Euristics take G ∶ [ 0 , 1 ] → R with compact support in ( 0 , 1 ) , ∇ p x − 1 ( Nt ) r x ( Nt ) ⎛ ⎞ ⎛ ⎞ = ∑ ∇ V ′ ( r x ( Nt )) d N ∑ 1 ⎜ p x ( Nt ) ⎟ ⎜ ⎟ G ( x / N ) G ( x / N ) ⎝ ⎠ ⎝ ⎠ E x ( Nt ) ∇ [ p x ( Nt ) V ′ ( r x ( Nt )] dt x x p x ( Nt ) ⎛ ⎞ ∼ − 1 N ∑ G ′ ( x / N ) ⎜ V ′ ( r x ( Nt )) ⎟ ⎝ ⎠ p x ( Nt ) V ′ ( r x ( Nt ) x S. Olla - CEREMADE hyperbolic limits
Euristics take G ∶ [ 0 , 1 ] → R with compact support in ( 0 , 1 ) , ∇ p x − 1 ( Nt ) r x ( Nt ) ⎛ ⎞ ⎛ ⎞ = ∑ ∇ V ′ ( r x ( Nt )) d N ∑ 1 ⎜ p x ( Nt ) ⎟ ⎜ ⎟ G ( x / N ) G ( x / N ) ⎝ ⎠ ⎝ ⎠ E x ( Nt ) ∇ [ p x ( Nt ) V ′ ( r x ( Nt )] dt x x p x ( Nt ) ⎛ ⎞ ∼ − 1 N ∑ G ′ ( x / N ) ⎜ V ′ ( r x ( Nt )) ⎟ ⎝ ⎠ p x ( Nt ) V ′ ( r x ( Nt ) x assuming local equilibrium, we have p ( t , y ) ⎛ ⎞ ∼ − ∫ 0 G ′ ( y ) 1 ⎜ τ ( u ( t , y ) , r ( t , y )) ⎟ dy ⎝ ⎠ p ( t , y ) τ ( u ( t , y ) , r ( t , y )) Note that y ∈ [ 0 , 1 ] is the material (Lagrangian) coordinate. S. Olla - CEREMADE hyperbolic limits
Results with conservative stochastic dynamics ▸ To prove some form of local equilibrium we need to add stochastic terms to the dynamics (the deterministic non-linear case is too difficult). S. Olla - CEREMADE hyperbolic limits
Results with conservative stochastic dynamics ▸ To prove some form of local equilibrium we need to add stochastic terms to the dynamics (the deterministic non-linear case is too difficult). ▸ Random exchanges of velocities between nearest neighbor particles, conserve energy, momentum and volume, destroying all other (possible) conservation laws. It provides the right ergodicity property. S. Olla - CEREMADE hyperbolic limits
Results with conservative stochastic dynamics ▸ To prove some form of local equilibrium we need to add stochastic terms to the dynamics (the deterministic non-linear case is too difficult). ▸ Random exchanges of velocities between nearest neighbor particles, conserve energy, momentum and volume, destroying all other (possible) conservation laws. It provides the right ergodicity property. ▸ With such noise in the dynamics, for smooth solutions the HL is proven in: ▸ N. Even, S.O., ARMA (2014) (with boundary conditions), ▸ S.O., SRS Varadhan, HT Yau, CMP (1993) (periodic bc). S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain This is an example of a non-ergodic dynamics: V ( r ) = r 2 / 2 in fact it is a completely integrable dynamics : q x = p x , p x = ∆ q x = q x + 1 + q x − 1 − q x , ˙ ˙ S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain This is an example of a non-ergodic dynamics: V ( r ) = r 2 / 2 in fact it is a completely integrable dynamics : q x = p x , p x = ∆ q x = q x + 1 + q x − 1 − q x , ˙ ˙ Take here x = 1 ,..., N , f ( k ) = ∑ k ∈ { 0 , 1 / N ,..., ( N − 1 )/ N } ˆ f x e i 2 π kx x ω ( k ) = 2 ∣ sin ( π k )∣ dispersion relation: q ( k )∣ 2 + ∣ ˆ H = ∑ E E E x = 1 [ ω ( k ) 2 ∣ ˆ p ( k )∣ 2 ] 2 N ∑ x k S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain This is an example of a non-ergodic dynamics: V ( r ) = r 2 / 2 in fact it is a completely integrable dynamics : q x = p x , p x = ∆ q x = q x + 1 + q x − 1 − q x , ˙ ˙ Take here x = 1 ,..., N , f ( k ) = ∑ k ∈ { 0 , 1 / N ,..., ( N − 1 )/ N } ˆ f x e i 2 π kx x ω ( k ) = 2 ∣ sin ( π k )∣ dispersion relation: q ( k )∣ 2 + ∣ ˆ H = ∑ E E E x = 1 [ ω ( k ) 2 ∣ ˆ p ( k )∣ 2 ] 2 N ∑ x k ψ ( t , k ) ∶= ω ( k ) ˆ q ( t , k ) + i ˆ p ( t , k ) . ˆ S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain This is an example of a non-ergodic dynamics: V ( r ) = r 2 / 2 in fact it is a completely integrable dynamics : q x = p x , p x = ∆ q x = q x + 1 + q x − 1 − q x , ˙ ˙ Take here x = 1 ,..., N , f ( k ) = ∑ k ∈ { 0 , 1 / N ,..., ( N − 1 )/ N } ˆ f x e i 2 π kx x ω ( k ) = 2 ∣ sin ( π k )∣ dispersion relation: q ( k )∣ 2 + ∣ ˆ H = ∑ E E E x = 1 [ ω ( k ) 2 ∣ ˆ p ( k )∣ 2 ] 2 N ∑ x 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 ) ψ ( t , k ) = e − i ω ( k ) t ˆ ψ ( 0 , k ) d ˆ ˆ dt S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain: Quantum Dynamics p x = − i ∂ q x = − i ( ∂ r x + 1 − ∂ r x ) E E E x = 1 2 ( p 2 x + r 2 x ) a k = ψ ( k ) , k = ψ ( k ) ∗ 1 1 a ∗ ˆ ˆ ω ( k ) ω ( k ) q ( k )∣ 2 + ∣ ˆ [ ω ( k ) 2 ∣ ˆ p ( k )∣ 2 ] H = ∑ E E E x = 1 2 N ∑ x k = 1 ω ( k ) a ∗ 2 N ∑ k a k k S. Olla - CEREMADE hyperbolic limits
Harmonic Oscillators Chain: Quantum Dynamics p x = − i ∂ q x = − i ( ∂ r x + 1 − ∂ r x ) E E E x = 1 2 ( p 2 x + r 2 x ) a k = ψ ( k ) , k = ψ ( k ) ∗ 1 1 a ∗ ˆ ˆ ω ( k ) ω ( k ) q ( k )∣ 2 + ∣ ˆ [ ω ( k ) 2 ∣ ˆ p ( k )∣ 2 ] H = ∑ E E E x = 1 2 N ∑ x k = 1 ω ( k ) a ∗ 2 N ∑ k a k k dt A ( t ) = i [ H , A ( t )] Heisenber evolution d a k ( t ) = e − i ω ( k ) t a k , k ( t ) = e − i ω ( k ) t a ∗ a ∗ k . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Thermal Equilibrium (Classic case) Consider the chain in thermal equilibrium: initial distribution with covariances ⟨ ⟨ ⟨ r x ( 0 ) ; r x ′ ( 0 ) ⟩ ⟩ ⟩ = ⟨ ⟨ ⟨ p x ( 0 ) ; p x ′ ( 0 ) ⟩ ⟩ ⟩ = β − 1 δ x , x ′ , ⟨ ⟨ ⟨ q x ; p x ′ ⟩ ⟩ = 0 , ⟩ for some inverse temperature β , while in mechanical local equilibrium : ⟨ ⟨ ⟨ r [ Ny ] ( 0 ) ⟩ ⟩ ⟩ � → r ( 0 , y ) , ⟨ ⟨ ⟨ p [ Ny ] ( 0 ) ⟩ ⟩ ⟩ � → p ( 0 , y ) . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Thermal Equilibrium (classic case) thermal equilibrium is conserved by the dynamics: for any t ≥ 0 ⟨ ⟨ ⟨ r x ( t ) ; r x ′ ( t ) ⟩ ⟩ ⟩ = ⟨ ⟨ p x ( t ) ; p x ′ ( t ) ⟩ ⟨ ⟩ ⟩ = β − 1 δ x , x ′ , ⟨ ⟨ ⟨ q x ( t ) ; p x ′ ( t ) ⟩ ⟩ ⟩ = 0 , Proof. Thermal equilibrium is Fourier space is: ⟨ ⟨ ⟨ ˆ ψ ( k , 0 ) ∗ ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 2 β − 1 δ ( k − k ′ ) , ⟨ ⟨ ⟨ ˆ ψ ( k , 0 ) ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 0 . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Thermal Equilibrium (classic case) thermal equilibrium is conserved by the dynamics: for any t ≥ 0 ⟨ ⟨ ⟨ r x ( t ) ; r x ′ ( t ) ⟩ ⟩ ⟩ = ⟨ ⟨ ⟨ p x ( t ) ; p x ′ ( t ) ⟩ ⟩ ⟩ = β − 1 δ x , x ′ , ⟨ ⟨ ⟨ q x ( t ) ; p x ′ ( t ) ⟩ ⟩ = 0 , ⟩ Proof. Thermal equilibrium is Fourier space is: ⟨ ⟨ ⟨ ˆ ψ ( k , 0 ) ∗ ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 2 β − 1 δ ( k − k ′ ) , ⟨ ⟨ ⟨ ˆ ψ ( k , 0 ) ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 0 . Consequently ⟩ = e i ( ω ( k )− ω ( k ′ )) t ⟨ ⟨ ⟨ ⟨ ˆ ψ ( k , t ) ∗ ; ˆ ψ ( k ′ , t ) ⟩ ⟩ ⟨ ⟨ ˆ ψ ( k , 0 ) ∗ ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 2 β − 1 δ ( k − k ′ ) ⟩ = e − i ( ω ( k )+ ω ( k ′ )) t ⟨ ⟨ ⟨ ˆ ⟨ ψ ( k , t ) ; ˆ ψ ( k ′ , t ) ⟩ ⟩ ⟨ ⟨ ˆ ψ ( k , 0 ) ; ˆ ψ ( k ′ , 0 ) ⟩ ⟩ ⟩ = 0 . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Thermal Equilibrium implies Euler Equation limit r [ Ny ] ( Nt ) and p [ Ny ] ( Nt ) converge weakly to the solution of the linear wave equation ∂ t r ( y , t ) = ∂ y p ( y , t ) , ∂ t p ( y , t ) = ∂ y r ( y , t ) . This is the Euler equation for this system since here τ ( u , r ) = r . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Thermal Equilibrium implies Euler Equation limit r [ Ny ] ( Nt ) and p [ Ny ] ( Nt ) converge weakly to the solution of the linear wave equation ∂ t r ( y , t ) = ∂ y p ( y , t ) , ∂ t p ( y , t ) = ∂ y r ( y , t ) . This is the Euler equation for this system since here τ ( u , r ) = r . For the energy, because of the thermal equilibrium, for any t ≥ 0 : ⟩ = β − 1 + 1 ⟩ 2 + ⟨ ⟨ E ⟨ ⟨ E E x ( t ) ⟩ ⟩ 2 (⟨ ⟨ ⟨ p x ( t ) ⟩ ⟩ ⟨ ⟨ r x ( t ) ⟩ ⟩ ⟩ 2 ) → e ( y , t ) = β − 1 + 1 ⟨ ⟨ ⟨ E E E [ Ny ] ( Nt ) ⟩ ⟩ ⟩ � 2 ( p 2 ( y , t ) + r 2 ( y , t )) , ∂ t e ( y , t ) = ∂ y ( p ( y , t ) r ( y , t )) . S. Olla - CEREMADE hyperbolic limits
Quantum Harmonic Chain: Thermal Equilibrium Initial density matrix ρ β , define ⟨ ⟨ ⟨ A ⟩ ⟩ ⟩ = tr ( A ρ β )) , ⟨ ⟨ ⟨ A ; B ⟩ ⟩ ⟩ = ⟨ ⟨ AB ⟩ ⟨ ⟩ − ⟨ ⟩ ⟨ ⟨ A ⟩ ⟩ ⟩⟨ ⟨ ⟨ B ⟩ ⟩ ⟩ such that ⟨ r x ( 0 ) ; r x ′ ( 0 ) ⟩ ⟨ ⟨ ⟩ ⟩ = ⟨ ⟨ ⟨ p x ( 0 ) ; p x ′ ( 0 ) ⟩ ⟩ ⟩ = C β ( x − x ′ ) , ⟨ ⟨ ⟨ q x ; p x ′ ⟩ ⟩ = i ⟩ 2 δ ( x − x ′ ) N [ β − 1 + ∑ C β ( x ) = 1 e 2 π ikx ( e βω k − 1 + ω k 2 )] ω k (1) k ≠ 0 S. Olla - CEREMADE hyperbolic limits
Quantum Harmonic Chain: Thermal Equilibrium Initial density matrix ρ β , define ⟨ ⟨ ⟨ A ⟩ ⟩ ⟩ = tr ( A ρ β )) , ⟨ ⟨ ⟨ A ; B ⟩ ⟩ = ⟨ ⟩ ⟨ ⟨ AB ⟩ ⟩ ⟩ − ⟨ ⟨ ⟨ A ⟩ ⟩ ⟩⟨ ⟨ B ⟩ ⟨ ⟩ ⟩ such that ⟨ ⟨ ⟨ r x ( 0 ) ; r x ′ ( 0 ) ⟩ ⟩ ⟩ = ⟨ ⟨ p x ( 0 ) ; p x ′ ( 0 ) ⟩ ⟨ ⟩ ⟩ = C β ( x − x ′ ) , ⟨ ⟨ ⟨ q x ; p x ′ ⟩ ⟩ ⟩ = i 2 δ ( x − x ′ ) N [ β − 1 + ∑ C β ( x ) = 1 e 2 π ikx ( e βω k − 1 + ω k 2 )] ω k (1) k ≠ 0 ⟨ ⟨ ⟨ r [ Ny ] ( 0 ) ⟩ ⟩ ⟩ � → r ( 0 , y ) , ⟨ ⟨ ⟨ p [ Ny ] ( 0 ) ⟩ ⟩ ⟩ � → p ( 0 , y ) . ⟨ E ⟨ ⟨ E E [ Ny ] ⟩ ⟩ ⟩ � → e ( y ) = ¯ C ( β ) + 1 2 ( p 2 ( y ) + r 2 ( y )) , C ( β ) = ∫ 0 ω ( k )( e βω ( k ) − 1 + 1 2 ) dk 1 1 ∼ β → 0 β − 1 ¯ S. Olla - CEREMADE hyperbolic limits
Quantum Harmonic Chain: Thermal Equilibrium implies Euler Equation limit r [ Ny ] ( Nt ) and p [ Ny ] ( Nt ) converge weakly to the solution of the linear wave equation ∂ t r ( y , t ) = ∂ y p ( y , t ) , ∂ t p ( y , t ) = ∂ y r ( y , t ) . ⟨ ⟨ ⟨ E E E [ Ny ] ( Nt ) ⟩ ⟩ ⟩ � → e ( y , t ) = ¯ C ( β ) + 1 2 ( p 2 ( y , t ) + r 2 ( y , t )) , C ( β ) = ∫ 0 ω ( k )( e βω ( k ) − 1 + 1 2 ) dk 1 1 ∼ β → 0 β − 1 ¯ ∂ t e ( y , t ) = ∂ y ( p ( y , t ) r ( y , t )) . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain: Local Thermal Equilibrium is not conserved The argument fails dramatically if the system is not in thermal equilibrium, even local thermal Gibbs ⟩ = β − 1 ( x ⟨ ⟨ ⟨ r x ( 0 ) ; r x ′ ( 0 ) ⟩ ⟩ ⟩ = ⟨ ⟨ ⟨ p x ( 0 ) ; p x ′ ( 0 ) ⟩ ⟩ N ) δ x , x ′ , ⟨ ⟨ ⟨ q x ( 0 ) ; p x ′ ( 0 ) ⟩ ⟩ ⟩ = 0 (2) is not conserved, and correlations between p x ( t ) and r x ( t ) build up in time. No autonomous macroscopic equation for the energy! There are infinite many conservation laws. S. Olla - CEREMADE hyperbolic limits
Wigner distribution ξ ∈ R , k ∈ [ 0 , 1 ] , ψ ∗ ( Nt , k − ξ ̂ W N ( ξ, k , t ) ∶= ψ ( Nt , k + ξ N ⟨ 2 ⟨ ˆ ⟨ 2 N ) ˆ 2 N ) ⟩ ⟩ ⟩ ̂ W N ( y , k , t ) = ∫ y ∈ R , W N ( t ,η, k ) e − i 2 πξ y d η, S. Olla - CEREMADE hyperbolic limits
Wigner distribution ξ ∈ R , k ∈ [ 0 , 1 ] , ψ ∗ ( Nt , k − ξ ̂ W N ( ξ, k , t ) ∶= ψ ( Nt , k + ξ N ⟨ 2 ⟨ ˆ ⟨ 2 N ) ˆ 2 N ) ⟩ ⟩ ⟩ ̂ W N ( y , k , t ) = ∫ y ∈ R , W N ( t ,η, k ) e − i 2 πξ y d η, In the limit it decompose in a thermal and a mechanical part: W N ( ξ, k , t ) = ̂ ̂ W th ( ξ, k , t ) + ̂ W m ( ξ, t ) δ 0 ( dk ) lim (3) N → ∞ The mechanical part ̂ W m ( ξ, t ) is the Fourier transform of the mechanical energy ̂ W m ( ξ, t ) = ∫ 2 ( p 2 ( y , t ) + r 2 ( y , t )) e i 2 πξ y dy , 1 S. Olla - CEREMADE hyperbolic limits
Wigner distribution For the thermal Wigner distribution it holds the transport equation: ∂ t W th ( y , k , t ) + ω ′ ( k ) 2 π ∂ y W th ( y , k , t ) = 0 . S. Olla - CEREMADE hyperbolic limits
Wigner distribution For the thermal Wigner distribution it holds the transport equation: ∂ t W th ( y , k , t ) + ω ′ ( k ) 2 π ∂ y W th ( y , k , t ) = 0 . in fact for k ≠ 0 2 N )] Nt ̂ ̂ W N ( ξ, k , t ) ∶= e i [ ω ( k − ξ W N ( ξ, k , 0 ) 2 N )− ω ( k + ξ N → ∞ e − i ω ′ ( k ) ξ t ̂ W th ( ξ, k , 0 ) ∼ S. Olla - CEREMADE hyperbolic limits
Wigner distribution For the thermal Wigner distribution it holds the transport equation: ∂ t W th ( y , k , t ) + ω ′ ( k ) 2 π ∂ y W th ( y , k , t ) = 0 . in fact for k ≠ 0 2 N )] Nt ̂ ̂ W N ( ξ, k , t ) ∶= e i [ ω ( k − ξ W N ( ξ, k , 0 ) 2 N )− ω ( k + ξ N → ∞ e − i ω ′ ( k ) ξ t ̂ W th ( ξ, k , 0 ) ∼ 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 hyperbolic limits
Wigner distribution e ( y , t ) (i.e. temperature) evolves Consequently the thermal energy ˜ non autonomously following the equation e ( y , t ) + ∂ y J ( y , t ) = 0 , J ( y , t ) = ∫ ω ′ ( k ) W th ( y , k , t ) dk . ∂ t ˜ We say that the system is in local equilibrium if W th ( y , k ) = β − 1 ( y ) constant in k . Starting in thermal equilibrium means W th ( y , k , 0 ) = β − 1 and trivially W th ( y , k , t ) = β − 1 for any t > 0. But starting with local equilibrium, i.e. W ( y , k , 0 ) = β − 1 ( y ) e ( y , t ) . constant in k , we have a non autonomous evolution of ˜ S. Olla - CEREMADE hyperbolic limits
Harmonic Chain with Random Masses The problem with the harmonic chain is that thermal waves of wavenumber k move with speed ω ′ ( k ) , if they are not uniformed distributed (i.e. the system is not in thermal equilibrium), the temperature profile will not remain constant, as it should be following the Euler equations. S. Olla - CEREMADE hyperbolic limits
Harmonic Chain with Random Masses The problem with the harmonic chain is that thermal waves of wavenumber k move with speed ω ′ ( k ) , if they are not uniformed distributed (i.e. the system is not in thermal equilibrium), the temperature profile will not remain constant, as it should be following the Euler equations. If the masses are random, the thermal modes remains localized (frozen), by Anderson localization. This allows to close the energy equation, without local equilibrium . S. Olla - CEREMADE hyperbolic limits
Harmonic Chain with Random Masses (F. Huveneers, C. Bernardin, S.Olla, 2017) { m x } i.i.d. with absolutely continuous distribution, 0 < m − ≤ m x ≤ m + , m = E ( m x ) . q x ( t ) = p x ( t ) , p x ( t ) = ∆ q x ( t ) , x = 1 ,..., N m x ˙ ˙ with q 0 = q 1 and q N + 1 = q N as boundary conditions. S. Olla - CEREMADE hyperbolic limits
Gibbs States, Local Gibbs States The Gibbs states are characterized by three parameters: β > 0 and p , r ∈ R . Its probability density writes 2 ( px m ) 2 − β mx mx − p − β 2 ( r x − r ) 2 N ∏ e Z ( β, p , r , m x ) . x = 1 S. Olla - CEREMADE hyperbolic limits
Gibbs States, Local Gibbs States The Gibbs states are characterized by three parameters: β > 0 and p , r ∈ R . Its probability density writes 2 ( px m ) 2 − β mx mx − p − β 2 ( r x − r ) 2 N ∏ e Z ( β, p , r , m x ) . x = 1 We start with a local Gibbs state: ( px ) 2 − β ( x / N ) mx mx − p ( x / N ) − β ( x / N ) ( r x − r ( x / N )) 2 N ∏ 2 m 2 e Z ( β ( x / N ) , p ( x / N ) , r ( x / N ) , m x ) . x = 1 S. Olla - CEREMADE hyperbolic limits
Harmonic Chain with Random Masses: hydrodynamic limit Almost surely with respect to { m x } : < r [ Ny ] ( Nt ) > , < p [ Ny ] ( Nt ) > , < E E E [ Ny ] ( Nt ) > ⇀ ( r ( y , t ) , p ( y , t ) , e ( y , t )) ∂ t r ( t , y ) = 1 m ∂ y p ( t , y ) ∂ t p ( t , y ) = ∂ y r ( t , y ) ∂ t e ( t , y ) = 1 m ∂ y ( r ( t , y ) p ( t , y )) with initial conditions: β ( y )+ p 2 ( y ) 2 m + r 2 ( y ) r ( y , 0 ) = r ( y ) , p ( y , 0 ) = p ( y ) , e ( y , 0 ) = 1 . 2 S. Olla - CEREMADE hyperbolic limits
Random Masses: Localization of Thermal Modes Equation of motion can be written as r x = −(∇ ∗ M − 1 ∇ r ) x ( 1 ≤ x ≤ N − 1 ) , p x = ( ∆ M − 1 p ) x ( 1 ≤ x ≤ N ) , ¨ ¨ where M x , x ′ = δ x , x ′ m x . S. Olla - CEREMADE hyperbolic limits
Random Masses: Localization of Thermal Modes Equation of motion can be written as r x = −(∇ ∗ M − 1 ∇ r ) x ( 1 ≤ x ≤ N − 1 ) , p x = ( ∆ M − 1 p ) x ( 1 ≤ x ≤ N ) , ¨ ¨ where M x , x ′ = δ x , x ′ m x . M − 1 / 2 (− ∆ ) M 1 / 2 ϕ k = ω 2 k = 0 ,..., N − 1 . k ϕ k , ψ k = M − 1 / 2 ϕ k , M − 1 ∆ ψ k = ω 2 k ψ k S. Olla - CEREMADE hyperbolic limits
Random Masses: Localization of Thermal Modes Equation of motion can be written as r x = −(∇ ∗ M − 1 ∇ r ) x ( 1 ≤ x ≤ N − 1 ) , p x = ( ∆ M − 1 p ) x ( 1 ≤ x ≤ N ) , ¨ ¨ where M x , x ′ = δ x , x ′ m x . M − 1 / 2 (− ∆ ) M 1 / 2 ϕ k = ω 2 k = 0 ,..., N − 1 . k ϕ k , ψ k = M − 1 / 2 ϕ k , M − 1 ∆ ψ k = ω 2 k ψ k (⟨∇ ψ k , r ( 0 )⟩ cos ω k t + ⟨ ψ k , p ( 0 )⟩ sin ω k t )∇ ψ k N − 1 r ( t ) = ∑ , ω k ω k k = 1 (⟨ ψ k , p ( 0 )⟩ cos ω k t − ⟨∇ ψ k , r ( 0 )⟩ N − 1 p ( t ) = sin ω k t ) M ψ k . ∑ ω k k = 0 S. Olla - CEREMADE hyperbolic limits
Localization of Thermal Modes Localization length ξ k diverges with N : 2 k ∼ ( k N ) ∼ ω 2 ξ − 1 , k √ only the modes k > N are localized. S. Olla - CEREMADE hyperbolic limits
Localization of Thermal Modes Localization length ξ k diverges with N : 2 k ∼ ( k N ) ∼ ω 2 ξ − 1 , k √ only the modes k > N are localized. More precisely: for 0 < α < 1 2 E ⎛ x ′ ∣⎞ N − 1 ∣ ψ k ∑ ⎠ ≤ Ce − cN − 2 α ∣ x − x ′ ∣ . x ψ k ⎝ k = N 1 − α This estimate is enough to prove that thermal modes remains localized and do not move macroscopically. S. Olla - CEREMADE hyperbolic limits
Random masses: Larger time scales Assume for simplicity that we are in a mechanical equilibrium : ⟨ ⟨ ⟨ r x ( 0 ) ⟩ ⟩ ⟩ = 0 , ⟨ ⟨ ⟨ p x ( 0 ) ⟩ ⟩ = 0 , ⟩ (only thermal energy present) but not in thermal equilibrium, then, for any α ≥ 1 < E E E [ Ny ] ( N α t ) > � N to ∞ e ( 0 , y ) = ¯ C ( β ( y )) → NO evolution for the temperature profile at any scale! S. Olla - CEREMADE hyperbolic limits
Random masses: Larger time scales Assume for simplicity that we are in a mechanical equilibrium : ⟨ ⟨ ⟨ r x ( 0 ) ⟩ ⟩ ⟩ = 0 , ⟨ ⟨ ⟨ p x ( 0 ) ⟩ ⟩ ⟩ = 0 , (only thermal energy present) but not in thermal equilibrium, then, for any α ≥ 1 < E E E [ Ny ] ( N α t ) > � N to ∞ e ( 0 , y ) = ¯ C ( β ( y )) → NO evolution for the temperature profile at any scale! In particular, for α = 2 (diffusive scaling), thermal diffusivity is null. S. Olla - CEREMADE hyperbolic limits
Open questions for the quantum case ▸ In order to deal with the anharmonic interaction, in the classical case, conservative noise is added to obtain ergodicity of the infinite dynamics (unique characterization of the translational invariant stationary states) ( cf B. Nachtergaele, and H-T Yau, CMP 2003). How to add conservative noise in the quantum dynamics in order to obtain similar result? S. Olla - CEREMADE hyperbolic limits
Open questions for the quantum case ▸ In order to deal with the anharmonic interaction, in the classical case, conservative noise is added to obtain ergodicity of the infinite dynamics (unique characterization of the translational invariant stationary states) ( cf B. Nachtergaele, and H-T Yau, CMP 2003). How to add conservative noise in the quantum dynamics in order to obtain similar result? ▸ Boundary tension? More generally boundary conditions, thermostat etc. S. Olla - CEREMADE hyperbolic limits
entropy evolution ∂ t r = ∂ x p ∂ t p = ∂ x τ ∂ t e = ∂ x ( τ p ) p ( t , 0 ) = 0 , τ ( r ( 1 , t ) , u ( 1 , t )) = τ ( t ) U = e − p 2 / 2, β = ∂ S ∂ U , τ = − 1 ∂ S β ∂ r S. Olla - CEREMADE hyperbolic limits
entropy evolution ∂ t r = ∂ x p ∂ t p = ∂ x τ ∂ t e = ∂ x ( τ p ) p ( t , 0 ) = 0 , τ ( r ( 1 , t ) , u ( 1 , t )) = τ ( t ) U = e − p 2 / 2, β = ∂ S ∂ U , τ = − 1 ∂ S β ∂ r For smooth solutions: dt S ( u ( y , t ) , r ( y , t )) = β ( ∂ t e − p ∂ t p ) − βτ∂ t r d = β ( ∂ x ( τ p ) − p ∂ x τ − τ∂ x p ) = 0 The evolution is isoentropic in the smooth regime. S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, ▸ contact discontinuities: discontinuities in the entropy profile. When this happens we have to consider weak solution , that typically are not unique. S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, ▸ contact discontinuities: discontinuities in the entropy profile. When this happens we have to consider weak solution , that typically are not unique. In order to select the right physical solutions , various properties (maybe equivalent) have been introduced: S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, ▸ contact discontinuities: discontinuities in the entropy profile. When this happens we have to consider weak solution , that typically are not unique. In order to select the right physical solutions , various properties (maybe equivalent) have been introduced: S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, ▸ contact discontinuities: discontinuities in the entropy profile. When this happens we have to consider weak solution , that typically are not unique. In order to select the right physical solutions , various properties (maybe equivalent) have been introduced: ▸ entropy solutions S. Olla - CEREMADE hyperbolic limits
Shocks, contact discontinuities, weak solutions, entropy solutions Even starting with initial smooth profiles, hyperbolic non-linear systems develops discontinuities: ▸ shocks: discontinuities in the tension profile, ▸ contact discontinuities: discontinuities in the entropy profile. When this happens we have to consider weak solution , that typically are not unique. In order to select the right physical solutions , various properties (maybe equivalent) have been introduced: ▸ entropy solutions ▸ viscosity solutions S. Olla - CEREMADE hyperbolic limits
weak solutions Consider a hyperbolic system of conservation laws v t + f ( v ) x = 0 , a weak solution v ( t , y ) on an open set Ω ⊂ R 2 satisfies, for any function φ ( t , y ) ∈ C 1 with compact support in Ω ∬ Ω [ φ t v + φ y f ( v )] dy dt = 0 No continuity assumption is made on v . S. Olla - CEREMADE hyperbolic limits
weak solutions Consider a hyperbolic system of conservation laws v t + f ( v ) x = 0 , a weak solution v ( t , y ) on an open set Ω ⊂ R 2 satisfies, for any function φ ( t , y ) ∈ C 1 with compact support in Ω ∬ Ω [ φ t v + φ y f ( v )] dy dt = 0 No continuity assumption is made on v . In the Euler case, v = ( r , p , e ) , u = e − p 2 / 2 and ⎛ ⎞ p f ( v ) = − ⎜ τ ( u , r ) ⎟ ⎝ ⎠ p τ ( u , r ) Strictly Hyperbolic System: the Jacobian matrix Df has real distinct eigenvalues. S. Olla - CEREMADE hyperbolic limits
weak solutions: Cauchy initial problem A weak solution of v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , is a weak solution of the Cauchy initial data problem if t ∈ [ 0 , T ] → v ( t , ⋅ ) is continuous in L 1 loc . S. Olla - CEREMADE hyperbolic limits
weak solutions: Cauchy initial problem A weak solution of v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , is a weak solution of the Cauchy initial data problem if t ∈ [ 0 , T ] → v ( t , ⋅ ) is continuous in L 1 loc . unfortunately it may not be unique! Existence proved only for v 0 of bounded variation (Glimm,....). S. Olla - CEREMADE hyperbolic limits
Entropic weak solutions v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n S. Olla - CEREMADE hyperbolic limits
Entropic weak solutions v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n A C 1 function η ∶ R n → R is an entropy function with entropy flux q ∶ R n → R , if D η ( v ) ⋅ Df ( v ) = Dq ( v ) that implies for smooth solutions: η ( v ) t + q ( v ) x = 0 . S. Olla - CEREMADE hyperbolic limits
Entropic weak solutions v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n A C 1 function η ∶ R n → R is an entropy function with entropy flux q ∶ R n → R , if D η ( v ) ⋅ Df ( v ) = Dq ( v ) that implies for smooth solutions: η ( v ) t + q ( v ) x = 0 . ▸ n = 1: any convex non-linear η is an entropy function , S. Olla - CEREMADE hyperbolic limits
Entropic weak solutions v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n A C 1 function η ∶ R n → R is an entropy function with entropy flux q ∶ R n → R , if D η ( v ) ⋅ Df ( v ) = Dq ( v ) that implies for smooth solutions: η ( v ) t + q ( v ) x = 0 . ▸ n = 1: any convex non-linear η is an entropy function , ▸ n ≥ 3: ? It may nont exists S. Olla - CEREMADE hyperbolic limits
Entropic weak solutions v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n A C 1 function η ∶ R n → R is an entropy function with entropy flux q ∶ R n → R , if D η ( v ) ⋅ Df ( v ) = Dq ( v ) that implies for smooth solutions: η ( v ) t + q ( v ) x = 0 . ▸ n = 1: any convex non-linear η is an entropy function , ▸ n ≥ 3: ? It may nont exists ▸ For the Euler System: the thermodynamic entropy η ( v ) = S ( e − p 2 / 2 , r ) is an entropy function . S. Olla - CEREMADE hyperbolic limits
Entropic weak solution v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n An weak solution is entropy-admissible if η ( v ) t + q ( v ) x ≤ 0 as distribution, for any entropy pair ( η, q ) . S. Olla - CEREMADE hyperbolic limits
Entropic weak solution v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n An weak solution is entropy-admissible if η ( v ) t + q ( v ) x ≤ 0 as distribution, for any entropy pair ( η, q ) . This implies that total entropy ∫ η ( v ( t , y )) dy increase in time (with no b.c. here). S. Olla - CEREMADE hyperbolic limits
Entropic weak solution v t + f ( v ) x = 0 , v ( 0 , y ) = v 0 ( y ) , v ( t , y ) ∈ R n An weak solution is entropy-admissible if η ( v ) t + q ( v ) x ≤ 0 as distribution, for any entropy pair ( η, q ) . This implies that total entropy ∫ η ( v ( t , y )) dy increase in time (with no b.c. here). Existence is proven only under bounded variation initial conditions. The conjecture is that entropy-admissible solutions are unique . S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity solutions t + f ( v ε ) x = ε v ε v ε xx , or more general t + f ( v ε ) x = ε Λ ( v ε ) , v ε where Λ is a second order differential operator (eventually non-linear). S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity solutions t + f ( v ε ) x = ε v ε v ε xx , or more general t + f ( v ε ) x = ε Λ ( v ε ) , v ε where Λ is a second order differential operator (eventually non-linear). loc as ε → 0 + , this is called a vanishing viscosity If v ε converges in L 1 solution . S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity solutions t + f ( v ε ) x = ε v ε v ε xx , or more general t + f ( v ε ) x = ε Λ ( v ε ) , v ε where Λ is a second order differential operator (eventually non-linear). loc as ε → 0 + , this is called a vanishing viscosity If v ε converges in L 1 solution . Bianchini-Bressan (AoM, 2005) : if initial data are of small BV, limit exists unique and is BV and is an entropy solution, (for linear viscosity). S. Olla - CEREMADE hyperbolic limits
Isothermal Dynamics ▸ J. Fritz, Microscopic theory of isothermal elasticity , ARMA 2011, infinite volume S. Olla - CEREMADE hyperbolic limits
Isothermal Dynamics ▸ J. Fritz, Microscopic theory of isothermal elasticity , ARMA 2011, infinite volume ▸ S. Marchesani, S. Olla, Nonlinearity 2018, boundary conditions. The system is in contact with a heat bath that keeps it at a constant temperature β − 1 . Energy is not conserved anymore. Macroscopically we have a p-system: ∂ t r ( t , y ) = ∂ y p ( t , y ) ∂ t p ( t , y ) = ∂ y τ [ β, r ( t , y )] S. Olla - CEREMADE hyperbolic limits
MIcroscopic isothermal dynamics √ ⎧ dr 1 = Np 1 dt + N σ N ( V ′ ( r 2 ) − V ′ ( r 1 )) dt − 2 β − 1 N σ N d ̃ ⎪ w 1 ⎪ √ ⎪ ⎪ ⎪ ⎪ dr i = N ( p i − p i − 1 ) dt + N σ N ( V ′ ( r i + 1 ) + V ′ ( r i − 1 ) − 2 V ′ ( r i )) dt + 2 β − 1 N σ N ( d ̃ w i − 1 − d ̃ w i ) ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ dr N = N ( p N − p N − 1 ) dt + N σ N ( V ′ ( r N − 1 ) − V ′ ( r N )) dt + 2 β − 1 N σ d ̃ w N − 1 √ ⎨ , ⎪ dp 1 = N ( V ′ ( r 2 ) − V ′ ( r 1 )) dt + N σ N ( p 2 − p 1 ) dt − 2 β − 1 N σ N dw 1 ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ dp i = N ( V ′ ( r i + 1 ) − V ′ ( r i )) dt + N σ N ( p i + 1 + p i − 1 − 2 p i ) dt + 2 β − 1 N σ N ( dw i − 1 − dw i ) ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎩ dp N = N ( ¯ τ ( t ) − V ′ ( r N )) dt + N σ N ( p N − 1 − p N ) dt + 2 β − 1 N σ N dw N − 1 , S. Olla - CEREMADE hyperbolic limits
MIcroscopic isothermal dynamics √ ⎧ dr 1 = Np 1 dt + N σ N ( V ′ ( r 2 ) − V ′ ( r 1 )) dt − 2 β − 1 N σ N d ̃ ⎪ w 1 ⎪ √ ⎪ ⎪ ⎪ ⎪ dr i = N ( p i − p i − 1 ) dt + N σ N ( V ′ ( r i + 1 ) + V ′ ( r i − 1 ) − 2 V ′ ( r i )) dt + 2 β − 1 N σ N ( d ̃ w i − 1 − d ̃ w i ) ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ dr N = N ( p N − p N − 1 ) dt + N σ N ( V ′ ( r N − 1 ) − V ′ ( r N )) dt + 2 β − 1 N σ d ̃ w N − 1 √ ⎨ , ⎪ dp 1 = N ( V ′ ( r 2 ) − V ′ ( r 1 )) dt + N σ N ( p 2 − p 1 ) dt − 2 β − 1 N σ N dw 1 ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ dp i = N ( V ′ ( r i + 1 ) − V ′ ( r i )) dt + N σ N ( p i + 1 + p i − 1 − 2 p i ) dt + 2 β − 1 N σ N ( dw i − 1 − dw i ) ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎩ dp N = N ( ¯ τ ( t ) − V ′ ( r N )) dt + N σ N ( p N − 1 − p N ) dt + 2 β − 1 N σ N dw N − 1 , N = lim σ N 2 = 0 . σ N N lim N → +∞ N → ∞ S. Olla - CEREMADE hyperbolic limits
Isothermal dynamics, generator G ¯ ∶= NL ¯ + N σ N ( S N + ˜ S N ) . τ ( t ) τ ( t ) N N N − 1 N = ( p i − p i − 1 ) ∂ r i + ( V ′ ( r i + 1 ) − V ′ ( r i )) ∂ p i +( ¯ τ ( t )− V ′ ( r N )) ∂ p N , ∑ ∑ τ ( t ) L ¯ N i = 1 i = 1 S. Olla - CEREMADE hyperbolic limits
Isothermal dynamics, generator G ¯ ∶= NL ¯ + N σ N ( S N + ˜ S N ) . τ ( t ) τ ( t ) N N N − 1 N = ( p i − p i − 1 ) ∂ r i + ( V ′ ( r i + 1 ) − V ′ ( r i )) ∂ p i +( ¯ τ ( t )− V ′ ( r N )) ∂ p N , ∑ ∑ τ ( t ) L ¯ N i = 1 i = 1 N − 1 N − 1 S N ∶= − S N ∶= − ∑ ∑ D ∗ D ∗ ˜ ˜ i ˜ i D i , D i , i = 1 i = 1 D i ∶= − ∂ i ∶= p i + 1 − p i − β − 1 D i ∂ D ∗ , ∂ p i + 1 ∂ p i D i ∶= − ∂ i ∶= V ′ ( r i + 1 ) − V ′ ( r i ) − β − 1 ˜ ∂ D ∗ ˜ ˜ , D i . ∂ r i + 1 ∂ r i S. Olla - CEREMADE hyperbolic limits
Initial distribution with respect to µ N = µ N The density f N β, 0 , 0 solves the Fokker-Plank t equation ∗ ∂ f N ∂ t = (G ¯ ) τ ( t ) t f N t . N ∗ Here (G ¯ ) = − NL ¯ + N ¯ τ ( t ) p N + N σ ( S N + ˜ S N ) is the adjoint τ ( t ) τ ( t ) N N of G ¯ τ ( t ) with respect to µ N . N S. Olla - CEREMADE hyperbolic limits
Initial distribution with respect to µ N = µ N The density f N β, 0 , 0 solves the Fokker-Plank t equation ∗ ∂ f N ∂ t = (G ¯ ) τ ( t ) t f N t . N ∗ Here (G ¯ ) = − NL ¯ + N ¯ τ ( t ) p N + N σ ( S N + ˜ S N ) is the adjoint τ ( t ) τ ( t ) N N of G ¯ τ ( t ) with respect to µ N . N relative entropy H N ( f N t ) ∶= ∫ R 2 N f N t log f N t d µ N S. Olla - CEREMADE hyperbolic limits
Initial distribution with respect to µ N = µ N The density f N β, 0 , 0 solves the Fokker-Plank t equation ∗ ∂ f N ∂ t = (G ¯ ) τ ( t ) t f N t . N ∗ Here (G ¯ ) = − NL ¯ + N ¯ τ ( t ) p N + N σ ( S N + ˜ S N ) is the adjoint τ ( t ) τ ( t ) N N of G ¯ τ ( t ) with respect to µ N . N relative entropy H N ( f N t ) ∶= ∫ R 2 N f N t log f N t d µ N assume or the initial distribution H N ( f N 0 ) ≤ CN . S. Olla - CEREMADE hyperbolic limits
Limite Hydrodynamique G ( x / N )( r x ( t ) 0 G ( y )( r ( y , t ) p x ( t )) � p ( y , t )) dy N ∑ 1 1 → N → ∞ ∫ x L 2 -valued weak solution of ∂ t r ( t , y ) = ∂ y p ( t , y ) ∂ t p ( t , y ) = ∂ y τ β [ r ( t , y )] p ( t , 0 ) = 0 , τ ( r ( t , 1 )) = ¯ τ ( t ) , in the sense 0 ( r ( t , x ) ∂ t ϕ ( t , x ) − p ( t , x ) ∂ x ϕ ( t , x )) dx dt = 0 1 ∞ ∫ ∫ 0 0 ( p ( t , x ) ∂ t ψ ( t , x ) − τ β ( r ( t , x )) ∂ x ψ ( t , x )) dx dt = 0 1 ∫ ∞ ∫ 0 for all functions ϕ,ψ with compact support on R + ∖ { 0 } × ( 0 , 1 ) . NO information on initial and boundary conditions, no entropy condition. S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity The heat bath interaction in the dynamics plays the role of a microscopic viscosity , vanishing in the macroscopic limit. S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity The heat bath interaction in the dynamics plays the role of a microscopic viscosity , vanishing in the macroscopic limit. The corresponding viscous equations would be: ⎧ ⎪ ∂ t r ε ( t , x ) − ∂ x p ε ( t , x ) = ε∂ xx τ β ( r ε ( t , x )) x ∈ ( 0 , 1 ) ⎪ ⎨ ⎪ ∂ t p ε ( t , x ) − ∂ x τ β ( r ε ( t , x )) = ε∂ xx p ε ( t , x ) , ⎪ ⎩ with boundary conditions p ε ( t , 0 ) = 0 , τ ( r ε ( t , 1 )) = ¯ τ ( t ) , ∂ x p ε ( t , 1 ) = 0 , ∂ x r ε ( t , 0 ) = 0 , S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity The heat bath interaction in the dynamics plays the role of a microscopic viscosity , vanishing in the macroscopic limit. The corresponding viscous equations would be: ⎧ ⎪ ∂ t r ε ( t , x ) − ∂ x p ε ( t , x ) = ε∂ xx τ β ( r ε ( t , x )) x ∈ ( 0 , 1 ) ⎪ ⎨ ⎪ ∂ t p ε ( t , x ) − ∂ x τ β ( r ε ( t , x )) = ε∂ xx p ε ( t , x ) , ⎪ ⎩ with boundary conditions p ε ( t , 0 ) = 0 , τ ( r ε ( t , 1 )) = ¯ τ ( t ) , ∂ x p ε ( t , 1 ) = 0 , ∂ x r ε ( t , 0 ) = 0 , Note the non-linear viscosity term. S. Olla - CEREMADE hyperbolic limits
Vanishing viscosity The heat bath interaction in the dynamics plays the role of a microscopic viscosity , vanishing in the macroscopic limit. The corresponding viscous equations would be: ⎧ ⎪ ∂ t r ε ( t , x ) − ∂ x p ε ( t , x ) = ε∂ xx τ β ( r ε ( t , x )) x ∈ ( 0 , 1 ) ⎪ ⎨ ⎪ ∂ t p ε ( t , x ) − ∂ x τ β ( r ε ( t , x )) = ε∂ xx p ε ( t , x ) , ⎪ ⎩ with boundary conditions p ε ( t , 0 ) = 0 , τ ( r ε ( t , 1 )) = ¯ τ ( t ) , ∂ x p ε ( t , 1 ) = 0 , ∂ x r ε ( t , 0 ) = 0 , Note the non-linear viscosity term. As ε → 0 boundary layers may appear. S. Olla - CEREMADE hyperbolic limits
The p-system It is usually difficult to control bounds in the vanishing viscosity ε → 0, Bressan-Bianchini can do it for the BV if viscosity is taken linear. For the p-system with no boundaries ∂ t r ( t , y ) = ∂ y p ( t , y ) ∂ t p ( t , y ) = ∂ y τ β [ r ( t , y )] can be proven existence of L ∞ weak solutions (Di Perna), and L 2 valued solutions (Schearer, Serre-Shearer). S. Olla - CEREMADE hyperbolic limits
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