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On entropy production in quantum statistical


On entropy production in quantum statistical mechanics
V. Jak×i? ?? and C.-A. Pillet? ? c Department of Mathematics Johns Hopkins University 3400 N. Charles Street, 404 Krieger Hall Baltimore, MD 21218, USA PHYMAT Université de Toulon, B.P. 132 F-83957 La Garde Cedex, France CPT-CNRS Luminy, Case 907 F-13288 Marseille Cedex 9, France August 3, 2000
? ? ?

Abstract We propose a de?nition of entropy production in the framework of algebraic quantum statistical mechanics. We relate our de?nition to heat ?ows through the system. We also prove that entropy production is non-negative in natural nonequilibrium steady states.
leave from Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada
? On

1

2

observables of the physical system under consideration. The group speci?es their time evolution. A physical state of the system is described by a mathematical state on ? , that is, a positive linear functional such that ? ? ?. The set ?? ? of all states on ? is a convex, weak-? compact subset of the dual ?? A state is invariant under the group if ? ? for all ?. For our purposes we assume that in addition to ?? ? we are given a -invariant state . The triple ?? ? describes a physical system in a steady state. We are interested in effects of local perturbations on such system. A local perturbation is speci?ed by a selfadjoint element ? of ? and in what follows we ?x such a ? . The perturbed time evolution is given by

1 Introduction Let ?? ? be a ? -dynamical system, where ? is a ? -algebra with identity and a strongly continuous group of automorphisms of ? (strong continuity means that the map ?? ? ? is continuous in norm for each ? ?). The elements of ? describe
? ?

?

? ?

? ?

?

? ?·
?

?

?

?? ??

?? ? ??

?

?

?

?

?

??

??

?? ?

??

?? ?

The pair ?? ? ? is also a weak-? limit points of the set

?

? -dynamical

?
?

?

?

?

? ?

?

??? ?
?

system. Following Ruelle [Ru2], we call the

?

?

??

nonequilibrium steady states (NESS) of the locally perturbed system. The set ?· ? ? of ? NESS of ?? ? ? is a non-empty, compact subset of ?? ? whose elements are ? -invariant. Our ?rst assumption is: (A1) There exists a strongly continuous group ? ???-KMS state. of automorphims of
?? ?

? such that

is

?? (i.e. ? ). We denote by Let ? be the generator of ?? ? is a norm-dense ?-subalgebra of ? and for ? ?? ?, ?

the domain of ? .

? ?

?

?

?

?

?

?

?

?

?

? ?

·

?

?

?

Our second assumption is: (A2) ?

?

?? ?.

We de?ne the observable
? ?

?? ?

and for reasons which will soon become clear, we call
?? ? ? ?
?

?

3
the entropy production (with respect to the reference state ) of the perturbed system ?? ? ? in the state ? ???. Note that ? , and hence ?? ???, depend in a non-trivial way on the state . ? ? be the GNS representation of the algebra ? associated to , and let Let ?? ? be the set of -normal states on ?, that is, the states represented by density matrices ?? ??? which we denote by on ? . Any ? ? has a continuous extension to ? ? , we denote by ??? ? the relative entropy of with the same letter. For respect to . (We use the de?nition of relative entropy given in [BR2], De?nition 6.2.29. This de?nition differs by a sign and the order of factors from the original Araki’s de?nition [Ar].) Our main result, which justi?es the above de?nition of entropy production, is:

?

Theorem 1.1 Assume that (A1) and (A2) hold. Then, for any faithful state ? ??, one has that ???
???

??

such

?

? ?

?

???

?

?

?

?

?? ?

?

× ?

?

×

Remark. The same result (with the same proof) holds for ?? -dynamical systems. In the rest of this section we will discuss some elementary properties of ? ???. Let ? · ? ?· ? ? and ?? ? be such that ? ?
??
?

?
??

??

?

?

? ?

?

?

·

(1.1)

Then, with the particular choice
??
?

, Theorem 1.1 gives

?
??

???

?

?? ?

?

?? ?
?

?
??

??

?

?

× ?

?

?

??

×

?

?? ?

?

·

?

(1.2)

Since the relative entropy is non-positive, we immediately get Theorem 1.2 Assume that (A1) and (A2) hold. Then, for any NESS has
?? ?
? ?

·

? ?· ?
?

?,

one

·

?

?

With regard to (1.2), on physical grounds one expects that the ratio
???

?

? ?

?

?

?

becomes independent of the choice of the reference state following result holds:

as ?

?. More precisely, the

4
Proposition 1.3 Assume (A1) and that ? ? ? such that for ? ? ? ? ,
???

??
???

is faithful. Then there is a norm-dense set

?

? ?

?

?

?

? ?

? · ? ???

as ?

?. ??
?

One also expects that in thermal equilibrium the entropy production is zero, that is, if ? ? is a ? ? ? ?-KMS state then ?? ? ? ?. In fact, a much stronger result holds. Proposition 1.4 Assume (A1), (A2) and that
?? ? ?

is a faithful,

?

-invariant state. Then

Remark. Again, this result also holds for ? ? -dynamical systems. Let ? be the CAR algebra over ?? ? ? ? describing a free Fermi gas on the lattice ? . Using some technical results proven in [BM] it is easy to construct a large class of quasi· free states and local perturbations ? such that (A1)-(A2) hold, and that ? ? ? consists ? · · of a single state ? . In these examples, ?? ? ? ? can be computed perturbatively (similar · calculations are done in [HTP]), and one easily constructs examples where ? ? ? ? ?. ? In the next example we relate entropy production to heat ?ows. ?, ? ?, each of which is in thermal Consider two independent systems ?? is a ? ? ?-KMS state on ? where equilibrium at temperature ? . This means that ? ? ? . Let

? ?? ? ??

?

?

?

?

?

?

(? is the ? -tensor product, see Section 2.7.2 in [BR1]). Let ? be the generator of and ? the generator of . Obviously, ? ?? · ?? (here we write ?? for ?? ? , etc). Let ? ? ? be such that ? ? ?? ?. Then

?

?
where ¨ is de?ned by

? ?

?? ?

?
? ?

?

?? ?

?

?

× ?

?¨?

×

?¨?

?

? ?

?? ?

Obviously, ¨ ¨? ·¨? where ¨ ? ?? ? describes the energy ?ux out of the -th system. ? ? · ? ? and ? ?? ? . Therefore, Since the states are KMS, (A1) holds with ? (A1) and (A2) hold and It follows that in a NESS
·

?

??

?? ¨?

· ?? ¨?

?

?

·
?

? ?,
?

the energy ?uxes satisfy
?¨? ?

?

·

?¨? ?

·

??

·

??

?

?? ?

?

·

?

?

5
· · Since ? ?¨? ? · ? ?¨? ? ?, if ?? ?? , then ¨? ? and the heat ?ows from the hot to the cold reservoir. This calculation is easily generalized to the case where ? -level quantum system is coupled to several independent thermal resevoirs. We ?nish this section with the following remarks. In [JP1] we prove an analog of Theorem 1.1 for time-dependent perturbations and discuss the relation between entropy production and the second law of thermodynamics. In the forthcoming paper [JP2], we will study NESS, entropy production and heat ?ows for a model of an ? -level quantum system coupled to several independent free Fermi gas reservoirs (similar models have been studied in [D, Ru1]). The entropy production for quantum spin systems has been studied in the recent preprint [Ru2].

Acknowledgments. The research of the ?rst author was partly supported by NSERC. Part of this work has been performed during a visit of the second author to University of Ottawa which was also supported by NSERC.

2 Proofs
We assume that the reader is familiar with the basic results of Tomita-Takesaki modular theory as discussed, for example, in [BR1, BR2, H, OP]. We begin by setting the notation and recalling some well-known facts. ?? ? ? denotes the GNS representation of the algebra ? associated to . By ?? ??? . Moreover, (A1) implies that (A1), the vector ? is cyclic and separating for is injective. We respectively denote by ? , ? and ? the modular operator, the modular conjugation and the natural cone canonically associated to the pair ? ? ?. We also adopt ?? ? and ? ? ? ? . Note that the shorthands ?

?

?

?

?

? ??

?? ?

? ?

? ??
??

With a slight abuse of notation, we write , ? Tomita-Takesaki theorem, ? ? ?
?? ?

? ? ? ? and
? ? ? ?
? ??

? ??

for

?

?.

By the (2.3)

The Liouvillean that

?

of the system
?

?

?
?

?

is the unique self-adjoint operator on
??

?

such

? ??
??

? ?

? ??

(2.4)
?

and one easily shows that
??

?
?

?
?

??

(2.5)
??

??

6
The self-adjoint operator
?? ?

·

?? ?

?

?

?? ??

is uniquely speci?ed by the following two requirements:
?
? ?

? ??

???

? ?

? ???

? ???

? ?

(2.6)

The dynamical groups and ? have natural extensions to notation. Note also that ? ?? · ?? ? ?, and therefore
??? ? ? ???

? for which we use the same
(2.7)

A state

??

has a unique vector representative ?
? ? ? ? ? ??? ?

? ? . Relations (2.6) yield that
(2.8) is de?ned as
? ?

The relative entropy of two faithful states
??? ? ??

??

?? ?

where ? is the relative modular operator. Relative entropy is more conveniently ex× as pressed in terms of the Radon-Nikodym cocycle
??? ? ?? × ? ?
× ×

? ??

(2.9)

Proof of Theorem 1.1. Let us denote by
? ???

? ??

???·

?? ??

the propagator in the interaction representation.
?
? ? ???

? ???

is the unique solution of

?

??

?? ??? ???

with initial data ? ???
? ???

?. It has a norm convergent Dyson expansion
? ? ?? ??



?
?

?

?

?

??

???

?? ?

?

?

??

?

???

?? ??

???

?

???

?? ??

from which we conclude that ? ??? ? ?? ?. With a slight abuse of notation, we will write ? for ? ? . A simple calculation shows that ? ??? ? ?? ? and
?

?

???? ?? ????

?

?

? ?? ????? ???

?

?

?

?



?

?

??

×

(2.10)

7
We claim that
? ??
??? ? ???

?? ????

(2.11)

To prove this fact, note that after differentiation with respect to ? both sides satisfy the same differential equation with the same initial condition at ? ?. ? To compute the relative entropy ??? ? ? ?, we will use Equation (2.9) and the fact that the Radon-Nikodym cocycle can be expressed as

?
??

? × ?

?

×

?× ? ? ? ?? ? ?

By de?nition of the relative modular operator, for any
? ? ? ? ?

? ? we have

? ? ?

(2.12)

? ? ?

? ? ? ?

?

? ? ? ?

Using Relations (2.6) and (2.8), we further obtain
??

? ? ? ? ?

? ? ?

? ???

?

? ? ???

?

It follows that
??

? ? ? ? ?

? ? ?

? ???

? ?

? ??

? ??? ? ???

? ?

? ? ? ? ? ?
? ? ? ?

??

? ?? ? ?? ?

?

? ???

? ? ? ? ???

where we used (2.7). Since for
? ? ? ? ? ?

?
? ? ?

?

is a core for ?

? ?

and

?

?

? ? ? ?

is a core

? ? ?

, we derive the relation
? ? ? ?

? ? ?

? ???

We now deal with ?

. First, for any
??

? ?,
? ?

?

???

(2.13)

? ?

? ? ?

? ? ? ?

Equations (2.8), (2.11) and (2.4) yield that

and since

?? ? ????
??

? ?? , we derive
? ? ?? ???? ? ?

? ? ? ?

?

?

? ??? ?? ?????

? ?

?

?

????
?

? ?

? ????
?

? ???? ? ? ? ? ? ? ???? ? ? ???? ? ? ? ? ? ?? ????? ? ????

?

8
Since ? is a core of conclude that

?

? ? ?

and
? ? ?

?? ? ????

?

?

?

? ? ? ?

is a core of

? ? ?

? ? ?

we

?

? ?? ?????

?? ????

(2.14)

Going back to (2.12), we derive from Equations (2.13) and (2.14)

?

? × ?

? × ? ??? ? × ?? ????? ?? ???? ? ? × ? ? ?? ? × ?? ????? ? ??? ? ? ?? ????
× ???

???

? ? ?? × ? × ?? ???? ? ?

???

where we used (2.11) and (2.5). Since × ?? ? ???? ? , it commutes with another application of (2.11) (together with Relation (2.12) at ? ?) gives

?

?? ? ????

and

?
We can therefore write Since ? ??? ?
??? ?

? × ?

×

?? ? ?? ????? ??? ? ?

×

?

(2.15)

?

? ?

? ?? ? , ???
×

?

? × ? ?

?

???

?

?

???

×

?? ????

? ???

?

×

? ?

(2.16)

the estimate
? ??? ? ? ·×
??? ?

?

?? ????

?

???? ?? ????

? ???

? · ??×?

holds in the norm of ? as ×
??? ?

?.

Furthermore, Equation (2.10) is easily rewritten as
? ??? ?
?

?

???? ?? ????

?

? ? ?

?

??

?

Equation (2.16) leads to the estimate

?

? ?

?

?

? × ? ?

?

× ?

? ?
? ? ? ?

?
as × ?. Since the cocyle gives the result. ?

×

?

??
×

??

×

? ?

?

· ??×?

(2.17)

is strongly continuous, insertion in Equation (2.9)

Proof of Proposition 1.3. For any self-adjoint ? by of
? ??

?

? ? we de?ne a group of automorphisms
? ??? ·? ?

?

???

·? ?

Araki’s perturbation theory yields that there is a state ? ? ? which is a ? ? ???-KMS state. Let ? ? be the set of all states obtained in this manner. It is well-known that ?? is dense in ? (see, e.g., [R]). By the result of Araki (see Proposition 6.2.32 in [BR2]),
???

?

? ?

??

???

?

? ?

?· ?

? ?

?? ??

? ??

?? ·? ? ?

?

?

9
The statement follows from this relation, the obvious estimate ?? ·? ? ? ? ?. ? fact that ? Proof of Proposition 1.4. Since
?

?

? ?

?? ??

?

and the

?
?
? ? ?

? ?

, Relation (2.17) yields that for all ×
??
×

?,

?

??

?

? ?

?

????

Taking ×

?

we get that for all ?,
?

?

?

?

?

?

?

?

?

?

?

and so

?? ? ?

?.

?

References
[Ar] [BM] Araki, H.: Relative entropy of states of von Neumann algebras, Pub. R.I.M.S., Kyoto Univ. 11, 809 (1976). Botvich, D.D., Malyshev, V.A.: Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas, Commun. Math. Phys. 91, 301 (1983).

[BR1] Brattelli, O, Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer-Verlag, Berlin, second edition (1987) [BR2] Brattelli, O, Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer-Verlag, Berlin, second edition (1996). [D] [H] Davies, E.B.: Markovian master equations, Commun. Math. Phys. 39, 91 (1974). Haag, R.: Local Quantum Physics. Springer-Verlag, Berlin (1993).

[HTP] Haag, R., Trych-Pohlmeyer E.: Stability properties of equilibrium states, Commun. Math. Phys. 56, 213 (1977). [JP1] [JP2] [OP] [R] c Jak×i? , V., Pillet, C.-A.: On entropy production in quantum statistical mechanics II. Time-dependent perturbations, preprint. c Jak×i? , V., Pillet, C.-A.: In preparation. Ohya, M., Petz, D.: Quantum Entropy and its Use, Springer-Verlag, Berlin (1993). Robinson, D.W.: ? -algebras in quantum statistical mechanics, in ? -algebras and their Applications to Statistical Mechanics and Quantum Field Theory, (D. Kastler editor), Amsterdam, North-Holand (1976).‘ Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98, 57 (2000). Ruelle, D.: Entropy production in quantum spin systems, preprint.

[Ru1] [Ru2]


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