Entropy production and steady states in quantum statistical - PowerPoint PPT Presentation

Entropy production and steady states in quantum statistical mechanics Vojkan Jaksic and Claude-Alain Pillet McGill University, Universit´ e de Toulon March 20, 2014

Statistical mechanics away from equilibrium is in a formative stage, where general concepts slowly emerge. David Ruelle (2008) 1

ENTROPY PRODUCTION OBSERVABLE Hilbert space H , dim H < ∞ . Hamiltonian H . Observables: O = B ( H ) . � A, B � = tr( A ∗ B ) . State: density matrix ρ > 0 . ρ ( A ) = tr( ρA ) = � A � . Time-evolution: ρ t = e − i tH ρ e i tH O t = e i tH O e − i tH . The expectation value of O at time t : � O t � = tr( ρO t ) = tr( ρ t O ) 2

”Entropy observable” (information function): S = − log ρ. Entropy: S ( ρ ) = − tr( ρ log ρ ) = � S � . Average entropy production over the time interval [0 , t ] : ∆ σ ( t ) = 1 t ( S t − S ) . Entropy production observable σ = lim t → 0 ∆ σ ( t ) = i[ H, S ] . � t ∆ σ ( t ) = 1 0 σ s d s. t 3

The entropy production observable = ”quantum phase space contraction rate”. Radon-Nikodym derivative=relative modular operator ∆ ρ t | ρ ( A ) = ρ t Aρ − 1 . ∆ ρ t | ρ is a self-adjoint operator on O and tr( ρ ∆ ρ t | ρ ( A )) = tr( ρ t A ) 4

log ∆ ρ t | ρ ( A ) = (log ρ t ) A − A log ρ �� t � = log ∆ ρ | ρ ( A ) + 0 σ − s d s A. � d � d t log ∆ ρ t | ρ ( A ) � t =0 = σA. 5

BALANCE EQUATION Relative entropy S ( ρ t | ρ ) = tr( ρ t (log ρ t − log ρ )) = � ρ 1 / 2 , log ∆ ρ t | ρ ρ 1 / 2 � ≥ 0 . t t � t 1 t S ( ρ t | ρ ) = � ∆ σ ( t ) � = 1 0 � σ s � d s. t 6

OPEN QUANTUM SYSTEMS R 1 R 2 R k S R M 7

Hilbert spaces H k , k = 0 , · · · , M . Hamiltonians H k . Initial states ρ k = e − β k H k /Z k . Composite system: H = H 0 ⊗ · · · ⊗ H M ρ = ρ 0 ⊗ · · · ⊗ ρ M � H fr = H k , H = H fr + V. 8

Energy change of R k over the time interval [0 , t ] : ∆ Q k ( t ) = 1 t (e i tH H k e − i tH − H k ) . The energy flux observable Φ k = − lim t → 0 ∆ Q k ( t ) = i[ H k , H ] = i[ H k , V ] . � t ∆ Q k ( t ) = − 1 0 Φ ks d s. t 9

The balance equation takes the familiar form: � S = − β k H k � ∆ σ ( t ) = − β k ∆ Q k ( t ) � σ = − β k Φ k � � ∆ σ ( t ) � = − β k � ∆ Q k ( t ) � ≥ 0 . Heat flows from hot to cold. 10

GOAL I � t � ∆ σ ( t ) � = 1 0 � σ s � d s. t TD= Thermodynamic limit. Existence of the limit (steady state entropy production): � σ � + = lim t →∞ lim TD � ∆ σ ( t ) � � σ � + ≥ 0 . Strict positivity: � σ � + > 0 . 11

GOAL II More ambitious: non-equilibriium steady state (NESS). TD leads to C ∗ quantum dynamical system ( O , τ t , ρ ) . � t 1 0 ρ ( τ s ( A ))d s. ρ + ( A ) = lim t t →∞ � σ � + = ρ + ( σ ) . Structural theory: σ + > 0 ⇔ ρ + ⊥ ρ. 12

THE REMARK OF RUELLE D. Ruelle: ”How should one define entropy production for nonequi- librium quantum spin systems?” Rev. Math. Phys. 14,701- 707(2002) The balance equation � � ∆ σ ( t ) � = − β k � ∆ Q k ( t ) � . can (should?) be written differently. 13

H \ k = � j � = k H j . State of the k -th subsystem at time t : ρ kt = tr H \ k ρ t . ∆ S k ( t ) = 1 t ( S ( ρ kt ) − S ( ρ k )) . ∆ σ k ( t ) � = 1 t S ( ρ kt | ρ k ) � ∆ � S ( t ) = ∆ S k ( t ) � ∆ � σ ( t ) = ∆ σ k ( t ) . Obviously, ∆ � σ ( t ) ≥ 0 . � S ( ρ k ) = S ( ρ ) = S ( ρ t ) and by the sub-additivity: ∆ � S ( t ) ≥ 0 . 14

One easily verifies � ∆ σ ( t ) � = ∆ � S ( t ) + ∆ � σ ( t ) . Clausius type decomposition. Set TD ∆ � Ep + = lim t →∞ lim S ( t ) ∆ � σ + = lim t →∞ lim TD ∆ � σ ( t ) . 15

OPEN PROBLEMS Mathematical structure of the decomposition � σ � + = Ep + + ∆ � σ + . The existence of Ep + and ∆ � σ + in concrete models (to be dis- cussed latter). When is ∆ � σ + = 0 ? Ruelle: Perhaps when the boundaries be- tween the small system and the reservoirs are allowed to move to infinity. This limit is more of less imposed by physics, but seems hard to analyze mathematically. Another possibility: adiabatically switched interaction (quasi-static process)? 16

XY SPIN CHAIN Λ = [ A, B ] ⊂ Z , Hilbert space H Λ = � x ∈ Λ C 2 . Hamiltonian � � � H Λ = 1 σ (1) σ (1) x +1 + σ (2) σ (2) J x x x x +1 2 x ∈ [ A,B [ � + 1 λ x σ (3) . x 2 x ∈ [ A,B ] � � � � � � 0 1 0 − i 1 0 σ (1) σ (2) σ (3) = , = , = . x x x 1 0 i 0 0 − 1 17

− M − N N M R L C R R Central part C (small system S ): XY-chain on Λ C = [ − N, N ] . Two reservoirs R L/R : XY-chains on Λ L = [ − M, − N − 1] and Λ R = [ N + 1 , M ] . N fixed, thermodynamic limit M → ∞ . Decoupled Hamiltonian H fr = H Λ L + H Λ C + H Λ R . 18

The full Hamiltonian is H = H Λ L ∪ Λ C ∪ Λ R = H fr + V L + V R , � � V L = J − N − 1 σ (1) − N − 1 σ (1) − N + σ (2) − N − 1 σ (2) , etc . − N 2 Initial state: � ρ = e − β L H Λ L ⊗ ρ 0 ⊗ e − β R H Λ R Z, ρ 0 = 1 / dim H Λ C . Fluxes and entropy production: Φ L/R = − i[ H, H L/R ] , σ = − β L Φ L − β R Φ R . 19

Araki-Ho, Ashbacher-Pillet ∼ 2000, J-Landon-Pillet 2012: NESS exists and � � σ � + = ∆ β E sinh(∆ βE ) R | T ( E ) | 2 d E > 0 . cosh β L E cosh β R E 4 π 2 2 ∆ β = β L − β R . Landauer-B¨ uttiker formula. σ + does not depend where the boundary N is set. 20

Steady state heat fluxes: � Φ L � + + � Φ R � + = 0 � σ � + = − β L � Φ L � + − β R � Φ R � + . � � Φ R � + = 1 E sinh(∆ βE ) R | T ( E ) | 2 d E. cosh β L E cosh β R E 4 π 2 2 21

Idea of the proof–Jordan-Wigner transformation. O is transformed to the even part of CAR( ℓ 2 ( Z )) generated by { a x , a ∗ x | x ∈ Z } acting on the fermionic Fock space F over ℓ 2 ( Z ) . Transformed dynamics: generated by d Γ ( h ) , where h is the Jacobi matrix u ∈ ℓ 2 ( Z ) . hu x = J x u x +1 + J x − 1 u x − 1 + λ x u x , Φ R (and similarly Φ L , σ ) is transformed to − i J N J N +1 ( a ∗ N a N +2 − a ∗ N +2 a N ) − i J N λ N +1 ( a ∗ N a N +1 − a ∗ N +1 a N ) . 22

Decomposition ℓ 2 ( Z ) = ℓ 2 (] −∞ , − N − 1]) ⊕ ℓ 2 ([ − N, N ]) ⊕ ℓ 2 ([ N +1 , ∞ [) , h fr = h L + h C + h R , h = h fr + v L + v R , v R = J N ( | δ N +1 �� δ N | + h.c ) The initial state ρ is transformed to the quasi-free state gener- ated by 1 1 1 1 + e β L h L ⊕ 2 N + 1 ⊕ 1 + e β R h R . 23

The wave operators w ± = s − t →±∞ e i th e − i th fr 1 ac ( h fr ) lim exist and are complete. The scattering matrix: s = w ∗ + w − : H ac ( h fr ) → H ac ( h fr ) � � A ( E ) T ( E ) s ( E ) = . T ( E ) B ( E ) 24

� T ( E ) = 2i π J − N − 1 J N � δ N | ( h − E − i0) − 1 δ − N � F L ( E ) F R ( E ) F L/R ( E ) = Im � δ L/R | ( h L/R − E − i0) − 1 δ L/R � , δ L = δ − N − 1 , δ R = δ N +1 . T ( E ) is non-vanishing on the set sp ac ( h L ) ∩ sp ac ( h R ) . J x = const , λ x = const (or periodic) | T | = χ σ ( h ) 25

Assumption: h has no singular continuous spectrum Open question: The existence and formulas for Ep + and ∆ � σ + . Open question: NESS and entropy production if h has some singular continuous spectra. Transport in quasi-periodic struc- tures. 26

HEISENBERG SPIN CHAIN The Hamiltonian H of XY spin chain is changed to H P = H + P where � P = 1 K x σ (3) σ (3) x +1 . x 2 x ∈ [ − N,N [ The central part is now Heisenberg spin chain � 1 J x σ (1) σ (1) x +1 + J x σ (2) σ (2) x +1 + K x σ (3) σ (3) x x x x +1 2 x ∈ [ − N,N [ � + 1 λ x σ (3) . x 2 x ∈ [ − N,N ] 27

Initial state remains the same. h is the old Jacobi matrix. Fluxes and entropy production: Φ L/R = − i[ H P , H L/R ] σ = − β L Φ L − β R Φ R . TD limit obvious. τ P denotes the perturbed C ∗ -dynamics. 28

Assumption For all x, y ∈ Z , � ∞ |� δ x , e i th δ y �| d t < ∞ . 0 Denote � ∞ |� δ x , e i th δ y �| d t, ℓ N = sup 0 x,y ∈ [ − N,N [ K = 6 6 1 1 ¯ . 7 6 24 N ℓ N 29

Theorem. Suppose that | K x | < ¯ sup K. x ∈ [ − N,N [ Then for all A ∈ O , t →∞ ρ ( τ t ρ + ( A ) = lim P ( A )) exists. 30

Comments: No time averaging. The constant ¯ K is essentially optimal. With change of the constant ¯ K the result holds for any P depending on finitely many σ (3) : x � � K x i 1 ··· x ik σ (3) x i 1 · · · σ (3) P = x ik . The NESS ρ + is attractor in the sense that for any ρ -normal initial state ω , t →∞ ω ◦ τ t lim P = ρ + . 31

The map ( { K x } , β L , β R ) �→ � σ � + = ρ + ( σ ) is real analytic. This leads to the strict positivity of entropy pro- duction. Green-Kubo linear response formula holds for thermodynamical force X = β L − β R (J-Pillet-Ogata) Bosonization Central Limit Theorem holds (J-Pautrat-Pillet) 32

OPEN PROBLEM The existence (and properties) of NESS � t 1 0 ρ ( τ s ρ + ( A ) = lim P ( A ))d s t t →∞ for all { K x } ∈ R 2 N . This is an open problem even if P = K 0 a ∗ 0 a 0 a ∗ 1 a 1 . Dependence of � σ � + on N ? 33