Amelia Carolina Sparavigna

Lidia Dastrù

The Pantheon, eye of Rome, and its glimpse of the sky

Keywords: Archaeoastronomy, Zenith Passage of the Stars, Stellarium

The Pantheon, eye of Rome, and its glimpse of the sky Amelia Carolina Sparavigna, Lidia Dastrù

The Pantheon is an ancient temple in Rome. The present building, used today as a church, was built on the site of an earlier temple commissioned by Marcus Agrippa during the reign of Augustus. The Pantheon of Agrippa was destroyed in a huge fre in the year 80 AD. Domitan rebuilt a temple, which was burnt again in 110 AD. The new Pantheon was completed by the Emperor Hadrian and probably dedicated about 126 AD [START_REF] Terenzio | Pantheon. Enciclopedia Italiana[END_REF]. The design is circular, having a portco with columns of Corinthian granite. The Pantheon dome is made of concrete [START_REF] Sparavigna | Some Notes on Ancient Concrete[END_REF]. At the top of the dome, we fnd an Oculus, a round opening which is the main source of natural light inside the temple (Figs. 1 and2). The Oculus has a diameter of 9 m, whereas the building is 43 m tall [START_REF] Terenzio | Pantheon. Enciclopedia Italiana[END_REF]. The Pantheon is the subject of several works discussing the link between its architecture and astronomy [START_REF] Masi | The Pantheon as an Astronomical Instrument[END_REF][START_REF] Belardi | Il Pantheon: storia, tecnica e restauro[END_REF][START_REF] Lim | The Formal Analysis of Pantheon in Rome in Relaton to the Solar Angles[END_REF][START_REF] De Fine Licht | The Rotunda in Rome: a study of Hadrian's Pantheon[END_REF][START_REF] Joost-Gaugier | The iconography of sacred space: A suggested reading of the meaning of the Roman pantheon[END_REF][START_REF] Macdonald | The Pantheon: design, meaning, and progeny[END_REF][START_REF] Meeks | Pantheon paradigm[END_REF][START_REF] Nicoleta | The Urban Set of the Pantheon and the Mausoleum of August in Rome, between Architectural and Astronomical Symbolism[END_REF][START_REF] Jones | Who Built the Pantheon? Agrippa, Apollodorus, in Hadrian: Art, Politcs and Economy[END_REF][START_REF] Swetnam-Burland | Pantheon, Rome. The Encyclopedia of Ancient History[END_REF][START_REF] Wilkins | The Pantheon as a globe-shaped concepton[END_REF][START_REF] Sperling | Das Pantheon in Rom, Abbild und Mass des Kosmos[END_REF][START_REF] Alibert | Geometrical Analysis of the Cofers of the Pantheon's Dome in Rome[END_REF][START_REF] Sperling | Early Orchestraton: The Pantheon as a resonance element[END_REF][START_REF] Sperling | The "Quadrivium" in the Pantheon of Rome[END_REF][START_REF] Martnes | The relatonship between Architecture and Mathematcs in the Pantheon[END_REF][START_REF] Hannah | The Pantheon as a Timekeeper[END_REF][START_REF] Hannah | The role of the sun in the Pantheon's design and meaning[END_REF][START_REF] Pantheon | svelato il segreto della cupola[END_REF].

In a short pamphlet [START_REF] Masi | The Pantheon as an Astronomical Instrument[END_REF], Fausto Masi proposed the Pantheon as an astronomical instrument, to determine the alttude of the sun. The pamphlet is containing sketches showing the positon of the shaf of sunlight passing through the roof at the solstces and the equinoxes (the Figure 3 shows a drawing inspired by a sketch in [START_REF] Masi | The Pantheon as an Astronomical Instrument[END_REF], concerning equinoxes). Among the given references, other artcles are concerning the sun and the light entering the temple. In partcular, in [START_REF] Belardi | Il Pantheon: storia, tecnica e restauro[END_REF] we fnd mentoned the role on the sun on April 21, the Birthday of Rome. In [START_REF] Belardi | Il Pantheon: storia, tecnica e restauro[END_REF], we can read that "Un ulteriore riferimento va fato alle condizioni che si verifcano nel Pantheon il giorno 21 aprile, Natale di Roma. Infat alle 12.00 (ora solare) all'interno del monumento, la penetrazione del sole crea una sensazione eccezionale. I raggi del sole abbagliano il visitatore all'ingresso, il quale perde la percezione materica sovrastante e riceve una immagine davvero suggestva". That is, we have to consider the conditons that occur in the Pantheon on 21 April, Birthday of Rome. In fact, at noon, inside the monument, the rays of the sun create an exceptonal sensaton. The rays dazzle the visitor at the entrance, who loses the percepton of the overlying structure of the building. This leads us to argue -Ref. 4 tells -that, according to the period and the complexity of the constructon, in choosing a North orientaton of the entrance of the building, the architects could have considered astronomical references and positoned the axis of the door and other architectural elements accordingly [START_REF] Belardi | Il Pantheon: storia, tecnica e restauro[END_REF].

Let us note that the ancient Romans were sure that April 21, the day of the festval sacred to Pales, goddess of shepherds, was also the day of the foundaton of Rome. Therefore, any link of this day to architectures comes as no surprise. The role of the sun in the Pantheon on April 21 is also mentoned in [START_REF] Hannah | The role of the sun in the Pantheon's design and meaning[END_REF]. The authors are proposing that the symbolic acton of the sun on that day was that of putng "Rome among the Gods". "If we suppose, as seems likely, that the emperor was celebratng this precise day there, then his entrance "together with the sun" would have been a symbolic link between the people and the Gods" [START_REF] Hannah | The role of the sun in the Pantheon's design and meaning[END_REF].

The Pantheon that we see today was built at the tme of Hadrian, but it seems reasonable to imagine that some rituals existed -and related hierophanies linking the Pantheon to the sun -even at the tme of Augustus. As told in [START_REF] Pantheon | svelato il segreto della cupola[END_REF], some recent discoveries tell that, in the area in front of the Pantheon, there are the original stairs of the previous Augustan temple. According to Eugenio La Rocca, Accademia Nazionale dei Lincei, these fndings testfy that the Pantheon rebuilt by Hadrian, that is the temple that we see today, has preserved the original North orientaton of the temple of Augustus. Also the monumental bronze door is that of the Augustan building. La Rocca contnues explaining that there are all the reasons to suppose that any phenomenon concerning the rays of the sun existed also in the Pantheon of Augustus, a temple that had a similar conformaton, not so complex but with a similar façade. "Why this mise-en-scène?" -Reference 21 is asking. "It is the politcal program of Augustus -highlights La Rocca -to restore his appearance as the new founder of the city, in the sign of peace", in an authentc "solar theater" [START_REF] Pantheon | svelato il segreto della cupola[END_REF].

As we can see from the An observer in the temple was able to see the zenith passage of these stars during the year, and easily measure the tme, during the night. From the Figures 6 and7, we can see Vega, Deneb, Mirphak and Capella passing close to the zenith, in the frame of the pupil of the temple (due to the precession of the Earth's axis, today Capella is not a zenith star, Figure 8). Therefore, inside this temple, which could be imagined as a huge eye, the gods had the possibility to guard the moton of the world of which the city of Rome was the head, that is, the Caput Mundi. Like a human eye, the Pantheon has a feld of view (Fig. 9) which is larger than that given in the Figure 5, and then other stars and related asterisms were and are visible (in a previous discussion [START_REF] Sparavigna | The Zenith seen through the Oculus of the Pantheon[END_REF], we have considered only the Zenith). Let us remember that Capella is Alpha Aurigae, the brightest star in the constellaton of Auriga. It is the sixth-brightest star in the night sky, and the third-brightest in the northern celestal hemisphere afer Arcturus and Vega. Beside the constellaton of the Auriga, in the frame of the Oculus of the Pantheon, other constellatons were passing. We had Perseus, of which the brightest star is Mirphak. Also the mirror of the vain queen Cassiopeia was passing at the center of the Oculus (Fig. 10). Then we have the Cygnus, the northern constellaton lying on the plane of the Milky Way. Deneb is the bright star in the tail of the constellaton (Fig. 10). Let us consider again Figures 1 and5, which are showing the black appearance of the Oculus. It is like the pupil located in the center of the iris of an eye, which appears black because the light entering the pupil is either absorbed by the tssues inside the eye or absorbed afer some difusion within the eye, so that it mostly misses exitng the narrow pupil. Therefore, as we imagine the Pantheon seen from above, to the gods in the heavens the building was like an eye at the center of the Campus Martus in Rome. Let us note that it was from this place that Romulus ascended to heaven, and that it "could not have been coincidental that this very area was chosen by Agrippa as the site of the Pantheon, which functoned as a dynastc temple honoring the celestal gods and Augustus' deifed father, Julius Caesar" [START_REF] Pollini | From Republic to Empire: Rhetoric, Religion, and Power in the Visual Culture of Ancient Rome[END_REF].

Let us conclude our discussion on the Pantheon of Rome with the following suggeston. In our opinion, the possibility exists that the human eye was the model for the architecture of the Pantheon. Actually, examples of parts of the human body used as models for architectures exist [START_REF] Kausel | Evocaton Of The Head And Brain In Old Vaulted Buildings[END_REF], and among them we fnd the well-known case of the colonnades designed by Gian Lorenzo Bernini for St Peter's Square. As told by Bernini himself, the colonnades were designed to surround people as "the maternal arms of Mother Church", in his Baroque theatricality vision of the architecture [START_REF] Sparavigna | Light and Shadows in Bernini's Oval of Saint Peter's Square[END_REF]. Therefore, also the architect of the ancient Rome that planned the Pantheon could have had the same symbolic vision of architecture. We can imagine that the he planned the building like a human eye, not only inspired by the functon of the light in it, but also by its anatomy. In this manner he created a temple linking Rome to the heavens, in the form of an eye that could observe the moton of the universe.

Figure 1 -

 1 Figure 1 -The cofered concrete dome of the Pantheon in Rome, with a central opening (Oculus) to the sky, as seen from satellite (courtesy: Google Maps).

Figure 2 -

 2 Figure 2 -The Oculus in the dome (courtesy Wikipedia, User Livioandronico2013).

Figure 3 -

 3 Figure 3 -An example of drawings that we can see in Ref.3.

Figure 4 -

 4 Figure 4 -The entrance of the building is not perfectly aligned along the cardinal directon (courtesy: Bing Maps).

Figure 4 ,

 4 the entrance of the Pantheon is not perfectly aligned along the cardinal directon. There is a diference of about three degrees. Probably, this is due to the fact that the entrance of the Pantheon of Hadrian was oriented towards the Mausoleum of Augustus[START_REF] Hannah | The role of the sun in the Pantheon's design and meaning[END_REF]. Let us add the possibility that, any tme an emperor entered the Pantheon on April 21, he was imagined accompanied by the spirit of Augustus coming from his mausoleum to celebrate the foundaton of Rome again. Besides the sun and related rituals, let us also consider the Oculus and the night sky. The temple, like a huge human eye, had its pupil which was observing the stars passing at the Zenith of Rome. Let us remember that the sun and the moon, at the lattude of Rome, cannot reach the zenith positon. If we are precisely at the center of the foor of the Pantheon, through the Oculus we can see an angle 10 degrees wide, as given in the Figure5. Therefore, let us investgate by means of the planetarium sofware Stellarium what were the stars observed by the Roman emperors.

Figure 5 -

 5 Figure 5 -The Pantheon is 43 m tall. The diameter of the dome is of 43 meters. The Oculus has a diameter of 9 m. If we are at the center of the foor, we can see an angle 10 degrees wide.

Figure 6 -

 6 Figure 6 -Here we can see, simulated by Stellarium, some stars passing close to the Zenith of Rome. On the lef, 20 May, 126 AD (Vega); on the right, the same night, afer two hours (Deneb).

Figure 7 -

 7 Figure 7 -On the lef, 20 September, 126 AD (Mirphak); on the right, the same night, afer one hour and 35 minutes (Capella).

Figure 8 -

 8 Figure 8 -Today, Capella has not a zenith passage.

Figure 9 -Fig 10 -

 910 Figure 9 -The feld of view is larger than the angle of Fig.5.


N Garnier

S Ciliberto

Nonequilibrium fluctuations in a resistor

Keywords: numbers: 05.40.-a, 05.70.-a, 07.50.-e, 84.30.Bv

In small systems where relevant energies are comparable to thermal agitation, fluctuations are of the order of average values. In systems in thermodynamical equilibrium, the variance of these fluctuations can be related to the dissipation constant in the system, exploiting the Fluctuation-Dissipation Theorem (FDT). In non-equilibrium steady systems, Fluctuations Theorems (FT) additionally describe symmetry properties of the probability density functions (PDFs) of the fluctuations of injected and dissipated energies. We experimentally probe a model system: an electrical dipole driven out of equilibrium by a small constant current I, and show that FT are experimentally accessible and valid. Furthermore, we stress that FT can be used to measure the dissipated power P = RI 2 in the system by just studying the PDFs symmetries.

Introduction

In the last decade, Fluctuation Theorems (FT) [START_REF] Evans | [END_REF]2] appeared in nonequilibrium statistical physics. These new theorems relate the asymmetry of fluctuations of energies (or powers) with the dissipated power required to maintain a nonequilibrium steady state of a system. Thus FT give a measure of the distance from equilibrium. Indeed, FT can be extremely useful to probe nonequilibrium states in nanophysics and biophysics where energies involved are typicaly small and thermal agitation cannot be neglected: in those systems, the variance of fluctuations is of the order of the average values. Moreover, standard Fluctuation-Dissipation Theorem (FDT) [3,4,5] is derived for equilibrium systems only and so it may fail to describe states far from equilibrium. FT are a generalization of FDT out of -possibly far from -equilibrium. In order to safely apply FT in complex systemse.g. in biophysics or nanotechnologies -it is important to test them in all simple cases where theoretical predictions can be accurately checked. In spite of the large theoretical effort, just a few experiments have been conducted on this topic [6,7]. For this reason, we test in this letter their use on a simple electrical system. Electrical systems are particuliarly interesting, because they are one of the first where FTD was formalized: Johnson [8] and Nyquist [9] related equilibrium fluctuations of voltage U across a dipole with the resistive part of this dipole. Moreover, all parameters in the setup can easily be controled. Thus, out of equilibrium, electrical circuits are a key example to stress the differences between FT and FDT, and to verify the validity of FT.

Our system is an electrical dipole constituted of a resistance R in parallel with a capacitor C (Fig. 1). We drive it out of equilibrium by making a constant current I flow in it. Noting k B the Boltzmann constant and T the absolute temperature, the injected power is typically of some k B T per second, of the same order as in biophysics or nanoscale physics experiments. This represent the fun- damental case of a system in contact with two different (electrons-) reservoirs, one of the simplest and most fundamental problems of nonequilibrium physics [10]. Using a powerful analogy with a forced Langevin equation [11], a precise formulation of FT was recently given [12] in this case. Nyquist FDT as well as FT are checked experimentally, by looking at the injected power and the dissipated heat in the system.

Experimental setup

The circuit we use is composed of a resistor in parallel with a capacitor, as depicted on Fig. 1. The resistance is a standard metallic one of nominal value R = 9.52 MΩ. In parallel, we have an equivalent capacitor of value C = 280 pF. This accounts for the capacitance of the all set of coaxial connectors and cables that we used. The time constant of the circuit is τ 0 ≡ RC = 2.67ms. Using a 50 GΩ resistance, we inject in the circuit a constant current I ranging from 0 to 6 × 10 -13 A. This current corresponds to an injected power I 2 R ranging from 0 to 1000k B T /s. Experiments are conducted at room temperature T = 300 K. The typical values of the injected energy for, e.g., I = 1.4×10 -13 A and τ = 10τ 0 are of order of a few hundreds of k B T , which is small enough to ensure that the resistance is not heating; expected changes of temperature are estimated to be less than 10 -14 K over a one hour experiment. Moreover, the resistance is thermostated: all the heat dissipated by Joule effect is absorbed by the thermal bath.

The fluctuating voltage U across the dipole is measured with a resolution of 10 -11 V sampled at 819.2 Hz. This is achieved by first amplifying the signal by 10 4 , using a FET amplifier, with a 4 GΩ input impedance, a voltage noise level of 5 nV/ √ Hz and a current noise of 10 -15 A/ √ Hz. The signal is then digitized with a 24-bits data acquisition card at frequency 8192 Hz, and decimated by averaging ten consecutive points.

Fluctuation-Dissipation theorem

The electrical dipole of Fig. 1 is a pure resistance R in parallel with a capacitor C; its complex impedance reads Z(f ) = 1/(1/R + i2πRCf ) where i 2 = -1 and f is the frequency. The effective dissipative part is the real part ℜ(Z) of Z. It was experimentally observed by Johnson [8], and then demonstrated by Nyquist [9] that the potential difference U across the dipole fluctuates with a stationary power spectral density S(f )df such that

S(f )df = 4k B T ℜ(Z)df , (1)
At equilibrium I = 0, in average no current is flowing in the circuit, and mean U is zero. Integrating over all positive frequencies, one gets the variance of U :

U 2 = k B T R τ 0 = k B T C . (2)
Eqs. ( 1) and ( 2) are the expressions of the Fluctuation-Dissipation theorem (FDT) for electrical circuits.

The exact value of our capacitance was determined by fitting the power density spectrum of equilibrium fluctuations (at imposed I = 0 A) by a Lorentzian low-pass transfer function (eq.( 1)), as illustrated on Fig. 2b. Application of FDT leads with a very good accuracy to the determination of R, in perfect accordance with the measured nominal value (Fig. 2). When I = 1.4 × 10 -13 A, we found the same power spectral density for U , and performing the same treatments gave the same estimates of R and C. We therefore conclude that FDT is still holding in our system driven out of equilibrium.

Fluctuation theorems

The power injected in the circuit is P in = U I, but only the resistive part dissipates, so the dissipated power is P diss = U i R , where i R is the current flowing in the resistor (Fig. 1). As already noted [13], in average, one expects P in = P diss ≡ P, where the brackets stand for time average over sufficiently long times compared to τ 0 . This is very well checked in our experiment. P in and P diss fluctuate in time because U itself is fluctuating. If one assumes that fluctuations of U have a Gaussian distribution, which is the case at equilibrium when I = 0, then P in has also a Gaussian distribution, because I is constant. On the contrary, i R fluctuates, as we see from Kirchoff's laws:

I = i R + C dU dt , so P in = P diss + 1 2 C dU 2 dt , (3)
and therefore, the probability distribution of P diss is not Gaussian [12]. It is worth noting that for large current I, some orders of magnitude larger than the one we use, P in and P diss will be much larger than the conservative part C 2 dU 2 dt and therefore the probability distributions of both the injected and dissipated power will be practically equal and usually Gaussian, as it is expected in macroscopic systems.

We call g τ (t) = 1 Reasoning with energies instead of powers, we define W τ (t) = τ P in τ (t), the energy injected in the circuit during time τ , analogous to the work performed on the system (positive when received by the system). In the same way, we write Q τ (t) = τ P diss τ (t), the energy dissipated by Joule effect during time τ , analogous to the heat dissipated by the system (positive when given by the system to the thermostat). We used values of τ spanning from fractions of τ 0 up to hundreds of τ 0 .

For a given value of I, we measure U (t) and compute P in and P diss . We then build the probability density functions of cumulated variables W τ and Q τ using 10 6 points; their typical distributions are plotted on Fig. 3. As expected, fluctuations of W τ are Gaussian for any τ whereas heat fluctuations are not for small values of τ . They are exponential for small τ : large fluctuations of heat Q τ are more likely to occur than large fluctuations of work W τ .

Then we study the normalized symmetry function: where E τ stands for either W τ / W τ or Q τ / Q τ . If the Fluctuation Theorem for the work W τ holds, then one should have [11,12], for large enough τ , the relationship lim τ →∞ f W (τ, a) = a. In contrast, if the Fluctuation Theorem for the heat Q τ holds, then for τ → ∞, the asymptotic values of

S E (τ, a) ≡ k B T τ P ln p(E τ = a) p(E τ = -a) , (4)
S Q (τ, a) are S ∞ Q (a) = a for a ≤ 1, S ∞ Q (a) = 2 for a ≥ 3,
and there is a continuous parabolic connection for 1 ≤ a ≤ 3 that has a continuous derivative [11,12]. From histograms of Fig. 3, we compute the symmetry functions S W (τ, a) and S Q (τ, a) (Fig. 4).

Work fluctuations First, for any given τ we checked that the symmetry function S W (τ, a) is linear in a (Fig. 4a). We measured the corresponding proportionality coefficient σ W (τ ) such that S W (τ, a) = σ W (τ )a. This coefficient σ W (τ ) tends to 1 when τ is increased (see Fig. 5).

Heat fluctuations We found that S Q (τ, a) is linear in a only for a < 1, as expected [11,12]. Again, as τ → ∞, the limit slope of the symmetry function is 1 whereas for FIG. 5: (a): dependence on τ of the slope of SW in Fig. 4. Slope is converging from above. (b): distance between SQ(τ ) for finite τ and theoretical prediction of SQ for infinite τ , for several values of a = Q/ Q . for a < 3, convergence is from below, whereas it is from above for a > 3.

a > 3, S Q (τ, a) tends to two.

Asymptotic symmetry functions and convergence In [11,12], expressions for the convergence towards the asymptotic limits S ∞ W (a) and S ∞ Q (a) are given in terms of τ . We can check these predictions with our data. The convergence for the work W τ is very well reproduced by these predictions, as can be seen on Fig. 4a: continuous straight lines are theoretical predictions for small values of τ , using eqs. ( 9) and ( 10) from [12], with no adjustable parameters. On Fig. 5a the slope σ W (τ ) of the experimental symmetry function is plotted. The continuous line is the prediction of [12], which perfectly agrees with our data.

For the heat, we distinguish two regimes for the convergence towards the asymptotic symmetry function. For a = Q τ / Q < 3, we find that when τ is increased, symmetry functions are converging to the asymptotic function from below (Fig. 4), which is the opposite of what is observed for the work. On the contrary, for a > 3, convergence to the asymptotic function is from above, thus enhancing the peculiarity of the point a = 3. On Fig. 5, we have plotted the evolution of S Q (τ, a) versus τ for several fixed values of a. The convergence from above for a ≥ 3 and from below for a < 3 is clear. For increasing values of τ , only smaller and smaller values of a are accessible because of the averaging process. Therefore the accessible values of a are quickly lower than 1, and only the linear part of S Q (τ, a) can be experimentally tested. Nevertheless, for intermediate time scales τ we see in Fig. 4 that the data converge towards the theoretical asymptotic nonlinear symmetry function [11,12] (smooth curve in Fig. 4b).

We observed that the convergence reproduced on Fig. 5 depends on the injected current I, as pointed out in [12]. Other experiments with a larger current ( P = 186k B T /s) give a faster convergence; corresponding results will be reported elsewhere.

Conclusions

We have shown experimentally that the asymmetry of the probability distribution functions of work and heat in a simple electrical circuit driven out of equilibrium by a fixed current I, is linked to the averaged dissipated power in the system. The recently proposed Fluctuation Theorems for first order Langevin systems are then experimentally confirmed. Exploiting formula (4), FT can be used to measure an unknown averaged dissipated power P = lim τ →∞ Qτ τ by using only the symmetries of the fluctuations, i.e. computing S W or S Q and measuring their asymptotic slope.

We operated with energies of order of k B T in order to have strong fluctuations compared to the averaged values. It is worth noting that as the driving current is increased up to macroscopic values, the fluctuations become more and more negligible, therefore Fluctuation Theorems become harder and harder to use, and therefore less relevant.

FIG. 1 :

 1 FIG.1: Model circuit : an electrical dipole is composed of a resistive part R and a capacitive part C. Due to thermal fluctuations of charges positions, a fluctuating voltage U is observed. We drive the system away from equilibrium by imposing a constant flux of electrons, via a constant current I.

FIG. 2 :

 2 FIG. 2: (a): For I = 0 A, Johnson-Nyquist noise has a Gaussian distribution. Relation (2) is verified. (b): The noise is white up to the cutoff frequency f0 of the RC dipole. We use FDT to extract from the energy of the noise the value of the resistive, dissipative part of the circuit R: for low frequency, the noise level spectral density is constant, equal to √ 4kBT R in a band [f ; f + df ]. A Lorentzian fit of the spectrum (thin line) additionally gives τ0 = RC.

  ′ )dt ′ the time-averaged value of a function g over a time τ .

FIG. 3 :

 3 FIG. 3: Histograms of Wτ and Qτ when a current I = 1.4 × 10 -13 A is flowing in the dipole. This corresponds to Wτ = Qτ = τ P with P = RI 2 = 45kBT /s.

FIG. 4 :

 4 FIG.4: Normalized symmetry functions SW and SQ for Wτ and Qτ when I = 1.4 × 10 -13 A (same conditions as Fig.3). (a): For any τ , SW is a linear function of a = Wτ / Wτ . Straight lines are theoretical predictions from[12]. For τ → ∞, SW (a) tends to have a slope 1. (b): On the contrary, SQ(a) tends to a limit function (continuous black curve) which is a straight line of slope 1 for a < 1 only.