Thermal conductivity for a chain of harmonic oscillators in a - PowerPoint PPT Presentation

. . Thermal conductivity for a chain of harmonic oscillators in a magnetic field . . . . . Makiko Sasada The University of Tokyo IHP, June 15, 2017 Stochastic Dynamics Out of Equilibrium Joint work with Keiji Saito and Shuji Tamaki (Keio University)

Introduction Microscopic Model : System of oscillators (Hamiltonian dynamics) + magnetic field + stochastic noise Goal : Understand • Behavior of macroscopic energy diffusion • In particular, the anomalous diffusion in d = 1 , 2 • In this talk, we only consider the order of the divergence of thermal conductivity • Role of “momentum conservation” • Role of “sound velocity”

Transport of energy T T R L j N Thermal conductivity in a stationary non-equilibrium state: NJ ( T L − T R ) ∼ N α κ N = J : current per a particle Normal transport : α = 0, κ N → κ < ∞ Fourier’s law : j ( x , t ) = − κ∂ x T ( x , t ) Diffusion equation : ∂ t T ( x , t ) = κ c ∆ T ( x , t ) Anomalous transport : 0 < α < 1 (or κ N ∼ log N ) Diffusion equation: ∂ t T ( x , t ) = − c ( − ∆) c α T ( x , t ) ?? (Ballistic transport : α = 1)

Model : system of harmonic oscillators (periodic b.c.) • Z d N = Z d / N Z d N ( d ∗ is not necessarily equal to d ) • q x , p x ∈ R d ∗ , x ∈ Z d x { | p x | 2 | q x − q y | 2 • H = ∑ + ∑ } =: ∑ x E x | y − x | =1 2 4 Hamiltonian dynamics (deterministic) :  dq k x x H = p k ( k = 1 , . . . , d ∗ ) dt = ∂ p k   x  (0) dp k x x H = (∆ q k ) x ( k = 1 , . . . , d ∗ )  dt = − ∂ q k   | y − x | =1 ( F y − F x ) for F : Z d where (∆ F ) x = ∑ N → R * The energy transport is ballistic for the deterministic dynamics

Magnetic field for d ∗ = 2 Consider the system in a magnetic field with strength B and its direction is orthogonal to the plane where oscillators move. Model (I) : Uniform • Each oscillator has a uniform charge • Operator : G I = ∑ x ( p 2 x − p 1 x ∂ p 1 x ∂ p 2 x ) • Generator of the deterministic part : L = A + BG I Model (II) : Alternative • Assume N is even and d = 1 • Each oscillator has a charge with uniform absolute value but its sign is alternative in x • Operator : G II = ∑ x ( − 1) x ( p 2 x − p 1 x ∂ p 1 x ∂ p 2 x ) • Generator of the deterministic part : L = A + BG II . Remark . . . We can also consider the term G comes from Coriolis force in Model (I). . . . . .

Alternative Uniform p Chain of Oscillators in a magnetic field ( d = 1 , d ∗ = 2)

Model : system of harmonic oscillators in a magnetic field with uniform charge • B ̸ = 0: strength of the magnetic field N ( d ∗ ≥ 2) • q x , p x ∈ R d ∗ , x ∈ Z d x { | p x | 2 | q x − q y | 2 • H = ∑ + ∑ } | y − x | =1 2 4 Hamiltonian dynamics + magnetic field (deterministic) : dq k  x x H = p k ( k = 1 , . . . , d ∗ ) dt = ∂ p k  x      dp 1   x x H + Bp 2 x = (∆ q 1 ) x + Bp 2 dt = − ∂ q 1   x  ( I ) dp 2 x x H − Bp 1 x = (∆ q 2 ) x − Bp 1  dt = − ∂ q 2  x      dp k   x x H = (∆ q k ) x ( k = 3 , . . . , d ∗ )  dt = − ∂ q k 

Model : system of harmonic oscillators in a magnetic field with alternative charge • B ̸ = 0: strength of the magnetic field • N : even, d = 1 • q x , p x ∈ R d ∗ , x ∈ Z N ( d ∗ ≥ 2) x { | p x | 2 | q x − q y | 2 • H = ∑ + ∑ } | y − x | =1 2 4 Hamiltonian dynamics + magnetic field (deterministic) : dq k  x x H = p k ( k = 1 , . . . , d ∗ ) dt = ∂ p k  x      dp 1   x x H +( − 1) x Bp 2 x = (∆ q 1 ) x +( − 1) x Bp 2 dt = − ∂ q 1   x  ( II ) dp 2 x x H − ( − 1) x Bp 1 x = (∆ q 2 ) x − ( − 1) x Bp 1  dt = − ∂ q 2  x      dp k   x ( k = 3 , . . . , d ∗ ) x H = (∆ q k ) x dt = − ∂ q k  

Conserved quantities • E x := | p x | 2 | q x − q y | 2 + ∑ : energy of x 2 | y − x | =1 4 • H = ∑ x E x x p k x ( k = 1 , 2 , . . . , d ∗ ) Model (0): ∑ x E x , ∑ x p k x ( k = 3 , . . . , d ∗ ), ∑ x ( p 1 x − Bq 2 Model (I) : ∑ ∑ x E x , x ), x ( p 2 x + Bq 1 ∑ x ) x p k x ( k = 3 , . . . , d ∗ ), Model (II) : ∑ x E x , ∑ x : even ( p 1 x + p 1 x +1 − Bq 2 x + Bq 2 x : even ( p 2 x + p 2 x +1 + Bq 1 x − Bq 1 ∑ x +1 ), ∑ x +1 ) x p 1 x p 2 • ∑ x and ∑ x are not conserved for both ( I ) and ( II ) • The number of conserved quantities are same for all models • Precisely, there are infinitely many conserved quantities without the stochastic noise

Micro-canonical state space and micro-canonical measure • Ω N , E := { ( q x , p x ) ∈ ( R 2 d ∗ ) N d ; ∑ x E x = E N d } x q x = 0 , ∑ x p x = 0 , ∑ • µ N , E : Uniform measure on Ω N , E . • ⟨·⟩ N , E : Expectation w.r.t. µ N , E • Ω N , E and µ N , E are invariant for Model (0) and (I) • For Model (II), we do not consider the micro-canonical state space for simplicity. • For the coordinates ( q x , p x ) with periodic b.c., ∫ exp( − β H ( q , p )) dqdp = ∞ for any β > 0, so we can not consider the canonical measure.

System of harmonic oscillators (periodic b.c.) for ( r , p )-coordinates • d = 1 • r x , p x ∈ R d ∗ , x ∈ Z N • Change the coordinates with r x = q x +1 − q x formally in the dynamics (but q x + N = q x does not hold here) Hamiltonian dynamics (deterministic) :  dr k dt = p k x x +1 − p k ( k = 1 , . . . , d ∗ )   x  (0) dp k ( k = 1 , . . . , d ∗ )  dt = r k x x − r k   x − 1

System of harmonic oscillators in a magnetic field for ( r , p )-coordinates with uniform charge • B ̸ = 0 • d = 1 • r x , p x ∈ R d ∗ , x ∈ Z N ( d ∗ ≥ 2) Hamiltonian dynamics + magnetic field (deterministic) : dr k  ( k = 1 , . . . , d ∗ ) dt = p k x x +1 − p k  x      dp 1   dt = r 1 x − r 1 x − 1 + Bp 2 x   x  ( I ) dp 2 dt = r 2 x x − r 2 x − 1 − Bp 1    x     dp k   dt = r k x x − r k ( k = 3 , . . . , d ∗ )   x − 1

System of harmonic oscillators in a magnetic field for ( r , p )-coordinates with alternative charge • B ̸ = 0 • N : even, d = 1 • r x , p x ∈ R d ∗ , x ∈ Z N ( d ∗ ≥ 2) Hamiltonian dynamics + magnetic field (deterministic) : dr k  ( k = 1 , . . . , d ∗ ) dt = p k x x +1 − p k  x      dp 1   dt = r 1 x − r 1 x − 1 +( − 1) x Bp 2 x   x  ( II ) dp 2 dt = r 2 x x − r 2 x − 1 − ( − 1) x Bp 1    x     dp k   dt = r k x x − r k ( k = 3 , . . . , d ∗ )   x − 1

Conserved quantities x ( k = 1 , 2 , . . . , d ∗ ), ∑ x ( k = 1 , 2 , . . . , d ∗ ) Model (0): ∑ x E x , ∑ x p k x r k x ( k = 3 , . . . , d ∗ ), ∑ x ( k = 1 , 2 , . . . , d ∗ ) x p k x r k Model (I) : ∑ x E x , ∑ x p k x ( k = 3 , . . . , d ∗ ), ∑ x r k x ( k = 1 , 2 , . . . , d ∗ ), Model (II) : ∑ x E x , ∑ x : even ( p 1 x + p 1 x +1 + Br 2 x : even ( p 2 x + p 2 x +1 − Br 1 ∑ x ), ∑ x ) x ( p 1 x − Bq 2 x ( p 2 x + Bq 1 • ∑ x ) and ∑ x ) are not functions of ( r , p ) • The number of conserved quantities are different between Model (0),(II) and Model (I) • If d = 1 and d ∗ = 2, Model (0) and (II) have five conserved quantities, but Model (I) has only three conserved quantities

Canonical state space and canonical measure • Ω N := { ( r x , p x ) ∈ ( R 2 d ∗ ) N } = R 2 d ∗ N 1 • µ N ,β ( drdp ) = Z β exp( − β ∑ x E x ) drdp = x ) 2 +( p k x ) 2 2 π exp( − β ( r k β Π x Π d ∗ ) dr k x dp k x for β > 0. k =1 2 • ⟨·⟩ N ,β : Expectation w.r.t. µ N ,β • µ N ,β is invariant for Model (0),(I) and (II) • The measure is product because d = 1

Stochastic noise (momentum exchange) For each k ∈ { 1 , . . . , d ∗ } and a pair x , y ∈ Z d N satisfying | x − y | = 1, exchange p k x ↔ p k y with rate γ > 0. • Every conserved quantity is also conserved by the stochastic noise • Micro-canonical (resp. canonical) state spaces and measures are still invariant with the stochastic noise Full generator of our dynamics: L = A + BG + γ S where G ( I ) = G ( II ) = ∑ ( p 2 x − p 1 ∑ ( − 1) x ( p 2 x − p 1 x ∂ p 1 x ∂ p 2 x ) , x ∂ p 1 x ∂ p 2 x ) x x d ∗ ∑ ∑ ∑ ( f ( q , p x , y , k ) − f ( q , p )) Sf = k =1 x | y − x | =1 d ∗ ∑ ∑ ∑ ( f ( r , p x , y , k ) − f ( r , p )) or k =1 x | y − x | =1

Thermal conductivity Infinite system (Formal argument) • S ( x , t ) := ⟨ ( E x ( t ) − E )( E 0 (0) − E ) ⟩ where E = ⟨E 0 ⟩ • ⟨·⟩ : expectation w.r.t. some shift-invariant equilibrium measure • κ k , l := lim t →∞ 1 x ∈ Z d x k x l S ( x , t ) ∑ 2 E 2 t Green-Kubo formula : ∫ t ∫ t 1 κ k , l = lim ∑ j 0 , e l ( s ′ ) ds ′ ) ⟩ ⟨ ( j x , x + e k ( s ) ds )( 2 t E 2 t →∞ 0 0 x ∈ Z d ∫ ∞ = 1 ⟨ j x , x + e k ( t ) j 0 , e l (0) ⟩ dt = δ k , l κ 1 , 1 = κδ k , l ∑ E 2 0 x ∈ Z d • j x , x + e k ( t ) : energy current from x to x + e k at time t • If the energy fluctuation diffuses normally, 0 < κ < ∞ • By the symmetry (even for B ̸ = 0), κ k , k = κ for any k = 1 , 2 , . . . , d