Fluctuation-dissipation theorem for the microcanonical ensemble

Fluctuation-dissipation theorem for the microcanonical ensemble

Marcus V. S. Bonança

*

Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany 共Received 1 April 2008; published 5 September 2008兲

A derivation of the fluctuation-dissipation theorem for the microcanonical ensemble is presented using linear response theory. The theorem is stated as a relation between the frequency spectra of the symmetric correlation and response functions. When the system is not in the thermodynamic limit, this result can be viewed as an extension of the fluctuation-dissipation relations to a situation where dynamical fluctuations determine the response. Therefore, the relation presented here between equilibrium fluctuations and response can have a very different physical nature from the usual one in the canonical ensemble. These considerations imply that the fluctuation-dissipation theorem is not restricted to the context of the canonical ensemble, where it is usually derived. Dispersion relations and sum rules are also obtained and discussed in the present case. Although analogous to the Kramers-Kronig relations, they are not related to the frequency spectrum but to the energy dependence of the response function.

DOI:10.1103/PhysRevE.78.031107 PACS number共s兲: 05.70.Ln, 05.40.⫺a, 05.20.⫺y, 05.30.⫺d

I. INTRODUCTION

The relation between the fluctuations occurring in a sys- tem at equilibrium and dissipation effects dates back to Ein- stein关1兴and his theory on Brownian motion. After that, Ny- quist 关2兴derived a relation between the electrical resistance and voltage fluctuations in linear electrical systems. It was realized then by Callen and Welton 关3兴 that such a relation could be proven for general linear dissipative systems using quantum mechanics. At that moment, the intuition of the authors, as described in the last paragraph of their Introduc- tion, was that the relationship between equilibrium fluctua- tions and irreversibility would provide a method for a gen- eral approach to a theory of irreversibility and, indeed, this was the way pursued by Kubo 关4兴to achieve the theory of linear response. It is well established now that linear re- sponse theory gives a general proof of the fluctuation- dissipation theorem 共FDT兲 which states that the linear re- sponse of a given system to an external perturbation is expressed in terms of the fluctuation properties of the system in thermal equilibrium.

Because of this deep relation between the FDT and linear response theory, it is worth noting that the response, as for- mulated by that theory, is given for any equilibrium en- semble. In other words, the response function can, in prin- ciple, be known not only when the system is initially in thermal equilibrium but also in another equilibrium state such as, for example, the microcanonical one. Therefore, the theory is quite general in the sense that the linear response of a system and its equilibrium fluctuations could be related to each other for any kind of equilibrium conditions. Indeed, fluctuation-response relations have been derived even in the context of stochastic systems关5,6兴and non-Hamiltonian de- terministic systems 关7兴using linear theory. Perhaps the very first work concerning different equilibrium conditions from the thermal one in Hamiltonian systems is Ref. 关8兴, where the author shows that Kubo’s formula can also be derived in

the classical microcanonical ensemble as long as the thermo- dynamic limit is considered. However, for many and differ- ent reasons, much more attention was given for the statistical mechanics in the canonical ensemble than in the microca- nonical one and the generality of linear response theory con- cerning different equilibrium conditions was not much ex- plored. Of course, one could argue that the equivalence of the ensembles in the thermodynamic limit would be a reason for focusing just on the canonical ensemble, but recent de- velopments have shown that there are indeed strong motiva- tions to consider different equilibrium situations. For ex- ample, a path integral representation for the quantum microcanonical ensemble关9兴presented a few years ago was motivated by situations where the microcanonical approach may be more appropriate, as for the description of systems at low temperatures or with a finite number of particles. The microcanonical ensemble has also been considered in rela- tions between fluctuation and response in systems far from equilibrium like the Crooks relation, where its microcanoni- cal version helps to understand the connection between vari- ous of those fluctuation theorems 关10兴. In Ref.关11兴, a deri- vation of a microcanonical quantum fluctuation theorem was presented. Considering the work performed by a classical force on a quantum system when it is initially prepared in the microcanonical state, the authors provide a relation that could be accessible experimentally to measure entropies. In the context of nanosystems, where the number of degrees of freedom constituting the environment is not always large enough to be considered in the thermodynamic limit, the microcanonical ensemble has also been considered. In Ref.

关12兴, a quantum master equation was derived describing the dynamics of a subsystem weakly coupled to an environment of finite heat capacity and initially described by a microca- nonical distribution. Finally, an analysis in the microcanoni- cal state has also contributed to the recent debate about the foundations of the canonical formalism 关13兴.

The microcanonical ensemble implies a description of an isolated system. Therefore, one might ask how a relation between fluctuations and dissipation can be possible in a situation where no energy can be dissipated. In the present work, our goal is to explore the relation between fluctuations

*marcus.bonanca@physik.uni-regensburg.de

and response in microcanonical equilibrium conditions through the framework of linear response theory. As will be explained later, mainly after the development of linear re- sponse theory, the name fluctuation-dissipation theorem was associated with some relations which are analogs of the re- sults presented here in the context of the microcanonical en- semble. That is the reason we took the freedom to call them also a FDT even in a situation where there is no physical mechanism for dissipation. The paper is organized as fol- lows. In Sec. II the derivation of a FDT using linear response theory is presented and its validity is verified in a simple example. In Sec. III different dispersion relations and sum rules are derived in analogy with the usual Kramers-Kronig ones and their meaning is discussed. They are different be- cause they are not derived in the frequency space, as are the usual ones. Conclusions are presented finally in Sec. IV.

II. DERIVATION OF THE FLUCTUATION-DISSIPATION THEOREM

We start by considering a system whose dynamics is given by a HamiltonianHˆ. An external forceK共t兲is applied to this system such that Hˆ is now perturbed by an external potential given by −Aˆ K共t兲. Following关4兴, the response func- tion of the system due to the external force measured through an observableBˆ is given, in linear response, by

␾BA共␭,t−t

⬘

^{兲= Tr}

冉

^{␳ˆe共␭兲}iប¹关Aˆ共0兲,Bˆ共t−t

⬘

^兲兴

冊

= Tr

冉

^{␳ˆe共␭兲i1}ប关Aˆ共t

⬘

兲,Bˆ共t兲兴

冊

^{, 共1兲}

where 关,兴 is the commutator and ␳^ˆe共␭兲 is the equilibrium density operator as a function of a macroscopic parameter␭. One can also define the following correlation function be- tweenAˆ andBˆ:

CBA共␭,t−t

⬘

^{兲= Tr}

冉

^{␳ˆe共␭兲1}2兵Aˆ共0兲,Bˆ共t−t

⬘

^兲其

冊

= Tr

冉

^{␳ˆe共␭兲12兵Aˆ共t}

^⬘

^{兲,Bˆ共t兲其}

冊

^{, 共2兲}

where 兵,其 is the anticommutator. This function gives the spectrum of equilibrium fluctuations when the system is un- perturbed. For the canonical ensemble, ␳^ˆe共␭兲=␳^ˆe共␤兲

=e^−␤Hˆ/Z共␤兲, where ␤⁼共kBT兲⁻¹, and the FDT establishes a relation between the spectra of␾BA andCBA. That means a relation between an equilibrium and a nonequilibrium quan- tity.

Our goal here is to show that there is also a relation be- tween␾BAandCBAin the microcanonical ensemble. First of all, let us start with the expression for the microcanonical density operator␳^ˆe共␭=E兲. Following关9兴, we take it as

␳^ˆe共E兲=␦共E−Hˆ兲

Z共E兲 , 共3兲

whereZ共E兲= Tr␦共E−Hˆ兲.

To derive the FDT, it is necessary to introduce an appro- priate representation of ␦共E−Hˆ兲like, for example关9兴,

␦共E−Hˆ兲= 1 2␲ⁱ

冕

␥−i⬁

␥+i⬁

dzexp关共E−Hˆ兲z兴. 共4兲 Expressions 共1兲and共2兲can be written now in the following way:

␾BA共E,t−t

⬘

兲= 1

Z共E兲Tr

冉

^21␲i

^冕

^{␥−i␥+i⬁⬁dz e共E−Hˆ兲z关Aˆ共t}

^⬘

^{iប兲,Bˆ共t兲兴}

冊

^,

共5兲 CBA共E,t−t

⬘

兲= 1

Z共E兲Tr

冉

^21␲i

^冕

^{␥−i␥+i⬁⬁dz e共E−Hˆ兲z兵Aˆ共t}

^⬘

^{兲,B2ˆ共t兲其}

冊

^.

共6兲 It is important to note that, since the integrals in the complex plane are always convergent, the trace and integral signs can be interchanged. Doing that, it is convenient to define the following new quantities: ␸BA共E,t−t

⬘

^{兲=Z共E兲␾BA共E,t−t}

⬘

^兲 andCBA共E,t−t

⬘

^{兲=Z共E兲CBA共E,t−t}

⬘

^{兲to obtain}

␸BA共E,t−t

⬘

兲= 1 2␲ⁱ

冕

␥−i⬁

␥+i⬁

dz e^Ez␹BA共z,t−t

⬘

兲, 共7兲

CBA共E,t−t

⬘

^{兲= 1}

2␲ⁱ

冕

_␥−i^␥+i_⬁^{⬁dz eEzFBA共z,t−t}

^⬘

^{兲, 共8兲}

where

␹BA共z,t−t

⬘

兲= Tr

冉

^{e−Hˆ z关Aˆ共t}

^⬘

iប^{兲,Bˆ共t兲兴}

冊

^{, 共9兲}

FBA共z,t−t

⬘

兲= Tr

冉

^{e−Hˆ z兵Aˆ共t}

^⬘

^{兲,B2ˆ共t兲其}

冊

^{. 共10兲}

Since␸BAandCBAare given as inverse Laplace transforms of

␹BAandF_BA, they also satisfy the following relations:

␹BA共z,␶兲=

冕

0

⬁

dE e^−Ez␸BA共E,␶兲, 共11兲 F_BA共z,␶兲=

冕

0

⬁

dE e^−EzCBA共E,␶兲, 共12兲 where ␶^=t−t

⬘

. We introduce now the Fourier transform of

␹BAandF_BA,

␹

˜_BA共z,␻兲= 1 2␲

冕

−⬁

⬁

d␶e^−i␻␶␹BA共z,␶兲, 共13兲

F˜

BA共z,␻兲= 1

2␲

冕

_−⬁^{⬁ d␶e−i␻␶FBA共z,␶兲, 共14兲}

and also the auxiliary function

SAB共z,␶兲= Tr关e^{−Hˆ z}Aˆ共t

⬘

^{兲Bˆ共t兲兴. 共15兲}

Noticing that e^{−Hˆ z}Aˆ共t

⬘

兲=Aˆ共t

⬘

^+izប兲e^{−Hˆ z}and using the cy- clic property of the trace, we obtain

Tr关e^{−Hˆ z}Bˆ共t兲Aˆ共t

⬘

^{+izប兲兴= Tr关e−Hˆ zAˆ共t}

⬘

^{兲Bˆ共t兲兴. 共16兲} Using

1 2␲

冕

−⬁

⬁

d␶e^−i␻␶Tr关e^{−Hˆ z}Bˆ共t兲Aˆ共t

⬘

⁺izប兲兴

= 1

2␲

冕

_−⬁^{⬁ d␶}

^⬘

^{e−i␻␶⬘Tr关e−Hˆ zBˆ共t兲Aˆ共t}

^⬙

^{兲兴ezប␻, 共17兲}

where t

⬙

^=t

⬘

^{+izប and} ␶

⬘

^=t−t

⬙

, one obtains from 共15兲 and 共16兲

S˜

AB共z,␻兲=˜S

BA共z,␻兲e^zប␻, 共18兲 where

˜S_BA共z,␻兲= 1

2␲

冕

_−⬁^{⬁ d␶}

^⬘

^{e−i␻␶⬘Tr关e−Hˆ zBˆ共t兲Aˆ共t}

^⬘

^{兲兴. 共19兲}

Using共18兲in the Fourier transforms of共13兲and共14兲yields

␹

˜_BA共z,␻兲= 1 iប关S˜

AB共z,␻兲−S˜

BA共z,␻兲兴=˜S

BA共z,␻兲共e^zប␻− 1兲 iប ,

共20兲 F˜

BA共z,␻兲=1 2关˜S

AB共z,␻兲+˜S

BA共z,␻兲兴=˜S

BA共z,␻兲共e^zប␻+ 1兲

2 .

共21兲 Finally, from共20兲and共21兲, we obtain

F˜

BA共z,␻兲=iប

2 coth

冉

^zប2^␻

冊

^{␹˜BA共z,␻兲, 共22兲}

which is our quantum FDT. In the classical limit ប→0, we obtain

F˜

BA共z,␻兲= i

z␻␹^˜BA共z,␻兲, 共23兲 which is our classical FDT. One easily realizes from共22兲and 共23兲that the replacement of zby␤in those equations leads precisely to the quantum and classical versions of the FDT in the canonical ensemble. However, the physical nature of共22兲 and 共23兲 can be quite different from that in the canonical case. Let us consider, for example, in the classical regime an ergodic and small system, small in the sense that it is not in the thermodynamic limit. Then the microcanonical ensemble averages in 共1兲 and 共2兲 can be replaced by time averages whose behaviors are given by the dynamics of the system.

Therefore, the fluctuations in this case happen due to the dynamics of the concerned system itself and not due to the coupling to a thermostat as in the canonical ensemble. From this point of view, it is surprising that there is a simple rela- tion between the FDT in the canonical and microcanonical ensembles. Indeed, if one wants to compare both cases, the inverse Laplace transform inzshould be performed on 共22兲 and共23兲 since the canonical FDT consists of a relation be-

tween␾^˜BA共␤^,␻兲andC˜

BA共␤^,␻兲, keeping the original macro- scopic parameter␤. For the classical case, this can be easily done using共23兲, leading to

C˜

BA共E,␻兲= i

␻

冕

0 E

dE

⬘

␸^˜BA共E

⬘

^,␻兲. 共24兲 For the quantum regime, the inverse Laplace transform should be performed on 共22兲. It is not hard to imagine how different the result will also be from the canonical case.

In addition to the pure meaning of the relation between response and fluctuations, one may wonder whether共22兲and 共23兲can be useful or not. We would say they can be useful in situations where the microcanonical ensemble can be applied and the thermodynamic limit is not satisfied. However, what we mean by usefulness is the possibility of applying the FDT in a context very different from the ones considered so far, to obtain response functions from correlation functions, and vice versa. If by useful one meant to go further and speak about, e.g., transport coefficients, then one would have to discuss more carefully the linear response theory in the mi- crocanonical ensemble, especially because van Kampen’s objections 关14兴can be trickier in this case. The first objec- tion, concerning the validity of the linearization, could still be answered as usual, we believe, by the argument of the stability of the distribution functions关7,15兴. The second ob- jection, concerning the origin of the decay of correlation functions which lead to finite transport coefficients, cannot be answered as is done sometimes in the context of the ca- nonical ensemble by coupling to an environment 关16,17兴.

The reason is simple: to use the microcanonical ensemble one assumes an isolated system. A possible answer in this case would be the instability of the dynamics关18,19兴. How- ever, the question of what “dissipation” would mean in the present context of the microcanonical ensemble would re- main. This is because, originally, the name fluctuation- dissipation theorem comes from the fact that part of the Fou- rier transform of the response function is related to the power dissipated by the system when a time-periodic perturbation is applied to it. But for an isolated system there will be no dissipated power. On the other hand, the fluctuation- dissipation theorem, mainly after linear response theory was developed, has been associated with an equation relating the frequency spectra of the response function and of the corre- sponding symmetric correlation function. In this sense, 共22兲 and共23兲are analogous to Eq.共6.16兲 of Ref.关4兴 for the mi- crocanonical ensemble and therefore we took the freedom of calling them fluctuation-dissipation theorems as well. Al- though beyond the scope of the present work, a general and deep discussion of the subtle points mentioned above as well as of the linear response theory for the microcanonical en- semble would be of great interest and value.

Example: The harmonic oscillator

As an example, we would like to check共22兲and共23兲for a simple system whose response and correlation functions are known directly. In order to do that, we choose a simple harmonic oscillator. We consider the caseAˆ=Bˆ=Xˆ, whereXˆ

is the position operator. To perform first the calculation in the classical regime, we define the classical analogs of 共5兲 and 共6兲as

␸共E,t−t

⬘

^兲=

冕

^{dx0dp0␦„E−H共x0,p0兲…兵x共t}

^⬘

^{兲,x共t兲其0,}

共25兲

C共E,t−t

⬘

兲=

冕

^{dx0dp0␦„E−H共x0,p0兲…x共t兲x共t}

^⬘

^{兲, 共26兲}

where兵, 其0is the Poisson bracket with respect to the initial conditions 共x0,p₀兲 and x共t兲 is the solution of the classical equations of motion for the position. The averages above can be easily performed, leading to

␸共E,␶兲= 2␲ m␻0

2sin共␻0␶兲, 共27兲

C共E,␶兲=2␲^E

m␻03cos共␻0␶兲. 共28兲 We can now calculate

␹

˜共z,␻兲= 1 2␲

冕

−⬁

⬁

d␶e^−i␻␶

冕

0

⬁

dE e^−Ez␸共E,␶兲, 共29兲

F˜共z,␻兲= 1

2␲

冕

_−⬁^{⬁ d␶e−i␻␶}

冕

0

⬁

dE e^−EzC共E,␶兲. 共30兲

The results are

␹

˜共z,␻兲= −i␻^{z 2}␲ m␻0

3˜g共␻兲

冕

0

⬁

dE e^−EzE, 共31兲

F˜共z,␻兲= 2␲ m␻03˜g共␻兲

冕

0

⬁

dE e^−EzE, 共32兲

where˜g共␻兲=共1/2␲兲兰_−⬁^⬁ d␶^e−i␻␶cos共␻0␶兲. Therefore, F˜共z,␻兲= i

z␻␹^˜共z,␻兲, 共33兲 which agrees with 共23兲.

Quantum mechanically, we can calculate directly共9兲and 共10兲for the harmonic oscillator using the energy eigenbasis

␹共z,␶兲=

兺

_n ^e−Enzsin_m^共_␻^␻␶兲

0

, 共34兲

F共z,␶兲=

兺

_n ^{e−EnzEncos共}_m␻^␻␶兲

0

2 , 共35兲

whereEnare the energy eigenvalues. Therefore, for the Fou- rier transform␹^˜共z,␻兲we obtain

␹

˜共z,␻兲=

兺

_n ^e−Enz_2mⁱ_␻

0

关␦共␻0+␻兲−␦共␻0−␻兲兴. 共36兲

Using 共22兲 and 共36兲, we obtain an expression for F˜共z,␻兲. Inverting the Fourier transform, we get

F共z,␶兲=

兺

_n ^e−Enzប₂ ^coth

冉

^zប2␻0

冊

^cosm共␻^␻0^0␶兲

. 共37兲

Since

兺

n

e^−EnzEn=ប␻0

2 coth

冉

^zប2^␻0

冊

2 sinh共zប¹ ␻0/2兲. 共38兲 Equation共37兲can be written as

F共z,␶兲=

兺

n

e^−EnzEn

cos共␻0␶兲

m␻02 , 共39兲 which agrees with 共35兲. This verification of 共22兲 for the quantum harmonic oscillator is the same as in the canonical ensemble case ifzis replaced by␤. However, here共22兲still has to be transformed back to energy.

III. DISPERSION RELATIONS AND SUM RULES In the canonical ensemble, it is possible to derive relations between the real and imaginary parts of the Fourier trans- form of the response function 关15,20兴. Those are the so- called Kramers-Kronig relations and mainly they express a causality property contained in the response function. In the present case, dispersion relations also hold in the z space because ␾BAandC_BAare defined for positive values of en- ergy. Equations 共11兲 and 共12兲 imply that ␹BA共z,␶兲 and FBA共z,␶兲 are analytic functions in the half plane Re共z兲艌␥^, where␥ is positive. Therefore, in this region

␹BA共z0,␶兲= 1

2␲ⁱ

冖

^{dz␹BAz−共z,z}0^␶兲

. 共40兲

Since lim_兩z兩→⬁兩␹BA共z,␶兲兩= 0, we can close the integration contour with a semicircle in the half plane where␹BA共z,␶兲is analytic and a line along Re共z兲=␥ and send the radius to infinity to obtain from 共40兲the relation

␹BA共y0,␶兲= 1

␲^iP

冕

_−⬁^{⬁ dy␹BAy−共y,y}0^␶兲

, 共41兲

where the choices z=␥^{+iy and z}0=␥^+iy0 were made. The right-hand side denotes the principal value of the integral.

Writing ␹BA in terms of its real and imaginary parts, ␹BA

=␹BA

⬘

⁺ⁱ␹BA

⬙

^{, Eq.} 共41兲 leads to the following dispersion rela- tions:

␹BA

⬘

共y0,␶兲= 1

␲^P

冕

_−⬁^{⬁ dy␹BA}

^⬙

^y−共y,y0^␶兲

, 共42兲

␹BA

⬙

共y0,␶兲= − 1

␲^P

冕

_−⬁^{⬁ dy␹BA}

^⬘

^y−共y,y0^␶兲

. 共43兲

As it is usually done关20兴, from the two relations above, it is possible to derive the moment sum rules, which, in this case,

are related to the energy dependence instead of the frequency spectrum. The derivation of such sum rules is sketched in the Appendix. The results for the first three moments are shown below, where the subscriptBAwas dropped for convenience:

␸共0,␶兲= 1

␲

冕

−⬁

⬁

dy␹

⬘

共y,␶兲, 共44兲

␸^共1兲共0,␶兲= − 1

␲

冕

−⬁

⬁

dy y

冉

^␹

^⬙

^{共y,␶兲+␸共0,y␶兲}

冊

^{, 共45兲}

␸^共2兲共0,␶兲= − 1

␲

冕

_−⬁^{⬁ dy y2}

冉

^␹

^⬘

^{共y,␶兲+␸共1兲共y0,␶兲}

冊

^{, 共46兲}

where

␸^共n兲共0,␶兲=

冏 冉

⳵^⳵Enn␸共E,␶兲

冊 冏

E=0

. 共47兲

The moment sum rules above are related to the asymptotic expansion of␹BAwith respect toz共which means low-energy behavior兲. For small values ofz共i.e., high-energy behavior兲, one obtains the following sum rules:

␸^共−1兲共0,␶兲= − 1

␲

冕

−⬁

⬁

dy␹

⬙

共0,␶兲

y , 共48兲

␸^共−2兲共0,␶兲= 1

␲

冕

−⬁

⬁

dy1

y²关␹

⬘

共0,␶兲+␸^共−1兲共0,␶兲兴, 共49兲

␸^共−3兲共0,␶兲= 1

␲

冕

−⬁

⬁

dy1

y³关␹

⬙

共0,␶兲+y␸^共−2兲共0,␶兲兴, 共50兲 where

␸^共−n兲共0,␶兲=

冕

0

⬁

dE₁

冕

E₁

⬁

dE₂¯

冕

E_n−1

⬁

dEn␸共E,␶兲. 共51兲

The procedure shown in the Appendix can be repeated as long as the derivatives ␸^共n兲 and the integrals ␸^{共−n兲 exist to} derive higher-order moment sum rules.

As for the sum rules in the frequency space, those above can be used to correct phenomenological expressions for

␸共E,␶兲. For example, if one assumes a functional form for the response function with some free parameters with respect to the energy dependence, one could determine them by im- posing the sum rules for high- or low-energy behavior. The way to do that in the frequency space is shown, for example, in关15,20兴. Since the relations above are valid for any value of ␶, one could also have dropped the␶dependence by set- ting ␶= 0. Then, it is easier to understand the meaning and the importance of the sum rules: thezspectrum of␹^{is given} in terms of static quantities like ␸^共n兲共0 ,␶= 0兲 and

␸^共−n兲共0 ,␶= 0兲, which could be calculated quantum mechani- cally in terms of the commutation relations betweenAˆ andBˆ 共see, for example,关4兴兲.

IV. CONCLUSIONS

Using linear response theory, we presented a derivation of the fluctuation-dissipation theorem in the microcanonical en- semble in both quantum and classical regimes. The theorem is stated as a relation between the Laplace-Fourier transforms of the response and symmetric correlation functions. Al- though this relation is very similar to the one derived in the canonical ensemble context, it is valid, for example, in a situation where the fluctuations are very different from ther- mal ones, namely, fluctuations of an isolated system that is not in the thermodynamic limit. Therefore, the fluctuation- dissipation theorem can be considered as a much more gen- eral relation and not constrained just to the context of the canonical ensemble. We believe this result can be very useful to calculate correlation functions from response functions 共and vice versa兲for systems in the microcanonical ensemble when they are not in the thermodynamic limit. In this sense, as mentioned in 关9,12兴 共see also the references in关12兴兲, the present work can be considered as an additional effort to apply statistical physics to small systems. Moment sum rules were also presented for the energy dependence and they could be useful to correct phenomenological expressions for the response functions.

ACKNOWLEDGMENTS

The author acknowledges the support of the Brazilian re- search agency CNPq and DFG 共GRK 638兲. The author is also grateful to J. D. Urbina, M. A. M. de Aguiar, and K.

Richter for their careful reading of the manuscript and valu- able suggestions.

APPENDIX: DERIVATION OF THE SUM RULES In this appendix we give a brief sketch of how to derive the sum rules presented in Sec. III. For a careful derivation and deeper discussion about the subject, we refer to关20兴. Our starting point is the function f共z兲defined by

f共z兲=

冕

0

⬁

dE e^−Ez␾共E兲, 共A1兲 where z=␥+iy is complex with its real part positive and

␾共E兲 is real. Therefore, f共z兲 is analytic in the half plane Re共z兲艌␥ and it satisfies the following dispersion relations:

f

⬘

共y₀兲= 1

␲^P

冕

_−⬁^{⬁ dyyf}

^⬙

^−共yy兲0

, 共A2兲

f

⬙

^共y0兲= − 1

␲^P

冕

_−⬁^{⬁ dyyf}

^⬘

^−共y兲y0

, 共A3兲

where f

⬘

^共y0兲and f

⬙

^共y0兲 are the real and imaginary parts of f共y0兲, respectively. From共A2兲, we can write

f

⬘

^共0兲=_␲^1P

冕

_−⬁^{⬁ dyf}

^⬙

^{共y兲y 共A4兲}

and from共A3兲multiplied by y₀we obtain

lim

y₀→⬁y₀f

⬙

共y₀兲= 1

␲

冕

_−⬁^{⬁ dy f}

^⬘

^{共y兲. 共A5兲}

To calculate the left-hand side of 共A5兲, we go back to共A1兲 and integrate by parts to obtain

f共z兲=␾共0兲 z +1

z

冕

0

⬁

dE e^−Ez␾^共1兲共E兲, 共A6兲

where

␾^共n兲共0兲=

冏 冉

^{dEdnn␾共E兲}

冊 冏

E=0

. 共A7兲

Therefore,

兩zlim兩→⬁zf共z兲=␾共0兲, 共A8兲

and, from 共A5兲and共A8兲

␾共0兲= 1

␲

冕

_−⬁^{⬁ dy f}

^⬘

^{共y兲, 共A9兲}

which is the first sum rule. To derive the next one, we define a new function

g共z兲=

冕

0

⬁

dE e^−Ez␾^共1兲共E兲, 共A10兲 which is analytic again for Re共z兲艌␥. Therefore, g共z兲 obeys the same dispersion relations as f共z兲. Integrating 共A10兲 by parts yields

g共z兲=␾^共1兲共0兲 z + 1

z

冕

0

⬁

dE e^−Ez␾^共2兲共E兲, 共A11兲

from which we obtain lim_兩z兩→⬁zg共z兲=␾^共1兲共0兲. By the same procedure as before,

␾^共1兲共0兲= 1

␲

冕

_−⬁^{⬁ dy g}

^⬘

^{共y兲. 共A12兲}

Since, from共A6兲and共A10兲, g共z兲=

冕

0

⬁

dE e^−Ez␾^共1兲共E兲=zf共z兲−␾共0兲 共A13兲

and

g

⬘

^{共y兲= −y f}

⬙

^共y兲−␾共0兲, 共A14兲 we obtain, from共A12兲and共A14兲, the second sum rule

␾^共1兲共0兲= − 1

␲

冕

_−⬁^{⬁ dy y}

冉

^f

^⬙

^{共y兲+␾共0兲}y

冊

^{. 共A15兲}

Repeating the same procedure again, we obtain

␾^共2兲共0兲= − 1

␲

冕

_−⬁^{⬁ dy y2}

冉

^f

^⬘

^{共y兲+␾共1兲y2共0兲}

冊

^{, 共A16兲}

and so on, as long as the ␾^共n兲共0兲exist.

A similar procedure can be applied to generate a different kind of sum rule关20兴. Starting again with共A1兲, we integrate by parts in a different way now:

f共z兲= −␾^共−1兲共0兲+z

冕

0

⬁

dE e^−Ez␾^共−1兲共E兲, 共A17兲

where

␾^共−n兲共0兲=

冕

0

⬁

dE₁

冕

E₁

⬁

dE₂¯

冕

E_n−1

⬁

dEn␾共E兲. 共A18兲

From 共A2兲,

f

⬘

共0兲= 1

␲

冕

_−⬁^{⬁ dyf}

^⬙

^{共yy兲, 共A19兲}

and from共A18兲,f

⬘

^{共0兲= −}␾^共−1兲共0兲, so

␾^共−1兲共0兲= − 1

␲

冕

−⬁

⬁

dyf

⬙

共y兲

y . 共A20兲

We again repeat the procedure, as before, defining from 共A20兲a new functiong共z兲,

g共z兲=

冕

0

⬁

dE e^−Ez␾^共−1兲共E兲= −␾^共−2兲共0兲 +z

冕

0

⬁

dE e^−Ez␾^共−2兲共E兲= f共z兲

z +␾^共−1兲共0兲

z .

共A21兲 Since g共z兲satisfies the same dispersion relations asf共z兲, we obtain

g

⬘

^{共0兲= −}␾^共−2兲共0兲= 1

␲

冕

_−⬁^{⬁ dyg}

^⬙

^{y共y兲. 共A22兲}

Inserting the imaginary part of the second line of 共A21兲 in 共A22兲leads to

␾^共−2兲共0兲= 1

␲

冕

_−⬁^{⬁ dyy12关f}

^⬘

^{共y兲+␾共−1兲共0兲兴. 共A23兲}

Repeating the same procedure again, we obtain

␾^共−3兲共0兲= 1

␲

冕

_−⬁^{⬁ dyy13关f}

^⬙

^{共y兲+y␾共−2兲共0兲兴, 共A24兲}

and so on, as long as the ␾^共−n兲共0兲exist.

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