On the foundations of non-equilibrium quantum statistical mechanics - PowerPoint PPT Presentation

On the foundations of non-equilibrium quantum statistical mechanics Vojkan Jaksic McGill University Joint work with L. Bruneau, Y. Ogata, R. Seiringer, C-A. Pillet September 27, 2016

1967: KMS CONDITION Haag-Hugenholtz-Winnick C ∗ -dynamical system ( O , τ t ) . A state ω on O is called ( τ, β ) - KMS, where β ∈ R , if for all A, B ∈ O F A,B ( t ) = ω ( Aτ t ( B )) F A,B ( t + i β ) = ω ( τ t ( B ) A ) The definition is the same in the W ∗ - case with ω normal. If O = B ( H ) , dim H < ∞ , τ t ( A ) = e i tH A e − i tH , then ω ( A ) = tr( A e − βH ) / tr(e − βH ) is the unique ( τ, β ) -KMS state. 1

1967: MODULAR THEORY Tomita-Takesaki ( M , Ω) , M von Neumann algebra on H , Ω cyclic and separat- ing vector. SA Ω = A ∗ Ω Polar decomposition: S = J ∆ 1 / 2 J anti-unitary involution (modular conjugation), ∆ ≥ 0 modular operator. The modular group: σ t ( A ) = ∆ i t A ∆ − i t . Natural cone: P = { AJAJ Ω : A ∈ M } cl . 2

Theorem (Tomita-Takesaki) J M J = M ′ , σ t ( M ) = M . Moreover, the vector state ω ( A ) = (Ω , A Ω) is ( σ, − 1) -KMS state. KMS Condition and Modular theory ⇒ Golden Era of algebraic quantum statistical mechanics (Bratteli-Robinson). 3

1974: DYNAMICAL STABILITY Haag–Kastler–Trych-Pohlmeyer C ∗ -dynamical system ( O , τ t ) , ω a stationary state. KMS condition ⇔ dynamical stability of ω under local perturba- tions V = V ∗ ∈ O . τ t = e tδ . τ t λ = e tδ λ , δ λ ( · ) = δ ( · ) + i λ [ V, · ] . Perturbed station- ary states: ω ± t →±∞ ω ( τ t λ ( A ) = lim λ ( A )) . We assume existence and ergodicity of ω ± λ . Ergodicity ⇒ ω + λ ⊥ ω − λ or ω + λ = ω − λ . The stability ω + λ = ω − λ in the first order of λ gives 4

Stability Criterion (SB) � ∞ −∞ ω ([ V, τ t ( A )])d t = 0 . Assumption L 1 ( O 0 ) asymptotic abelianness: � ∞ −∞ � [ V, τ t ( A )] � d t < ∞ for V, A in the norm dense ∗ -subalgebra O 0 . Theorem (Haag–Kastler–Trych-Pohlmeyer, Bratteli–Kishimoto- Robinson) Suppose in addition that ω is a factor state and that (SB) holds for V, A ∈ O 0 . Then ω is a ( τ, β ) -KMS state for some β ∈ R ∪ {±∞} . 5

DYNAMICAL INSTABILITY Same setup, but ω + λ ⊥ ω − λ Dynamical instability ⇔ Non-equilibrium Quantification of non-equilibrium (our main message): Degree of separation of the pair of mutually normal states λ , ω ◦ τ − t ( ω ◦ τ t λ ) as they approach the mutually singular limits ( ω + λ , ω − λ ) as t → ∞ . 6

PICTURE: OPEN QUANTUM STSTEMS R 1 R 2 R k S R M 7

RELATIVE MODULAR THEORY Araki ( H , π, Ω) GNS-representation of ( O , τ t , ω ) , M = π ( O ) ′′ , Ω cyclic and separating (assumption), ω ( A ) = (Ω , A Ω) , τ t λ ( A ) = e i tL λ A e − i tL λ , e − i tL λ P = P , Ω t = e − i tL λ Ω ∈ P the vector representative of ω ◦ τ t λ . S = J ∆ 1 / 2 SA Ω = A ∗ Ω t , , t ∆ t ≥ 0 is the relative modular operator of the pair of states ( ω ◦ τ t λ , ω ) . Non-commutative Radon-Nikodym derivative. 8

RENYI AND RELATIVE ENTROPY S t ( α ) = log(Ω , ∆ α t Ω) , Ent t = (Ω t , log ∆ t Ω t ) . S t (0) = S t (1) = 0 , α �→ S t ( α ) convex, we assume it is finite, S ′ t (1) = Ent t ≥ 0 . � R e − αts d P t ( s ) , S t ( α ) = log where P t is the spectral measure for − 1 t log ∆ t and Ω . Time-reversal invariance (TRI) ⇒ S t ( α ) = S t (1 − α ) . 9

BASIC OBJECTS 1 e ( α ) = lim t S t ( α ) . t →∞ Assumption Existence of limit and real-analyticity of e ( α ) . α �→ e ( α ) is convex, e (0) = e (1) = 0 . TRI ⇒ e ( α ) = e (1 − α ) . Entropy production of ( O , τ t λ , ω ) is 1 1 t S ′ Σ = lim t Ent t = lim t (1) . t →∞ t →∞ TRI ⇒ Σ = 0 iff e ( α ) ≡ 0 . 10

LARGE DEVIATIONS Rate function I ( θ ) = − inf α ∈ R ( αθ + e ( α )) . For any O ⊂ R open, 1 lim t log P t ( O ) = − inf θ ∈ O I ( θ ) . t →∞ TRI ⇒ I ( − θ ) = θ + I ( θ ) Quantum Gallavotti-Cohen Fluctuation Relation. 11

BACK TO TIME SEPARATION Shorthand ω t := ω ◦ τ t λ . ( ω t , ω − t ) → ( ω + , ω − ) as t → ∞ . t →∞ � ω t − ω − t � = � ω + − ω − � = 2 . lim D t = 1 2 − � ω t − ω − t � � � . 2 Quantum Neyman-Pearson Lemma � ω t ( T ) + ω − t ( 1 − T ) � D t = inf , T where inf is over all orthogonal projections T ∈ M . Quantum Hypothesis Testing 12

CHERNOFF EXPONENT Theorem (JOPS) 1 lim 2 t log D t = α ∈ [0 , 1] e ( α ) min t →∞ Proof: Based on the estimate 1 2 P t ( R − ) ≤ D t ≤ (Ω , ∆ α t Ω) , α ∈ [0 , 1] The difficult part is the upper-bound. α = 1 / 2 proven by Araki in 1973. In the case of matrices: 1 2 (Tr A + Tr B − Tr | A − B | ) ≤ Tr A 1 − α B α K. M. R. Audenaert, J. Calsamiglia, R. Munoz-Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete (2007). Simple proof: Ozawa (unpublished). General case: Ogata, JOPS. 13

STEIN EXPONENT ǫ ∈ ]0 , 1[ , � � � 1 t log ω − t ( T t ) � � ω t ( T t ) ≥ ǫ s ǫ = inf lim � � t →∞ { T t } Theorem (JOPS) s ǫ = − Σ 14

HOEFFDING EXPONENT r > 0 , � � � 1 1 t log ω − t ( 1 − T t ) � t log ω t ( T t ) < − r h ( r ) = inf lim � lim sup � � t →∞ { T t } t →∞ Theorem (JOPS) − rα − e ( α ) ψ ( r ) = − sup . 1 − α α ∈ [0 , 1[ 15

THE MEANING OF P t Consider a confined quantum system on H , dim H < ∞ . O = B ( H ) , τ t λ ( A ) = e i tH λ A e − i tH λ , H = H + λV. The state ω = density matrix on H , ω > 0 , ω ( A ) = tr( ωA ) , λ = e − i tH λ ω e i tH λ . ω t Assume TRI. Entropy observable S = − log ω. Confined open quantum systems: � S = β S H S + β k H k . k 16

� S = sP s Measurement at t = 0 yields s with probability tr( ωP s ) . State after the measurement: ωP s / tr( ωP s ) . State at later time t : e − i tH λ ωP s e i tH λ / tr( ωP s ) . Another measurement of S yields value s ′ with probability tr( P s ′ e − i tH ωP s e i tH ) / tr( ωP s ) . 17

Probability distribution of the mean change of entropy φ = ( s ′ − s ) /t tr( P s ′ e − i tH P s e i tH ) . � P t ( φ ) = s ′ − s = tφ S t ( α ) = log tr([ ω ] 1 − α [ ω t ] α ) = log e − αtφ P t ( φ ) . � φ S t ( α ) = S t (1 − α ) is equivalent to P t ( − φ ) P t ( φ ) = e − tφ . P t , spectral measure of − 1 t log ∆ t , is identified with so called full statistics of the energy/entropy change in a repeated mea- surement protocol described above. Thermodynamic limit gives physical interpretation of P t of extended systems.

CONCLUSION Equlibrium. KMS-condition, dynamical stability, equivalence of the two directions of time. Non-equilibrium. Dynamical instability, the directions of time are not-equivalent. The separation of time directions is quanti- fied by entropic exponents. The exponents are in turn related to LDP for suitable spectral measure of relative modular Hamilto- nian. This spectral measure is linked to full statistics of repeated measurements of energy/entropy. TRI implies Fluctuation Rela- tions. Σ , the Stein exponent, related to ex- Entropy production. pected value of heat/charge fluxes in non-equilibrium steady state. Σ = 0 for sufficently many V ′ ’s + AA ⇒ dynamical sta- bility and KMS condition (J, Pillet). 18

TOPICS NOT DISCUSSED (1) Concrete physically relevant models. (2) Onsager reciprocity relations, Fluctuation-Dissipation Theo- rem. (3) Host of other entropic functionals (4) Quantum transfer operators and Ruelle’s resonance picture of e ( α ) 19