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 S. Olla - CEREMADE thermal-scattering

thermal boundaries Thermal boundaries appear in macroscopic equations for the evolution of energy or temperatures profiles. T + T − S. Olla - CEREMADE 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: ∂ t u ( t , y ) = ∂ y ( D ( u ) ∂ y u ( t , y )) , y ∈ [ 0 , L ] u ( t , 0 ) = T + , u ( t , L ) = T − . S. Olla - CEREMADE thermal-scattering

Microscopic modeling: Langevin stochastic thermostats q ( t ) , p ( t ) positions and velocities of the system at time t : d q ( t ) = p ( t ) dt √ d p ( t ) = − ∂ q V ( q ( t )) − γ p ( t ) dt + 2 γ Td w ( t ) The equilibrium measure is the canonical Gibbs at temperature T : e − H ( p , q )/ T H ( p , q ) = p 2 2 + V ( q ) . , Z T S. Olla - CEREMADE thermal-scattering

Microscopic modeling: Langevin stochastic thermostats q ( t ) , p ( t ) positions and velocities of the system at time t : d q ( t ) = p ( t ) dt √ d p ( t ) = − ∂ q V ( q ( t )) − γ p ( t ) dt + 2 γ Td w ( t ) The equilibrium measure is the canonical Gibbs at temperature T : e − H ( p , q )/ T H ( p , q ) = p 2 2 + V ( q ) . , Z T What is the effect of the thermostat in the macroscopic evolution, after a rescaling of space and time? S. Olla - CEREMADE thermal-scattering

A kinetic limit Consider an infinite one dimensional chain of coupled harmonic oscillators, with a Langevin thermostat at site 0. p = { p x , x ∈ Z } , q = { q x , x ∈ Z } H( p , q ) ∶ = 1 2 ∑ x + 1 2 ∑ p 2 x , x ′ α x − x ′ q x q x ′ x S. Olla - CEREMADE thermal-scattering

A kinetic limit Consider an infinite one dimensional chain of coupled harmonic oscillators, with a Langevin thermostat at site 0. p = { p x , x ∈ Z } , q = { q x , x ∈ Z } H( p , q ) ∶ = 1 2 ∑ x + 1 2 ∑ p 2 x , x ′ α x − x ′ q x q x ′ x d q x ( t ) = p x ( t ) dt , x ∈ Z √ d p x ( t ) = −( α ∗ q ( t )) x +(− γ p 0 ( t ) dt + 2 γ Tdw ( t )) δ 0 , x , where { w ( t ) , t ≥ 0 } is a standard Wiener process. S. Olla - CEREMADE thermal-scattering

Coupled Harmonic Oscillators f ( k ) = ∑ k ∈ Π ∼ [ 0 , 1 ] f x e − i 2 π kx ˆ x α ( k ) ∈ C ∞ ( Π ) . √ ▸ ˆ ▸ ω ( k ) = α ( k ) : dispersion relation. S. Olla - CEREMADE thermal-scattering

Coupled Harmonic Oscillators f ( k ) = ∑ k ∈ Π ∼ [ 0 , 1 ] f x e − i 2 π kx ˆ x α ( k ) ∈ C ∞ ( Π ) . √ ▸ ˆ ▸ ω ( k ) = α ( k ) : dispersion relation. In the n.n. unpinned chain ( acoustic chain ) : ω ( k ) = ∣ sin ( π k )∣ . S. Olla - CEREMADE thermal-scattering

Coupled Harmonic Oscillators f ( k ) = ∑ k ∈ Π ∼ [ 0 , 1 ] f x e − i 2 π kx ˆ x α ( k ) ∈ C ∞ ( Π ) . √ ▸ ˆ ▸ ω ( k ) = α ( k ) : dispersion relation. In the n.n. unpinned chain ( acoustic chain ) : ω ( k ) = ∣ sin ( π k )∣ . ψ ( t , k ) ∶ = ω ( k ) ˆ q ( t , k ) + i ˆ p ( t , k ) . ˆ S. Olla - CEREMADE thermal-scattering

Coupled Harmonic Oscillators f ( k ) = ∑ k ∈ Π ∼ [ 0 , 1 ] f x e − i 2 π kx ˆ x α ( k ) ∈ C ∞ ( Π ) . √ ▸ ˆ ▸ ω ( k ) = α ( k ) : dispersion relation. In the n.n. unpinned chain ( acoustic chain ) : ω ( k ) = ∣ sin ( π k )∣ . ψ ( t , k ) ∶ = ω ( k ) ˆ q ( t , k ) + i ˆ p ( t , k ) . ˆ √ ψ ( t , k ) = − i ω ( k ) ˆ ψ ( t , k ) dt − i γ p 0 ( t ) dt + i 2 γ Tdw ( t ) d ˆ S. Olla - CEREMADE thermal-scattering

Coupled Harmonic Oscillators f ( k ) = ∑ k ∈ Π ∼ [ 0 , 1 ] f x e − i 2 π kx ˆ x α ( k ) ∈ C ∞ ( Π ) . √ ▸ ˆ ▸ ω ( k ) = α ( k ) : dispersion relation. In the n.n. unpinned chain ( acoustic chain ) : ω ( k ) = ∣ sin ( π k )∣ . ψ ( t , k ) ∶ = ω ( k ) ˆ q ( t , k ) + i ˆ p ( t , k ) . ˆ √ ψ ( t , k ) = − i ω ( k ) ˆ ψ ( t , k ) dt − i γ p 0 ( t ) dt + i 2 γ Tdw ( t ) d ˆ In integral form ψ ( t , k ) = e − i ω ( k ) t ˆ ψ ( 0 , k ) − i γ ∫ 0 e − i ω ( k )( t − s ) p 0 ( s ) ds t ˆ √ + i 0 e − i ω ( k )( t − s ) dw ( t ) 2 γ T ∫ t S. Olla - CEREMADE thermal-scattering

Wigner distribution η ∈ R , ψ ∗ ( ε − 1 t , k − εη ̂ W ε ( t ,η, k ) ∶ = ε 2 E [ ˆ 2 ) ˆ ψ ( ε − 1 t , k + εη 2 )] W ε ( t ,η, k ) ∶ = ̂ ̂ W ε ( t , − η, k ) ∗ and the inverse Fourier transform in η ̂ W ε ( t , y , k ) = ∫ W ε ( t ,η, k ) e i 2 πη y d η ∈ R , y ∈ R , S. Olla - CEREMADE thermal-scattering

Wigner distribution η ∈ R , ψ ∗ ( ε − 1 t , k − εη ̂ W ε ( t ,η, k ) ∶ = ε 2 E [ ˆ 2 ) ˆ ψ ( ε − 1 t , k + εη 2 )] W ε ( t ,η, k ) ∶ = ̂ ̂ W ε ( t , − η, k ) ∗ and the inverse Fourier transform in η ̂ W ε ( t , y , k ) = ∫ W ε ( t ,η, k ) e i 2 πη y d η ∈ R , y ∈ R , W ε ( t , y , k ) ⇀ ε → 0 W ( t , y , k ) ≥ 0 , as distribution When γ = 0 it is easy to prove that ∂ t W ( t , y , k ) + ω ′ ( k ) 2 π ∂ y W ( t , y , k ) = 0 S. Olla - CEREMADE thermal-scattering

Easy case: γ = 0, no thermostat From waves to particle: ψ ( t , k ) = ˆ ψ ( 0 , k ) e − i ω ( k ) t ˆ S. Olla - CEREMADE thermal-scattering

Easy case: γ = 0, no thermostat From waves to particle: ψ ( t , k ) = ˆ ψ ( 0 , k ) e − i ω ( k ) t ˆ ψ ∗ ( ε − 1 t , k − εη W ε ( t ,η, k ) ∶ = ε E [ ˆ 2 ) ˆ ψ ( ε − 1 t , k + εη 2 )] ˆ S. Olla - CEREMADE thermal-scattering

Easy case: γ = 0, no thermostat From waves to particle: ψ ( t , k ) = ˆ ψ ( 0 , k ) e − i ω ( k ) t ˆ 2 )] ε − 1 t ̂ ̂ W ε ( t ,η, k ) ∶ = e i [ ω ( k − εη W ε ( 0 ,η, k ) 2 )− ω ( k + εη W ( 0 ,η, k ) ∼ ε → 0 e − i ω ′ ( k ) η t ˆ S. Olla - CEREMADE thermal-scattering

Easy case: γ = 0, no thermostat From waves to particle: ψ ( t , k ) = ˆ ψ ( 0 , k ) e − i ω ( k ) t ˆ 2 )] ε − 1 t ̂ ̂ W ε ( t ,η, k ) ∶ = e i [ ω ( k − εη W ε ( 0 ,η, k ) 2 )− ω ( k + εη W ( 0 ,η, k ) ∼ ε → 0 e − i ω ′ ( k ) η t ˆ W ( t , y , k ) = W ( 0 , y − ω ′ ( k ) 2 π t , k ) Phonon of wave number k moves freely with velocity ω ′ ( k ) 2 π . S. Olla - CEREMADE thermal-scattering

γ > 0 Explicit solution (microscopic) J ( t ) = ∫ T cos ( ω ( k ) t ) dk , S. Olla - CEREMADE thermal-scattering

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 − λ t J ( t ) dt = ∫ T λ ˜ λ 2 + ω 2 ( k ) dk . 0 − 1 = ∫ ∞ ∣ ˜ g ( λ )∣ < 1 g ( λ ) = ( 1 + γ ˜ J ( λ )) e − λ t g ( dt ) . ˜ 0 φ ( t , k ) = ∫ 0 e i ω ( k ) τ g ( d τ ) � g ( − i ω ( k )) ∶ = ν ( k ) t t →∞ ˜ → ψ ( t , k ) = e − i ω ( k ) t [ ˆ ψ ( 0 , k ) − i γ ∫ 0 φ ( t − s , k ) e i ω ( k ) s p 0 t 0 ( s ) ds ˆ √ 2 γ T ∫ 0 φ ( t − s , k ) e i ω ( k ) s dw ( s )] t + i 0 ( i ) : moment of particle 0 under the free evolution for γ = 0. p 0 S. Olla - CEREMADE thermal-scattering

Results γ > 0 , W ( 0 , y , k ) = 0 ν ( k ) = ( 1 + γ ˜ J (− i ω ( k ))) − 1 T. Komorowski, L. Ryzhik, S.O., H. Spohn (2017). k Phonons creation at y = 0 with rate 2 γ T ∣ ν ( k )∣ 2 : ∂ t W ( t , y , k ) + ω ′ ( k ) 2 π ∂ y W ( t , y , k ) = 2 γ T ∣ ν ( k )∣ 2 δ ( y ) S. Olla - CEREMADE thermal-scattering

Results for γ > 0 ∂ t W ( t , y , k ) + ω ′ ( k ) 2 π ∂ y W ( t , y , k ) = T γ ∣ ν ( k )∣ 2 δ ( y ) + ∣ ω ′ ( k )∣ ( p + ( k ) − 1 ) W 0 ( − ω ′ ( k ) t , k ) δ ( y ) 2 π 2 π + ∣ ω ′ ( k )∣ p − ( k ) W 0 ( ω ′ ( k ) t , − k ) δ ( y ) , 2 π 2 π W ( 0 , y , k ) = W 0 ( y , k ) . S. Olla - CEREMADE thermal-scattering

Results for γ > 0 ∂ t W ( t , y , k ) + ω ′ ( k ) 2 π ∂ y W ( t , y , k ) = T γ ∣ ν ( k )∣ 2 δ ( y ) + ∣ ω ′ ( k )∣ ( p + ( k ) − 1 ) W 0 ( − ω ′ ( k ) t , k ) δ ( y ) 2 π 2 π + ∣ ω ′ ( k )∣ p − ( k ) W 0 ( ω ′ ( k ) t , − k ) δ ( y ) , 2 π 2 π W ( 0 , y , k ) = W 0 ( y , k ) . [ ω ′ ( k )] 2 γ 2 ∣ ν ( k )∣ 2 − 2 γπ Re ν ( k ) ∣ ω ′ ( k )∣ = ∣ 1 − γπ ν ( k ) 2 π 2 p + ( k ) ∶ = 1 + ∣ ω ′ ( k )∣∣ ≥ 0 . π 2 p − ( k ) ∶ = [ ω ′ ( k )] 2 γ 2 ∣ ν ( k )∣ 2 . S. Olla - CEREMADE thermal-scattering

Asymptotics for γ → ∞ S. Olla - CEREMADE thermal-scattering

Asymptotics for γ → ∞ ▸ creation rate: γ ∣ ν ( k )∣ 2 � γ →∞ 0 → S. Olla - CEREMADE thermal-scattering

Asymptotics for γ → ∞ ▸ creation rate: γ ∣ ν ( k )∣ 2 � γ →∞ 0 → S. Olla - CEREMADE thermal-scattering

∂ t W ( t , y , k ) + ω ′ ( k ) 2 π ∂ y W ( t , y , k ) = TS ( k ) δ ( y ) + 1 k > 0 {( T ( k ) − A ( k )) W ( t , 0 + , k ) + R ( k ) W ( t , 0 − , − k )} δ ( y ) + 1 k < 0 {( T ( k ) − A ( k )) W ( t , 0 − , k ) + R ( k ) W ( t , 0 + , − k )} δ ( y ) W ( 0 , y , k ) = W 0 ( y , k ) . S ( k ) + T ( k ) − A ( k ) + R ( k ) = 0 � ⇒ W ( t , y , k ) = T is a stationary solution. S. Olla - CEREMADE thermal-scattering