)t

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Easy case: γ = 0, no thermostat

From waves to particle: ˆ ψ(t,k) = ˆ ψ(0,k)e−iω(k)t ˆ Wε(t,η,k) ∶= εE[ ˆ ψ∗ (ε−1t,k − εη 2 ) ˆ ψ (ε−1t,k + εη 2 )]

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Easy case: γ = 0, no thermostat

From waves to particle: ˆ ψ(t,k) = ˆ ψ(0,k)e−iω(k)t ̂ Wε(t,η,k) ∶= ei[ω(k− εη

2 )−ω(k+ εη 2 )]ε−1t ̂

Wε(0,η,k) ∼

ε→0 e−iω′(k)ηt ˆ

W (0,η,k)

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Easy case: γ = 0, no thermostat

From waves to particle: ˆ ψ(t,k) = ˆ ψ(0,k)e−iω(k)t ̂ Wε(t,η,k) ∶= ei[ω(k− εη

2 )−ω(k+ εη 2 )]ε−1t ̂

Wε(0,η,k) ∼

ε→0 e−iω′(k)ηt ˆ

W (0,η,k) W (t,y,k) = W (0,y − ω′(k) 2π t,k) Phonon of wave number k moves freely with velocity ω′(k)

2π .

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γ > 0 Explicit solution (microscopic)

J(t) = ∫T cos(ω(k)t) dk,

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γ > 0 Explicit solution (microscopic)

J(t) = ∫T cos(ω(k)t) dk, The Laplace transform of J(t) is given by ˜ J(λ) = ∫

∞

e−λtJ(t)dt = ∫T λ λ2 + ω2(k)dk. ˜ g(λ) = (1 + γ ˜ J(λ))

−1 = ∫ ∞

e−λtg(dt). ∣˜ g(λ)∣ < 1 φ(t,k) = ∫

t 0 eiω(k)τg(dτ)

→

t→∞ ˜

g(−iω(k)) ∶= ν(k) ˆ ψ(t,k) = e−iω(k)t[ ˆ ψ(0,k) − iγ ∫

t 0 φ(t − s,k)eiω(k)sp0 0(s) ds

+i √ 2γT ∫

t 0 φ(t − s,k)eiω(k)s dw(s)]

p0

0(i) : moment of particle 0 under the free evolution for γ = 0.

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Results γ > 0,W (0,y,k) = 0

ν(k) = (1 + γ ˜ J(−iω(k)))

−1

• T. Komorowski, L. Ryzhik, S.O., H. Spohn (2017).

k Phonons creation at y = 0 with rate 2γT∣ν(k)∣2: ∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = 2γT∣ν(k)∣2δ(y)

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Results for γ > 0

∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = Tγ∣ν(k)∣2δ(y) +∣ω′(k)∣ 2π (p+(k) − 1)W0 (−ω′(k)t 2π ,k)δ(y) +∣ω′(k)∣ 2π p−(k)W0 (ω′(k)t 2π ,−k)δ(y), W (0,y,k) = W0(y,k).

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Results for γ > 0

∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = Tγ∣ν(k)∣2δ(y) +∣ω′(k)∣ 2π (p+(k) − 1)W0 (−ω′(k)t 2π ,k)δ(y) +∣ω′(k)∣ 2π p−(k)W0 (ω′(k)t 2π ,−k)δ(y), W (0,y,k) = W0(y,k). p+(k) ∶= 1 + π2 [ω′(k)]2 γ2∣ν(k)∣2 − 2γπRe ν(k) ∣ω′(k)∣ = ∣1 − γπ ν(k) ∣ω′(k)∣∣

2

≥ 0. p−(k) ∶= π2 [ω′(k)]2 γ2∣ν(k)∣2.

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Asymptotics for γ → ∞

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Asymptotics for γ → ∞

▸ creation rate:

γ∣ν(k)∣2 →

γ→∞ 0

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Asymptotics for γ → ∞

▸ creation rate:

γ∣ν(k)∣2 →

γ→∞ 0

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∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = TS(k)δ(y) + 1k>0{(T(k) − A(k))W (t,0+,k) + R(k)W (t,0−,−k)}δ(y) + 1k<0{(T(k) − A(k))W (t,0−,k) + R(k)W (t,0+,−k)}δ(y) W (0,y,k) = W0(y,k). S(k) + T(k) − A(k) + R(k) = 0

• ⇒

W (t,y,k) = T is a stationary solution.

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Multimodal dispersion relation ω(k)

If ω(k) is not unimodal, scattering is more complex and an incident phonon of wave number k can produce a scattered phonon of different k′ ≠ k according to the solution of the equation ω(k′) = ω(k)

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When other bulk scattering is present

If we add a (slow) random scattering in the bulk of the system: random flip of sign of velocities, random exchange of velocity of n.n. particles, with rates ∼ γ′ε. Conjecture ∂tW (t,y,k) + ω′(k) 2π ∂yW (t,y,k) = TS(k)δ(y) + 1k>0{(T(k) − A(k))W (t,0+,k) + R(k)W (t,0−,−k)}δ(y) + 1k<0{(T(k) − A(k))W (t,0−,k) + R(k)W (t,0+,−k)}δ(y) + γ′LW (t,y,k) LW (k) = ∫ R(k,k′)(W (k′) − W (k)). Basile, O. Spohn, ARMA 2009, case γ = 0.

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Diffusive and superdiffusive behavior for γ′ > 0, γ = 0

▸ For random flip of velocities sign: R(k,k′) = 1, diffusive

behaviour: W (λ2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = D γ′ ∂yyρ, D = 1 4π2 ∫ ω′(k)(−L)−1ω′(k)dk

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Diffusive and superdiffusive behavior for γ′ > 0, γ = 0

▸ For random flip of velocities sign: R(k,k′) = 1, diffusive

behaviour: W (λ2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = D γ′ ∂yyρ, D = 1 4π2 ∫ ω′(k)(−L)−1ω′(k)dk

▸ For random exchange of nearest neighbor velocities

R(k,k′) ∼ ∣k∣2 for k ∼ 0, then

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Diffusive and superdiffusive behavior for γ′ > 0, γ = 0

▸ For random flip of velocities sign: R(k,k′) = 1, diffusive

behaviour: W (λ2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = D γ′ ∂yyρ, D = 1 4π2 ∫ ω′(k)(−L)−1ω′(k)dk

▸ For random exchange of nearest neighbor velocities

R(k,k′) ∼ ∣k∣2 for k ∼ 0, then

▸ if ω′(k) ∼ k (optical chain) : D < +∞, diffusive behaviour

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Diffusive and superdiffusive behavior for γ′ > 0, γ = 0

▸ For random flip of velocities sign: R(k,k′) = 1, diffusive

behaviour: W (λ2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = D γ′ ∂yyρ, D = 1 4π2 ∫ ω′(k)(−L)−1ω′(k)dk

▸ For random exchange of nearest neighbor velocities

R(k,k′) ∼ ∣k∣2 for k ∼ 0, then

▸ if ω′(k) ∼ k (optical chain) : D < +∞, diffusive behaviour ▸ if ω′(k) ∼ 1 (acustic chain) : D = +∞, superdiffusive behaviour

(Jara-Komorowski-Olla AAP 2009, Basile-Bovier MPRF 2010): W (λ3/2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = c∣∂yy∣3/4ρ,

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Diffusive behaviour with thermal BC: γ′ > 0, γ > 0

with Giada Basile and Tomasz Kimorowki, in progress In the cases of bulk diffusive behavior: W (λ2t,λy,k) →

λ→0 ρ(t,y)

∂tρ = Dγ′∂yyρ − (Dγ′ − Kγ)(∂yρ(t,0+) − ∂yρ(t,0−))δ(y), ρ(t,0+) = T = ρ(t,0−) Kγ = 1 4π ∫ sign(k)[T(k) − A(k) − R(k)](−L)−1ω′(k)dk, Dγ′ = 1 4π2γ′ ∫ ω′(k)(−L)−1ω′(k)dk

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