Entropy production and steady states
in quantum statistical mechanics
Vojkan Jaksic and Claude-Alain Pillet
McGill University, Université 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). hA, Bi = tr(A∗B).
State: density matrix ρ > 0. ρ(A) = tr(ρA) = hAi.
Time-evolution:
ρt = e−itH ρeitH
Ot = eitH Oe−itH .
The expectation value of O at time t:
hOti = tr(ρOt) = tr(ρtO)
2
”Entropy observable” (information function):
S = − log ρ.
Entropy:
S(ρ) = −tr(ρ log ρ) = hSi.
Average entropy production over the time interval [0, t]:
∆σ(t) =
1
(St − S).
t
Entropy production observable
σ = lim ∆σ(t) = i[H, S].
t→0
1 t
∆σ(t) =
σsds.
t 0
Z
3
The entropy production observable = ”quantum phase space
contraction rate”.
Radon-Nikodym derivative=relative modular operator
∆ρt|ρ(A) = ρtAρ−1.
∆ρt|ρ is a self-adjoint operator on O and
tr(ρ∆ρt|ρ(A)) = tr(ρtA)
4
log ∆ρt|ρ(A) = (log ρt)A − A log ρ
= log ∆ρ|ρ(A) +
Z t
0
σ−sds A.
d
log ∆ρt|ρ(A)
= σA.
t=0
dt
5
BALANCE EQUATION
Relative entropy
S(ρt|ρ) = tr(ρt(log ρt − log ρ))
1/2
= hρt
1/2
, log ∆ρt|ρρt
i ≥ 0.
1
1 t
S(ρt|ρ) = h∆σ(t)i =
hσsids.
t
t 0
Z
6
OPEN QUANTUM SYSTEMS
R1
R2
S
Rk
RM
7
Hilbert spaces Hk , k = 0, · · · , M . Hamiltonians Hk .
Initial states
ρk = e−βk Hk /Zk .
Composite system:
H = H0 ⊗ · · · ⊗ HM
ρ = ρ0 ⊗ · · · ⊗ ρM
Hfr =
X
Hk ,
H = Hfr + V.
8
Energy change of Rk over the time interval [0, t]:
1 itH
∆Qk (t) = (e Hk e−itH − Hk ).
t
The energy flux observable
Φk = − lim ∆Qk (t) = i[Hk , H] = i[Hk , V ].
t→0
1 t
Φksds.
∆Qk (t) = −
t 0
Z
9
The balance equation takes the familiar form:
S=−
X
βk Hk
∆σ(t) = −
X
βk ∆Qk (t)
σ=−
X
βk Φk
h∆σ(t)i = −
X
βk h∆Qk (t)i ≥ 0.
Heat flows from hot to cold.
10
GOAL I
1 t
hσsids.
h∆σ(t)i =
t 0
Z
TD= Thermodynamic limit. Existence of the limit (steady state
entropy production):
hσi+ = lim limh∆σ(t)i
t→∞ T D
hσi+ ≥ 0. Strict positivity:
hσi+ > 0.
11
GOAL II
More ambitious: non-equilibriium steady state (NESS). TD leads
to C ∗ quantum dynamical system (O, τ t, ρ).
1 t
ρ+(A) = lim
ρ(τ s(A))ds.
t→∞ t 0
Z
hσi+ = ρ+(σ).
Structural theory:
σ+ > 0 ⇔ ρ+ ⊥ ρ.
12
THE REMARK OF RUELLE
D. Ruelle: ”How should one define entropy production for nonequilibrium quantum spin systems?” Rev. Math. Phys. 14,701707(2002)
The balance equation
h∆σ(t)i = −
X
βk h∆Qk (t)i.
can (should?) be written differently.
13
H\k =
N
j6=k Hj . State of the k-th subsystem at time t:
ρkt = trH\k ρt.
1
∆Sk (t) = (S(ρkt) − S(ρk )).
t
1
∆σk (t)i = S(ρkt|ρk )
t
b
∆S(t)
=
X
∆Sk (t)
b (t) =
∆σ
X
∆σk (t).
Obviously,
b (t) ≥ 0.
∆σ
P
S(ρk ) = S(ρ) = S(ρt) and by the sub-additivity:
b
∆S(t)
≥ 0.
14
One easily verifies
b
b (t).
h∆σ(t)i = ∆S(t)
+ ∆σ
Clausius type decomposition.
Set
b
Ep+ = lim lim ∆S(t)
t→∞ T D
b + = lim lim ∆σ
b (t).
∆σ
t→∞ T D
15
OPEN PROBLEMS
Mathematical structure of the decomposition
b +.
hσi+ = Ep+ + ∆σ
b + in concrete models (to be disThe existence of Ep+ and ∆σ
cussed latter).
b + = 0? Ruelle: Perhaps when the boundaries beWhen is ∆σ
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
N
Λ = [A, B] ⊂ Z, Hilbert space HΛ = x∈Λ C2.
Hamiltonian
1 X
(1) (1)
(2) (2)
Jx σx σx+1 + σx σx+1
HΛ =
2 x∈[A,B[
1 X
(3)
λxσx .
+
2 x∈[A,B]
(1)
σx
"
=
0 1
1 0
#
,
(2)
σx
"
=
0 −i
i 0
#
,
(3)
σx
"
=
1 0
0 −1
17
#
.
−M
N
−N
RL
C
M
RR
Central part C (small system S): XY-chain on ΛC = [−N, N ].
Two reservoirs RL/R : XY-chains on ΛL = [−M, −N − 1] and
ΛR = [N + 1, M ].
N fixed, thermodynamic limit M → ∞.
Decoupled Hamiltonian Hfr = HΛL + HΛC + HΛR .
18
The full Hamiltonian is
H = HΛL∪ΛC ∪ΛR = Hfr + VL + VR ,
J
(1)
(1)
(2)
(2)
VL = −N −1 σ−N −1σ−N + σ−N −1σ−N , etc.
2
Initial state:
ρ=e
−βL HΛ
L
−βR HΛ
⊗ ρ0 ⊗ e
R
.
Z,
ρ0 = 1/ dim HΛC .
Fluxes and entropy production:
ΦL/R = −i[H, HL/R ],
σ = −βLΦL − βR ΦR .
19
Araki-Ho, Ashbacher-Pillet ∼ 2000, J-Landon-Pillet 2012: NESS
exists and
∆β
E sinh(∆βE)
2
hσi+ =
dE > 0.
|T (E)|
βR E
βL E
4π R
cosh
cosh
Z
2
2
∆β = βL − βR . Landauer-Büttiker formula.
σ+ does not depend where the boundary N is set.
20
Steady state heat fluxes:
hΦLi+ + hΦR i+ = 0
hσi+ = −βLhΦLi+ − βR hΦR i+.
1
E sinh(∆βE)
2
hΦR i+ =
dE.
|T (E)|
βL E
βR E
4π R
cosh
cosh
Z
2
2
21
Idea of the proof–Jordan-Wigner transformation.
O is transformed to the even part of CAR(`2(Z)) generated
by {ax, a∗x | x ∈ Z} acting on the fermionic Fock space F over
`2(Z).
Transformed dynamics: generated by dΓ (h), where h is the
Jacobi matrix
hux = Jxux+1 + Jx−1ux−1 + λxux,
u ∈ `2(Z).
ΦR (and similarly ΦL, σ) is transformed to
−iJN JN +1(a∗N aN +2 − a∗N +2aN )
− iJN λN +1(a∗N aN +1 − a∗N +1aN ).
22
Decomposition
`2(Z) = `2(] − ∞, −N − 1]) ⊕ `2([−N, N ]) ⊕ `2([N + 1, ∞[),
hfr = hL + hC + hR ,
h = hfr + vL + vR ,
vR = JN (|δN +1ihδN | + h.c)
The initial state ρ is transformed to the quasi-free state generated by
1
1
1
⊕
⊕
.
1 + eβLhL 2N + 1 1 + eβRhR
23
The wave operators
w± = s − lim eithe−ithfr 1ac(hfr )
t→±∞
exist and are complete.
The scattering matrix:
∗ w : H (h ) → H (h )
s = w+
ac fr
ac fr
−
"
s(E) =
A(E) T (E)
T (E) B(E)
#
.
24
q
2i
T (E) = J−N −1JN hδN |(h−E −i0)−1δ−N i FL(E)FR (E)
π
FL/R (E) = Im hδL/R |(hL/R − E − i0)−1δL/R i,
δL = δ−N −1, δR = δN +1.
T (E) is non-vanishing on the set spac(hL) ∩ spac(hR ).
Jx = const, λx = const (or periodic)
|T | = χσ(h)
25
Assumption:
h has no singular continuous spectrum
b +.
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 structures.
26
HEISENBERG SPIN CHAIN
The Hamiltonian H of XY spin chain is changed to
HP = H + P
where
X
1
(3) (3)
P =
Kxσx σx+1.
2 x∈[−N,N [
The central part is now Heisenberg spin chain
X
1
(1) (1)
(2) (2)
(3) (3)
Jxσx σx+1 + Jxσx σx+1 + Kxσx σx+1
2 x∈[−N,N [
X
1
(3)
+
λxσx .
2 x∈[−N,N ]
27
Initial state remains the same. h is the old Jacobi matrix.
Fluxes and entropy production:
ΦL/R = −i[HP , HL/R ]
σ = −βLΦL − βR ΦR .
TD limit obvious. τP denotes the perturbed C ∗-dynamics.
28
Assumption For all x, y ∈ Z,
Z ∞
0
|hδx, eithδy i|dt < ∞.
Denote
`N =
Z ∞
0
sup
|hδx, eithδy i|dt,
x,y∈[−N,N [
66 1 1
K̄ = 6
.
7 24N `N
29
Theorem. Suppose that
sup
|Kx| < K̄.
x∈[−N,N [
Then for all A ∈ O,
ρ+(A) = lim ρ(τPt (A))
t→∞
exists.
30
Comments:
No time averaging. The constant K̄ is essentially optimal. With
change of the constant K̄ the result holds for any P depending
(3)
on finitely many σx :
P =
XY
(3)
(3)
Kxi ···xi σxi · · · σxi .
1
k
1
k
The NESS ρ+ is attractor in the sense that for any ρ-normal
initial state ω,
lim ω ◦ τPt = ρ+.
t→∞
31
The map
({Kx}, βL, βR ) 7→ hσi+ = ρ+(σ)
is real analytic. This leads to the strict positivity of entropy production.
Green-Kubo linear response formula holds for thermodynamical
force X = βL − βR (J-Pillet-Ogata)
Bosonization Central Limit Theorem holds (J-Pautrat-Pillet)
32
OPEN PROBLEM
The existence (and properties) of NESS
1 t
ρ+(A) = lim
ρ(τPs (A))ds
t→∞ t 0
Z
for all {Kx} ∈ R2N .
This is an open problem even if
P = K0a∗0a0a∗1a1.
Dependence of hσi+ on N ?
33
Idea of the proof: Jordan-Wigner transformation: τPt is generated by
X
1
Kx(2a∗xax − 1)(2a∗x+1ax+1 − 1).
dΓ (h) +
2 x∈[−N,N [
One proves that
γ +(A) = lim τ −t ◦ τPt (A)
t→∞
exists and is an ∗-automorphism of O. The starting point is the
Dyson expansion of τ −t ◦ τPt . One then proceeds with careful
combinatorial estimates of each term in the expansion.
The key ingredient is:
34
Theorem (Botvich-Maassen).
f k be two sequences of nonnegative numbers and g, g
e
Let mk , m
two integrable nonnegative functions on [0, ∞[. Denote by kgk
and kgek their L1-norms, set g0 = ge and gk = g for k > 0 and
define
∞
X
mk k
M (x) ≡
x ,
k!
k=0
∞ f
X
mk k
f (x) ≡
M
x .
k!
k=0
To any rooted tree T with root 0 and nodes 1, · · · , n associate
the weight (rj is the number of children of the node j)
f r0 mr1 · · · mrn
w(T ) = m
×
Z
n
Y
gT (j)(sT (j) − sj ) ds0 · · · dsn−1.
0=sn ≤sn−1 ≤···≤s0 j=1
35
Then, the sum
W =
∞
X
X
w(T )
n=1 T ∈Tn
is finite if and only if the equation
M (kgkx) = x
has a positive solution x such that
f (kg
ekx) < ∞.
M
If x∗ denotes the least such solution, then
f (kg
ekx∗).
W =M
36
The transport theory of non-equilibrium quantum statistical mechanics leads to an insight regarding two basic questions of
spectral theory:
What is localization?
What is absolutely continuous spectrum?
37
WHAT IS LOCALIZATION?
Simplest setup. XY chain.
Jx = JC for x ∈ [−N, N [, Jx = JL, λx = λL for x < −N and
Jx = JR , λx = λR for x > N .
[a, b] = sp(hL) ∩ sp(hR ).
{λx}x∈[−N,N ] i.i.d. random variables.
hC = JC ∆ + λx
discrete Schrödinger operator on [−N, N ]. γ(E) its Lyapunov
exponent.
38
∆β b
E sinh(∆βE)
E(hσi+) =
E(|T (E)|2)
dE
β
E
β
E
R
L
4π a
cosh
cosh
Z
2
2
1
log E(hσi+) ∼ − inf γ(E).
N →∞ N
E∈[a,b]
lim
Physically natural characterization of localization.
39
Heisenberg chain, Kx = K for x ∈ [−N, N − 1].
hσ̄i+ = lim suph∆σ(t)i.
t→∞
Definition of exponential localization on [a, b]:
1
log E(hσ̄i+) < 0.
N →∞ N
lim sup
40
OPEN PROBLEM
Instead of Heisenberg chain chain, consider a single site interaction
P = Ka∗0a0a∗1a1.
Prove exponential localization in this case.
41
ELECTRONIC BLACK BOX MODEL
R2
R1
S
R3
42
ENTROPIC FLUCTUATIONS
• J., Ogata, Pautrat, Pillet:
”Entropic fluctuations in non-equlibrium quantum statistical
mechanics. An Introduction.”
In Quantum Theory from Small to Large Scales, Les Houches
Proceeding (2012)
• J., Pillet, Rey-Bellet:
”Entropic Fluctuations in Statistical Mechanics I. Classical
Dynamical Systems.” Nonlinearity (2011)
43
NAIVE FLUCTUATION RELATION FAILS
Finite dimensional setup. Time-reversal invariance.
Spectral resolution
X
1 t
∆σ(t) =
σsds =
λPλ.
t 0
Z
Time-reversal implies
dim Pλ = dim P−λ.
Entropy balance equation
X
1
S(ρt|ρ) = h∆σ(t)i =
λtr(ρPλ) ≥ 0.
t
44
Positive λ’s are favoured. Heat flows from hot to cold.
BAD NEWS: The fluctuation relation
tr(ρP−λ)
= e−tλ
tr(ρPλ)
FAILS.
Cummulant generating function:
enaive(α) = log tr(ρe−αt∆σ(t))
= log tr(e−S e−α(St−S)).
Equivalent form of bad news:
enaive(α) = enaive(1 − α)
FAILS.
45
QUANTUM ENTROPIC FUNCTIONAL I
Kurchan (2000), Tasaki-Matsui (2003)
efcs(α) = log tr(e−(1−α)S e−αSt ).
Renyi relative entropy:
efcs(α) = log tr(ρ1−α
ρα).
t
Time reversal invariance implies that the symmetry
efcs(α) = efcs(1 − α)
HOLDS.
46
Tasaki-Matsui relative modular operator interpretation.
O = B(H), hA, Bi = tr(A∗B). Ωρ = ρ1/2.
∆ρt|ρ(A) = ρtAρ−1.
efcs(α) = loghΩρ, ∆−α
Ω i
ρ |ρ ρ
t
= log
Z
R
e−αtς Pt(ς).
47
Atomic probability measure Pt is the spectral measure for the
operator
1
1
− log ∆ρt|ρ(A) = − log ∆ρ|ρ(A) − ∆σ(t)A
t
t
and Ωρ.
efcs(α) = efcs(1 − α) is equivalent to
Pt(−ς)
= e−tς .
Pt(ς)
48
Kurchan interpretation gives the physical meaning:
efcs(α) is the cummulant generating function for the full counting statistics (Levitov-Lesovik) of the repeated quantum measurement of S = − log ρ.
S=
X
sPs
Measurement at t = 0 yields s with probability tr(ρPs).
49
State after the measurement:
ρPs/tr(ρPs).
State at later time t:
e−itH ρPseitH /tr(ρPs).
Another measurement of S yields value s0 with probability
tr(Ps0 e−itH ρPseitH )/tr(ρPs).
The probability of measuring the pair (s, s0) is
tr(Ps0 e−itH ρPseitH )
50
Probability distribution of the mean change of entropy
ς = (s0 − s)/t
is the spectral measure of Tasaki-Matsui:
Pt(ς) =
X
tr(Ps0 e−itH PseitH ).
s0 −s=tς
efcs(α) is the cummulant generating function for Pt.
51
QUANTUM ENTROPIC FUNCTIONAL II
J-Ogata-Pautrat-Pillet.
evar (α) = log tr(e−(1−α)S−αSt ).
Time reversal implies
evar (α) = evar (1 − α)
Variational characterization:
evar (α) = − inf (αtr(ω(St − S)) + S(ρ|ω)) .
ω
Golden-Thompson:
evar (α) ≤ efcs(α).
52
Herbert Stahl (2011): Bessis-Moussa-Villani conjecture.
There exist probability measure Qt such that
evar (α) = log
Z
R
e−αtς dQt(ς).
evar (α) = evar (1 − α) implies
dQt(−ς)
= e−tς .
dQt(ς)
53
ALGEBRAIC BMV CONJECTURE
(M, τ t, Ω) W ∗-dynamical system on a Hilbert space H. Ω is
(τ, β)-KMS vector.
τ t(A) = eitLAe−itL.
V ∈ M selfadjoint, ΩV the β- KMS vector for perturbed dynamics
τVt (A) = eit(L+V )Ae−it(L+V ).
ΩV = e
− β2 (L+V )
Ω
54
The Pierls-Bogoluibov and Golden-Thompson inequality hold:
e−βhΩ,V Ωi/2 ≤ kΩV k ≤ ke−βV /2Ωk.
CONJECTURE:
There exists measure Q on R such that for α ∈ R,
kΩαV k2 =
Z
eαφdQ(φ).
R
Finite systems:
kΩαV k2 = tr(e−β(H+αV ))/tr(e−βH ).
55
INTERPOLATING FUNCTIONALS
For p ∈ [1, ∞),
− 1−α
S − 2α
St − 1−α
S
p
p
e
e p
e
ep(α) = log tr
= log tr
1−α 2α 1−α
ρ p ρpρ p
p/2
!p/2
t
.
• e2(α) = efcs(α).
• e∞(α) = limp→∞ ep(α) = evar (α).
• ep(α) = ep(1 − α)
56
• ep(0) = ep(1) = 0.
• α 7→ ep(α) is convex.
• e0p(0) = −S(ρt|ρ), e0p(1) = S(ρt|ρ).
• [1, ∞] 3 p 7→ ep(α) is decreasing (strictly): (Araki)-LiebThirring.
3
2
1
0.5
-0.5
-1
1.0
1.5
• Interpolating functionals motivated recent work: arXiv:1310.7178.
M.R. Audenaert, N. Datta: α-z-relative Renyi entropies.
For ν, ζ > 0, set
Sp,α(ν, ζ) = log tr
1−α 2α 1−α p/2
ν p ζpν p
.
Obtaining a single quantum generalization of the classical
relative Renyi entropy, which would cover all possible operational scenarios in quantum information theory, is a challenging (and perhaps impossible) task. However, we believe Sp,α is thus far the best candidate for such a quantity,
since it unifies all known quantum relative entropies in the
literature.
57
• Quantum transfer operators. Act on B(H). Specific norm:
1/p
1/p
p
kAkp = tr(|Aρ | )
.
Up(t)A = A−t
1S
1
−t − p S
p
e
e
.
Properties:
Up(t1 + t2) = Up(t1)Up(t2)
Up(−t)AUp(t) = At
kUp(t)Akp = kAkp.
Crucial property:
ep(α) = log kUp/α(t)1kpp.
58
GOALS
• Mathematical structure of finite time theory that deals directly with infinitely extended system within the framework
of algebraic quantum statistical mechanics. Modular theory
of W ∗-dynamical systems (Araki, Connes, Haagerup).
Critical role: Araki-Masuda theory of non-commutative Lpspaces.
Araki, H., Masuda, T. (1982). Positive cones and Lp-spaces
for von Neumann algebras. Publ. RIMS, Kyoto Univ. 18,
339–411.
• Benefit of unraveling the algebraic structure of entropic functionals: Quantum Ruelle transfer operators.
59
• Concrete models: Thermodynamic limit of the finite time
finite volume structures.
• The existence and regularity of
1
ep+(α) = lim ept(α).
t→∞ t
Difficult problem in physically interesting models. Link with
quantum Ruelle resonances.
ep+(α) inherits all the listed properties of ept(α).
3
2
1
0.5
-0.5
1.0
1.5
-1
e0p+(0) = −hσi+,
e0p+(0) = hσi+
60
• Implications. p = 2, the large deviation principle and central limit theorem for the full counting statistics of entropy/energy/charge
transport. The symmetry α → 1 − α in the linear regime
(small α, linear response) yields the Green-Kubo formulas
and Onsager reciprocity relations energy and charge fluxes.
The Fluctuation-Dissipation Theorem follows.
• p = ∞. The large deviation principle and central limit theorem for the BMV Qt. Quantum version of Gallavotti’s linear
response theory.
61
BACK TO XY CHAIN
Two additional functionals:
(I) (Evans-Searles) Fluctuations with respect to the initial state:
Ct(α) = lim log tr ρe−α
Rt
0 σs ds
M →∞
.
(II) (Gallavotti-Cohen) Steady state fluctuations: ρt = e−itH ρeitH .
lim log tr ρT e−α
T →∞ M →∞
Ct+(α) = lim
Rt
0 σs ds
.
Naive quantizations of the classical entropic functionals.
62
After the TD limit
Ct(α) = log ρ e−α
= log
Z
Rt
0 σs ds
e−αtς dPt(ς),
R
Ct+(α) = log ρ+ e−α
= log
Z
R
Rt
0 σs ds
e−αtς dPt+(ς).
Rt
1
Pt/t+ is the spectral measure for ρ/ρ+ and t 0 σsds.
63
THEOREM
Assumption: Jacobi matrix h has purely ac spectrum.
(1)
1
1
C(α) = lim Ct(α) = lim Ct+(α)
t→∞ t
t→∞ t
!
det(1 + Kα(E)) dE
=
log
,
det(1 + K0(E)) 2π
R
Z
∗ (E)k (E)s(E)−k (E)) k (E)/2
k
(E)/2
α(s
0
0
0
Kα(E) = e
e
e 0
,
"
k0(E) =
−βLE
0
0
−βR E
#
"
, s(E) =
A(E) T (E)
T (E) B(E)
64
#
(2)
!
1
det(1 + Kαp(E)) dE
ept(α) =
,
log
t→∞ t
det(1 + K0(E)) 2π
R
Z
ep+(α) = lim
p/2
k
(E)(1−α)/p
k
(E)2α/p
∗
k
(E)(1−α)/p
Kαp(E) = e 0
s(E)e 0
s (E)e 0
(3) The functionals C(α), ep+(α) are real-analytic and strictly
convex.
C(0) = ep+(0) = 0 and
C 0(0) = e0p+(0) = −hσi+.
(4)
E
1 ∞ D
00
00
C (0) = e2(0) =
(σt − hσi+)(σ − hσi+)
dt.
+
2 −∞
Z
(5) The function
[1, ∞] 3 p 7→ ep+(α)
is continuous and decreasing.
It is strictly decreasing unless h is reflectionless:
|T (E)| ∈ {0, 1}
∀E.
If h is reflectionless, then ep+(α) does not depend on p and
ep+(α) = C(α) =
cosh((βL(1 − α) + βR α)E/2) × (L → R)
1
dE.
2π sp(h)
cosh(βLE/2) cosh(βR E/2)
Z
Phenomenon: ”Entropic triviality.”
(6) If h is not reflectionless, C(1) > 0.
(7) The Central Limit Theorem and Large Deviation Principle
hold for measures Pt, Pt+, Pt, Qt.
Pt/t+ → δhσi+ , Pt, Qt → δhσi+ . Gärtner-Ellis theorem.
Pt(B) ' Pt+(B) ' e−t inf ς∈B I(ς),
Pt(B) ' e−t inf ς∈B I(ς),
Qt(B) ' e−t inf ς∈B J (ς)
I(ς) = − inf (ας + C(α)) ,
α∈R
I(ς) = − inf ας + e2+(α) ,
α∈R
J (ς) = − inf ας + e∞+(α)
α∈R
Fluctuation Relation implies
I(−ς) = ς + I(ς),
etc.
OPEN PROBLEM
Suppose that Jx = J > 0 for all x.
hux = J(ux+1 + ux−1) + λxux
discrete Schrödinger operator.
Davies-Simon (1978): h is called homogenuous if it is reflectionless and has purely a.c. spectrum.
We feel that the theory of homogeneous Hamiltonians
is worthy of further study.
Does there exist h with purely a.c. spectrum which is not reflectionless?
66
HEISENBERG CHAIN
K̄ = supx∈[−N,N ] |Kx|.
Given δ > 0 there exists > 0 such that if |K̄| < ,
ep+(α)
exists for p = 2, ∞ and α ∈] − δ, 1 + δ[.
({Kx}, α) 7→ ep+(α) is real analytic.
CLT and local LDP for Pt and Qt.
Proof: Combination of De Roeck-Kupianien dynamical polymer
expansion and combinatorial estimates of J-Pautrat-Pillet.
67
ELECTRONIC BLACK BOX MODEL
R2
R1
S
R3
68
MCLENNAN-ZUBAREV DYNAMICAL ENSEMBLES
Open systems:
ρt = e−S−t = e
−
P
R
βk (Hk + 0t Φk(−s )ds)
ρt– Gibbs state at inverse temperature 1 for
X
βk (Hk +
Z t
0
Φk(−s))ds.
69
TD limit: ρt is KMS-state for the dynamics generated by
δt(·) =
X
βk δk (·) +
Z t
0
[Φk(−s), ·]ds
NESS ρ+ is the KMS state for the dynamics generated by
δt(·) =
X
βk δk (·) +
Z ∞
0
[Φk(−s), ·]ds
Aschbacher-Pillet, Ogata-Matsui, Tasaki-Matsui.
70
XY-chain, Jx = const, λx = 0,
β = (βL + βR )/2,
γ = (βR − βL)/2.
ρ+ is β-KMS state for Hamiltonian
H+
δ
K
β
1 X
(1) (3)
(3) (2)
K = j(x − y)
σx σx+1 · · · σy−1σy − h.c.)
2i x<y
where j is the Fourier transform of | cos θ|.
71
e2t(α) = log tr(e−(1−α)S e−αSt ).
e∞t(α) = log tr(e−(1−α)S−αSt ).
In open systems:
e2t(α) = log tr e−(1−α)
P
e∞t(α) = log tr e−
βk Hk e−α
P
P
Rt
βk (Hk + 0 Φksds)
.
Rt
βk (Hk +α 0 Φksds.
.
Similarly for other ept(α). The entropic functionals can be viewed
as deformations McLennan-Zubarev dynamical ensembles.
72
ENTROPIC GEOMETRY (UNDER CONTRUCTION)
David Ruelle: Extending the definition of entropy to nonequilibrium steady states. Proc. Nat. Acad. Sci. 100 (2003).
Outside of equilibrium entropy has curvature.
An old idea. Ruppeiner geometry (1979).
G. Ruppeiner (1995): Riemannian geometry in thermodynamic
fluctuation theory. Reviews of Modern Physics 67 (3): 605659
Older:
B. Effron: Defining the curvature of the statistical problem. The
Annals of Statistics (1975), 1189-1242.
73
Even older:
Rao, C.R: (1945) Information and accuracy attainable in the estimation of statistical parameter. Bull. Calcutta Math. Soc. 37.
Information geometry. Monograph:
Amari-Nagaoka: Methods of information geometry (2000)
For our purposes:
G. Crooks (2007): Measuring thermodynamic length. PRL 99.
Parameter manifold: (β1, · · · , βM ).
e00p+(0) =
X
βj βk Lpjk .
j,k
This introduces a (possibly degenerate) metric on the tangent
space at (β1, · · · , βM ).
p = 2. metric is induced by the CLT variance of CLT for the full
counting statistics.
1 ∞
ρ+ (Φjs − ρ+(Φj ))(Φks − ρ+(Φk )) ds.
L2jk =
2 −∞
Z
74
In equilibrium β1 = · · · = βM , L2jk are Onsager transport
coefficents.
At p = ∞, twist to Bogoluibov-Kubo-Mari inner product.
The induced norms k · kp are monotone in p.
Crooks thermodynamical path out of equilibrium.
75
RELATIONS WITH QUANTUM INFORMATION THEORY
Landauer principle: the energy cost of erasing quantum bit of information by action of a thermal reservoir at inverse temperature
T is ≥ kT log 2 with the equality for quasi-static processes.
Full counting statistics, e2,+(α) = the Chernoff error exponent
in the quantum hypothesis testing of the arrow of time, i.e., of
the family of states {ρt, ρ−t}t>0.
J., Ogata-Pillet-Seiringer.: Quantum hypothesis testing and nonequilibrium statistical mechanics, Rev. Math. Phys, 24 (6) (2012),
1-67
Parameter estimation, Fisher entropies, entropic/information geometry?
76
STEADY STATE FLUCTUATION RELATIONS
Classical statistical mechanics: Dynamical system (M, φt, ρ).
R
Observable: f : M → R. ρ(f ) = M f dρ. Time evolution
ft = f ◦ φ t
ρt = ρ ◦ φ−t.
Phase space contraction:
∆ρt|ρ =
dρt
.
dρ
Entropy production observable
d
log ∆ρt|ρ
σ=
t=0
dt
77
log ∆ρt|ρ =
S(ρt|ρ) =
Z
M
Z t
0
σ−sds.
log ∆ρt|ρdρt =
Z t
0
ρ(σs)ds.
Classical open systems:
σ=−
X
βk Φk
Φk = {Hk , V }.
78
Evans-Searles entropic functional:
et(α) = log
Z
e−α
Rt
0 σs ds dρ.
M
Time-reversal invariance.
Evans-Searles fluctuation relation:
et(α) = et(1 − α).
Let Pt be the probability distribution of
1 t
σs
t 0
Z
with respect to ρ.
79
dPt(−ς)
= e−tς .
dPt(ς)
1
e+(α) = lim et(α).
t→∞ t
CLT and LDP are with respect to ρ.
Important: The classical counterpart of the theory of quantum
entropic fluctuations described so far is the Evans-Searles fluctuation relation.
Gallavotti-Cohen fluctuation relation: Related but also very different.
80
NESS: weak limit
ρ+(f ) = lim ρt(f ).
t→∞
ρ+(σ) > 0 ⇔ ρ+ ⊥ ρ.
Gallavotti-Cohen entropic functional:
ebt(α) = log
Z
R
−α 0t σs ds
dρ
e
M
+.
Finite time fluctuation relation
ebt(α) = ebt(1 − α)
does not hold.
It is even possible that ebt(1) = ∞ for t > 0 (chain of harmonic
oscillators).
81
1
ebt(α).
t→∞ t
eb+(α) = lim
Gallavotti-Cohen fluctuation relation: for Anosov diffeomprhisms
of compact manifolds the symmetry
eb+(α) = eb+(1 − α)
is restored. In this case
eb+(α) = e+(α).
General Gallavotti-Cohen fluctuation relation.
82
Principle of regular entropic fluctuations.Exchange of limits:
Rt
1
e+(α) = lim log
e−α 0 σsdsdρ
t→∞ t
M
Z
R
1
−α uu+t σs ds
= lim log
dρ
e
t→∞ t
M
Z
Rt
1
= lim log
e−α 0 σsdsdρu
t→∞ t
M
Z
R
1
−α 0t σs ds
= lim lim log
dρu
e
t→∞ u→∞ t
M
Z
R
1
−α 0t σs ds
= lim log
e
dρ+
t→∞ t
M
Z
= eb+(α).
83
OPEN PROBLEM
Gallavotti-Cohen fluctuation relation in quantum statistical mechancs.
Two obvious routes:
Tasaki-Matsui: (H, π, Ωρ) GNS-representation induced ρ,
log ∆ρt|ρ = log ∆ρ|ρ +
e2t(α) = log
Z
R
Z t
0
σ−sds.
e−αtς Pt(ς)
= loghΩρ, ∆α
ρt |ρ Ωρ i
Rt
α log ∆ρ|ρ − 0 σs ds
= loghΩρ, e
Ωρi
84
(H+, π+, Ωρ+ ),
α log ∆ρ |ρ
+ +
eb2t(α) = loghΩρ+ , e
= log
Z
R
Rt
− 0 σs ds
Ωρ+ i
b (ς).
e−αtς dP
t
1
eb2+(α) = lim eb2t(α).
t→∞ t
XY chain:
eb2+(α) = e2+(α).
Same for locally interacting case.
85
Missing: Physical interpretation. Repeated quantum measurement procedure incompatible with NESS structure.
One possibility: Indirect measurements.
Bauer M., Bernard D., Phys. Rev. A84, (2011) Convergence
of repeated quantum non-demolition measurements and wave
function collapse.
M. Bauer, T. Benoist, D. Bernard, Repeated Quantum Non-Demolition
Measurements: Convergence and Continuous Time Limit, Ann.
Henri Poincar 14 (2013) 639 67
86
One difficulty in quantum optic experiments is to measure a system without destroying it. For example to count a number of
photons usually one would need to convert each photon into an
electric signal. To avoid such destruction one can use non demolition measurements. Instead of measuring directly the system, quantum probes interact with it and are then measured.
The interaction is tuned such that a set of system states are
stable under the measurement process.
This situation is typically the one of Serge Haroche’s (2012 Nobel prize in physics) group experiment inspired the work I will
present. In their experiment they used atoms as probes to measure the number of photons inside a cavity without destroying
them.
Abstract of T. Benoist talk at McGill (2014)
87
p = ∞. Slightly more satisfactory.
e∞t = − inf
S(ω|ρ) + α
ωρ
eb∞t = − inf
ωρ+
Z t
0
S(ω|ρ+) + α
ω(σs)ds .
Z t
0
ω(σs)ds .
Again, in the cases where one can compute (XY, etc):
eb∞+(α) = e∞+(α).
88
TOPICS NOT DICSUSSED
Weak coupling limit (Davies 1974, Lebowitz-Spohn 1978)
Repeated interactions systems:
Bruneau, Joye, Merkli: Repeated interactions in open quantum
systems, to appear in JMP.
Pauli-Fierz systems (finite level atom coupled to bosonic reservoirs).
Classical statistical mechanics.
89
CONCLUSIONS
90