Energy Diffusion in a System of An-harmonic Oscillators Stefano - PowerPoint PPT Presentation

Energy Diffusion: heat equation Energy Diffusion in a System of An-harmonic Oscillators Stefano Olla – CEREMADE, Paris Makiko Sasada – Keio University, Tokyo Kochi, December 6, 2011 S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Chain of Anharmonic oscillators p i , q i ∈ R , i ∈ Λ, ∣ Λ ∣ = N or Λ = Z . H = ∑ [ p 2 2 + V ( q i − q i − 1 ) + U ( q j )] i i = ∑ e i i S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Chain of Anharmonic oscillators p i , q i ∈ R , i ∈ Λ, ∣ Λ ∣ = N or Λ = Z . H = ∑ [ p 2 2 + V ( q i − q i − 1 ) + U ( q j )] i i = ∑ e i i dq i = p i dt dp i = − ∂ q i H dt S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Chain of Anharmonic oscillators p i , q i ∈ R , i ∈ Λ, ∣ Λ ∣ = N or Λ = Z . H = ∑ [ p 2 2 + V ( q i − q i − 1 ) + U ( q j )] i i = ∑ e i i dq i = p i dt dp i = − ∂ q i H dt β = T − 1 > 0 dQ β = e − β H dpdq Z β S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation e i = p 2 2 + V ( q i − q i − 1 ) + U ( q i ) i Energy of atom i . S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation e i = p 2 2 + V ( q i − q i − 1 ) + U ( q i ) i Energy of atom i . e i = (  i − 1 , i −  i , i + 1 ) ˙ local conservation of energy. S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation e i = p 2 2 + V ( q i − q i − 1 ) + U ( q i ) i Energy of atom i . e i = (  i − 1 , i −  i , i + 1 ) ˙ local conservation of energy.  i , i + 1 = − p i V ′ ( q i + 1 − q i ) hamiltonian energy currents S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Non-stationary behavior We would like to prove that G ( i / N ) e i ( N 2 t ) � N →∞ ∫ G ( y ) u ( t , y ) dy N ∑ 1 → i with u ( t , y ) solution of the nonlinear heat equation: ∂ t u = ∂ y D ( u ) ∂ y u with the thermal conductivity defined by the Green-Kubo formula : D ( u ) = χ − 1 ⟨  i , i + 1 ( t )  0 , 1 ( 0 )⟩ β dt , β = β ( u ) ∞ β ∑ i ∈ Z ∫ 0 S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Non-stationary behavior We would like to prove that 1 N ∑ G ( i / N ) e i ( N 2 t ) � N →∞ ∫ G ( y ) u ( t , y ) dy → i with u ( t , y ) solution of the nonlinear heat equation: ∂ t u = ∂ y D( u ) ∂ y u with the thermal conductivity defined by the Green-Kubo formula : D( u ) = χ − 1 ∞ ⟨  i , i + 1 ( t )  0 , 1 ( 0 )⟩ β dt , β = β ( u ) β ∑ i ∈ Z ∫ 0 Not clear under which initial conditions such limit would be true S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Equilibrium Fluctuations: Linear response Here is a theorem that has a clear and precise mathematical statement: S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Equilibrium Fluctuations: Linear response Here is a theorem that has a clear and precise mathematical statement: Consider the system in equilibrium at temperature T = β − 1 , and perturbe it at time 0 in atom 0 by adding some energy there: S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Equilibrium Fluctuations: Linear response Here is a theorem that has a clear and precise mathematical statement: Consider the system in equilibrium at temperature T = β − 1 , and perturbe it at time 0 in atom 0 by adding some energy there: so we start with the measure β = e 0 dQ ′ < e 0 > β dQ β We want to study the time evolution of β, t = < e i ( t ) e 0 ( 0 ) > < e i ( t ) > Q ′ β = ∫ e i dQ ′ < e 0 > S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linear response Assuming that the corresponding limits exist, we have that β 2 χ ( β ) = < e 0 > β i 2 < e i ( t ) > Q ′ D = 1 κ t ∑ χ ( β ) lim t →∞ β i ∈ Z with χ ( β ) = ∑ i (< e i e 0 > β − < e i > β < e 0 > β ) . S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linear response Assuming that the corresponding limits exist, we have that β 2 χ ( β ) = < e 0 > β i 2 < e i ( t ) > Q ′ D = 1 κ t ∑ χ ( β ) lim t →∞ β i ∈ Z with χ ( β ) = ∑ i (< e i e 0 > β − < e i > β < e 0 > β ) . In fact, using stationarity and translation invariance i 2 < e i ( t ) > Q ′ i 2 < ( e i ( t ) − e i ( 0 )) e i ( 0 ) > β < e 0 > β ∑ β = ∑ i ∈ Z i ∈ Z = 2 ∫ 0 ds ∫ t 0 d τ ∑ s ⟨  i , i + 1 ( s − τ )  0 , 1 ( 0 )⟩ i � ∞ ⟨  i , i + 1 ( s )  0 , 1 ( 0 )⟩ ds ∑ t →∞ 2 ∫ → 0 i S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linearized heat equation Define C ( i , j , t ) =< e i ( t ) e j ( 0 ) > β − ¯ e 2 S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linearized heat equation Define C ( i , j , t ) =< e i ( t ) e j ( 0 ) > β − ¯ e 2 Conjecture: N →∞ ( 2 π D) − 1 / 2 exp (−( x − y ) 2 NC ([ Nx ] , [ Ny ] , N 2 t ) � ) → 2 t D S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linearized heat equation Define C ( i , j , t ) =< e i ( t ) e j ( 0 ) > β − ¯ e 2 Conjecture: N →∞ ( 2 π D) − 1 / 2 exp (−( x − y ) 2 NC ([ Nx ] , [ Ny ] , N 2 t ) � ) → 2 t D i.e. the limit follows the linearized heat equation ∂ t C = D ∂ xx C S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Linearized heat equation Define C ( i , j , t ) =< e i ( t ) e j ( 0 ) > β − ¯ e 2 Conjecture: N →∞ ( 2 π D) − 1 / 2 exp (−( x − y ) 2 NC ([ Nx ] , [ Ny ] , N 2 t ) � ) → 2 t D i.e. the limit follows the linearized heat equation ∂ t C = D ∂ xx C this is more challenging than proving existence for D . S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation How to prove this? Define, for a good choice of a sequence of smooth local functions F n Φ n =  0 , 1 − D( e 1 − e 0 ) − L F n with L the generator of the dynamics, S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation How to prove this? Define, for a good choice of a sequence of smooth local functions F n Φ n =  0 , 1 − D( e 1 − e 0 ) − L F n with L the generator of the dynamics, and pick a nice test function G ( x ) : G ( i N ) F ( j N )[ C ( i , j , N 2 t ) − C ( i , j , 0 )] N ∑ 1 i , j G ( i N ) F ( j N )⟨( e i ( N 2 t ) − e i ( 0 )) e j ( 0 )⟩ = 1 N ∑ i , j S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation How to prove this? Define, for a good choice of a sequence of smooth local functions F n Φ n =  0 , 1 − D( e 1 − e 0 ) − L F n with L the generator of the dynamics, and pick a nice test function G ( x ) : G ( i N ) F ( j N )[ C ( i , j , N 2 t ) − C ( i , j , 0 )] N ∑ 1 i , j G ( i N ) F ( j N )⟨( e i ( N 2 t ) − e i ( 0 )) e j ( 0 )⟩ = 1 N ∑ i , j = ∫ ∇ G ( i N ) F ( j N )⟨  i , i + 1 ( N 2 s ) e j ( 0 )⟩ ds 0 ∑ t i , j S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation ∆ G ( i N ) F ( j N )D ⟨ e i ( N 2 s ) e j ( 0 )⟩ ds t N ∑ = ∫ 1 0 i , j + ∫ N 2 ∑ ∇ G ( i N ) F ( j N )⟨( N 2 L ) τ i F n ( N 2 s ) e j ( 0 )⟩ ds t 1 0 i , j + ∫ ∇ G ( i N ) F ( j N )⟨ τ i Φ n ( N 2 s ) e j ( 0 )⟩ ds 0 ∑ t i , j S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation ∆ G ( i N ) F ( j N )D ⟨ e i ( N 2 s ) e j ( 0 )⟩ ds t N ∑ = ∫ 1 0 i , j + ∫ N 2 ∑ ∇ G ( i N ) F ( j N )⟨( N 2 L ) τ i F n ( N 2 s ) e j ( 0 )⟩ ds t 1 0 i , j + ∫ ∇ G ( i N ) F ( j N )⟨ τ i Φ n ( N 2 s ) e j ( 0 )⟩ ds 0 ∑ t i , j ∆ G ( i N ) F ( j N )D NC ( i , j , N 2 t ) ds ∼ ∫ t N 2 ∑ 1 0 i , j + 1 ∇ G ( i N ) F ( j N )⟨ τ i (F n ( N 2 t ) − F n ( 0 )) e j ( 0 )⟩ ds N 2 ∑ i , j + ∫ F ( j N )∇ G ( i N )⟨ 1 τ l Φ n ( N 2 s ) e j ( 0 )⟩ ds 0 ∑ t ∑ 2 k i , j ∣ i − l ∣≤ k S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Φ n =  0 , 1 − D( e 1 − e 0 ) − LF n 2 k ∑ Φ n , k = 1 ˆ τ j Φ n ∣ j ∣≤ k By Schwarz we can bound the square of the last term by e 2 ⟨(∫ G ′ ( i 2 ∥ F ∥ 2 ¯ 0 N ∑ N ) τ i ˆ Φ n , k ( N 2 s ) ds ) ⟩ t i G ′ ( i 2 = C ⟨(∫ N 2 t N ) τ i ˆ Φ n , k ( s ) ds ) ⟩ N ∑ 1 0 i S. Olla - CEREMADE Energy diffusion

Energy Diffusion: heat equation Φ n =  0 , 1 − D( e 1 − e 0 ) − LF n 2 k ∑ Φ n , k = 1 ˆ τ j Φ n ∣ j ∣≤ k By Schwarz we can bound the square of the last term by e 2 ⟨(∫ G ′ ( i 2 ∥ F ∥ 2 ¯ 0 N ∑ N ) τ i ˆ Φ n , k ( N 2 s ) ds ) ⟩ t i G ′ ( i 2 = C ⟨(∫ N 2 t N ) τ i ˆ Φ n , k ( s ) ds ) ⟩ N ∑ 1 0 i We are left to prove that this is negligeable as N → ∞ , k → ∞ and n → ∞ . S. Olla - CEREMADE Energy diffusion