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Sporadic randomness, Maxwell’s Demon and the Poincare recurrence times

Gerardo Aquino1, Paolo Grigolini1,2,3 and Nicola Scafetta 3

1Dipartimento di Fisica dell’Universita di Pisa and INFM,

Piazza

Torricelli 2, 56127 Pisa, Italy2Istituto di Biofisica CNR, Area della Ricerca di Pisa, Via Alfieri 1, San Cataldo 56010 Ghezzano-Pisa, Italy

3Center for Nonlinear Science, University of North Texas,

P.O. Box 5368, Denton, Texas 76203

(February 1, 2008)

In the case of fully chaotic systems the distribution of the Poincare recurrence times is an exponentialwhose decay rate is the Kolmogorov-Sinai (KS) entropy. We address the discussion of the sameproblem, the connection between dynamics and thermodynamics, in the case of sporadic randomness,using the Manneville map as a prototype of this class of processes. We explore the possibility ofrelating the distribution of Poincare recurrence times to “thermodynamics”, in the sense of the KSentropy, also in the case of an inverse power law. This is the dynamic property that Zaslavsly (G.M. Zaslavsky, Physics Today 52(8), 39 (1999)) finds to be responsible for a striking deviationfrom ordinary statistical mechanics under the form of Maxwell’s Demon effect.We show that this way of establishing a connection between thermodynamics and dynamics is validonly in the case of strong chaos, where both the sensitivity to initial conditions and the distributionof the Poincare recurrence times are exponential. In the case of sporadic randomness, resulting atlong times in the Levy diffusion processes, the sensitivity to initial conditions is initially a powerlaw, but it becomes exponential again in the long-time scale, whereas the distribution of Poincarerecurrence times keeps, or gets, its inverse power law nature forever, including the long-time scalewhere the sensitivity to initial condition becomes exponential.We show that a non-extensive version of thermodynamics would imply the Maxwell’s Demon effectto be determined by memory, and thus to be temporary, in conflict with the dynamic approach toLevy statistics. The adoption of heuristic arguments indicate that this effect is possible, as a formof genuine equilibrium, after completion of the process of memory erasure.

05.45.+b,03.65.Sq,05.20.-y

I. INTRODUCTION

In a lucid discussion Lebowitz [1] has recently restatedthe point of view of Boltzmann to establish the micro-scopic origin of irreversible macroscopic behavior. In hisview the adoption of the laws of big numbers is essen-tial, and the role of deterministic chaos becomes impor-tant only if it applies to a macroscopic number of non-interacting particles. According to Lebowitz mixing andergodicity are notions “unnecessary, misguided and mis-leading”. In other words, this opinion reflects the convic-tion, mirrored by the handbooks of statistical mechanics,that the unification of mechanics and thermodynamicsrests on N → ∞, where N denotes the number of de-grees of freedom of the system under study.

This point of view has to be contrasted with a newview about the origin of thermodynamics [2] producedby the increasing interest in nonlinear dynamics. Withinthis new perspective the sensitivity to the initial condi-tions, rather than the large number of degrees of freedominvolved, is playing the crucial role of generator of ther-modynamics. Zaslavsky [3] has recently discussed thephenomenon of the Poincare recurrences without usingthe traditional argument that these recurrences occur ina so extended time scale as to lie beyond the observa-tion range. Using the Bernouilli shift map, as a dynamic

prototype of strong chaos, Zaslawsky proves [3] that thedistribution of Poincare recurrence times, PR(t), obeysthe following prescription:

PR(t) ∝ exp(−hKSt), (1)

where hKS denotes the Kolmogorov-Sinai (KS) entropy[2]. Zaslavsky makes the interesting remark that Eq.(1)establishes a nice connection between mechanics, the lefthand side of this equation, and thermodynamics, theright hand side of the same equation. This is so be-cause hKS has an entropic significance [2]. It is also wellknown that the KS entropy is related to the Lyapunovcoefficient and thus to the sensitivity to initial conditions,as a consequence of the Pesin theorem [4]. To make theconnection between PR(t) and the Lyapunov coefficientperspective more transparent, it is convenient to intro-duce also the property:

ξ(t) ≡ lim∆x0→0

|∆x(t)

∆x(0)|, (2)

where ∆x(t) denotes the difference between the spacecoordinates of two trajectories, moving from two distinctinitial conditions at a distance |∆x(0)| the one from theother. It is evident that after identifying hKS with theLyapunov coefficient λ, we can rewrite Eq.(1) in the form

1

PR(t) ∝ ξ(−t) =1

ξ(t), (3)

where

ξ(t) = exp(λt). (4)

At a first sight, this point of view seems to conflictwith that of Lebowitz. In this paper we shall argue, witha heuristic approach, that it is not so, and that a sort ofcompromise between the two views can be found. Herewe want to limit ourseleves to noticing that the observa-tion frequently made that the occurrence of strong chaosis exceptional, does not necessarily mean that thermody-namics can only rest on the (N → ∞)-perspective. Itmight only imply that the process of transition to equi-librium and the ensuing statistics are not ordinary. Thesame aspect might emerge also from within the adoptionof the (N → ∞)-perspective. We have two examplesin mind, where the thermodynamic level, if it exists, isclosely related to anomalous rather than ordinary sta-tistical mechanics. The first is the dynamical processrecently discussed by Latora et al. [5]. This is a Hamil-tonian system of N rotors, each rotor being coupled toall the other rotors of the system with the same cou-pling. The authors show that in the limit N → ∞ fromthis Hamiltonian picture a process of superdiffusion un-der the form of a Levy walk emerges. On the other hand,we have in mind a second example, of Hamiltonian na-ture, resulting in Levy diffusion. This example is affordedby the class of billiards discussed in the recent book ofZaslavsky [3]. These are systems with only one degreeof freedom, thereby seemingly departing from the condi-tion N → ∞ that, according to Lebowitz [1], is essentialfor the emergence of thermodynamics. In this paper weshall try to prove that the point of view advocated byLebowitz [1] is compatible with that sustained by Za-slavsky [3,6,7] in the case of fully chaotic systems, wherethe dynamical approach to statistical mechanics yieldsthe same ordinary form of statistical mechanics as the(N → ∞)-perspective, thereby making apparently use-less the adoption of a dynamic view. However, we shallargue that the role of dynamics might become importantwhen the system under study is characterized by persis-tent memory. We make also the conjecture that a kind ofstatistical equivalence exists between extended memoryand long-range interactions.

For the time being, we would like to stress that in hisvery interesting book [3], as well as in the original papers[6,7], Zaslavsky focuses his attention on the anomalousthermodynamic nature of these generators of anomalousdiffusion. He shows that two chaotic billiards, coupled toone another through a small hole in the wall separatingone billiard from the other, result in a so strong viola-tion of the condition of equal distribution as to suggestthe occurrence of a Maxwell’s Demon effect. Maxwell’sDemon was described by Maxwell in his book ”Theory

of Heat” (1871) and represented a rather simple device.Two chambers, A and B, are separated by a divisionwith a small hole. Positioned at the hole the demonallows swifter molecules to pass from chamber A to Band slower molecules to pass from B to A. After a whilechamberA will contain mainly swifter molecules, whereaschamber B will contain mainly slower molecules. Thisimplies the appearance of temperature difference with-out expenditure of work and contradicts the Second Lawof Thermodynamics. The version of Maxwell’s Demonrealized in practice in Refs. [3,6,8,7] is equivalent to thecase where particles with the same kinetic energy tendsto spend more time in one of the two chambers even ifthe volume available is the same.

According to Zaslavsky this surprising behavior is pro-voked by the fact that the distribution density PR(t) de-viates from the thermodynamic condition of Eq.(1). Thisis so because even in the phase space of seemingly chaoticsystems there might exist stable islands. The surface ofseparation between the chaotic sea and the stable islandsis characterized by fractal properties and these proper-ties make these surfaces very sticky. Consequently, in thelong-time limit the function PR(t) becomes proportionalto ψ(t), denoting the distribution of waiting times at theborder between chaotic sea and stability island. Usingrenormalization group arguments it is shown [3] that

limt→∞

ψ(t) =const

tµ, (5)

where µ > 2, in accordance with the Kac theorem [9].This means that the first moment of the waiting timedistribution is finite. After revealing the sources of devi-ations from the ordinary prescriptions of thermodynam-ics and statistical mechanics, Zaslasky does not rule outthe possibility that even in this case a thermodynamicperspective applies. He states [8] that all this is obligingus to rethink the foundation of thermodynamics.

We see that there are several issues to discuss:(i) Is there a connection between the condition of

strong chaos leading to thermodynamics in the sense ofEq. (1) and the Boltzmann condition N → ∞? Is there,in general, a connection between the perspective basedon deterministic chaos of low-dimensional systems andthe perspective based on N → ∞?

(ii) As far as the connection between ψ(t) and the sen-sitivity to initial condition is concerned, we see that inthe case where the exponential picture does not applyEq.(3) splits into the following two different possibilities:

ψ(t) ∝ ξ(−t) (6)

and

ψ(t) ∝1

ξ(t). (7)

Which is the correct property?

2

(iii) Is there a form of thermodynamics involved in thiscase? If it is so, which is the nature of this form of ther-modynamics? This would answer the fundamental ques-tion raised by Zaslavsky about the origin of his Maxwell’sDemon effect [3,8,6].

(iv) Is it possible to find a general analytical expres-sion for ψ(t), fitting the inverse power law of Eq.(5), andcollapsing, for a critical value of a given parameter, intothe exponential form of Eq. (1) with the same coefficienthKS?

To give the reader a preliminary understanding of ques-tion (iv) we have to make some remarks. We shall showthat the Manneville map [10] is a dynamic process generalenough as to reproduce both the exponential property ofEq.(1) and the inverse power law of Eq.(5), dependingon the value of a given control parameter z. We shallstudy this dynamic system while keeping in mind thenon-extensive thermodynamics introduced by Tsallis in1988 [11], which is now becoming more and more popular[12]. From a formal point of view, within this new formof statistical mechanics a crucial role is played by the socalled q-exponential. If we express the function ψ(t) as aq-exponential we obtain

ψ(t) = [1 − (1 − q)hqt]1/(1−q). (8)

The parameter q is called entropic index. We see immedi-ately that with q > 1 the function ψ(t) is an inverse powerlaw. At the critical value q = 1 the function ψ(t) wouldmake an abrupt transition from the inverse power lawEq.(5) to the exponential form of Eq. (1). An importantmotivation for this paper has been to assess whether ornot the non-extensive thermodynamics of Tsallis mightbe a satisfactory solution of question (iii) as well as of(iv).

This paper is organized as follows. In Section II wediscuss the dynamical properties of the Manneville map.We shall discuss theoretically and numerically the wait-ing time distribution ψ(t) in this specific case. In Sec-tion III we illustrate very simple arguments supportingthe choice of Eq.(7). Section IV is devoted to exploringthe possibility that the Tsallis non-extensive thermody-namics might shed light on the thermodynamic nature ofthe Maxwell’s Demon effect. Section V shows that theextended regime of transition from dynamics to thermo-dynamics corresponds to the steady process of memoryerasure accompanying the dynamic realization of Levyprocesses. Finally Section VI is devoted to making abalance on the answers to questions (i)-(iv) that we aregiving throughout this paper.

II. THE MANNEVILLE MAP

In this section we illustrate some key dynamic prop-erties of the Mannneville map [10]. The reader can find

surprising that after mentioning the key issues of SectionI, we decide to focus our attention on a non-Hamiltoniansystem, with only one variable. We feel therefore theneed of justifying our choice. First of all, we want to re-mark that, as proved by the authors of Ref. [5], the useof a Hamiltonian system with infinitely many degrees offreedom does not rule out the possibility that the re-sulting dynamics, as far as a few collective variables areconcerned, is equivalent to a random walk process thatcan be obtained dynamically also from a map [13]. Forthe emergence of low-dimensional chaos from a Hamilto-nian picture with a large number of degrees of freedomwe refer to the discussion of Tennyson et al. [14].

Then there is another reason why the choice of theManneville map is convenient to mimic the Hamiltonianproperties responsible for the birth of the Levy diffusion.According to Zaslavsky, the Levy superdiffusion in oneof his two-dimensional systems is generated by the factthat the states of uniform motion last with a waitingtime distribution that has an inverse power law nature.This means that any condition of uniform motion cor-responds to a stable island whose border with the sur-rounding chaotic sea is fractal. A trajectory sticks to thisborder with an inverse power law distribution of wait-ing times. For the whole sojourn time of the trajectoryon this border, in the laboratory frame of reference thesystem undergoes a uniform motion in a given direction.From time to time the trajectory leaves the fractal regionand after a short sojourn in the chaotic sea it sticks againto another fractal region. The particle sticking to a givenfractal region moves in the laboratory frame of referencewith constant velocity. The change from one sticking con-dition to another might have the effect of changing themotion direction, but it does not affect the velocity mod-ulus. Note that the velocity modulus is constant becausethe trajectory motion occurs in the constant energy sur-face of the phase space. A nice dynamical model sharingall these properties is the egg-carton model of Geisel et

al. [15], where the trajectory moves of uniform motionalong a given channel of the egg-crate landscape and at agiven time, randomly selected from an inverse power lawdistribution, begins moving in one of the two oppositedirections of a orthogonal channel. Also this direction ofmotion is randomly selected. Notice that randomly hererefers to a property of deterministic chaos.

The one-dimensional version of this generator ofanomalous diffusion corresponds to a velocity ξ with onlytwo possible values, W and −W . These two states musthave a distribution of sojourn times with an inverse powerlaw. Thus, we see immediately that the one-dimensionalversion of the Hamiltonian systems studied by Zaslavskybecomes similar to the one-dimensional map of Ref. [13].This map has been widely used in the recent past toderive Levy processes [16–19]. On the other hand, forthe purpose of the discussion of the present paper, asalready done in the earlier work of Ref. [20], it is con-

3

venient to shift our attention from the bimodal maps ofRefs. [13,16–19] to the case of the Manneville map, whichhas only one laminar region. The Kolmogorov complexityof the Manneville map has been already analyzed yearsago by Gaspard and Wang [21]. The complexity of a se-quence equivalent to that of the bimodal maps of Refs.[13,16–19] was studied recently by Buiatti et al. [?]. It isstraightforward to show that the Kolmogorov complexityof the Manneville map is proportional to the complexityof the bimodal maps generating Levy processes [23]. Weare interested in sporadic randomness, and for our pur-poses it is irrelevant whether after exiting from one lami-nar region and crossing the chaotic region, the trajectoryis injected back into the original laminar region, or intoa new laminar region equivalent to the original [23]. Ifwe make the arbitrary decision that crossing the chaoticregion always implies a transition from one laminar re-gion to the other, the resulting complexity is the sameas that of the Manneville map, with only one laminarregion, and consequently no uncertainty associated withits choice. In the bimodal case, the injection back to theoriginal laminar region and the jump into the other lam-inar region are random processes, with the same prob-ability. This implies a Kolmogorov complexity which istwice as that of the Manneville map [23].

In conclusion, the reason for the choice of the Man-neville is that this map mimics very well the conditionof sporadic randomness that, according to Zaslavsky [3],is responsible for Maxwell’s Demon effect. Therefore, weare convinced that a numerical treatments of the Hamil-tonian systems with sporadic randomness would lead tothe same conclusions as those reached by us here, on thebasis of the Manneville map.

The Manneville map reads:

xn+1 = Φ(xn) = xn + xzn(mod.1)(1 ≤ z). (9)

We note that at z = 1 the Manneville map becomesequivalent to the Bernouilli shift map used by Zaslavslyto prove the fundamental result of Eq. (1). For z > 1the interval [0, 1] is divided in two regions, the laminarregion, [0, d(z)], and the chaotic region, [d(z), 1], withd(z) defined by

d(z) + d(z)z = 1 (10)

We review here the arguments used by Geisel andThomae [13] to derive an analytical expression for thedistribution of the times of sojourn in the laminar re-gion. First of all we assume that the injection point x0

is so close to x = 0 as to replace eq.(9) with:

dx/dt = xz . (11)

Thus we obtain the following time evolution:

x(t) = [x1−z0 + (1 − z)t]1/(1−z). (12)

Hence the time necessary for the trajectory to get theborder x = d(z) is given by

t = T (x0) ≡ (1

xz−10

−1

dz−1)

1

z − 1. (13)

Note that in the special case where the initial conditon isso close to x = 0 as to fulfill the condition x0 << d, theexit time T (x0) can be satisfactorily approximated by

T (x0) ≈1

xz−10

1

z − 1, (14)

which, as we shall see in Section III, can be used to de-fine the time at which we lose control of the trajectoriesdeparting from a region of the map very close to x = 0.

Note that the distribution function ψ(t) is related tothe injection probability p(x0) by

ψ(t)dt = p(x0)dx0. (15)

Assuming equiprobability for the injection process wehave:

p(x0)dx0 =1

d(z)|dx0

dT|dT. (16)

Thus we obtain:

ψ(t) =1

d(z)|dx0

dt|. (17)

By differentiating t with respect to x0 we finally arriveat

ψ(t) = dz−1[1 + dz−1(z − 1)t]−z/(z−1). (18)

It is important to observe that the mean waiting timeTav is given by

Tav =1

dz−1

1

2 − z, (19)

with Tav = ∞ for 2 ≤ zWe see that Eq.(18) implies that the region correspond-

ing to z > 2 is characterized by a diverging first moment.This region is in conflict with the Kac theorem [9] andfor this reason we do not take it into account. The regionpertaining to the interval 1.5 < z < 2 is characterized bya finite second moment and a diverging second moment.This is the region of interest for us, since it correspondsto that generating Levy diffusion according to the recentwork of Ref. [20]. The region 1 < z < 1.5 is characterizedby a finite second moment. Of course, the ideal condi-tion z = 1 implies all the moments to be finite. This isa region which would correspond, in the perspective ofRef. [20], to ordinary Gaussian diffusion. We note thatin the region 1 < z < 1.5 the waiting function distri-bution must make a transition from the inverse powerlaw behavior of Eq.(18) to the exponential regime, where

4

the arguments of Zaslavsky leading to Eq.(1) apply. Infact, at z = 1, the Manneville map becomes identical tothe Bernouilli shift map. In that case, the theoreticalremarks of Zaslavsky yield:

ψ(t) ∝ exp(−tln2). (20)

We are now in a position to make a preliminary ob-servation about question (iv) of Section I. The analyticalexpression of Eq.(18) can be expressed in the form of theq-exponential of Eq.(8), with

q = 1 + (z − 1)/z. (21)

However, this would not fit completely the request ofbeing an exact representation of the waiting function inthe fully chaotic case. In fact q = 1 would correctly implyz = 1, but this would make Eq.(18) collapse into

ψ(t) ∝ exp(−t). (22)

rather than in the exact expression of Eq.(20).This, in principle, would not rule out the possibility

of finding a different analytical expression fulfilling therequirement of question (iv). To explore more deeply is-sue (iv) we use a numerical treatment of the Mannevillemap of Eq. (9). For computational purposes we foundto be more convenient to evaluate the population of thelaminar region, M(t), rather than the waiting time dis-tribution ψ(t). The two functions are related the one tothe other by

M(t) = 1 −

∫ t

0

ψ(t′)dt′. (23)

Thus, the analytical expression of Eq.(18) yields

M(t) =[

1 + dz−1(z − 1)t]−1/(z−1)

. (24)

The explicit expression of M(t) in the fully chaotic case(z = 1) is derived from Eq.(1). In fact, Zaslavsky proves[3] that Eq.(1) becomes exact in the case of the Bernouillimap, with hKS = ln2. In the normalized form, Eq.(1)becomes:

ψ(t) = hKSexp(−hKSt). (25)

Let us plug Eq.(25) into Eq.(23). We obtain

M(t) = exp(−tln2). (26)

We know that Eq.(26) is exact and we expect Eq.(24)to become exact for z → ∞. In conclusion, as to theimportant function ψ(t), we know its exact expressionat z = 1 and in the asymptotic limit z → ∞. It isworth exploring the intermediate region with a numericaltreatment.

Figs. 1 and 2 illustrate the results of our numericaltreatment.

1 10 100 1000t

10−9

10−7

10−5

10−3

10−1

M(t

)

FIG. 1. M(t) as a function of time. The meaning of

the six full lines is as follows. The lowest full line is the function

M(t) = exp(−tln2). All the other full lines denote the long-time

inverse power law M(t) = 1/t1

z−1 . The dotted lines are the nu-

merical result. All the full lines but the lowest have been shifted

to the right to make them distinguishable from the numerical re-

sult. The value of the parameter z, from the bottom to the top

is: z = 1, 1.1, 1.2, 1.3, 1.4, 1.5

1 10 100 1000 10000t

10−5

10−4

10−3

10−2

10−1

100

M(t

)

FIG. 2. M(t) as a function of time. The meaning of the four

pair of lines is as follows. The full lines denote the function M(t) of

Eq.(24) and the dotted lines denote the numerical results. To make

he full lines distinguishable from the dotted lines we shifted them to

the right by the quantity ǫ = 0.1. In the logarithmic representation

adopted, this is equivalent to replacing t of M(t) with texp(−ǫ). The

value of the parameter z from the bottom to the top changes as

follows: z = 1.5, 1.6.1.7, 1.8, 1.9

In Fig. 1 we illustrate the result of the numerical cal-culation with the parameter z in the interval [1, 1.5]. Wesee that the long-time limit fits for all values of z con-sidered but z = 1, the theoretical prescription of the in-

verse power law t−1

z−1 . However, this inverse power lawregime is reached after an extended transition regime,which seems to be exponential-like. The duration of thistransition regime become more and more extended withz coming closer and closer to z = 1. At z = 1 this tran-sition regime becomes infinitely extended and coincidentwith the theoretical prediction of Eq.(26).

5

In Fig. 2, devoted to studying the relaxation of theManneville map with z in the interval [1.5, 2], we seethat the prediction of Eq.(24) is very accurate and tendsto become exact with increasing the value of z.

In conclusion, these numerical results prove that theq-exponential of Eq.(8) cannot satisfactorily fit the con-ditions set by question (iv). However, this is not yet astrong reason to rule out the non-extensive thermody-namics of Tsallis [11,12]. Let us make another attemptat establishing this non-extensive thermodynamics as theproper form of thermodynamics that Zaslavsky is look-ing for. We now set a request more flexible than theone earlier adopted. This is based on disregarding theextended regime of transition to the inverse power law.In other words, we now accept an analytical form of q-exponential which at q = 1 becomes identical to Eq.(20)while at z > 1 results in the same inverse power law asthe Manneville map. We try with the following possibil-ity:

ψ(t) = [1 + (1 − q)hKSt]1/(1−q), (27)

supplemented by Eq.(21). To a first sight, this possibilitylooks very attractive. In fact, at q = 1, the prescriptionof Eq.(21) yields z = 1. This means the Bernouilli map,and, consequently [2]

hKS = ln2. (28)

Thus, the requirement of recovering Eq.(20) is fulfilled,while the request of producing the correct power law atz > 1 is obviously satisfied due to the Eq.(21).

However, this is not physically correct. We let thereader know that years ago, Gaspard and Wang [21]evaluated the Lyapunov coefficient of the Mannevillemap and found it to be given with a good accuracy byhKS = (2 − z)zln2. This means that hKS vanishes atz = 2, thereby yielding a strong disagreement betweenEq.(27) and Eq.(18).

One might think that the departure of Eq.(27) fromEq.(18) is caused by the fact that the coefficient hq ap-pearing in Eq.(8) cannot be identified with the ordinaryLyapunov coefficient. In fact, according to the Pesin the-orem [4] we have

hKS = λ ≡

∫ 1

0

dxρ(x)ln(dΦ/dx). (29)

One might think that the adoption of Tsallis non-extensive thermodynamics implies that the ordinary def-inition of Lyapunov coefficient cannot be maintained andthat consequently hq cannot be identified with the prop-erty of Eq.(29), which is in fact the property discussedby Gaspard and Wang [21]. In Section IV B we shallprove that also this possibility has to be ruled out. Atthe same time we shall show that the correct answer toquestion (ii) cannot be given by Eq.(6). The derivationof the correct answer to question (ii) is given in SectionIII.

III. SENSITIVITY TO INITIAL CONDITIONS

The sensitivity to initial conditions of the Mannevillemap has been studied in Ref. [20]. The authors of thispaper evaluated the function ξ(t) defined by Eq.(2), usingthe same assumptions as those adopted in Section II toderive Eq. (18). The result is given by:

ξ(t) = [1 − (z − 1)xz−10 t]−z/(z−1). (30)

This function is, in a sense, faster than the exponential,since it diverges at the time Tdiv given by

Tdiv ≡1

(z − 1)xz−10

, (31)

while the case of exponential sensitivity would yield finitevalues of ξ(t) for any finite value of time.

It would be tempting to conclude that the Mannevillemap is characterized by a form of sensitivity to initialcondions stronger than the usual exponential sensitivity,but it would be wrong. Let us explain why it is so usingheuristic arguments. Let us divide the phase space ofthe Manneville map into two regions. The first is veryclose to x = 0, and it is characterized by the the localLyapunov coefficients

λ(x0) = ln(1 + zxz−10 ) ≈ zxz−1

0 . (32)

The second region includes the whole chaotic region plusa portion of the laminar region and it is characterized bylocal Lyapunov coefficients of the order of

λ ≈ ln(1 + z). (33)

Let us now define the time at which we lose control ofa trajectory, in the case of exponential sensitivity, as atime proportional to 1/λ, where λ is the Lyapunov coef-ficient. We see that the time at which we lose control ofthe trajectories departing from the first region is of theorder of the time scale Tdiv, of Eq.(31), while the time atwhich we lose control of the trajectories departing fromthe second region is of the order of unity. We note thatfor values of x0 very close to x = 0 we can fulfill theinequality:

Tav << Tdiv, (34)

where Tav is the waiting mean time defined by Eq.(19).Note also that for x0 so small as to fit the condition ofEq.(34), the time Tdiv can be interpreted in two distinctways. The former is the time at which ξ(t) diverges, andthe second is the inverse of the local Lyapunov coeffi-cient of Eq.(32). We see, in conclusion, that the com-plexity of the Manneville map is decided by the timescales 1/log(1 + z) and Tav rather than by Tdiv. This isthe reason why the randomness detected by the method

6

of Kolmogorov complexity [21] is strongly determined bythe influence of the chaotic region of the Manneville map.

This is a key point worth of a comment. The functionξ(t) with the form of Eq.(30) cannot be used for the sta-tistical interpretation of the dynamic process under studyfor times larger than the exit times, and consequentlycannot have any relevance when the time Tav of Eq.(19)is finite, namely for z < 2. In this case, in the time scalet >> Tav the analytical form of the function ξ(t) changesand becomes exponential [21]. At this stage there are nomore possible connections between ψ(t) and ξ(t) left. Infact,the function ψ(t) has a meaning completely differentfrom that of ξ(t). The function ψ(t) signals the proba-bility of observing sojourn times of any length, includingthose exceeding the time scale Tav. Thus, the functionψ(t) maintains its inverse power law nature even in thetime scale where ξ(t) is exponential.

In accordance with this remarks we find a connectionbetween the function ψ(t) and the function ξ(t) at theexit time T using Eq.(7). The exit time depends on theinitial condition of the trajectory within the laminar re-gion and is given by Eq. (13). Let us express x0 ofEq.(30) in terms of this exit time, and let us replace itinto Eq. (30). The resulting expression has to be inter-preted as the value that the function ξ(t) gets at the exittime. Its explicit expression is:

ξ(T ) = dz [d1−z + (z − 1)T ]z/(z−1). (35)

Comparing Eq.(35) to Eq.(18) we reach the conclusionthat:

ψ(T ) ∝1

ξ(T ). (36)

Thus with very simple arguments we are led to concludethat the generalization of Eq.(3) implies the choice ofEq.(7). The meaning of this conclusion is that the valueof ψ(T ) is determined by that of the function ξ(t) with tbeing a given exit time. This very simple argument set-tles question (ii) of Section 1. We have to address nowthe much more delicate question of the connection withthe thermodynamic perspective. Before ending this sec-tion, we would like to notice that the condition of Eq.(36)does not apply only in the region z > 1.5 which favorsthe continuous approximation behind the analytical re-sults of Eqs.(30) and (18). It applies very well also tothe long-time limit in the region 1 < z < 1.5, as clearlyillustrated by Fig. 1.

IV. THERMODYNAMIC APPROACH TO THEWAITING TIME DISTRIBUTION

Here we show that the entropic arguments of thenon-extensive thermodynamics [12] would naturally yieldEq.(6) rather than Eq.(7), creating a conflict with the

conclusions of Section III. Thus the entropic argumentsyield a wrong conclusion, which by itself does not nec-essarily imply that the Tsallis non-extensive thermody-namics is ill founded. If the Tsallis non-extensive ther-modynamics were well founded, on the other hand, thiswrong conclusion would certainly imply that this form ofthermodynamics [11,12] is not permanent. In Section Vwe shall show, on the other hand, that the Maxwell’s De-mon effect is permanent and takes place in a later timescale .

A. Entropic derivation of ψ(t)

The dynamic derivation of Levy processes from inter-mittent maps [18,19] has established that probability forthe particle to make a jump of length |x| in a time t,Π(x), is related to ψ(t) by

Π(x) = ψ(x/W )/W. (37)

To make immediately evident to the reader the expla-nation of the form of Eq.(37) it is enough for us to lethim/her know that the dynamic derivation of of Levyprocesses [18,19] rests on a one-dimension motion witha fluctuating velocity. The particle velocity fluctuatesamong the values W and −W . The particle sojourns inthese two states with a distribution of waiting times givenby ψ(t), which has the same structure as that of Eq.(18).This accounts for Eq.(37). On the other hand, on thebasis of the increasing interest for the subject of non-extensive entropy [12], and along lines similar to thoseadopted by earlier work [25–27], we decide to maximizethe Tsallis entropy [11] to determine the form of Π(x).Therefore we define

Sq[Π(x)] = −1 −

∫

∞

−∞Π(x)

qdx

(1 − q), (38)

and we maximize it under the constraint of assigning afixed value to the first moment of the escort distribu-tion with entropic index q. This means that, as done inRef. [20], we have to maximize the functional form Fq(Π)defined by

Fq(Π) ≡1 −

∫

∞

−∞Π(x)

qdx

(1 − q)− β

∫

∞

−∞

|x|Π(x)qdx+ α

∫

∞

−∞

Π(x)dx.

(39)

As a result of the entropy maximization we get [20]

Π(x) =A

[1 + β(q − 1)|x|ν ]1/(q−1). (40)

It is evident that for this entropic argument to be com-patible with the statistical derivation of ψ(t) as result-ing from Eq.(18) we are forced to accept the relation of

7

Eq.(21). In the region 1 ≤ z ≤ 1.5, as shown in Fig. 1,the time regime of validity of the inverse power law ful-filling Eq.(21) is confined to regions of larger and largertimes as z → 1, since the size of the early time regionwith exponential-like behavior becomes more and moreextended.

We notice that the Tsallis information approach af-fords a formal justification for the birth of an inversepower law. However, the special value of Eq.(21) is ap-parently established in such a way as to make the entropicargument compatible with the dynamic argument, withad hoc arguments, rather than as a result of a theoreti-cal prediction. The role of the entropy of Eq.(38) wouldbecome much more important if it were possible to useit with no recourse to ad hoc arguments. In Section IVB we shall show that perhaps it is possible to establish adynamic approach to this non-extensive thermodynam-ics, and consequently to a q > 1 established with no ad

hoc assumption, with the caution, however, of interpret-ing this thermodynamics as temporary. We shall discussalso in which sense the entropic index given by Eq.(21)as to be thought of as “magic”.

B. A transient form of Kolmogorov-Sinai entropy

The results found years ago by Gaspard and Wang[21] that the ordinary Lyapunov coefficient is finite inthe range 1 ≤ z ≤ 2 seem to rule out the possibility ofapplying to the present case the prediction of Eq.(1). Infact, since hKS is finite, we can use Eq.(1), but this wouldlead to an exponential ψ(t) in deep contrast with the factthat is an inverse power law instead. The breakdown ofthis important theoretical prediction is certainly due tothe fact that the demonstration made by Zaslavsky [3] isincompatible with the existence of sporadic randomness.

Here we plan to discuss whether this connection be-tween ψ(t) and the KS entropy can be extended providedthat a form of non-extensive KS entropy is used to exam-ine the regions of times shorter than the mean waitingtime Tav. To discuss this important possibility we mustreview the recent theoretical result of Ref. [24]. This pa-per provides arguments for the non-extensive generaliza-tion of the Pesin theorem [4]. As shown in Ref. [22], theproper non-extensive generalization of the Kolmogorov-Sinai entropy should rest on the calculation of

Hq(N) ≡1 −

∑

ω0...ωN−1p(ω0...ωN−1)

q

q − 1, (41)

where p(ω0...ωN−1) is the probability of finding the cylin-der corresponding to the sequence of symbols ω0...ωN−1

[2]. In the case where we set q = 1 the entropy of Eq.(41)becomes the ordinary Kolmogorov-Sinai (KS) entropy.This entropy expression affords a rigorous way of definin-ing the earlier introduced concept of true entropic index

Q. If it exists, Q is the value of the entropic index qmaking the entropy of Eq.(41) increase linearly in time.

To point out the the importance of the work of Ref.[24], we have to mention the difficulties with the numer-ical calculation of Ref. [22]. These authors tried to eval-uate the dependence on N of the entropy of Eq.(41) toestablish Q in the case of a text of only two symbols, withstrong correlations. They have been forced to maintaintheir numerical study within the range of windows of sizeN = 10. The case of many more symbols would be be-yond the range of the current generation of computers.However, when the sequence of symbols is generated bydynamics, as in the case here under study, and the func-tion ξ(t, x) of Eq. (2) is available (for convenience, wekeep now explicit the dependence on the initial conditionx), it is possible to adopt the prescription of Ref. [24],which writes Hq(t) of Eq.(41) as

Hq(t) ≡1 − δq−1

∫

dxp(x)qξ(t, x)1−q

q − 1, (42)

where the symbol t denotes time regarded as a continuousvariable. In fact, when the condition N >> 1 applies, itis legitimate to identify N with t. The function p(x)denotes the equilibrium distribution density and δ thesize of the partition cells: According to Ref. [24] the phasespace, a one-dimensional interval, has been divided intoM = 1/δ cells of equal size, with M >> 1.

If the property of Eq.(30) held true with no dependenceon the initial condition, namely in the form

ξ(t) = [1 − (z − 1)λzt]−z/(z−1), (43)

one would reach the conclusion that the entropy ofEq.(42) increase linearly in time if we adopt the entropicindex of Eq.(21). At the same time the theoretical pro-posal of Eq.(6) would turn out to be correct. This is thereason why we call this special value of q ‘magic” and wedenote it with the capital letter Q thus writing

Q = 1 + (z − 1)/z. (44)

What about the case under study, where the local Lya-punov coefficient depends on x? A possible way outwould be given by the following procedure. We assign tothe entropic index q of Eq.(42) the value prescribed byEq.(44). This has the nice effect of making the entropyHq(t) linear dependent on time. Unfortunately the distri-bution function p(x) raised to the power of 1+(z−1)/z,results in a divergent contribution as it can be easilychecked noticing [21] that p(x) ∝ 1/xz−1. We noticethat it would have been very appealing to establish the“thermodynamic” foundation of ψ(t) on the relation

ψ(t) =< (ξ(−t))1−Q >1/(1−Q)Q , (45)

with the mean value < A(x) >Q defined by

8

< A(x) >Q≡

∫

dxp(x)QA(x). (46)

This possibility is ruled out by the divergency stemmingfrom p(x)Q.

We have to point out another important property sug-gested by Eq.(42). We see that this formula shows thatthe only possible way of making Hq(t) independent ofthe cell size δ is to set q = 1. In Ref. [28] it has beenpointed out that in the multifractal case it is convenientto carry out the average over the distribution of powerindices rather than on p(x). This has the nice effect ofmaking Hq(t) independent of the cell size. In the spe-cific case where the invariant distribution exists and isnot multifractal, as in the case here under study, we arecompelled to use q = 1. This is line with the fact thatwhen the invariant distribution of the Manneville mapexists, namely if z < 2, the average Lyapunov coefficientis finite, the Pesin theorem work and the ordinary KSentropy can be defined [21].

Therefore we are left with the possibility of applyingthe prescription of Eq.(42), with the associated averageof Eq.(46), to the case where p(x) is not the invariantmeasure, but it is a given out of equilibrium distribution.This would correspond to giving up the assumption thatthe Tsallis non-extensive thermodynamics has an equi-librium significance, in line with the numerical results ofRef. [29].

On the other hand, it is evident, on the basis of theconclusions of section III, that the function ξ(t, x) doesnot keep its power-law nature forever, and that in thetime scale t >> Tav, with Tav given by Eq.(19), it is ex-pected to recover an exponential form. Thus, are back tothe point that this non-extensive form of thermodynam-ics must be temporary.

We would be tempted to conclude this section withthe following answer to question (iii) of Section I. TheMaxwell’s Demon thermodynamics is the non-extensiveTsallis thermodynamics, and this is a form of out ofequilibrium thermodynamics, implying that also theMaxwell’s Demon effect is temporary.

This conclusion, to a first sight, seems to be reasonable.However, it yields a conflict with the result of the earlierwork of Ref. [22]. The authors of Ref. [22] studied the en-tropy of Eq.(41) with a symbolic method, and, as earliersaid, they have been forced to use short-time windows.The adoption of short-time windows is imposed by thefact they did not make any direct recourse to dynamics asin the calculation yielding Eq.(43). The case studied inRef. [22] refers to a dynamic system with the same com-plexity as the Manneville map here under study. Proba-bly the main difference with the analysis here illustratedis that the method adopted by the authors of Ref. [22]is equivalent to studying the Geisel map rather than theManneville map, with a repartition of the space into onlytwo cells. The conflict between the analysis here made

and the result of the work of Ref. [22] is probably due tothe fact that here we adopt a much finer repartition ofthe phase space.

This kind of disagreement between two different theo-retical predictions about the entropic index of the samedynamic process is not yet a reason for the rebuttal of thenon-extensive thermodynamics as the explanation of theMaxwell’s Demon effect. In Section V we shall see thatthe Maxwell’s Demon effect seems to be caused by theemergence in chamber A, made more densely populatedby the Demon, of a diffusion equation with a “diffusioncoefficient” weaker than that of the diffusion equationemerging in chamber B. This dynamic effect is explainedusing the dynamic derivation of Levy statistics, which im-plies memory erasure and consequently emerge in a timescale where the temporary non-extensive thermodynamicregime, if it ever exists, is over.

V. LEVY PROCESSES AND MEMORY ERASURE

Lebowitz [1] states that the Universe time evolution ischaracterized by the transition from a condition of lowerto a condition of larger entropy without implying a de-parture from microscopic reversibility. Let us focus ourattention on the diffusion process:

dx

dt= v(t). (47)

Let us assume that the stochastic velocity v(t) fluctuatesamong the two values W and −W , and that the distri-bution of times of sojourn in one of these two velocitystates, the same for the two states, is given by Eq.(18).It is shown [30] that the distribution density p(x, t) obeysthe following equation of motion

∂

∂tp(x, t) =< v2 >

∫ t

0

Φv(t′)∂2

∂x2p(x, t− t′)dt′, (48)

where < v2 > is the equilibrium mean quadratic value ofv, and Φv(t) is the normalized correlation function of thevariable v. For details, and especially to understand whythis equation in the long-time limit yields the process ofLevy diffusion, the interested reader should consult Ref.[30].

Here we limit ourselves to noticing a few aspects. Firstof all, the distribution density p(x, t) can be interpretedas a probability concerning a very large number of par-ticles imbedded in the same space phase, for instance,the phase space of the billiards of Zaslavsky [3]. Thus,in a sense, the condition N → ∞ is fulfilled. This hasto do with question (i) of Section I. If condition N → ∞is essential for the birth of statistical mechanics, what isthe role of mixing, and that of ensuing ergodicity?

We try to answer this question by making a second,very important, remark. This is that Eq.(48) is exactunder the following conditions:

9

(a) The variable v(t) is dichotomus, with only two pos-sible values, either W or −W .

(b) All the particles are initially located in the sameposition, x = 0.

We note that condition (a) is that mentioned in Sec-tion IV A to make easier for the reader to understandthe meaning of the key relation of Eq.(37), leading tothe entropic derivation of the waiting time distributionψ(t). Both conditions are relevant for our main purposeof comparing the point of view of Zaslavsly [3] to thatof Lebowitz [1]. The latter property certainly realizesan initial condition whose probability is extremely low,thereby making impossible the regression to the initialcondition. In the ideal case of a time inversion, this wouldbe possible, in the irrealistic case of no perturbation ofany kind, ranging from the round-off errors to the en-vironmental fluctuations. This is again in line with theobservations of Lebowitz on the birth of statistical me-chanics.

The former property, however, serves in our opin-ion a purpose which apparently conflicts with Lebowitz,namely, that of stressing the importance of mixing, asa key property leading to an invariant distribution. Forthe reader to appreciate this aspect it is necessary thathe/she goes through the derivation of Eq.(48) [19]. Theprojection method adopted in Ref. [19] to derive Eq.(48)rests on a projection operator, which, in turns, impliesthe existence of an invariant distribution for the variablev. In the specific case of deterministic chaos with mixing,the prototype of which is given in fact by the Bernouillishift map (as we have shown, the Manneville map withz = 1), an invariant distribution exists and is quicklyreached from any off-equilibrium condition.

The waiting function ψ(t) is proportional to the sec-ond derivative of the correlation function Φv(t). Thismeans that the departure from the exponential conditionof Eq.(1) has dramatic effects on the process of transitionto statistical mechanics, as birth of a diffusion process. Infact, in the region 1.5 ≤ z < 2 it is still possible to fulfillthe condition necessary for the existence of the invariantdistribution according to the Kac theorem [9]. However,in this specific case the correlation function Φv satisfiesthe asymptotic property

limt→∞Φv(t) = const/tβ , (49)

with

β =z

z − 1− 2. (50)

This means that in the interval 1.5 ≤ z < 2 the indexβ runs between β = 1, at the border with the basin ofGaussian attraction, and β = 0, corresponding to thecritical condition of no relaxation of the velocity correla-tion function.

In conclusion the condition of strong chaos, withstrongly mixing properties, has the effect of producing

ordinary Brownian diffusion. It is not clear to us whatwould be the consequence of a dynamic condition with noform of mixing. We think that in principle it is possibleto produce a fast decay of the correlation function Φv(t)as a result of a mere superposition of infinitely manynormal modes (again in action the condition N → ∞).However, we are not aware of any treatment proving thata natural equilibrium distribution is reached also in thatcase, if no arbitrary statistical assumptions are made.

We have the impression that in the case of ordinarystatistics it might be difficult to support the dynamicview against the (N → ∞)-perspective, or the latteragainst the former, without using very subtle argumentssuch as the request of neither explicit nor implicit statis-tical assumptions, which, in turn, quite probably wouldbe questioned. For all practical purposes, the two viewsare expected to provide equivalent results. For instance,in the dynamic perspective adopted by Zaslavsly thePoincare recurrences are frequent and those with veryshort time duration are more probable than those of verylarge time duration (see Eq.(1)). This seems to conflictwith Boltzmann’s idea that the Poincare recurrences ofsystems with very large number of freedoms become ex-ceedingly large, so as to exceed the possibility of anydirect observation. However, if we make the assump-tion that the experimental observation is made on a largenumber of non-interacting particles, all of them moving inone of the billiards studied by Zaslavsky [3,8,7], we reacha different conclusion. The phase space under study isnot more that of the two-dimensional billiard: It becomesa 2N-dimension phase space. The distribution densitydriven by Eq.(48) becomes a 2N -dimensional trajectory,whose return to the initial condition takes place, in ac-cordance with Boltzmann’s view, virtually after an in-finitely long time. In Zaslavsky’s picture the process cor-responding to the 2N -dimension trajectory leaving theinitial condition and returning to it after an extremelylong time would correspond to the following scenario.Let us imagine, for instance, that the initial distributiondensity is different from zero only in a small fraction ofthe whole 2-dimension phase space, where it is assumedto be constant. As a result of mixing, this initial dis-tribution, throughout its time evolution, would undergoa fragmentation process that, within a kind of coarse-grained perspective, would correspond to the volume ofthis initial condition growing till to becoming identical tothe volume of the whole phase space. The Poincare re-currences of the Boltzmann-Lebowitz perspective wouldcorrespond to this dilated distribution shrinking to theinitial condition, a process that is beyond the range of theexperimental observation as the return of the Boltzmann-Lebowitz trajectory.

We think that the relevance of dynamics for statisti-cal mechanics cannot be easily ruled out in the case ofprocesses with long-time memory as those here under dis-cussion. Quite on the contrary, here we show that the dy-

10

namic approach to Levy diffusion in the form discussed inRef. [30] can account for the Maxwell’s Demon effect. Inthis moment, it is not clear to us how this interesting ef-fect might be explained using the (N → ∞)-perspective.The authors of Ref. [30] prove that in a time scale muchlarger than Tav Eq.(48) as an effect of the memory erasureprovoked by sporadic randomness becomes equivalent toan equation of motion that in the Fourier representationreads

∂

∂tp(k, t) = −b|k|β+1p(k, t), (51)

with

b ≡ Γ(−β)W β+1 T βav β

(β+1) cos(β

2π). (52)

Note that p(k, t) is the Fourier transform of p(x, t) andthat this is the Fourier representation of the well knownprocess of Levy diffusion. The element of interest of theresult of the dynamic approach of Ref. [30] is that thecoefficient b of Eq.(52) plays the role of a diffusion coeffi-cient. The reader can easily realize that the intensity ofthis coefficient can be different in the two chambers of theexperiment on the Maxwell’s Demon effect [3,8,7] even ifthe velocity W is the same. The possibility of realizingtwo different values for b using the same kinetic energybecomes still wider if we refer to a generator of fluctu-ations with an inverse power law distribution of waitingtimes more general than the Manneville map [23]. It isevident that two different diffusion equations in the twochambers can produce a breakdown of the condition ofequal population. The trajectory, in fact, will tend tospend more time in the regions with a smaller “diffusioncoefficient”.

In conclusion, according to this heuristic interpreta-tion, the Maxwell’s Demon Effect is permanent and itcan show up after the long time process of transition fromdynamics to diffusion. This rules out the non-extensivethermodynamics of Tsallis as a possible theoretical inter-pretation of this effect. In fact, in Section IV B we haveseen that if the non-extensive approach applies, it doesas a form of temporary thermodynamics.

We can also rule out the non-extensive nature of en-tropy as the basic property behind the Maxwell’s DemonEffect. Let us see why. We have two chambers, A and B,separated by a wall, with a little hole. A can be a Sinaibilliard and B a Cassini billiard [7]. The physical entropyof this regime, S, expressed in terms of the conventionalGibbs prescription, might not be extensive. The entropicindicator advocated by Tsallis [12] is not additive, and itviolates the property

S(A+B) = S(A) + S(B) (53)

even when the two subsystems A and B are totally un-correlated the one from the other. In the case of the

dynamic approach to Levy diffusion the breakdown ofthe additivity condition might be dictated by memoryeffects. To explain why it is so with intuitive arguments,we can refer ourselves to the dynamical experiment itselfused by Zaslavsly and Edelman [7] to reveal the existenceof Maxwell’s Demon effect. A particle that moves from Ainto B might still have memory of its initial condition inA, if sporadic randomness did not act for a sufficiently ex-tended time. This means that the transition process thatin the long-time scale will produce Levy statistics is notextensive in nature. However, we expect that when Levystatistics are finally established, the usual additive condi-tion is restored, as a consequence of the memory erasureprocess. Using the earlier arguments we can imagine thepossibility that the two chambers might correspond totwo different difusion equations, even if the velocity in-tensity of the particle does not change with moving fromthe one to the other chamber, thereby resulting in theMaxwell’s Demon effect after the process of memory era-sure.

VI. CONCLUDING REMARKS

Let us summarize our conclusion about the questionsraised in Section I.

Question (i). Our arguments concerning question (i)are not compelling and rigorous. However, we hope thatthey might make plausible the following statement: Bothdeterministic chaos, whose role is well understood onlyin the case of low-dimension systems, and the conditionN → ∞ give rise to the birth of statistical mechanics.We are convinced that the joint use of these two per-spectives will contribute a deeper understanding of thedynamic origin of statistical mechanics. It is interestingto notice that the exponential of Eq.(1), implying a fre-quent return to the initial condition, does not conflictwith the irreversible character of thermodynamics andstatistical mechanics if we perceive the Gibbs ensemblemethod as being equivalent to the (N → ∞)-perspective.

Question(ii). We found the proposal of Eq.(7), validat the exit time of a trajectory from the laminar region,to be the correct connection between the sensitivity toinitial conditions and the distribution of waiting timesin the laminr region. This means that in the case of thedynamical systems studied by Zaslavsky, the distributionof the Poincare recurrence times can be related to the KSentropy only in the absence of intermittency. When anintermittent process is present, the long-time form of thedistribution PR(t) changes from the exponential form ofEq.(1) to an inverse power law form. In the region 1.5 ≤z ≤ 2, corresponding to the emergence of Levy processes,the KS entropy is still finite and the adoption of Eq.(1)would conflict with the observed inverse power law natureof this distribution. The non-extensive thermodynamicsof Tsallis [12] might help, provided that it is viewed as

11

temporary and provided that the conflict between theprediction about whether Q > 1 or Q < 1 is settled.

Question (iii). The dynamic approach to Levy diffu-sion of Ref. [30] proves that the Maxwell’s Demon effect ispermanent and it takes place in a long-time scale. Thismeans that it is not necessary to use memory proper-ties to explain it, and consequently, not even the non-extensive nature of entropy.

Question (iv). The condition of ordinary statisticalmechanics does not imply an abrupt transition when weset z = 1. However, for even infinitesimally small de-viations from z = 1 in the long time limit the waitingtime distribution ψ(t) appears to be characterized by aninverse power law form. This does not have strong con-sequences, though, because in the region z < 1.5 thesecond moment of the waiting time distribution is finite,and consequently the system is attracted by the Gaussbasin of attraction. Nevertheless, the waiting time dis-tribution in the region where z is very close to 1, is anexponential at short time and an inverse power law atlong time, a property which makes it incompatible withthe structure generated by the method of entropy maxi-mization resting on the non-extensive indicator of Tsallis[12].

It seems to us that the long-standing problem of athermodynamic approach to Levy processes [31] is notyet solved, and that the Maxwell’s Demon effect of Refs.[8,3,7] can be regarded as a paradigmatic case challeng-ing the Boltzmann perspective advocated by Lebowitz[1]. We plan to do further research work along these di-rections.

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[23] We note that the work of Ref. [22] is based on ran-domly drawing a number x in the interval [0, 1], withthe Bernouilli map. With a nonlinear transformation thisnumber is changed into a number y with an inversepower law distribution. Then they toss a coin and decidewhether to adopt the symbol + or the symbol −. Theyconsider the integer part of y, [y], and fill [y] sites of aninfinite seuence with either the symbol + or the symbol−, according to the indication of the tossed coin. Thework of Ref. [?] is devoted to the study of the complex-ity of the resulting sequence. The process of coin tossingand number drawing is carried out at regular time inter-vals. This is equivalent to using a time N , correspondingto the number of times the process of coin tossing andnumber drawing has been done. The entropy increase isS(N) = 2Nlog2. In fact, log2 is the uncertainty relatedto both coin tossing and to a time step of the Bernouillimap. If the sequences of +′s and −

′ is fixed without toss-ing a coin and alternating the +′s with −

′s, then the en-tropy increase is S(N) = Nlog2. The complexity of theManneville map depends on the connection between the“physical” time Y (N) = [y]1+[y]2 . . . [y]N and the “inter-nal” time N . The connection between these two times isthe essential property determining the Kolmogorov com-plexity of both kinds of maps, the difference bing only afactor of 2. The essential aspect of the complexity of theManneville map studied by Gaspard and Wang [21] is thetransition from a non vanishing KS entropy with z > 2to a vanishing KS entropy with z < 2. For the meaningof z see Eq.(9).

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13