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A note on the entropy production formula
Vojkan Jaksic, Claude-Alain Pillet
To cite this version:
Vojkan Jaksic, Claude-Alain Pillet. A note on the entropy production formula. Contemporary math-ematics, American Mathematical Society, 2003, 327, pp.175-180. �hal-00184182�
Vojkan Jakˇsi´c
1and Claude-Alain Pillet
2,3 1Department of Mathematics and Statistics
McGill University
805 Sherbrooke Street West
Montreal, QC, H3A 2K6, Canada
2
PHYMAT, Universit´e de Toulon
B.P. 132, 83957 La Garde Cedex, France
3
FRUMAM
CPT-CNRS Luminy, Case 907
13288 Marseille Cedex 9, France
September 11, 2002
Abstract
We give an elementary derivation of the entropy production formula of [JP1]. Using this derivation we show that the entropy production of any normal, stationary state is zero.
1
Introduction
LetO be a C∗- algebra,E(O) the set of all states on O and ω ∈ E(O). We assume that there
exists a referenceC∗- dynamicsσt
ω on O such that ω is a (σω, −1)-KMS state. We denote by
δω the generator of σωt (i.e. σωt = etδ ω
) and by D(δω) its domain. Let (Hω, πω, Ωω) be the
GNS-representation of the algebraO associated to the state ω.
A stateη ∈ E(O) is called ω- normal if there exists a density matrix ρη onHωsuch that, for all
A ∈ O, η(A) = Tr(ρηπω(A)). Let Nωbe the set of allω- normal states on O.
2
For η ∈ Nω, we denote by Ent(η|ω) the relative entropy of Araki [Ar1, Ar2]. (We use the
notational convention for relative entropy of [BR, Don].) Ifη 6∈ Nω, we setEnt(η|ω) = −∞.
For unitaryU ∈ O and η ∈ E(O), we denote by ηU the stateηU(A) ≡ η(U∗AU). The main
result of this note is:
Theorem 1.1 For any unitaryU ∈ O ∩ D(δω) and any η ∈ E(O),
Ent(ηU|ω) = Ent(η|ω) − iη(U∗δω(U)). (1.1)
As we shall explain below, Theorem 1.1 is a natural generalization of the entropy production formula derived in [JP1, JP2]. The method of proof we will use in this note, however, is quite different from the one in [JP1]. We will reduce the proof of Theorem 1.1 to a fairly elementary application of some well known identities in Araki’s theory of perturbation of KMS structure. The proof in [JP1], based on Araki-Connes cocycles, was technically more involved and re-stricted to faithful statesη ∈ Nω.
We now relate Equ. (1.1) to the entropy production formula of [JP1, JP2]. Assume that there exists aC∗- dynamics τt on O and that ω is τ - invariant. Let V (t) be a time-dependent local
perturbation, that is,V (t) is norm-continuous, self-adjoint, O- valued function on R (the time-independent case of [JP1] of course follows by settingV (t) ≡ V ). The perturbed time evolution
is the strongly continuous family of∗- automorphisms of O given by the formula
τVt(A) ≡ τt(A) +X n≥1 in Z t 0 dt1 Z t1 0 dt2· · · Z tn−1 0 dtn[τtn(V (tn)), [· · · , [τt1(V (t1)), τt(A)]]].
In the interaction representation,τt
V is given by
τVt(A) = ΓtVτt(A)Γt∗V,
whereΓt
V ∈ O is a family of unitaries satisfying the differential equation
d dtΓ
t
V = iΓtVτt(V (t)), Γ0V = 1.
Theorem 1.1 then has the following immediate corollary (see also Theorem 4.8 in [JP2]):
Corollary 1.2 Assume thatω is τ - invariant and that Γt
V ∈ D(δω). Then, for any η ∈ E(O),
From now on we will consider the time-independent case V (t) ≡ V . If V ∈ D(δω), then Γt V ∈ D(δω) and d dtΓ t Vδω(Γt∗V) = −iτVt(δω(V )). (1.3)
Hence, (1.2) reduces to the entropy production formula of [JP1]:
Ent(η ◦ τVt|ω) = Ent(η|ω) − Z t
0
η ◦ τVs(δω(V )) ds. (1.4)
We emphasize that the above derivation of (1.4) allows for non-faithfulη.
The entropy production of a stateη ∈ E(O) was defined in [JP1, JP2] by EpV(η) ≡ η(δω(V )),
see also [OHI, O1, O2, Ru, Sp]. On physical grounds, it is natural to conjecture that ifη is
ω-normal andτV- invariant, then EpV(η) = 0. For faithful η this was proven in [JP1]. Here, we
establish this result in full generality.
Theorem 1.3 Assume thatω is τ - invariant, that V ∈ D(δω) and that η is τV- invariant and
ω- normal. Then,
EpV(η) = 0.
Remark. If Ent(η|ω) > −∞, then Theorem 1.3 is an immediate consequence of Equ. (1.4).
The caseEnt(η|ω) = −∞ requires a separate and somewhat delicate argument.
The results of this note were announced in the recent review [JP2] where the interested reader may find additional information and references about entropy production and its role in non-equilibrium quantum statistical mechanics.
Acknowledgment. The research of the first author was partly supported by NSERC. A part of
this work has been done during the visit of the second author to the McGill University which was supported by NSERC.
2
Proof of Theorem 1.1
We assume that the reader is familiar with basic results of Tomita-Takesaki modular theory as discussed, for example, in [BR, DJP, Don, OP].
Let Mω ≡ πω(O)′′ be the enveloping von Neumann algebra. Sinceω is (σω, −1)-KMS state,
the vectorΩωis separating for Mω, and we denote byP, J, ∆ωthe corresponding natural cone,
modular conjugation and modular operator. We recall that∆ω = eLω, whereLω is the unique
self-adjoint operator onHω such that
πω(σtω(A)) = eitL ω
4
In particular,σt
ωextends naturally to aW∗- dynamics on Mω which we again denote byσωt. In
this contextσt
ω is called modular dynamics.
Any stateη ∈ Nω has a unique normal extension to Mω which we denote by the same letter.
Obviously, η is ω- normal iff ηU is ω- normal for all unitaries U ∈ O and so, in the proof of
Theorem 1.1, we may restrict ourselves toω- normal η’s.
We will use the fact that ifγ : Mω 7→ Mωis a∗- automorphism, then
Ent(η ◦ γ|ω ◦ γ) = Ent(η|ω).
In particular,
Ent(ηU|ω) = Ent(η|ωU∗).
LetΨU∗ be the unique vector representative of the stateωU∗ in the coneP. A simple
computa-tion shows that
ΨU∗ = πω(U∗)Jπω(U∗)Ωω.
We will considerP ≡ πω(−iU∗δω(U)) as a local perturbation of the modular group σtω. Letαt
be the locally perturbedW∗- dynamics,
αt(A) ≡ eit(Lω+P )Ae−it(Lω+P )= ΘtPσωt(A)Θt∗P,
whereeit(Lω+P )
e−itLω ≡ Θt
P ∈ Mω is a family of unitaries satisfying
d dtΘ
t
P = iΘtPσωt(P ), Θ0P = 1. (2.5)
Letψ be the unique (α, −1)-KMS state on Mω. By the Araki theory,Ωω ∈ D(e(Lω+P )/2) and
the unique vector representative ofψ in the natural cone P is
Ψ = e (Lω+P )/2 Ωω ke(Lω+P )/2Ω ωk .
Another fundamental result of Araki’s theory is the relation
Ent(η|ψ) = Ent(η|ω) + η(P ) − log ke(Lω+P )/2Ωωk2, (2.6)
which holds for allω- normal states η. (For η faithful, this relation was proven in [Ar1, Ar2], see
also [BR]. Its extension to generalη was obtained in [Don], see also the next section). Hence,
to finish the proof it suffices to show thate(Lω+P )/2Ω
ω = ΨU∗.
We setTt≡ U∗σt
ω(U) and observe that
d dtT
t= iTtσt
Comparison with Equ. (2.5) immediately leads toπω(Tt) = ΘtP and therefore
eit(Lω+P )
Ωω = πω(Tt)eitLωΩω
= πω(U∗)eitLωπω(U)Ωω.
(2.7)
Since the vector-valued functionz 7→ eiz(Lω+P )Ωωis analytic inside the strip−1/2 < Im z < 0
and strongly continuous on its closure, analytic continuation of the identity (2.7) toz = −i/2,
yields
e(Lω+P )/2
Ωω = πω(U∗)∆1/2ω πω(U)Ωω
= πω(U∗)Jπω(U∗)Ωω
= ΨU∗,
which is the desired relation.
3
Proof of Theorem 1.3
LetΩη be the vector representative ofη in the natural cone P. The standard Liouvillean
associ-ated to the dynamicsτt
V isLV = L + πω(V ) − Jπω(V )J, where L is the standard Liouvillean
associated toτt. We recall thatL and L
V are uniquely specified by
πω(τt(A)) = eitLπω(A)e−itL, LΩω = 0,
and
πω(τVt(A)) = eitLVπω(A)e−itLV, LVΩη = 0.
We denote bysη the support of the stateη and set s′η = JsηJ. Obviously
sηΩη = s′ηΩη = Ωη,
and sinceη is τV- invariant
eitLVsη = sηeitLV, eitLVs′η = s′ηeitLV.
Let∆ω|η be the relative modular operator. We recall thatKer ∆ω|η = Ker s′η,
J∆1/2ω|ηAΩη = s′ηA∗Ωω,
for allA ∈ Mωand that∆ω|ηis essentially self-adjoint on MωΩη + (1 − s′η)Hω. Hence
∆ω◦τt V|η◦τ t V = e −itLV∆ ω|ηeitLV,
6
and sinceη is τV- invariant,
eitLV
∆ω|ηe−itLV = ∆ω◦τ−t
V |η = ∆ωU∗|η,
whereU∗ ≡ Γt V.
As in the proof of Theorem 1.1, we setP ≡ πω(−iU∗δω(U)) and denote by α the perturbation
of the modular dynamicsσω byP . It follows that ωU∗ is the unique(α, −1)-KMS state. Since
also ke(Lω+P )/2
Ωωk = 1, the basic perturbation formula of Araki-Donald (see Lemma 5.7 in
[Don]) yields
s′ηlog ∆ωU∗|η = s′ηlog ∆ω|η − s′ηP.
Hence,
eitLVs′ηlog ∆ω|ηe−itLV = s′ηlog ∆ω|η− s′ηP,
and we conclude that for any real numberλ 6= 0, eitLV s′ηlog ∆ω|η + iλ
−1
e−itLV = s′ηlog ∆ω|η − s′ηP + iλ
−1 .
Sincee−itLVΩ
η = Ωη, the second resolvent identity yields that for all realλ 6= 0,
(Ωη, (s′ηlog ∆ω|η+ iλ)−1s′ηP (s′ηlog ∆ω|η − s′ηP + iλ)−1Ωη) = 0.
Since s − lim λ→∞ iλ s ′ ηlog ∆ω|η + iλ −1 = 1, and s − lim λ→∞ iλ s ′ ηlog ∆ω|η− s′ηP + iλ −1 = 1, we derive that (Ωη, P Ωη) = (Ωη, s′ηP Ωη) = 0.
On the other hand, using Equ. (1.3), we get
P = πω(−iU∗δω(U)) = −
Z t
0
πω(τVs(δω(V ))) ds,
and sinceη is τV- invariant we conclude that
0 = (Ωη, P Ωη) = −
Z t
0
η ◦ τVs(δω(V )) ds = −tη(δω(V )),
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