Linear Response Theory and the KMS Condition - Project Euclid

Commun. math. Phys. 44, 87—99 (1975) © by Springer-Verlag 1975

Linear Response Theory and the KMS Condition J. Naudts University of Antwerpen, Antwerpen, Belgium

A. Verbeure and R. Weder* University of Leuven, Heverlee, Belgium** Received November 15, 1974 Abstract. The response, relaxation and correlation functions are defined for any vector state ω of a von Neumann algebra 9Ji, acting on a Hilbert space Jf, satisfying the KMS-condition. An operator representation of these functions is given on a particular Hilbert space jj?. With this technique we prove the existence of the static admittance and the relaxation function. Finally we generalize the fluctuation-dissipation theorem and other relations between the above mentionned functions to infinite systems.

I. Introduction

In conventional statistical mechanics an equilibrium state of a finite system is given by a Gibbs state. It is well known that the states of infinite continuous systems are no longer of this type. It has been suggested and now widely accepted that an equilibrium state of an infinite system should be described by a state satisfying the KMS-condition [1]. This is also the point of view of this paper. The problem of non-equilibrium statistical mechanics is to explain the occurrence of an equilibrium state. This problem can be tackled in different ways. There is a direction where people study the problem by placing the system in a larger one. This leads to the study of open systems, where topics like the master equation are widely studied [2,3], some aspects of the theory have recently been made rigorous [4-6]. Also a lot of rigorous work has been done on models, such as harmonic oscillators and lasers (see e.g. [7]). Another way of studying the problem is to consider small perturbations of the system and to wait for the behaviour after a long time (see e.g. [8, 9]). Linear response theory must be situated in this direction and the principal purpose of the present paper is to prove and generalize to infinite systems rigorously some aspects of linear response theory, as introduced by Kubo [10] and Mori [11]. In Section II we introduce a new scalar product on the set of observables and define a new Hilbert space jft, and construct explicity a unitary operator from J^ to the KMS-Hilbert space. We prove that it is equivalent with the scalar product of the Kubo-Mori theory, and we give some other characterizations of this scalar product. * Onderzoeker ΠKW, Belgium. ** Postal address: Institute for Theoretical Physics, Celestijnenlaan 200 D, B-3030 Heverlee, Belgium.

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In Section III we define the correlation, response, relaxation and admittance functions of linear response theory. We give an operator representation of this functions on jft. This technique enables us to prove in an easy way the existence of the static admittance and the relaxation function for all pairs of observables, and also to give their explicit forms. Other results which we want to mention are the proof of the fluctuation-dissipation theorem for infinite-system equilibrium states, and the proof of other relations between the above-mentioned functions. Finally we mention the proof of the Bogoliubov inequality for infinite system equilibrium states, as a straightforward application of the techniques, which are used. The formalism developed here, for linear response theory is applied in a subsequent paper [19], where the importance of this theory is indicated in the study of cluster properties.

II. The KMS-Condition and the Hubert Space # Let 9K be a von Neumann algebra on a Hubert space jff let t-+ Ut be a strongly continuous map from the real numbers 1R into the group of unitaries on Jf, then there exists a self-adjoint operator H on ffl such that Ut=εxpitH and let xf = UtxUf\ furthermore let ω be any vector state on SCR i.e. ω(x) = (Ω, xΩ) for all x e 501 with Ω a cyclic element of J f the state ω is an equilibrium state if it satisfies the following definition. Definition 11.1. The state ω on Sϋΐ satisfies the KMS-condition at inverse temperature β= 1/kT if for any pair (x, y) of 9Jt, there exists a complex function FXty{z), defined, bounded and continuous on the strip - β^ Imz^O, and analytic inside, with boundary values: Fx,y{t)=ω{xty), Fxy(t-iβ)

= ω(yxt).

Let us first quote some consequences of the KMS-condition [12]. If ω satisfies the properties of Definition II. 1, then the vector Ω is also separating for 90Ϊ, furthermore Ω is Ut-invariant i.e. UtΩ = Ω for all ίelR. There exists an operator A = exp(-βH), given by

where E(λ) is the spectral family of H [12, p. 69]. This operator is called the modular operator. It can be written as Δ = FS where S is the closure of the conjugate linear operator, mapping xΩ into x*Ω with domain SEΓiΩ and F the adjoint of S. Let 5 = JΔ1/2 be the polar decomposition of S then J is an anti-unitary operator and F = Δ~1/2J; furthermore JΔit =Δit3 for all ί e l R a n d Ut = Δu, JΩ = Ω, ΔΩ = Ω. Finally we quote the following property which is used afterward: FXty{βz) = {x*Ω,ΔizyΩ)

if

O^Imz^—1/2, if

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Linear Response Theory and the KMS Condition

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In the following we will always assume that ω is a KMS-state and for notational convenience we put β= 1. Define the unbounded operator T by )

E(dλ).

As is easily checked T is a positive, self-adjoint operator, and as, for λ-+ — 112 ~λ\»

λ 1/2 ι ί-e~ e \\

the domain ^(zl 1 / 2 ) of A112 is contained in the domain &{T) of Γ and We define the following sesquilinear form on 9JΪ: ( x , ^ = ( 7 x 0 , TyΩ)

x,yeWl.

Lemma Π.2. The sesquilinear form ( , )~ on 3DΪ is non degenerated. Proof. Let (x, X)L=O then Tx£2 = 0. If we prove that zero is not an eigenvalue of T9 using the fact that Ω is separating for 9M, it follows that x = 0. Let W(λ) be the spectral family associated with the operator ( - T). Then [18].

for any Borel subsets of the real line, where ψ~1(S) denotes the inverse image of S under the function ψ(λ)= - f

λ In particular W({ — ε,0]) = E((b9 oo]), where ε > 0 , and b is the inverse image under ψ of — ε. Clearly

then lim W{{ - ε, 0]) = lim £((&, oo]) = 0 . So that, 0 is not an eigenvalue of T, because it is not an eigenvalue of (— T). Q.E.D. It follows from this Lemma that the closure of 9K with respect to the sesquilinear form ( , )L is a Hubert space, which we denote by 3$. Following Ref. [12] let $ be the linear space spanned by elements f*g, where / and g are continuous functions of a real variable with compact support, and where (f*g){t)=^f(s)g(t-s)ds is the convolution product. Denote by S the subalgebra of SOΪ generated by the set {f(H)x\feί,xeW.}, f(H)=^a>f{λ)E(dλ)

[12, p. 67].

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93 is called the set of analytic elements of SCR and has the following properties (i) 93 is invariant under zΓ, α e f, then there exists a sequence {xn}n in 9JI such that limx^x. As 9Jϊί2C^(T), there exists a sequence {ynj}ι in 93 such that (i) UmynflΩ = xnΩ, (ii) UmTynJΩ=TxnΩ. Let ε>0 be any positive number, choose n such that ||x —xM||^I|L