c -dynamical systems and nonequilibrium...

C∗-Dynamical Systems

andNonequilibrium Quantum Statistical Mechanics

Algebraic Approach to the Thermodynamicsof

Open Quantum Systems

C.-A. Pillet

Centre de Physique Theorique – UMR 6207

Universite du Sud – Toulon-Var

ESI–June 2008 – p. 1

Overview


Overview

• Review of some recent results in the Nonequilibrium Quantum StatisticalMechanics of open systems:Joint program (1997-2008) with Vojkan Jakšic, McGill University – Montreal.


Overview


• Nonequilibrium Steady States (NESS).


Overview



• Entropy Production in NESS.


Overview




• [Structural Properties of NESS.]


Overview





• Linear Response Theory (with Yoshiko Ogata, Fukuoka University).


Overview






• Central Limit Theorem (with Yan Pautrat, Orsay).


Overview







• [Landauer-Büttiker approach (with Walter Aschbacher, TU-München and YanPautrat).]


Overview








• Models: Quasifree and locally interacting fermions. N -levels system coupled tofree Fermi reservoirs.


Overview








• Models: Quasifree and locally interacting fermions. N -levels system coupled tofree Fermi reservoirs.

• [Repeated Interactions (with Laurent Bruneau, Cergy).]


0. Mathematical Framework

Algebraic Quantum Statistical Mechanics as described in "the Bible":O. Bratteli D. Robinson: Operator Algebras in Quantum Statistical Mechanics vol. 1 & 2




Dynamical system (M,φ, µ) C∗-dynamical system (O, τ, ω)





Group G ∋ t 7→ φt : M →M






Observables f ∈ C(M) Observables A ∈ O, a C∗-algebra






Observables f ∈ C(M) Observables A ∈ O, a C∗-algebraτ t(f) = f φt Group of ∗-automorphisms τ t(A)







Measure µ⇒ expectation µ(f) State ω ⇒ expectation ω(A)








Invariant measure µ(τ t(f)) = µ(f) Invariant state ω τ t = ω









Koopman space Hµ = L2(M,µ) GNS representation (Hω , πω ,Ωω)










eitLψ = ψ φt, L1 = 0 eitLπω(A)Ωω = πω τ t(A)Ωω

eitLfe−itL = τ t(f) eitLπω(A)e−itL = πω(τ t(A))












ν ≪ µ ω-normal states: η(A) = trρπω(A)













Lebesgue decomposition ν = νac + νsing η = ηnormal + ηsing














Koopmanism: ergodic prop. ⇔ sp(L) Quantum-Koopmanism














Koopmanism: ergodic prop. ⇔ sp(L) Quantum-Koopmanismµ(fτ t(g)) = µ(τ t(g)f) (τ, β)-KMS states ω(Aτ t+iβ(B)) = ω(τ t(B)A)


1. Example: The Ideal Fermi Gas



• Simplest model for electrons in a (semi-)conductor.




• 1-particle Hilbert space h (e.g L2(M),M ⊂ Rd or ℓ2(M),M ⊂ Zd).





• 1-particle Hamiltonian H (e.g. H = −∆ + V ).






• 1-particle propagator U(t) = e−itH solves the Schrödinger equation i∂tψ = Hψ.







• n-particles Hilbert space and propagator, taking Pauli into account

Γ−n (h) = h∧ · · · ∧ h, Γn(U(t)) = U(t) ⊗ · · · ⊗ U(t).








Γ−n (h) = h∧ · · · ∧ h, Γn(U(t)) = U(t) ⊗ · · · ⊗ U(t).

• Second quantization: With Γ−0 (h) = C, Γ0(U(t)) = I,

Γ−(h) =

∞M

n=0

Γ−n (h), Γ(U(t)) =

∞M

n=0

Γn(U(t)).








Γ−n (h) = h∧ · · · ∧ h, Γn(U(t)) = U(t) ⊗ · · · ⊗ U(t).

• Second quantization: With Γ−0 (h) = C, Γ0(U(t)) = I,

Γ−(h) =

∞M

n=0

Γ−n (h), Γ(U(t)) =

∞M

n=0

Γn(U(t)).

• Creation/annihilation operators

a∗(f)f1 ∧ · · · ∧ fn =√n+ 1f1 ∧ · · · ∧ fn ∧ f,

a(f)f1 ∧ · · · ∧ fn =1√n

nX

j=1

(−1)n−j(f, fj)f1 ∧ · · ·fj · · · ∧ fn.



• Canonical Anticommutation Relations (CAR):

a(f), a(g) = a(f)a(g) + a(g)a(f) = 0,

a(f), a∗(g) = a(f)a∗(g) + a∗(g)a(f) = (f, g)I.




a(f), a(g) = a(f)a(g) + a(g)a(f) = 0,


• CAR(h) = C∗-algebra generated by a(f)|f ∈ h.




a(f), a(g) = a(f)a(g) + a(g)a(f) = 0,



• C∗-dynamical system (CAR(h), τ) with

τ t(a(f)) ≡ Γ(U(t))∗a(f)Γ(U(t)) = a(eitHf),




a(f), a(g) = a(f)a(g) + a(g)a(f) = 0,





• Unique (τ, β)-KMS state

ωβ(a∗(g1) · · · a∗(gn)a(fm) · · · a(f1)) = δnm det(fi, T gj),

with T = (I + eβH)−1 (Gauge-invariant quasi-free state ⇒ Araki-Wyss).




a(f), a(g) = a(f)a(g) + a(g)a(f) = 0,





• Unique (τ, β)-KMS state

ωβ(a∗(g1) · · · a∗(gn)a(fm) · · · a(f1)) = δnm det(fi, T gj),

with T = (I + eβH)−1 (Gauge-invariant quasi-free state ⇒ Araki-Wyss).

• Local perturbation

V =K

X

k=1

nkY

j=1

a∗(gkj)a(fkj),

leads to locally interacting Fermi gas.


2. Open Quantum Systems

S

R1

R2

V2

V1



S

R1

R2

V2

V1

Reservoir Rj : Extended ideal quantum gas (Oj , τtj = etδj ).



S

R1

R2

V2

V1


Small system S: Finite quantum system (O0, τt0 = etδ0 ) (may be absent!).



S

R1

R2

V2

V1



(O, τ) = ⊗j(Oj , τj) with multi-KMS reference state ω−→β

= ⊗jωβj.



S

R1

R2

V2

V1




= ⊗jωβj.

Coupling V =P

j>0 Vj : Local perturbation Vj ∈ O0 ⊗Oj .



S

R1

R2

V2

V1




= ⊗jωβj.

Coupling V =P

j>0 Vj : Local perturbation Vj ∈ O0 ⊗Oj .

Coupled system (O, τ tV = et(

P

j δj+i[V, · ])).


2.1 NESS of Open Quantum Systems

[Ruelle ’00]For all A ∈ O and all ω−→

β–normal state η

limt→∞

η τ tV (A) = ω−→

β+(A).





limt→∞

η τ tV (A) = ω−→

β+(A).

Return to equilibrium: β1 = β2 = · · · = β =⇒ ω−→β

+ is (τV , β)-KMS;

[Jakšic-P ’96], [Bach-Fröhlich-Sigal ’00], [Derezinski-Jakšic ’03], [Fröhlich-Merkli ’04].





limt→∞

η τ tV (A) = ω−→

β+(A).




Out of equilibrium: βi 6= βj =⇒ ω−→β

+ is ω−→β

-singular.

⇓No density matrix in GNS space Hω−→

β:

Hilbert space approach ?





limt→∞

η τ tV (A) = ω−→

β+(A).




Out of equilibrium: βi 6= βj =⇒ ω−→β

+ is ω−→β

-singular.

⇓No density matrix in GNS space Hω−→

β:

Hilbert space approach ?

Basic Problem 1. Existence of NESS.


2.2 NESS: The Scattering Approach

C∗-scattering theory: [Hepp ’70], [Robinson ’73], [Botvich-Malyshev ’78].




NESS construction: [Dirren-Fröhlich-Graf ’98], [Ruelle ’00], [Fröhlich-Merkli-Ueltschi ’03].





τ -invariant reference state ω: ω τ tV = ω τ−t τ t

V ,






V ,

Møller morphism: γ+(A) = limt→∞

τ−t τ tV (A),






V ,


τ−t τ tV (A),

limt→∞

ω τ tV = ω γ+ = ω+






V ,


τ−t τ tV (A),

limt→∞

ω τ tV = ω γ+ = ω+

O+ ≡ Ranγ+ ⇒ (O+, τ |O+ , ω|O+ ) ∼ (O, τV , ω+)






V ,


τ−t τ tV (A),

limt→∞

ω τ tV = ω γ+ = ω+

O+ ≡ Ranγ+ ⇒ (O+, τ |O+ , ω|O+ ) ∼ (O, τV , ω+)

Theorem. [Aschbacher-Jakšic-Pautrat-P ’06] If the C∗-dynamical system(O+, τ |O+ , ω|O+ ) is mixing then

limt→∞

η τ tV (A) = ω+(A), (∗)

for all A ∈ O and all ω-normal states η. Moreover (O, τV , ω+) is mixing.

Remark. The scattering approach does not directly provide information on the rate ofconvergence in (∗).






V ,


τ−t τ tV (A),

limt→∞

ω τ tV = ω γ+ = ω+

O+ ≡ Ranγ+ ⇒ (O+, τ |O+ , ω|O+ ) ∼ (O, τV , ω+)

Application to open systems. Show that:

1. (OR, τR, ωR) ≡ ⊗j>0(Oj , τj , ωβj) is mixing (the easy part).

2. γ+ exists.

3. O+ = OR (⇒ ω+ independent of ω|O0).



Strategy. Ideas from Hilbert space scattering theory.




1. If α+(A) = limt→∞ τ−tV τ t(A) exists for all A ∈ OR ⇒ γ+ α+(A) = A

↓OR ⊂ O+.





↓OR ⊂ O+.

2. Cook’s method:

τ−tV τ t(A) = A+

Z t

0

d

dsτ−sV τs(A) ds = A+

Z t

0τ−sV (i[V, τs(A)]) ds.





↓OR ⊂ O+.

2. Cook’s method:


Z t

0

d


Z t


3. If δ0 has only pure point spectrum and limt→∞ ‖[X, τ tV (A)]‖ = 0 for all X ∈ O0

↓O+ ⊂ OR.





↓OR ⊂ O+.

2. Cook’s method:


Z t

0

d


Z t



↓O+ ⊂ OR.

4. Schwinger-Dyson expansion

τ−t τ tV (A) = A+

X

n>0

Z

−t<tn<···<t1<0i[τ tn (V ), i[· · · , i[τ t1 (V ), A] · · · ]] dt1 · · · dtn.





↓OR ⊂ O+.

2. Cook’s method:


Z t

0

d


Z t



↓O+ ⊂ OR.

4. Schwinger-Dyson expansion

τ−t τ tV (A) = A+

X

n>0

Z

−t<tn<···<t1<0i[τ tn (V ), i[· · · , i[τ t1 (V ), A] · · · ]] dt1 · · · dtn.

5. CAR ⇒ diagrammatic expansion, rooted trees combinatorics.


2.3 NESS: The Liouvillean Approach

[Jakšic-P ’02], [Merkli-Mueck-Sigal ’06].Work in GNS representation (Hω , πω ,Ωω) of reference state.

ESI–June 2008 – p. 10



Assume that the reference state ω is τ -invariant and modular (e.g. ω = ω−→β

):

πω(O)′Ωωcl

= Hω .

ESI–June 2008 – p. 10




):

πω(O)′Ωωcl

= Hω .

↓S : πω(A)Ωω 7→ πω(A∗)Ωω ,

densely defines a closable anti-linear operator on Hω .

ESI–June 2008 – p. 10




):

πω(O)′Ωωcl

= Hω .



Tomita-Takesaki modular theory↓

Polar decomposition S = J∆1/2ω with ∆ω ≡ S∗S > 0.

ESI–June 2008 – p. 10




):

πω(O)′Ωωcl

= Hω .



Tomita-Takesaki modular theory↓

Polar decomposition S = J∆1/2ω with ∆ω ≡ S∗S > 0.

J = J∗ = J−1: modular conjugation: J∆ωJ = ∆−1ω , Jπω(O)′′J = πω(O)′,

σtω(A) ≡ ∆it

ωA∆−itω : modular group on πω(O)′′.

ESI–June 2008 – p. 10


πω(τ tV (A)) = eiLV tπω(A)e−iLV t with L∗

V Ωω = 0,

uniquely defines the C-Liouvillean LV (non-selfadjoint !).

ESI–June 2008 – p. 11



V Ωω = 0,


LV = L+ πω(V ) − Jσi/2ω (πω(V ))J.

ESI–June 2008 – p. 11



V Ωω = 0,



Strategy: Gelfand triplet K ⊂ Hω ⊂ K′, dense subalgebra O ⊂ O s.t. πω(O)Ωω ⊂ K.

if ∃Ψ ∈ K′ s.t. w∗ − limt→∞

e−iL∗tφ = (Ψ, φ)Ωω for all φ ∈ K, (∗)⇓

limt→∞

η τ tV (A) = ω+(A) for all A ∈ O and all ω-normal state η,

ω+(A) = (Ωω , πω(A)Ψ) for A ∈ O.

ESI–June 2008 – p. 11



V Ωω = 0,



Strategy: Gelfand triplet K ⊂ Hω ⊂ K′, dense subalgebra O ⊂ O s.t. πω(O)Ωω ⊂ K.

if ∃Ψ ∈ K′ s.t. w∗ − limt→∞

e−iL∗tφ = (Ψ, φ)Ωω for all φ ∈ K, (∗)⇓

limt→∞

η τ tV (A) = ω+(A) for all A ∈ O and all ω-normal state η,

ω+(A) = (Ωω , πω(A)Ψ) for A ∈ O.

Under sufficient regularity assumptions on V spectral deformation techniques allow tocontrol resonances of L and prove (∗).

ESI–June 2008 – p. 11


sp(L∗|K) sp(e−itL∗

|K)

0

1

ESI–June 2008 – p. 12


sp(L∗|K) sp(e−itL∗

|K)

0

1

Imaginary part of resonances control rate of convergence

|η τ tV (A) − ω+(A)| ≃ e−γt

ESI–June 2008 – p. 12

3. Entropy Production

Φj = energy flux out of Rj = −rate of change of energy in Rj .

ESI–June 2008 – p. 13


Φj = energy flux out of Rj = −rate of change of energy in Rj .Total energy is conserved:

X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(H0 + V ) + i[H0, H0 + V ] + i[V,H0 + V ].

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).

Φj = δj(V ) = δj(Vj).

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).


Steady state entropy production rate = − total entropy flux

Ep(ω−→β

+) ≡ ω−→β

+(σ), σ ≡ −X

j>0

βjΦj .

The observable σ plays in this context a similar role as the phase-space contraction ratein dissipative classical dynamical systems.

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).



Ep(ω−→β

+) ≡ ω−→β

+(σ), σ ≡ −X

j>0

βjΦj .

Theorem. [Ruelle ’01], [Jakšic-P ’01] If η is ω−→β

-normal then

Ep(ω−→β

+) = − limt→∞

Ent(η τ tV |ω−→

β)

t≥ 0.

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).



Ep(ω−→β

+) ≡ ω−→β

+(σ), σ ≡ −X

j>0

βjΦj .

Basic Problem 2. Strict positivity of entropy production

Ep(ω−→β

+) > 0.

ESI–June 2008 – p. 13



X

j>0

Φj =d

dtτ tV (H0 + V )

˛

˛

˛

˛

t=0

=X

j>0

δj(V ).



Ep(ω−→β

+) ≡ ω−→β

+(σ), σ ≡ −X

j>0

βjΦj .

Theorem. [Jakšic-P ’02] If Ep(ω−→β

+) > 0 then ω−→β

+ is not ω−→β

-normal. Reciproquely, if

ω−→β

+ is not ω−→β

-normal and

lim supt→∞

˛

˛

˛

˛

Z t

0

“

ω−→β

τsV (σ) − ω−→

β+(σ)

”

ds

˛

˛

˛

˛

<∞,

then Ep(ω−→β

+) > 0.

ESI–June 2008 – p. 13

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).

Transport coefficients (Onsager matrix):Lkj = −∂βj

ω−→β

+(Φk)|−→β =

−→β equ

.

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.

• Existence: ω−→β

depends singularly on−→β .

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.



• Green-Einstein-Kubo Formula (1):

Lkj = β−1

Z β

0dθ

Z ∞

0dt ω−→

β equ

(τ t(Φk)τ iθ(Φj)).

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.




Lkj = β−1

Z β

0dθ

Z ∞

0dt ω−→

β equ


• Green-Einstein-Kubo Formula (2): Lkj =1

2

Z ∞

−∞

dt ω−→β equ

(Φkτt(Φj)).

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.




Lkj = β−1

Z β

0dθ

Z ∞

0dt ω−→

β equ



2

Z ∞

−∞

dt ω−→β equ

(Φkτt(Φj)).

• Onsager reciprocity relations: Lkj = Ljk.

ESI–June 2008 – p. 14

4. Linear Response

−→β equ = (β, β, . . . , β).

ω−→β

+ = limt→∞

ω−→β

τ tV , (|−→β −−→

β equ| < ǫ ).


ω−→β

+(Φk)|−→β =

−→β equ

.

Basic Problems 3.




Lkj = β−1

Z β

0dθ

Z ∞

0dt ω−→

β equ



2

Z ∞

−∞

dt ω−→β equ

(Φkτt(Φj)).

• Onsager reciprocity relations: Lkj = Ljk.

Remark. Order of limits: 1st thermodynamic limit, 2nd long time limit, 3rd weak forcinglimit.

ESI–June 2008 – p. 14

5. Quantum Dynamical Central Limit Theorem

[Hepp-Lieb ’73], [Derezinski ’ 85], [Goderis-Verbeure-Vets ’88], [Matsui ’02]

ESI–June 2008 – p. 15



C ⊂ O real subspace of self-adjoint elements is CLT-admissible ifZ ∞

−∞

|ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→

β+(B)| dt <∞ for A,B ∈ C.

ESI–June 2008 – p. 15




−∞

|ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→


At ≡ 1√t

Z t

0(τs

V (A) − ω−→β

+(A)) ds,

L(A,B) ≡Z ∞

−∞

“

ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→

β+(B)

”

dt.

ESI–June 2008 – p. 15




−∞

|ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→


At ≡ 1√t

Z t

0(τs

V (A) − ω−→β

+(A)) ds,

L(A,B) ≡Z ∞

−∞

“

ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→

β+(B)

”

dt.

Symplectic form on C: ς(A,B) ≡ 1

2i

Z ∞

−∞

ω−→β

+([τ tV (A), B]) dt.

ESI–June 2008 – p. 15




−∞

|ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→


At ≡ 1√t

Z t

0(τs

V (A) − ω−→β

+(A)) ds,

L(A,B) ≡Z ∞

−∞

“

ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→

β+(B)

”

dt.


2i

Z ∞

−∞

ω−→β

+([τ tV (A), B]) dt.

Fluctuation Algebra W: C ∋ A 7→W (A) s.t.:

W (−A) = W (A)∗ and W (A)W (B) = e−iς(A,B)/2W (A+B).

W = CCR-algebra over (C, ς).

ESI–June 2008 – p. 15




−∞

|ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→


At ≡ 1√t

Z t

0(τs

V (A) − ω−→β

+(A)) ds,

L(A,B) ≡Z ∞

−∞

“

ω−→β

+(Aτ tV (B)) − ω−→

β+(A)ω−→

β+(B)

”

dt.


2i

Z ∞

−∞

ω−→β

+([τ tV (A), B]) dt.

Fluctuation Algebra W: C ∋ A 7→W (A) s.t.:

W (−A) = W (A)∗ and W (A)W (B) = e−iς(A,B)/2W (A+B).

W = CCR-algebra over (C, ς).

Quasi-free state µL(W (A)) = e−L(A,A)/2

⇓regular GNS representation (HL, πL,ΩL) with πL(W (A)) = eiφL(A).

ESI–June 2008 – p. 15


Basic Problem 4. Simple Quantum Dynamical CLT:

limt→∞

ω−→β

+“

eiAt

”

= e−L(A,A)/2, for any A ∈ C.

ESI–June 2008 – p. 16



limt→∞

ω−→β

+“

eiAt

”


Theorem. [Jakšic-Pautrat-P ’07] If SQD-CLT holds for C andZ ∞

−∞

‖[A, τ tV (B)]‖ dt <∞

for A,B ∈ C then the Quantum Dynamical CLT

limt→∞

ω−→β

+“

eiA1t · · · eiAnt

”

=

exp

0

@−1

2L

0

@

nX

j=1

Aj ,

nX

j=1

Aj

1

A − iX

1≤j<k≤n

ς(Aj , Ak)

1

A ,

holds too.

ESI–June 2008 – p. 16



limt→∞

ω−→β

+“

eiAt

”


Theorem. [Jakšic-Pautrat-P ’07] If SQD-CLT holds for C andZ ∞

−∞

‖[A, τ tV (B)]‖ dt <∞

for A,B ∈ C then the Quantum Dynamical CLT

limt→∞

ω−→β

+“

eiA1t · · · eiAnt

”

=

exp

0

@−1

2L

0

@

nX

j=1

Aj ,

nX

j=1

Aj

1

A − iX

1≤j<k≤n

ς(Aj , Ak)

1

A ,

holds too.

Theorem. [Jakšic-Pautrat-P ’07] If QD-CLT holds for C then

limt→∞

ω−→β

+“

f1(A1t) · · · fn(Ant)”

= µL(f1(φL(A1)) · · · fn(φL(An))),

holds for all bounded Borel functions f1, . . . , fn.

ESI–June 2008 – p. 16

7. Models

Classified by techniques:

ESI–June 2008 – p. 17

7. Models


• Quasifree fermions/bosons and XY spin chains (→ independent particlesmodels): C∗- scattering reduces to Hilbert space scattering leading toLandauer-Büttiker formalism,[Araki-Ho ’00], [Aschbacher-P ’02], [Avron-Elgart-Graf-Sadun-Schnee ’02],[Cornean-Jensen-Moldoveanu ’04], [Barbaroux-Aschbacher ’06],[Aschbacher-Jakšic-Pautrat-P ’07], [Nenciu ’07], [Avron-Bachmann-Graf-Klich ’07].

ESI–June 2008 – p. 17

7. Models



• N -level system coupled to ideal quantum gases: Liouvillean approach,[Jakšic-P ’02], [Abou Salem-Fröhlich ’05], [Merkli-Mueck-Sigal ’06].

ESI–June 2008 – p. 17

7. Models




• Quantum spin systems: C∗ scattering approach,[Ruelle ’00].

ESI–June 2008 – p. 17

7. Models





• Locally interacting Fermi gases: C∗ scattering approach,[Dirren-Fröhlich-Graf ’89], [Fröhlich-Merkli-Ueltschi ’03], [Jakšic-Ogata-Pautrat-P’06].

ESI–June 2008 – p. 17

7. Models





• Locally interacting Fermi gases: C∗ scattering approach,[Dirren-Fröhlich-Graf ’89], [Fröhlich-Merkli-Ueltschi ’03], [Jakšic-Ogata-Pautrat-P’06].

• Repeated interactions: Liouvillean approach [Attal-Pautrat ’06],[Bruneau-Joye-Merkli ’06], [Attal-Joye ’07].

ESI–June 2008 – p. 17

Example 1: The Spin-Fermion Model

Simplest example of N -level system coupled to ideal quantum gases.

ESI–June 2008 – p. 18



S is a 2-level system: O0 = Mat(2,C), τ t0(A) = eitH0Ae−itH0 , H0 = σz .

ESI–June 2008 – p. 18




Rj is a free Fermi gas: One particle Hilbert space hj and Hamiltonian hj .Oj = CAR(hj), C∗ algebra generated by creation/annihilation operators aj(f), a∗j (f),

f ∈ hj .

τ tj (aj(f)) = aj(e

ithj f) Bogoliubov automorphism.

ESI–June 2008 – p. 18





f ∈ hj .



Coupling Vj = λσx ⊗ (a(αj) + a∗(αj)), αj ∈ hj .

ESI–June 2008 – p. 18





f ∈ hj .



Coupling Vj = λσx ⊗ (a(αj) + a∗(αj)), αj ∈ hj .

Assumptions. For all j’s:

(A1) hj = L2(R+, ds) ⊗ Kj and hj = s (spectral representation).

(A2) For some δ > 0 and all a: e−asαj(|s|) ∈ H2(|Ims| < δ) ⊗ Kj (analyticity).

(A3) ‖αj(2)‖Kj6= 0 (effective coupling).

ESI–June 2008 – p. 18






ESI–June 2008 – p. 19






Theorem. [Jakšic-P ’02], [Jakšic-Ogata-P, ’06] Assume (A1)-(A3) and let 0 < γ1 < γ2 begiven. Then there exists Λ > 0 such that, for all 0 < |λ| < Λ and γ1 < βj < γ2:

1. There exists a NESS ω−→β

+.

2. If the βj ’s are not all equal then ω−→β

+ is not ω−→β

-normal and Ep(ω−→β

+) > 0.

3. The Green-Einstein-Kubo formulas (1) and (2) hold as well as the Onsagerreciprocity relations.

ESI–June 2008 – p. 19

Example 2: Locally Interacting Fermi Gases

R1 R3

R2

Intera tion regionESI–June 2008 – p. 20


No small system S (included in reservoirs).

ESI–June 2008 – p. 20


No small system S (included in reservoirs).Rj is a free Fermi gas: One particle Hilbert space hj and Hamiltonian hj .

Oj = CAR(hj), C∗ algebra generated by creation/annihilation operators aj(f), a∗j (f),

f ∈ hj .



ESI–June 2008 – p. 20




f ∈ hj .



Gauge-invariant coupling/interaction

V = λ

KX

k=1

nkY

j=1

a∗(ujk)a(vjk).

ESI–June 2008 – p. 20




f ∈ hj .



Gauge-invariant coupling/interaction

V = λ

KX

k=1

nkY

j=1

a∗(ujk)a(vjk).

Assumptions. Set h ≡ ⊕jhj , h ≡ ⊕jhj and D0 ≡ ujk, vjk:

(B1) There is a dense subspace D ⊂ h containing D0 and such thatZ ∞

−∞

|(f, eithg)| dt <∞,

for all f, g ∈ D.

(B2) hD0 ⊂ D.

(B3) There is a complex conjugation on h which commute with h and such that ujk, vjk

are real (Time reversal invariance).

ESI–June 2008 – p. 20




−∞


for all f, g ∈ D.

(B2) hD0 ⊂ D.



ESI–June 2008 – p. 21




−∞


for all f, g ∈ D.

(B2) hD0 ⊂ D.



Theorem. [Jakšic-Ogata-P ’06], [Jakšic-Pautrat-P ’07] If (B1) holds then there exists Λ > 0

such that, for 0 < |λ| < Λ:

1. There exists a NESS ω−→β

+.

2. If (B2) also holds then the Green-Einstein-Kubo formula (1) holds.

3. If, in addition, (B3) holds then the Green-Einstein-Kubo formula (2) and theOnsager reciprocity relations hold. Moreover QDCLT holds for C = Φ1,Φ2, . . ..

ESI–June 2008 – p. 21

c -dynamical systems and nonequilibrium...

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