Sylvain Lebreton

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Epithets, divine (Greece and Rome)

SYLVAIN LEBRETON Divine epithets (eponumiai, epikleseis; cognomina, cognomenta) are a common feature of some ancient polytheisms, though the Hellenic world has recently benefited from far more studies than any other culture. The incredible variety of epithets, both in Greece and Rome, calls for a typology.

A first categorization has to be made according to their use: Pausanias already distinguished epithets given by poets or local cult-epithets from those known by everyone (7.21.7); accordingly, modern scholars usually differentiate cult-epithets or poetic epithets from those that designate iconographic types (respectively, ATHENA Polias, glaukopis, and Promachos). However, these categories are not absolute [START_REF] Parker | The problem of the Greek cult epithet[END_REF]: some epithets can be used both in poetry and in cult, not to mention cases when they appear out of their original context, for example, in the work of lexicographers [START_REF] Brulé | La Banque de données des épiclèses (BDDE) du Crescam: sa philosophie[END_REF]). Yet, among these categories, cult-epithets undoubtedly remain the richest field of experimentation about ancient polytheisms. Their combination with theonyms forms a dual system of divine denomination [START_REF] Brulé | Le langage des épiclèses dans le polythéisme hellénique[END_REF]) which designates a divine figure (MARS Ultor), namely, the actualization of a divine power (Mars) in a specific context. Cult-epithets aim at marking out a divine figure from other ones, or fixing it in a given space [START_REF] Parker | The problem of the Greek cult epithet[END_REF], in order to specify which divine protection is expected.

Cult-epithets can be divided into two main categories, according to toponymy and function. The innumerable "toponymic" epithets are built on a place-name, which in most cases is the same as the cult-place (APOLLO Palatinus on the Palatine). "Transferred" toponymic epithets reveal the diffusion of a widespread cult (ARTEMIS Ephesia) or the transfer of a local cult to another place (DIONYSOS Eleuthereus in Athens). "Functional" epithets form a wider category. Strictly speaking, they identify the field in which the help of the deity is expected: the epithet can be the name of the field itself, a derivative, or an agent-name (respectively, ZEUS Keraunos, Keraunios, and Keraunobolos); but other epithets designating a dynamic of intervention (Soter/Soteira) can also be categorized as functional.

The other categories actually prove to be variants of toponymic or (mostly) functional epithets [START_REF] Parker | The problem of the Greek cult epithet[END_REF]: some refer to ritual facts (Apollo Hekatombaios); others clearly derive from a theonym and closely bind a deity to another one (Athena Areia, Zeus Ares; cf. Parker 2005); "individual epithets," derived from personal names, link a god to the founder of a cult and often to his progeny [START_REF] Wallensten | Personal protection and tailor-made deities: the use of individual epithets[END_REF]; some emphasize the power of the deity (JUPITER Optimus Maximus) or its faculty to respond to the worshipers' wishes (Epekoos). Finally, some quite frequent "topographic" epithets belong to both functional and toponymic categories, since they can refer both to landscapes favored by a god and to a peculiar place through a generic denomination: so, Zeus Akraios can be a generic Zeus of mountaintops; but for Magnesians, he was also the Zeus whose sanctuary was at the top of nearby Mount Pelion.

Cult-epithets are a precious tool for the understanding of ancient polytheisms. On a larger scale, the quantitative distribution of cultepithets among gods and goddesses is really instructive, since it is far from being homogeneous (cf. Banque de données des épiclèses grecques for Greece; Carter 1898 for Rome): some deities are very rich in double designations (Zeus above all; then come Apollo, Artemis, Athena, and so on); other gods bear only few epithets (ARES), or have none (HEPHAISTOS). A similar observation can be made on a qualitative level: again, Zeus intervenes in many aspects of ancient Greeks' life, whereas most of DEMETER'S epithets refer to agriculture and plant growth. Likewise, some epithets are attributed to many different gods (Soter/Soteira), while other ones are exclusive to a single deity (POSEIDON Asphaleios), not to mention those that are shared by divine pairs (Zeus Phratrios and Athena Phratria).

On a closer scale, for a given deity, cultepithets reflect the features attributed by a given community, at a given time and in a given place, for a specific purpose. Although certain matches appear perfect (dedications to HERMES Agoraios by agoranomoi), the meaning of cultepithets does not always fit their literal signification, or at least seems not restricted to it: a dedication from a Macedonian woman to Artemis Elaphebolos ("who shoots deer"), most probably for the sake of her daughter (SEG 43.399A), shows that this Artemis was not (only) worshiped by (male) hunters and emphasizes the connection between this goddess, hinds, and youngsters (cf. Eur. IT). Such a case raises the issue of the coherence of a divine power despite its fragmentation in many figures, each of them being referred to by a single epithet (cf. Versnel 2011: 60-87). But as far as it can be representative of a whole, a dedication from the area of Nikaia (I.Iznik 1085), addressed to three distinct Zeuses (Bronton, Karpodotes, and Eucharistos) who are then merged into a single theos, shows that the articulation of these two levelsunity and distinctiondoes not seem to have raised many problems to ancient Greeks and Romans. SEE ALSO: Chthonic deities, Greece and Rome; Olympian deities; Religion, Greek; Religion, Roman; Twelve Gods.

The Encyclopedia of Ancient History. Edited by Roger S. Bagnall, Kai Brodersen, Craige B. Champion, Andrew Erskine, and Sabine R. Huebner. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd. DOI: 10.1002/9781444338386.wbeah30175


Laurent Nottale
email: laurent.nottale@obspm.fr

Solution to the cosmological constant problem from dark energy of the quark vacuum

We solve the cosmological constant puzzle by attributing its origin to gravitational self-energy of quark vacuum quantum fluctuations and to Mach-Einstein's principle of the relativity of all inertia. The Dirac sea of negative energy quarks and its fluctuations, which should be subjected to the expansion of the universe, is actually frozen because of quark-antiquark pair creation from the QCD confinement energy, which is well described by a string potential. When the members of a pair separate beyond a critical distance, the string breaks and a new pair is created, so that the density remains constant under adiabatic expansion, as required for a Lorentz invariant vacuum. We argue that the energy scale at which the vacuum no longer follows the expansion is therefore given by the effective mass of quarks in the neutral pion. This yields a predicted value of the reduced cosmological constant Ω Λ h 2 (pred) = 0.31152 ± 0.00006, which is 100 times more precise while in excellent agreement with its observational value recently determined by the 2018 Planck mission reanalysis, Ω Λ h 2 (obs) = 0.3153 ± 0.0065.

Introduction. The cosmological constant has been introduced in gravitational field equations by Einstein in 1917 in order to satisfy to Mach's principle of the relativity of inertia [START_REF] Einstein | Sitzungsberichte der Preussichen Akad. d. Wissenschaften[END_REF]. Then it was demonstrated by Cartan in 1922 [START_REF] Cartan | Oeuvres Complètes[END_REF] that the Einstein field tensor including a cosmological constant Λ, E µν = R µν - 1 2 Rg µν + Λg µν , is the most general tensor in Riemannian geometry having null divergence like the energy momentum tensor T µν . This theorem has set the general form of Einstein's gravitational field equations as E µν = κT µν and established from first principles the existence of Λ as an unvarying true constant.

The cosmological constant problem dates back to the realisation that it is equivalent to a vacuum energy density [START_REF] Lemaitre | Proc. Nat. Acad. Sci[END_REF][START_REF] Ya | [END_REF], while theoretical expectations of this "dark energy" from modern theories of elementary particles seem to exceed its observational value by some 120 orders of magnitude.

One of the main consequences in cosmology of a positive cosmological constant is an acceleration of the expansion of the universe. Such an acceleration has been first detected in 1981 in the Hubble diagram of infrared elliptical galaxies (which already reached at that time redshifts z ≈ 1), yielding a positive value close to the presently measured one, but with still large uncertainties [START_REF] Nottale | L'Univers, Comptes-Rendu de l'Ecole de Goutelas[END_REF]. Accurate measurements of the acceleration of the expansion since 20 years [6,7,8,9] have reinforced the problem. Indeed, all analyses so far support that the "dark energy" which drives this acceleration owns every properties of a cosmological constant: no variation of its value is needed to account for the precision observations, suggesting that it is a true constant, and the coefficient of its state equation is w = -1 within uncertainties, as expected for a Lorentz invariant vacuum that is equivalent to a cosmological constant (the 2018 reanalysis of Planck mission [START_REF] Aghanim | Planck 2018 results. VI. Cosmological parameters[END_REF] obtains w = -1.028 ± 0.032, and a flat universe Ω k = 0.0007 ± 0.0019 to high accuracy).

We show in this paper that the problem is solved both in principle and quantitatively by attributing the emergence of a cosmological constant to the effect of quark confinement on the self-gravitational energy of quantum fluctuations. Through evaluating this energy from the quark effective mass in the neutral pion (which is a material implementation of this process), we predict a theoretical value (Ω Λ h 2 ) pred = 0.31152 ± 0.00006. As early as 2004 the combination of WMAP and SDSS data yielded a best fit value close to this prediction, (Ω Λ h 2 ) obs = 0.317 ± 0.038 [10]. The WMAP nine years mission has obtained (Ω Λ h 2 ) obs = 0.341 ± 0.012 [8]. More recently, the Planck mission has yielded (Ω Λ h 2 ) obs = 0.318 ± 0.011 [9], still improved by a factor of 2 in its 2018 re-analysis to (Ω Λ h 2 ) obs = 0.315 ± 0.006 [START_REF] Aghanim | Planck 2018 results. VI. Cosmological parameters[END_REF], which agrees remarkably well with the theoretical prediction.

Elements of the problem. As remarked by Weinberg [START_REF] Weinberg | [END_REF], anything that contributes to the energy density of the vacuum acts just like a cosmological constant. Therefore the problem amounts not only to find which contribution yields the observed value and why it does, but also to understand why and how the other particles / fields do not contribute.

The vacuum energy varies with the length scale r as r -1 , and therefore the vacuum energy density as r -4 . Relating it to the Planck scale, one obtains ρ V (r) = ρ P × (r P /r) 4 . With today's precisely measured value of the cosmological constant [START_REF] Aghanim | Planck 2018 results. VI. Cosmological parameters[END_REF], one obtains a factor 3.4 × 10 121 from the Planck scale contribution, but even electron-positron virtual pairs provide a far too high contribution, by a factor 1.07 × 10 32 .

The cosmological constant Λ, as it appears in Einstein's equations, is a curvature (it is expressed in units of m -2 ). As such, besides being an energy density, it is also the inverse of the square of an invariant cosmic length L [START_REF] Nottale | Fractal Space-Time and Microphysics[END_REF],

Λ = 8πGρ V c 2 = 1 L 2 . ( 1
)
The recently measured value of Λ, derived from the best Planck solution (reduced cosmological constant Ω Λ = 0.6889 ± 0.0056 and reduced Hubble constant h = 0.6766 ± 0.0042), yields a cosmic length-scale L = (3.08 ± 0.03) Gpc. Its ratio with the Planck scale is an invariant pure number, K = L/r P = (5.878 ± 0.060) × 10 60 (whose square just yields the 10 121 discrepancy). Note that, while the status of the cosmological constant as an energy density has been often emphasized, its complementary (and original) status as an invariant cosmic length has been underestimated. The solution proposed here is just based on this double meaning [START_REF] Nottale | Fractal Space-Time and Microphysics[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]. A vacuum is a perfect fluid that must be Lorentz invariant, and therefore it has the equation of state ρ V + p V /c 2 = 0, where ρ V is its density and p V its pressure. In Einstein's field equations, such a fluid contributes to the stress-energy tensor T µν = (ρ + p/c 2 )u µ u ν + pg µν by a mere negative pressure term p V g µν = -ρ V c 2 g µν which accelerates the expansion.

Another essential property of such a vacuum, as emphasized by Carroll et al. [START_REF] Carroll | [END_REF], is that its density must remain constant if a volume of this fluid is adiabatically compressed or expanded: a work p V dV provides exactly the amount of energy to fill the new volume dV to the same density ρ V . Therefore the energy density remains a true constant, as required for its identification with a general relativistic cosmological constant. As we shall see, this property plays a central role in the solution proposed here.

Gravitational self-energy density of quantum fluctuations. A first step toward a solution to this problem, which has been pointed out by many authors [START_REF] Ya | [END_REF][START_REF] Weinberg | [END_REF][START_REF] Carroll | [END_REF], is that energy is defined only up to a constant and plays a role in physics only by its differences, i.e., it can always be renormalized as < E >= 0. Therefore the vacuum energy density by itself has no reason to contribute as a source for the gravitational field. Arrived at that point, the cosmological problem is reversed, since the theory now predicts ρ V = 0, as was supported by Hawking's quantum gravity argument [13], while observations instead tell us that it is the dominant energy contribution in the universe.

A second step has been made by Zeldovich [START_REF] Ya | [END_REF], who remarked that, although the energy density can be taken identically equal to zero, the energy of the gravitational interaction of the virtual particle pairs contained in the vacuum cannot vanish because of Heisenberg's inequality. A quantum fluctuation ±∆E of the vacuum at length scale r allows the appearance of a pair of virtual particles of mass m = ∆E/c 2 ∝ 1/r. The gravitational potential self-energy of this pair is φ G = -Gm 2 /r = -G < ∆E 2 > /c 4 r, so that the corresponding energy density φ G /r 3 is proportional to r -6 . Relating it once again to the Planck scale, one obtains

ρ G = -ρ P × r P r 6 . (2)
The main point here is that, while the mean energy of a fluctuation can be renormalized (< E >= 0) as already pointed out, its square <E 2 > cannot vanish according to Heisenberg's inequality.

Arrived at that point, two problems arise : (1) the sign of this expression is the opposite of the observed energy density; (2) there is no reason for a gravitational potential energy to become a source for gravitation. We suggest hereafter a solution to both problems.

Consequences of Mach-Einstein's principle. The 'postulate of relativity of all inertia', which has been named by Einstein 'Mach principle' [START_REF] Einstein | Sitzungsberichte der Preussichen Akad. d. Wissenschaften[END_REF] allows one to solve this puzzle. Concerning matter, it has been expressed by Sciama [17] in a very simple way. According to this principle, the total energy of any body of mass m should be zero (i.e., ultimately, any body should be free). In the rest frame of the body, it is the sum of its mass energy and of the potential energy of its gravitational coupling with all the other bodies of the universe, i.e., m I c 2 + (-Gm p i m i /r i ) = 0. In agreement with the weak equivalence principle, the inertial mass and the passive gravitational mass of the body are equal, m I = m p , and we are therefore left with a black hole-like relation for the whole universe,

G c 2 M U R U = 1. (3)
Recall that Einstein's goal when introducing the cosmological constant was precisely to implement such a relation M/R = cst for the Universe, in order to render general relativity 'Machian' [START_REF] Einstein | Sitzungsberichte der Preussichen Akad. d. Wissenschaften[END_REF]. Since non static models of the universe have M U ∝ ρ a 3 = cst but R ≈ c/H variable with time (where H is the Hubble constant), Einstein suggested that the universe should be static with R = 1/ √ Λ = c/H, a solution which has been subsequently invalidated by the discovery of the expansion of the universe.

However, one can make nowadays an important remark. The very introduction of a cosmological true constant Λ = 1/L 2 , which is now supported by astronomical observations, introduces an invariant length-scale L at the cosmological scales (which has no reason to be identified with the time-varying c/H(t)) even in non static model (it applies to the whole general relativity theory). Therefore the problem of having a 'Machian' universe is actually solved in principle for any cosmological model, since there exists a prime integral M U = cst for each model, so that a relation

M U /R U = cst is implemented in general with R U = L.
The new element in the present work consists of now applying this Mach-Einstein-Sciama principle, not only to real matter, but also to virtual particles, i.e. to the vacuum itself. In the same way as the total energy of matter should vanish, the total energy of the vacuum is also expected to be zero. Therefore, since we have found that a negative gravitational self-energy density ρ G = -ρ P × (r P /r) 6 must be created by quantum fluctuations of the vacuum, we expect that it must be cancelled by a positive energy density ρ V such that ρ V + ρ G = 0. And in the same way as the rest energies mc 2 of material bodies are sources of gravitation, we also expect this positive energy density,

ρ V = ρ P × r P r 6 , (4)
to be a source for gravitation and to enter the stress-energy tensor in Einstein's field equations.

Cosmological constant as a transition scale. In order for the above formula to yield the correct numerical value of the cosmological constant Λ = r -2 P (ρ V /ρ P ), quantum vacuum fluctuations must have been frozen at a transition length scale r 0 given by ρ V = ρ P × (r P /r 0 ) 6 . Introducing again the cosmic length L = 1/ √ Λ, it yields the simple 'large number' relation r 3 0 = r 2 P L [14, p. 303]. This result is remarkable, since the question of calculating the vacuum energy density ρ V = Λc 2 /8πG has been replaced by the problem of identifying the transition scale r 0 (or equivalently the corresponding mass scale m 0 = /r 0 c). It is no longer necessary to calculate explicitely the vacuum energy density in order to obtain the cosmological constant, since it is now given directly by a ratio of length scales:

Λ = 1 L 2 = r 4 P r 6 0 , (5)
from which one can easily derive the observed parameter Ω Λ = Λc 2 /3H 2 0 . This is an important point for an explicit numerical calculation of the cosmological constant since the value of the energy density depends on numerical constants which are difficult to estimate and may change in dependence of the models.

Quark vacuum Dirac sea. We have suggested more than 25 years ago that the cosmological constant could be a relic of the quark-hadron transition and derived at that time a value Ω Λ h 2 = 0.36 [START_REF] Nottale | Mach's principle, Dirac's large numbers and the cosmological constant problem[END_REF], [14, p. 305], in satisfying agreement with the observational values measured after 1998 (see also Beck [22]). However, this initial result was based on the observation of the convergence of fundamental elementary particle scales around ≈ 70 MeV, the classical radius of the electron, the effective mass of quarks in the neutral and charged pion and the QCD scale for 6 quark flavors (which is today estimated to a larger value, 89 ± 6 MeV [31]). This leads to a reduced cosmological constant in the observed range [0.31 -0.39], but an explicit mechanism of transition was still lacking. It was suggested in [15, p. 547] that such a mechanism is just quark confinement, which imposes a largest possible value for the interdistance between two quarks (or a quark and an antiquark) given by the effective Compton length of quarks in the neutral pion. It is this proposed solution that we elaborate here.

Under such a view, the solution to the cosmological constant problem lies in the thermal history of the primordial Universe. It is now well known that, when the temperature of the Universe was larger than about 150 MeV, it was dominated by a plasma of quarks and gluons [23]. Because of asymptotic freedom, the quarks and gluons are almost free and deconfined at high temperature. However, when the Universe cools because of its expansion, a transition occurs at an epoch of about 10 µs, since quarks and gluons are forced to be confined as colorless hadrons at low temperature. Simulations of three-flavor QCD on a lattice have given a fairly well-defined temperature of T c = (154±8) MeV [18,19,20]. Just after the transition, the Universe is dominated by pions, which rapidly decay, leaving only the residual nucleons (plus electrons, positrons, neutrinos and photons).

However, besides this history of matter, we are more closely concerned here with the parallel history of the quantum vacuum and of its fluctuations. Our suggestion is that the clue to the solution of the cosmological constant problem lies in the understanding of the behavior of the vacuum under the action of the expansion of the Universe.

In order to be more specific, let us consider a Dirac sea representation of the quark vacuum. In the Dirac sea model, the vacuum is filled by all the possible quark states. A quantum fluctuation extracts a quark from the vacuum, leaving a hole which is seen as an antiquark, thus yielding a virtual quark-antiquark pair.

The structure of the QED Dirac sea has been elaborated by e.g. [24,25]. The quark Dirac sea has been specifically addressed by Molodtsov and Zinovyev [28,29,30], who find that its ground state is identical to a BCS state, therefore involving a condensate of quark-antiquark pairs into bosons. They find that the quasiparticles obtained are identical to the pion and that their derived dynamical mass depends only on the coupling constant.

The question addressed now is what happens to the Dirac sea of quarks and to its fluctuations when the universe expands. The Dirac vacuum is not empty, but filled by an infinity of virtual particles which endow it with a rich structure. As for the quark Dirac sea, it is therefore a quark-gluon plasma of negative energy states.

Pair creation is usually described in terms of a momentum-energy representation, but a more complete description is possible in terms of position and momentum [24]. One can therefore consider the effect on the vacuum and on its fluctuations of the increase of the interdistance between the negative energy quarks due to expansion of the universe. If, because of quantum fluctuations, a negative energy quark from the Dirac sea jumps to positive energy, leaving a hole viewed as an antiparticle, the quark-antiquark interdistance is expected to follow the same evolution.

Concerning real particles, it is known that if one pulls on the two quarks forming a pion, the energy needed to break it is larger than the energy of creation of light q q pairs. In other words, this operation results in two pions (four confined quarks) instead of two free quarks. This confinement of quarks is well described by a linear + Coulomb-like string potential, V (r) = σr + µ + β/r + .., which agrees with lattice results up to large r (see e.g. [START_REF] Sumino | Perturbative QCD Potential and String Tension[END_REF]). The dominant term V = σr is directly proportional to distance, and it therefore yields a constant force between the two quarks and a potentially infinite binding energy.

The string breaking has been observed in lattice simulations of QCD [START_REF] Bali | [END_REF], yielding a typical string breaking distance r c = (1.25 ± 0.06) fm for two sea quark flavors, slightly increasing to r c = (1.27 ± 0.08) fm if a third (strange) quark is added. This corresponds to an energy of 155 MeV, i.e. precisely the QH transition temperature. Extrapolated to real QCD, this value is reduced to (1.13 ± 0.12) fm, including the uncertainty on the extrapolation and on the physical scale of reference.

Let us consider the structure of the quark vacuum and of its fluctuations around this Fermi scale. At length scales smaller than ∆r ≈ 1.25 fm, energy fluctuations ∆E > /∆r ≈ 150 MeV allow the creation of quark-antiquark pairs of mass m q ≈ 70 MeV, which is the effective mass of quarks in the pion. The new question here concerns the behavior of these virtual particle pairs under the effect of the expansion of the universe. The interdistance between the quark and the antiquark is expected to increase due to this expansion. But they should be submitted to exactly the same string potential as the real pairs in the pion. As a direct consequence, when the interdistance reaches the stringbreaking value, a new virtual quark-antiquark pair is expected to appear and to contribute to the vacuum fluctuations. Contrarily to the case of real matter, where this pair manifests itself as a pion, the virtual particles remain as a quark and an antiquark, as required by the fact that they are part of the Dirac quark vacuum sea and its fluctuations. Therefore the volume density of the pairs become frozen at this scale, since the density decrease expected from the expansion of the universe is continuously compensated by particle creation from the 'confinement' energy field (which is probably just a manifestation of the 'large' scale non-perturbative QCD field, as now supported by lattice QCD simulations). The freezing of the pair density (i.e. of the scale of the vacuum fluctuations of the Dirac quark sea) finally yields a freezing of the gravitational self-energy density ρ G , then of the dark energy ρ V = -ρ G . This is exactly the kind of process expected for implementing a vacuum energy that remains invariant under adiabatic expansion (a property, as already mentioned, explicitly required for a dark energy to be identical to a cosmological constant [START_REF] Carroll | [END_REF]).

Two complementary problems are solved by this proposal: (i) we understand the order of size of the cosmological constant, linked to the typical confinement scale ≈ 70 MeV; (ii) we understand why other fields do not contribute to dark energy, since the quark confinement field is the only one which owns this property of being able to cancel the expansion of the universe by continuous pair creation involving such a freezing process. The other fields are diluted as r -6 in the course of the universe expansion and have finally a vanishing contribution.

Predicted accurate numerical value of the cosmological constant. The last step amounts to obtain a precise theoretical determination of the numerical value of the cosmological constant from this process. A difficulty could be the uncertainty on the precise numerical constants which intervene in it. But this difficulty is actually shunted by the use of the Planck scale as reference scale. Only scale ratios intervene in this calculation, not the dimensioned scales themselves, and we just have to check that the same definition is used for the reference (Planck) scale and for the transition scale. Defining the Planck length scale as r P = G/c 3 means that it is defined as the 'Compton length' of a Planck mass (instead of its wave length based on h), and therefore that it should be related to the transition scale r 0 also defined as a Compton length, r 0 = /m 0 c. Arrived at this point, we postulate a full similarity between the real (matter) particle properties and the virtual (vacuum fluctuation) particles, as supported by experimental data and by QED and QCD theoretical calculations. We assume that the lightest meson, the neutral pion of mass m π 0 , is a matter manifestation of the largest possible volume density of quark-antiquark pairs (the charged pions being excluded because they contain an electromagnetic contribution to their mass-energy), and that it is reflected in the vacuum and its fluctuations with the same property. The Compton length corresponding to the individual quarks in the pion is therefore r 0 = 2 /m π 0 c.

With this value, the transition scale is r 0 = (2.92386±0.00001) fm from m q = m π 0 /2 = 67.48850 ± 0.00025 MeV [31], and one obtains (assuming no other contribution)

Ω Λ h 2 (pred) = 0.311524 ± 0.000064. ( 6
)
This expectation is in excellent agreement with the observational value determined by the Planck mission, Ω Λ h 2 (obs) = 0.318 ± 0.011 [9], improved by a recent re-analysis [START_REF] Aghanim | Planck 2018 results. VI. Cosmological parameters[END_REF] to

Ω Λ h 2 (obs) = 0.3154 ± 0.0065, (7)
from their best fits Ω Λ = 0.6889(56) and h = 0.6766(42). The theoretical prediction being still ≈ 100 times more precise than the observational measurement, its validity will be testable in possible future observational improvements. The value of the corresponding theoretically predicted cosmic length-scale L = Λ -1/2 is L = (3.10109 ± 0.00032) Gpc, (8) to be compared to the observational Planck 2018 value L = (3.08 ± 0.03) Gpc. The vacuum dark energy density of the cosmological constant is

ρ v = Λc 2 8πG ≈ 5.91 × 10 -27 kg.m -3 , (9)
i.e. ≈ 49 quarks by m 3 (taking an effective mass m q = m π0 /2 for quarks). The density compensation of the expansion by quark energy creation from QCD field is given by δρ/ρ v = 3H(t)δt, corresponding to one quark by m 3 every 100 million years at the present epoch.

Toward a full first principle theoretical solution. The above predicted value of the cosmological constant is not fully theoretical since it remains derived from the experimentally measured mass of the neutral pion and on the hypothesis that the pion, which is the lightest two quark meson, is a strict equivalent at the level of matter of the quark vacuum behavior. This assumption is supported by the fact that only a small fraction of its mass (5%) comes from the constituent quarks. Most of its mass finds its origin in spontaneous chiral symmetry breaking of the QCD Lagrangian by the quark condensate induced by non perturbative strong interactions, i.e., therefore, ultimately from QCD vacuum energy density. The Goldstone bosons resulting from this process are just the three neutral and charged pions (in terms of the two lightest quarks, u and d) [31].

The resulting neutral pion mass is given to lowest order by the relation m 2 π 0 = B(m u + m d ), where B is an unknown constant, which applies also in a consistent way to similar relations for the other Goldstone bosons (K and η) derived from three light quarks u, d, and s [31].

This has opened the possibility of predicting the pion mass from fundamental principles. Recent lattice non perturbative QCD calculations have brought this goal closer: the mass of the proton and of other light hadrons has been derived from that of the pion [32]. The mass of the pion has been extracted to 10% accuracy by Molodtsov et aL. [28] and more recently a pion mass differing only by less than 3% of its physical mass has been reversely derived in a self-consistent way with the proton mass as input [33]. A new step has been taken by the recent decomposition of the proton mass [34], showing that ≈ 70% of its mass comes from quark energy and glue field energy, and by the now most precise derivation of quark masses from lattice QCD [35]. Molodtsov and Zinovjev [29] underlign the remarkable fact that the quasiparticle size they obtain in their BCS-like model of the quark Dirac sea is close to the pion size and does not depend on the scale but on the mere coupling constant.

The proton mass result is naturally extended to the pion mass, so that we can conclude that most of the effective mass of quarks in the neutral pion comes from the quark and gluon field energy, not from their rest mass, and that it fairly represents the limit vacuum energy which gives rise to the cosmological constant. More profoundly, one can expect that the quark masses themselves find their origin in gauge fields energy. One can hope that, in a near future, a full theoretical solution of the cosmological problem be obtained from non perturbative lattice QCD, although the uncertainty on such a future possible achievement should reach 0.5 MeV on the theoretical pion mass in order to attain the present observational error on the measurement of the cosmological constant.

Conclusion and prospect.

Let us sum up the various results obtained in the present work and some of their implications.

(1) With the proposed solution of the cosmological constant problem from gravitational self-energy of the quark vacuum, there is no longer any contradiction between an explanation of the expansion acceleration in terms of cosmological constant or of dark energy: they become the same.

(2) The ∼ 10 120 discrepancy is solved by accounting for the explicit scale dependence of the vacuum gravitational self-energy, which varies as r -6 : there is no reason to attribute at cosmological scales the value obtained at the Planck scale. This well-known factor 10 120 is just the result of the cosmological constant being the inverse of the square of a length, while the ratio of this cosmic length to the Planck scale is ∼ 10 60 .

(3) While the energies of the various quantum fields remain too large to make the observed dark energy since they vary as r -4 (or can be naturally normalized to < E >= 0), the gravitational self-energy of their fluctuations both has the correct order of size thanks to its r -6 variation and cannot be made vanishing because of Heisenberg's relation.

(3) The Eddington-Dirac large number coincidence is no longer an approximate assumption but becomes a precise exact and well understood relation between the neutral pion scale and the cosmological constant length-scale. The ∼ 10 120 factor is just the ratio of Planck mass over pion mass, ∼ 10 20 , to the above power six. This is achieved without variation of constant, which is no longer needed.

(4) Mach-Einstein's principle (i.e., the principle of the relativity of all inertia) is implemented, not only for a specific cosmological model, but at the fundamental level of the general relativity theory (as was initially hoped by Einstein in his foundation of the theory), both for matter and for the quantum vacuum.

(5) The process of continuous extraction of energy from the confinement (i.e. probably QCD) field compensates in real time the cooling of the universe due to expansion. It ressembles, but now applied to the vacuum, Hoyle's model of continuous creation leading to a stationary universe (but here, only the dark energy remains stationary, while the universe is expanding).

(6) This process is still at work now, although it has been established first in the primordial universe, at the end of the quark-hadron transition. The cosmological constant therefore appeared at this epoch as a relic of the previous hot phase, in a way similar to the transitions giving rise to the emergence of nucleons, then nuclei, then light atoms and the CMB radiation at the (re)-combination epoch. (7) Finally, the fact that an apparently purely general relativistic and then physical "object" of gravitational nature be made from the QCD field points toward an underlying profound unity of field theories in physics that remains to be unveiled, not only at the Planck energy scale where the quantum, gravitational and gauge field effects are expected to become of the same order, but here, now and in a fundamental way.

It is fascinating that the length-scale of the cosmological constant, which is at some level a largest distance and horizon for our universe, be connected to (and derived from) the largest possible distance between quarks in the QCD inner "universe" of hadrons. These two interconnected distances play a similar role, one for gravitation and the other for QCD, while the largest one finds its origin in a continuous extraction of the quasiinfinite QCD confinement energy.


A K Nguyen

C B Trang

E Blond
email: eric.blond@univ-orleans.fr

E De Bilbao

T Sayet

A Batakis

Presqu'île de Giens (Var) A 2D self-organized percolation model for capillary impregnation

Keywords: Impregnation, porous media, gradient percolation, simulation

A two-dimension extension of the Self-organized Gradient Percolation (SGP) method initially developed for the one-dimensional simulation is proposed. The initialization in the two directions is considered as the analytic solution of the 2D (homogeneous) diffusion equation. The evolution of the saturation front is assumed to be the evolution of both standard deviations in each direction. The validation of the implementation is done by comparisons between SGP and finite element results.

Introduction

To simulate the unsaturated (non-reactive) impregnation in a porous medium, classical numerical methods (FEM, theta-method, etc.) have often been employed to solve the Richards' equation [START_REF] Bergamaschi | Mixed finite elements and Newton-type linearizations for the solution of Richards' equation[END_REF], [START_REF] Kodešová | Estimation of the Dual-Permeability Model Parameters using Tension Disk Infiltrometer and Guelph Permeameter[END_REF]. Yet, the CPU time and spurious oscillations become prohibitive, as for instance, when the impregnation is coupled with chemical reactions. That is why a drastically different, and more reliable approach, called Self-organized Gradient Percolation (SGP) method, was proposed [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF].

The initialization of the algorithm is driven by an analytic solution of the (homogeneous) diffusion equation. It is a convolution between a Probability Density Function (PDF) and a smoothing function [4]. The evolution of the capillary pressure profiles with time is reproduced by the self-evolution of the standard deviation of the PDF. To test this model, the capillary pressure profiles and the mass gain curve have been confronted with those obtained by FEM and experimental measurement, respectively [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF].

The main goal of the present work is to extend the SGP method to the 2D unsaturated impregnation only considering the non-reactive case. The two-dimensional SGP algorithm has the same structure than the one-dimensional one. On the mathematical side, the analytic form is a convolution between a PDF of two directions and a smoothing function. In addition, the evolution of the saturation front is proposed to be the evolution of the two standard deviations of the PDF in each direction. The visualizations and comparisons between the numerical results from SGP method and FEM are given.

Self-organized Gradient Percolation (SGP) method for impregnation

The gradient percolation method is a probabilistic approach to predict a spatial evolution. The idea for using this method here is to avoid the numerical difficulties of the resolution of the Richards' equation derived from the mass balance combined with Darcy's law [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]. Richards' equation is a nonlinear Partial Differential Equation (PDE), which requires a small-time increment and an appropriate fine space discretization to make sure the accuracy of the results. However, a small-time increment leads to a high computational cost and, moreover, a very small time-increment can give rise to spurious oscillations that impact the accuracy of the results [START_REF] Vermeer | An accuracy condition for consolidation by finite elements[END_REF]. The main goal for the development of this method is to reduce the CPU time and the number of parameters actually used for classical models. The Self-organized Gradient Percolation (SGP) method, which is based on the gradient percolation method, has been developed to reach these goals.

Two fundamental types of percolation are broadly used to simulate physical phenomena, the bond percolation and the site percolation [START_REF] Grimmett | Percolation[END_REF]. The site percolation can be applied to the simulation of the nonreactive impregnation process for the determination of the saturation, first considered as a state variable describing the state ("occupied" or "empty") of each local pore of the media.

Considering a random network defined by numerous cells/squares, function 𝑋𝑋(𝑧𝑧), where 𝑧𝑧 is the position of the cell, is a random value uniformly distributed on the interval [0,1]. A site declared to be "occupied" or to be "empty" by the liquid is defined in comparison with constant probability 𝑝𝑝 as:

• site 𝑧𝑧 is "occupied" when 𝑋𝑋(𝑧𝑧) ≥ 𝑝𝑝;

• site 𝑧𝑧 is "empty" otherwise.

In classical percolation model, probability 𝑝𝑝, which indicates the state of a site, is independent of the site, it is the same value for the whole domain. However, physics implies that the state of a site is given by capillary pressure, which is not constant on the lattice.

From physics point of view, the Capillary Pressure Curve (CPC) is at equilibrium only under the gravity field. The capillary pressure is the equilibrium pressure, which is determined by the difference between the non-wetting and the wetting phases. In the capillary rising test, the saturation profile along the vertical direction (Figure 1), considered as a function of the level (height) with respect to the liquid regarding Jurin's law, is an image of the CPC at each time 𝑡𝑡. That is why we propose to transform probability 𝑝𝑝 into a function of the location of the site. This can be done by adapting gradient percolation theory developed by Pierre Nolin [START_REF] Nolin | Critical exponents of planar gradient percolation[END_REF]. Choosing 𝑝𝑝 as a function of the location allows reproducing the impregnation gradient, but the results will not respect the continuity of the liquid phase. Moreover, it is impossible to manage the boundary conditions. It is thus proposed to define a convolution product to satisfy the continuity and the possibility to take into account various boundary conditions. Finally, the local saturation is obtained through two steps: (1) local state 𝑋𝑋 is determined by the classical percolation and (2) employing the convolution product gives:

𝑆𝑆(𝑧𝑧) = 𝑋𝑋(𝑧𝑧) * 𝛿𝛿(𝑧𝑧) (1)
where 〈 * 〉 is the convolution operator and 𝛿𝛿 is a smoothing function (spatial weighted average). The saturation profile at initial time 𝑡𝑡 0 is then proposed by the following expression [4]:

𝑋𝑋�𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 , 𝑡𝑡 0 � = 𝑆𝑆 𝑟𝑟 + (𝑆𝑆 𝑠𝑠 -𝑆𝑆 𝑟𝑟 )𝑒𝑒𝑒𝑒𝑝𝑝 �- �𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 -𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐,𝑡𝑡 0 𝑆𝑆 𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑚𝑚 𝑚𝑚𝜎𝜎 𝑚𝑚 � (2)
where 𝑆𝑆 𝑟𝑟 and 𝑆𝑆 𝑠𝑠 mean the residual and maximum saturation, respectively, 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 is the local capillary pressure, 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐,𝑡𝑡 0 𝑆𝑆 𝑚𝑚𝑚𝑚𝑚𝑚 is the minimum pore pressure to initiate the impregnation, 𝜎𝜎 is the standard deviation of the PDF, 𝑚𝑚 is the empirical parameter to switch between probability distributions to fit the CPC (e.g. 𝑚𝑚 = 1 and 𝑚𝑚 = 2 designate Laplace and Gaussian distributions, respectively). From the physics point of view, 𝑚𝑚 is a dimensionless parameter which depends on the porosity network (such as porosity, tortuosity, etc.).

To reproduce the physics of impregnation, it is necessary that the model can evolve autonomously with time following the physical laws of the problem. In order to ensure this self-evolution along time, a relationship between the standard deviation and the capillary pressure is proposed [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]:

𝜎𝜎 𝑛𝑛 �𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 � = 𝜎𝜎 𝑛𝑛-1 �𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 � + 𝐴𝐴 � 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐,𝑡𝑡 0 𝑆𝑆 𝑃𝑃 ℎ,𝑛𝑛 -1� (3)
where 𝑛𝑛 designates the time increment and 𝐴𝐴 designates a kinetic constant (𝑚𝑚 • 𝑠𝑠 -1 ).

Finally, there are only three parameters required for the SGP algorithm: 𝐴𝐴 (𝑚𝑚 • 𝑠𝑠 -1 ), 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐,𝑡𝑡 0 𝑆𝑆 (Pa) and 𝑚𝑚 (dimensionless). To identify these three parameters, the inverse identification method [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF] is employed by fitting the mass gain curve obtained from the SGP method with the experimental one and then is validated by comparison with FEM simulation if the capillary pressure profiles are known. Then, 𝐴𝐴 is closed to the intrinsic permeability (but not fully explained today) while 𝑚𝑚 is directly linked to the shape of the capillary pressure curve and can be taken as the inverse of the parameter of van Genuchten's model. Therefore, only unknown 𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐,𝑡𝑡 0 𝑆𝑆 is the minimum pore pressure to initiate impregnation.

SGP method for 2D problem

Since the SGP algorithm presented above for the (quasi) one-dimensional case permits to reduce the CPU time and to avoid the spurious oscillations [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]. Its extension to the 2D case thus keeps working on and is straightforward to construct: the main parameter will no longer be the height but the total pressure. It is to note that the Richards' equation, where the height parameter is replaced by the total pressure parameter since from the physics point of view in the general case, is written as [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]:

ϕ 𝜕𝜕𝑆𝑆 𝜕𝜕𝑡𝑡 = 𝑑𝑑𝑑𝑑𝑑𝑑 � 𝐾𝐾(𝑆𝑆) 𝜂𝜂 𝑔𝑔𝑔𝑔𝑔𝑔𝑑𝑑(𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 )� ( 4
)
where 𝜙𝜙 is the porosity (dimensionless), 𝑆𝑆 is the saturation (dimensionless), 𝐾𝐾(𝑆𝑆) is the nonlinear function of saturation (𝑚𝑚 2 ), 𝜂𝜂 is the dynamic viscosity (𝑃𝑃𝑔𝑔 • 𝑠𝑠), 𝜌𝜌 𝑊𝑊 is the mass density of the wetting liquid (𝑘𝑘𝑔𝑔 • 𝑚𝑚 -3 ); 𝑔𝑔𝑔𝑔𝑔𝑔𝑑𝑑(𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 ) = 𝑔𝑔𝑔𝑔𝑔𝑔𝑑𝑑�𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 + 𝜌𝜌 𝑊𝑊 𝑔𝑔𝑧𝑧� is not a spatial description, but rather depends on the direction of gravity, where 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 is the total pressure (𝑃𝑃𝑔𝑔), 𝑔𝑔 is acceleration due to the gravity (𝑚𝑚 • 𝑠𝑠 -2 ) and 𝑧𝑧 is the height along the gravity direction (𝑚𝑚).

Stepwise modelling procedure

Three main steps of the 2D SGP algorithm are presented as follows.

Step 1: Based on the main idea of solving the 1D (homogeneous) diffusion equation [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF], the analytic solution of the 2D case, even in the general case, is pointed out to be a convolution between a Probability Density Function and a smoothing function. To be physically meaningful, the initialization is thus taken to be the analytic solution of the (homogeneous) diffusion equation (only considering initial time 𝑡𝑡 0 ) in the vertical and horizontal directions (𝑧𝑧 1 , 𝑧𝑧 2 ) as the following:

𝑋𝑋(𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 (𝑧𝑧 1 ), 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 (𝑧𝑧 2 ), 𝑡𝑡 0 ) = 𝑆𝑆 𝑟𝑟 + (𝑆𝑆 𝑠𝑠 -𝑆𝑆 𝑟𝑟 )𝑒𝑒𝑒𝑒𝑝𝑝 �- �𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 (𝑧𝑧 1 ) -𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡,𝑡𝑡 0 𝑆𝑆 𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑚𝑚 𝑚𝑚𝜎𝜎(𝑧𝑧 1 ) 𝑚𝑚 - �𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 (𝑧𝑧 2 ) -𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡,𝑡𝑡 0 𝑆𝑆 𝑚𝑚𝑚𝑚𝑚𝑚 � 𝑚𝑚 𝑚𝑚𝜎𝜎(𝑧𝑧 2 ) 𝑚𝑚 � (5)
where 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 , 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡,𝑡𝑡 0 𝑆𝑆 𝑚𝑚𝑔𝑔𝑒𝑒 mean the total pressure and the minimum pore pressure to start the impregnation assuming to be the same in each direction, respectively, 𝑚𝑚 is the empirical parameters defining the shape of the distribution in each direction and 𝜎𝜎(𝑧𝑧 𝑖𝑖 ) is the standard deviation function along the 𝑑𝑑-direction.

To take into account Eq. ( 1) and Eq. ( 5), we employ the dedicate function "norminv" in Matlab and then a multistate gradient percolation model is then given.

Step 2: In the next time steps, we assumed that the evolution of the capillary front along each direction is the evolution of each standard deviation, respectively. In particular, standard deviation 𝜎𝜎 𝑛𝑛 (𝑧𝑧 𝑖𝑖 ) along 𝑑𝑑-direction at timestep 𝑛𝑛 can be written as:

𝜎𝜎 𝑛𝑛 (𝑧𝑧 𝑖𝑖 ) = 𝜎𝜎 𝑛𝑛-1 (𝑧𝑧 𝑖𝑖 ) + 𝛼𝛼�𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛 (𝑧𝑧 𝑖𝑖 )� (6)
where 𝛼𝛼�𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛 (𝑧𝑧 𝑖𝑖 )� means the evolution of the standard deviation at timestep 𝑛𝑛 along 𝑑𝑑-direction. The link with the Poiseuille's equation has been pointed out in the 1D capillary rising case, which gives:

𝛼𝛼�𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛 (𝑧𝑧 𝑖𝑖 )� = 𝛾𝛾𝐷𝐷 2 32𝜂𝜂 𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑛𝑛 (𝑧𝑧 𝑖𝑖 ) (7)
where 𝐴𝐴 = 𝛾𝛾𝐷𝐷 2 32𝜂𝜂 ⁄ depends on capillary diameter 𝐷𝐷 (𝑚𝑚) and 𝛾𝛾 is the surface tension (𝑁𝑁 𝑚𝑚 ⁄ ).

Step 3: In the final step, note that the motion of liquid here is due to driving force 𝑔𝑔𝑔𝑔𝑔𝑔𝑑𝑑�𝑃𝑃 𝑐𝑐𝑐𝑐𝑐𝑐 + 𝜌𝜌 𝑊𝑊 𝑔𝑔𝑧𝑧�. Obviously, only in the capillary rising case, the simulation will stop when there is no difference between capillary pressure and hydrostatic pressure [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]. Nevertheless, in the general case, the simulation will not be able to stop until there is no more liquid on the boundary of the source. For example, the horizontal impregnation is never stopped, since there is no effect of the gravity in this direction. In the numerical practice, we thus suggest comparing the difference between the mass of liquid in the container (denoted Π) and the mass gain in time in the porous medium (denoted 𝑚𝑚 𝑡𝑡 ). It leads to that the SGP algorithm will stop when there is no difference in this comparison.

Boundary conditions

𝜕𝜕Ω 𝑖𝑖 corresponds to the three types of boundary conditions [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]. At the initial timestep, depending on the treated problem, it is considered that almost one surface of the porous sample is in contact with the surface of liquid. This leads to impose that concerning row (first row) of the matrix is fully filled with liquid. A specific boundary condition will be applied on each interface. The choice of the boundary conditions depends on the interpretation of the physical phenomena.

Application

In order to numerically simulate the test using FEM and the SGP algorithm, we decided to build the two models implemented both in (2D) Abaqus standard (v. 6.14) and Matlab (v. R2019a) software, respectively. To compare the CPU time of the two methods, it is essential that both models have to be built with the same mesh size, time increment, dimensions and boundary conditions.

Description of the test

For the application of the SGP algorithm extended to the 2D simulation of the impregnation, considering a numerical example, a cylindrical porous sample is in (partial) contact with a liquid at the bottom surface. The height and width of the sample are 0.04 m and 0.035 m (in Figure 3) respectively. The porous material is Alumina 99% and the impregnated liquid is glycerine. Due to the lack of experiment data, the validation of the performance of the 2D SGP algorithm is made by comparisons of the local saturation of each square of the model using FEM with the one of each square with the same coordinates of the SGP model. Obviously, the number of the squares of each model is equal to the number of the squares in row multiplied with the one in the column. It leads to compare a very large number of squares in the case of very smooth mesh. To be simple in our ongoing study, we decided to consider the model with a coarse grid, but, of course, must respect the condition between time step and mesh size to avoid the spurious problems [START_REF] Vermeer | An accuracy condition for consolidation by finite elements[END_REF].

For the boundary conditions in both models, there is an imposed pressure on boundary surface 𝜕𝜕𝜕𝜕 and the free draining on boundary surfaces 𝜕𝜕𝐴𝐴 used in the simulation (Figure 3). In particular, for the first one, the liquid is in contact with surface 𝜕𝜕𝜕𝜕, so it corresponds to an imposed total pressure; for the second one, to make sure that the liquid can be free to flow out of these surfaces, the drainage-only boundary condition is imposed on surface 𝜕𝜕𝐴𝐴. The "expression" of these boundary conditions in the SGP model is done thanks to the convolution procedure [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF]. The computational costs for the test for the SGP method and FEM are obtained by running the software on 2.60 GHz, x64-based PC with 32 GB of RAM. Obviously, the computational time of the SGP algorithm is significantly lower than the one of FEM (Table 1). This variation in the ratio is due to the difference in the evolution of time step between both methods [START_REF] Nguyen | Self-organized gradient percolation method for numerical simulation of impregnation in porous media[END_REF].

Results and comparison

Conclusion

In this work, our ongoing study of the extending 2D SGP method to the impregnation process is proposed. The preliminary visualizations from the FEM and SGP method (Figure 4 (a) to (d)) seems to indicate that the behaviour of the evolution of the saturation fronts from the two methods are the same at the first time steps (Figure 4 (e) and (f)).

Hence, there is still a huge work to do in order to study the mesh sensitivity, the impact of the time step, the impact of each of the parameters required for the SGP model and their link with the physics (properties of the material). For the perspectives, the SGP algorithm is expected to adapt itself to taking into account the kinetics of the liquid motion in respect with the gravity direction. Then, to validate the results in this case, it is necessary to fit the kinetics of the liquid motion along the porous sample by comparing with the mass gain curves from FEM and/or experimental data at each separate zone regarding the gravity direction of the porous sample.

Figure 1 -

 1 Figure 1 -The saturation profile as a probability density function

Figure 2 -

 2 Figure 2 -Flowchart of the SGP algorithm for the 2D problem.

Figure 3 -

 3 Figure 3 -The specific dimensions and boundary conditions of the numerical test are used both for the models using FEM and SGP method. The mesh size is 0.03 m (along surface 𝜕𝜕𝜕𝜕), 0.025 m and 0.0267 m (along surfaces 𝜕𝜕𝐴𝐴 in the horizontal and vertical directions, respectively).

Figure 4 -

 4 Figure 4 -Simulations using FEM and SGP method for the evolution of the saturation fronts at different time from (a) to (d). The absolute errors (e) and (f), calculated by the absolute minus of the local saturation at a square

Table 1 -

 1 CPU time and time ratios between FEM and SGP method.

	Durations	CPU time (in seconds)	Time
	(in seconds)	FEM	SGP method	ratios
	142	15	0.075	200
	4254	48.3	0.5367	90


Expressing ill-being in the life narratives of possessors of child pornography videos

Frédéric Pugniere-Saavedra Université Bretagne Sud, Laboratoire Prefics What follows is the translation of an article written in French, arising from a presentation made at the 6 th World Congress of French Linguistics that took place at the University of Mons, Belgium, 9-13 July 2018. The article in French can be found on the Web site SHS Web Conf., Volume 46, 2018, at: https://doi.org/10.1051/shsconf/20184601007 Résumé. À travers des récits de vie qui reconfigurent des réalités sociales vécues de locuteurs qui détiennent et qui diffusent des documents vidéos pédopornographiques, nous organiserons notre étude selon le continuum suivant : après une présentation de ce corpus dit sensible et du groupe de paroles en tant que dispositif mobilisé, nous aborderons le coeur de notre étude avec deux focales portant sur les régularités énonciatives d'une part et d'autre part sur les organisateurs de discours « que/parce que, mais » qui forgent la textualité de ce discours et qui participent à exprimer le mal être des locuteurs.

Abstract. We use life narratives that reconstruct the lived social realities of French speakers who possess and distribute child pornography videos. The study is structured as follows: first, we present the investigation approach used, namely the sensitive corpus used and the discussion group; then, we focus upon both the enunciative regularities and the discoursestructuring terms que/parce que, mais (that/because, but) that create the textuality of this discourse and contribute to the verbalization of the speakers' ill-being.

This study is based upon the analysis of a corpus of self-disclosing life narratives that are viewed neither as factual data of biographical accounts [START_REF] Bertaux | Le récit de vie[END_REF] 2005) nor as subjective viewpoints on these factual data [START_REF] Beaud | L'usage de l'entretien en sciences sociales[END_REF]. Instead, they are viewed as discursive reconstructions of the world, what [START_REF] Ricoeur | Temps et récit. Tome I : L'intrigue et le récit historique[END_REF] calls "emplotments" (mises en intrigue), namely transforming a host of ad hoc and disparate events into a coherent and meaningful "story". Life narratives i are considered as activities that (re)construct the speakers' lived social realities through the use of specific patterns of syntax, lexis, and enunciative forms [START_REF] Nossik | Les récits de vie comme corpus sociolinguistique : une approche discursive et interactionnelle[END_REF]. These life narratives were collected in France between 2015 and 2017 in discussion groups of pedophiles, defined by the World Health Organization ii (WHO) as persons showing a "sexual preference for children, whether boys or girls or both, usually at the pre-puberty or early puberty stage iii ". Volunteer participants in these discussion groups are pedophile victims iv , aggressors who are on trial or have received sentences, possessors and users of child pornography material, men who experience sexual impulses but do not act on these, close relatives of aggressors, and sexually abstinent pedophiles. This study is focused solely upon possessors of child pornography videos v who talk about themselves in discussion groups where they put into words their family and transgenerational history, culture, beliefs, and their own pedophilia-related history. Discussion groups are a "listening standard" vi [START_REF] Fassin | Des maux indicibles. Sociologie des lieux d'écoute[END_REF][START_REF] Bass | Au fil de la parole, des groupes pour dire[END_REF] through which child pornography users, as they shape their narrative, can express their illbeing.

1. An investigative framework for this discourse of self-disclosure 1.1 Specific interactional configuration of this corpus Talking about oneself is a difficult activity for various reasons: there is the risk of transgressing the laws of modesty and in particular, the risk of revealing oneself through talk, of making oneself vulnerable by opening up to others and sharing confidences [START_REF] Goffman | Ed. française : Les rites d'interaction[END_REF] or even secrets. Vion (1994: 217) asserts that "conventionally masked, rarely mentioned and shared, suffering is not expressed (the law of privacy), and to describe it is for people an exercise of linguistic quasi-improvisation". Self-disclosure and selfnarrative entail taking risks, which explains the numerous forms of downplay (i.e. the process whereby the part of subjectivity engaged in an interaction is reduced through the "various forms of euphemisms, mitigation, circumlocutions, cautious phrasing, prudent wording, indirect acts, preambles, justifications, self-corrections, etc.") and of tension (i.e. the reverse process involving the "various forms of hyperbole, outbursts, marked and vivid wording, direct acts, etc.") [START_REF] Vion | Analyse des interactions Hachette Supérieur[END_REF](Vion, -2000: 244): 244). Talking about oneself though discussion groups is also a special communication contract (Charaudeau 1984). The discussion group members meet once a month at the home of the (woman) president of the Blue Angel Association vii . About 15 people sharing their pedophilia-related experience meet for three or four hours on Saturday afternoons, sitting around a large table with drinks, fruit, and cakes. Overall, participants are relaxed; some find it easy to talk about themselves, others find it more difficult to put their story into words, but a convivial atmosphere (and sometimes a questionable sense of humor) appears to facilitate the storytelling. Participants may attend several times depending upon their story, their progress (or lack thereof), or the timing of the legal proceedings (see Example 1).

Example 1

Oscar's story "[…] In fact, it was a long time ago, in 1969; Um ... I have come a long way since this event, but um each time I talk about it, um, I feel a strong emotion, what um, the first times I came to the discussion group, in fact after a while I stopped coming because it was too upsetting, well, maybe not enough, um and I dealt with all this".

Next to the large table is a space with about eight observers (students, psychologists, members of the participants' family, academics, etc.) who never speak. From an interactional perspective, the president of the association takes on the role of moderator; she places herself in a specific structuring position rather than that of a master, she knows the history of each participant viii , and she gives them the floor in such a way that each participant may respond to the others' stories. Speakers are free to talk about themselves; they are not given any preliminary instruction regarding time or narrative form, and they can stress whatever they deem to be important. The interactional situation is complex. Charaudeau (1984) stresses that the speakers talk about themselves to present third parties -namely the moderator president, other speakers sharing the same experience (e.g. sharing the experience of being a victim), other speakers with a different profile (e.g. aggressors talking in front of victims), and the group as a whole -and to absent third parties -namely the observers called absent because they cannot take part in the dialogue although they are present physically and witness the discussions. In these discussion groups, participants speak in turn and listen to the others; the purpose of the discussions is for people to use this opportunity to find the necessary support and to make pedophiles, victims, and users of child pornography videos meet so that some may be able to go beyond self towards others; these speakers can show the other participants that they possess the necessary inner resources to understand, and potentially face, the Other who could, for instance, represent a victim's aggressor. There are limits to this approach, however. This is a type of discourse whereby speakers are entirely free; it is far removed from the individual interviews (whether structured or semistructured) as practiced in sociology or sociolinguistics. The moderator makes little use of the restatement strategy, does not request speakers to elaborate on the post-rape personal changes or, for instance, the dissemination of child pornography material, and she refrains from asking participants to undertake any introspective articulation of the social consequences of actions. The chosen place for these discussions (i.e. the home of the president of the association) is another issue to consider: does it help people tell their story? Does it liberate people's voice? Finally, and most importantly, the discussion group brings together only people the president thought would benefit from the encounter and who could bring something to the others. The president thus acts as a filter that does not exclude those aggressors who have done a considerable amount of self-work and who, therefore, can contribute to the discussions, but does exclude entirely the repeat perverse pedophiles.

Favorable technology-related conditions making for new practices in the dissemination of child pornography videos

The level of IT equipment (computer-tablet-smartphone) is steadily growing x . People's computer equipment level has been relatively stable since 2012 (82%) whereas the proportion of home Internet connections is in constant growth (85%, +2 points). In particular, the mobile equipment of touchscreen tablets and smartphones is expanding apace, respectively +5 points and +7 points in 2016. Four out of 10 French citizens now possess a tablet, and 65% own a smartphone, which implies new Internet mobility patterns and the development of mobile phone-based digital practices. At the same time, Web 2.0 (also called the participative or social Web) brings together "a wide variety of platforms such as blogs (Bloggers, Thumbler), Wikis (Wikipedia, Wikitravel), digital social networks (Facebook, LinkedIn), microblogs (Twitter, Jaiku), collective bookmarking, and media content sharing sites for photos, music, and videos" (Proulx et al., 2012 : 4).

Finally, the high rate of IT equipment and new uses are boosted by the spread of very high speed broadband: 42% of mobile phone owners claim they use 4G as opposed to 14% in 2014, which indicates a three-fold increase over two years. In addition, the reduced cost of Internet access has helped expand private access to pornography and child pornography xi sites: "though it is difficult to quantify the presence of pornography on the Internet, there is no doubt that it is significant" (Dubois, 2014: 48). Moreover, the phenomenon of grooming has emerged; the term refers to those people who use instant messaging to communicate with minors, sometimes under the guise of being a youngster of similar age, to manipulate them and drive them into performing acts of a sexual nature.

The corpus and the theoretical framework

The sensitive corpus and transcription conventions

The corpus contains about 70 hours of recording. To our knowledge, this emerging sensitive corpus (Paveau & Pérea, 2012/2015) is not part of the institutionalized research practices (Maingueneau, 1991: 11); it deals mainly with issues of gender and/or security (public health, home security, cyber defense, confidentiality of sensitive, military, nuclear data, etc.). It also involves a number of prohibitions, which necessarily raises questions regarding the researchers' practices and research focus (see the work of Perea xii and Damian xiii ). Paveau's work on pornographic discourse starts from the notion that Talking about pornography involves considering texts and speech from the perspective of form, hence describing words, expressions, and ways of saying but also textual traditions, script construction, and the ways in which this discourse is situated, whether it is ignored, accepted, or rejected in the set of social discourses.

(2014: 25) The corpus of recordings has been entirely transcribed, using the transcription conventions selected for the CFPP2000 project (12)(13)(14)) xiv as the latter appeared to share the same concerns regarding corpus processing and analysis methods.

Any transcription is necessarily an awkward compromise between respecting the distinctive features of speech and being intelligible. Since the corpus gives access to the oral version, we have adopted an orthographic transcription in order to facilitate reading and simplify the use of correlation software. The transcription code is mainly that of DELIC xv , which has been slightly adapted. We transcribe words using accepted spelling without correcting deviations from the norm when these correspond to a known French morpheme.

The corpus involves exclusively male speech; a correspondence factor analysis (CFA) classified the men into three groups. xvi Members of Group 1 (Édouard and Tom) focus upon their difficult relationships with women, their need to use marijuana, and their perceptions of children who are the only ones who care for them. Members of Group 2 (Albert and Alain) stress the time they spend on the Internet and their inability to understand the problem caused by their attraction to children. Members of Group 3 (Thomas, Louis, and Jules) call into question the label "pedophile" because they are attracted to images of young girls with no assault whatsoever (see Table 3 showing the positive and negative characteristics and the frequency of some nouns and verbs, placed in the Appendix).

Conceptual framework

We assume that speech is based upon the social product of a language but displays its individual use (Dubois et al., 1994 : 267); it is a matter of discourse as defined by [START_REF] Foucault | Les mots et les choses : une archéologie des sciences humaines[END_REF], both language and materiality, symptom and sign, surface and core, and by bringing together words, objects, and practices (Mol, 2009 : 30), speaking begins a process of care because it refers to the willingness to "get involved in practical activities" (Mol, 2009 : 148). Speaking is the bottom line shared by the participants to the discussion groups; hence, it can be seen as a legitimate vector for the analysis of the speakers' mechanisms of experience. In this study, the words uttered by the participants and then transcribed and made anonymous are called discourse. We believe that "discourse is not just constituted of sets of words and phrases; it establishes places where standards and values are defined along with regulations and prohibitions, likes and dislikes, qualities and defects, identities, legitimacy, glory and disgrace" (Paveau, 2014: 25). The French form of discourse analysis is used to analyze the speakers' discourse; it is based upon the sets of utterances and the assumption that there are invariants, linguistic and discursive specific properties that make it possible to observe distinct configurations of this type of narrative. In addition, the argumentative dimension of these narratives is highlighted, and an instrumented textometric approach to corpora analysis is used to combine quantitative syntheses and text-based analyses [START_REF] Lebart | Statistique textuelle[END_REF]. [START_REF] Barbet | Débats pour l'Élysée[END_REF] asserts that even less so than the frequency of a word, the recurring presence of segments cannot be naively attributed to chance: either it signals a syntactic constraint, or it indicates a determination or semantic option. [START_REF] Mayaffre | Nicolas Sarkozy[END_REF] further claims that through textometry, it is possible

• to analyze a large number of textual documents that neither eyes nor memory could take into account in such a flood of words, • to call attention to the features that are revealed only in terms of a large set,

• to provide the precision, systematicity, comprehensiveness, or objectivity that are necessary for a scientific description, • to open avenues for historical interpretation that are distant from those induced by contemporary events, our own assumptions, our favorite topics, since the tool itself feels neither like nor dislike. (2012, 14-16)

Putting into words the ill-being of users of child pornography videos

The speakers' personal stories fall within a particular quest, the search for information to understand what made them fall into this type of attraction to these types of images. Two kinds of patterns are investigated in this study: enunciative regularities of speech (see 3.1) and two discourse markers, que/parce que/that/because and mais/but, which shape the textuality of this discourse type (see 3.2).

Salient enunciative regularities of speech

Personal stories lie within a deictic construction whereby the speaker speaks as an empirical speaking subject (i.e. a man who voices utterances), as a discourse-producing speaker (i.e. a person invited by the president of the association), and as an enunciator (i.e. a person who, through his personal history, is given the status of possessor and user of child pornography videos) who is responsible for the discourse (Ducrot, 1983). This enunciator takes responsibility for his words through the first-person markers Je, me, moi/ I, me, myself (1360 occurrences), but he also constructs an escalation through the process of emphasizing, using the tonic pronoun, or stressed pronoun (i.e. moi/me; in this case, an English equivalent would be the structure "as for me") to strengthen the subject pronoun je/I, namely moi je/me I (31 occurrences).

Example 3

Édouard's story But all this to say that in the end, children, me I cannot stand them, really.

Louis' story

And me I saw myself as a monster, I saw myself as the monster who was seeking children.

This enunciative phenomenon is complemented by special nominative phenomena. To name is not merely to represent the world as it is, or facts as they are; rather, it is to provide a specific viewpoint, to take a particular position regarding the subject through designating it or characterizing it in a certain way by adding nuances of evaluation, distance, domination, discrimination, proximity, or judgement, depending on one's perceptions of that subject [START_REF] Siblot | Nommer, c'est déjà prédiquer[END_REF]. Table 2 shows that designations used for the video users fall into the categories of hypernym (aggressor) and objective hyponyms (pedophile, sex offender, abuser) and subjective designations (monster, cybercriminal, criminal). Some designations of the minor (the child) are objective (young girl, boy, minor, victim) while others are more subjective (kid/youngster). The designations of the deviant act and its coreferents deliberately refer to a vague temporal space [START_REF] Brunner | Le vague / die Vagheit : du mot au concept, pragmatique et folk linguistique[END_REF]) without starting or ending markers but deictically constructed (my story, this event, in my memory) and to plurisignifying abstract designations (thing) and cataphoric pronouns (ça/this). Enunciators thus put into words the way in which they refer to the circumstances that drove them to child pornography by using vague designations, hence not naming them explicitly, as Example 4 shows.

Example 4

Jules' story On the last page of the paper, there was a naked little girl and me, for a long time, I thought that it was this that prompted me a little to go and look for well child pornography sexual characters.

Alain's story I mean, in fact, our relationship has not changed from what it was before, before this event. So, from that moment on, I could really start work, which now has ... helped me.

Designations that are both signifying (self-designation and designation of the child) and vague (designation of a moment viewed as a reference or awareness-raising point) and that are constructed with enunciative people markers indicate that the discourse is fully appropriated by the speaker, which contributes to reaffirming and making acceptable and useful for himself and for others his contribution to the discussion group.

Two discourse organizers

This section presents the two most common discourse organizers (i.e. que/parce que/that/because and mais/but) xviii that shape the textuality of the discourse, namely what makes a text into a signifying linguistic set, "a coherent set that progresses towards an end and that presents successfully a complete meaning" (Détrie, Siblot, & Verine, 2001: 349). They also contribute to clarifying the causal relationship linking the speakers to their actions. The argumentative dimension of the discourse thus constituted is not meant to persuade us that the speakers are aggressors or not, nor is it to persuade us that they are telling the truth. Instead, we should view this dimension according to the broader notion of argumentation xix as stated by Amossy (2000: 37): "the verbal resources called upon in a wording designed to act upon the receivers to make them accept an argument, to modify or strengthen the representations and opinions held, or merely to influence their ways of thinking or to raise questions regarding a specific issue."

The discourse organizer que/parce que/that/because

The analysis of speech connectors throughout the corpus shows that the use of que/parce que/that/because in personal narratives introduces strong nominalization or adjectivalization (Figure 2 shows 790 occurrences), the meaning of which is to express the speakers' self-justification (que je, parce que/that I, because). The most frequent conjunction in the corpus is one that is used to express cause and/or selfjustification. Argument using cause and effect. Usually, several causes are suggested; in their narratives, speakers use the notion of addiction (mainly to marijuana, naturism, pornography, information technology) to talk about the moment when justice takes its course (seizure of computers, complaints, temporary custody, detention).

Jules' story

Like others here, I got into child pornography through print publications, then Internet sites for naturists because I liked seeing naked children, that got me off.

Argument refuting causal links. Through this type of refutation, speakers reject the responsibility of a repeat offence onto the current prison institution.

Alain's story

Avoiding going to prison that's also important because I don't think that the prison environment err helps you change [...] Well I thought I was really in the shit because I knew I was doing it again, I was looking too much at naked kids.

Argument using low self-esteem

Thomas' story If a girl comes, and speaks to me, well at a push I'm going to push her away because me I don't deserve her.

The discourse organizer mais/but

The speakers' narratives also construct an asymmetrical discourse [START_REF] Ducrot | Les mots pour le dire, les éditions de minuit[END_REF] expressed through the conjunctive mais/but. Figure 4 shows that it refers to different semantic values xx such as opposition (in 29% of cases), restriction (in 22% of cases), refutation (in 7% of cases), justification (in 5% of cases), and explanation (in 4% of cases). It also refers to different discourse features such as markers of oral expression (in 15% of cases) and of thematization (in 9% of cases). These effects operate from a pivot word or phrase that is the reference point. When constructing x but y, y represents a stronger argument that guides towards the author's conclusion, at the expense of x.

Opposition value

The conjunctive but constructed with an antonym placed afterwards (e.g. minor vs. major) emphasizes the deviant nature of the relationship and contributes to the speaker's awareness.

Albert's story

The very young chick who pretended to be 18 because of course they can't write it underneath [on the sites] but who in fact was a minor and so then I stopped the thing.

In the next example showing the notion of opposition (the boss vs. the others), the main characteristic of the others, which is to ignore Édouard's activities, involves a kind of relief.

Édouard's story

In fact, I'm working in an organization where the boss knows that I use child pornography films but the others don't know it and that's good because otherwise they would make me pay for it.

Restriction value

The category girls then young girls constructs a lesser situation for the speaker than the youngest [girls] from 10 to 13 years of age, which is much more explicit in this discourse situation and is emphasized by the intensity marker but especially.

Alain's story I've always been attracted to girls, young girls, but especially to the youngest from 10 to 13 years of age.

Explanation value

In the next two examples, the value of the pivot words ugly or suicide is lessened by complements placed after the conjunctive but, hence constructing a specifying explanation.

Oscar's story I wasn't worth anything, I felt ugly, not physically but inside.

[…] I didn't try to commit suicide, not really, but I had suicidal behaviors.

Refutation value

In the following examples, the refutation of shaking leads to emphasizing the intensity of the body's reaction.

Albert's story OK, so, I'm taking my clothes off I'm shaking but not shivering, I mean real shaking like an illness, athetosis, well I was terrified.

[…] At a push, I can understand the effect of watching the child who is masturbating, but why are you sharing the video afterwards? That's what's I'm wondering about, in fact (question from a participant).

Justification value

The speaker justifies himself by downplaying the significance of his acts in two ways: first it is (only) a matter of photos of naked children (hence, he does not believe there is any transgression), and second, in a naturist environment, there is a kind of normalcy or social acceptability to watching naked bodies.

Alain's story

In fact, it was only a matter of photos of naked children, but in a naturist environment, it was the excuse, so to speak, it's true, I was aroused.

Markers of closure of a syntactic unit in speech

The conjunctive but associated with the exclamations well and hey (but well, but hey) indicates a short statement that signals the end of a sequence.

Marker of thematization

The conjunctive but may be a thematic marker referring to elements that are not strictly speaking related to the cotext but that depend upon a shared frame of reference.

Albert's story

Watching porn is the only way for me to climax; sorry to be so crude, but well, it's really the only way.

Louis' story

We were talking about people, the Anonymous on the Internet who set traps, but that's it, they made pedophiles pay, and they wanted to get pedophiles.

At different moments in the narrative, these two markers construct a multiplicity of values (e.g. overkill, refutation, opposition, restriction, justification, consequence, emphasis) that place the speaker within a situation of moral instability constructed with emotion-laden markers emphasizing ill-being (Pugniere-Saavedra to appear) accompanied by numerous elements of reiteration.

Conclusion

Talking about oneself as user or even distributor of child pornography videos mainly involves describing circumstances (e.g. one's history with children, one's relation to sex among adults, to games, to technology) and the speaker's psychology (e.g. constructing one's sexuality, self-image, self-confidence).

Using the specific process of discussion groups, the analysis shows the various discourse patterns used, both lexical and semantic (through designations, characterizations), syntactic, or enunciative (deictic markers). 1 Note: Translating some morphemes such as the French "ne" or "l'" would not be relevant to the discussion. Hence, we decided not to translate the items in the table. The author will willingly discuss these and any other aspect of the article in answer to individual queries. The contact address is: Frederic.pugniere-saavedra@univ-ubs.fr

i The term life narrative is chosen over confidence, which, etymologically, refers to fidelity, the trust given to someone in whom one confides in the private sphere. We believe that confidences involve self-disclosure that is expected in some types of interactions and even compulsory in some institutional contexts (e.g. the confessional, doctor's or psychanalyst's office, job interview, testifying in court). In addition to those revealed in the private sphere, some confidences are disclosed in the public sphere where secrecy is broken by the number of potential recipients, such as autobiographies, memoirs, political interviews published in newspapers, confessions given on the couch of a TV studio or some TV reality shows, and more recently in blogs or dating websites. The interactional situation in the present study is different, even though sometimes, at some points, these life narratives involve a certain element of self-disclosure.

ii The International Classification of Diseases (ICD) maintained by the World Health Organization (WHO) classifies pedophilia in the category of Disorders of sexual preference (F65), itself included in the Disorders of adult personality and behavior (F60-F69). Pedophilia (F65.4) is defined as "A sexual preference for children, boys or girls or both, usually of prepubertal or early pubertal age".

iii The victim's age brings different designations: Nepiophilia: sexual preference for infants and toddlers (aged 0-12 months) Infantophilia: sexual preference for infants (aged 1-6 years) Pedophilia: sexual preference for prepubertal children Hebephilia: sexual preference for pubescent youths (aged 12-14 years). Ephebophilia: sexual preference for adolescents. iv The total corpus comprises victims [35%], aggressors [20%], and people who possess and use child pornography material [26%], men who have sexual impulses but do not act on those [4 %], close relatives of aggressors [3 %], and sexually abstinent pedophiles [12 %]. The first three profiles account for 81% of participants in these discussion groups. v The Convention on the Rights of the Child defines child pornography as "any representation, by whatever means, of a child engaged in real or simulated explicit sexual activities or any representation of the sexual parts of a child for primarily sexual purposes" and cyber child pornography as recording, possessing, and distributing pornographic images representing minors on the Internet. Child pornography faces sanctions under Article 227-23 of the Criminal code whereby fixing, recording, or transmitting the pornographic image of a minor and distributing that image is punishable by five years' imprisonment and a fine of 75,000 euros. When the image or the representation involves a 15-year-old minor, punishment is meted out, even if the distribution of this image or representation was not intended. vi The social policies of the 1990s have led to the emergence of support centers to mitigate forms of suffering through giving people a voice for therapeutic purposes [START_REF] Fassin | Des maux indicibles. Sociologie des lieux d'écoute[END_REF]. Listening is widely acknowledged to provide a political response that is both humanistic and immediately visible. Listening to the Other becomes an act. As (Fassin 2004:73) puts it, "beyond the requirements of the selected procedure, the interest shown in people's histories and the attention given to their misfortunes and their woes are part of a particular way of dealing with social relations through a kind of intimist concern that creates a space for special relationships. Sometimes, these places are the only ones where people can talk about themselves and what means the most to them. This is how, beyond the traditional circles of socialization constituted by family, friends, co-workers, or neighbors, close links are constructed in an institutional -though also personal -relationship of support and solidarity". vii http://ange-bleu.com/en/support-us viii The president may be contacted by email and telephone, and she receives all the calls from speakers in distress (victims, aggressors or possessors and/or users of child pornography material); from these conversations with the speakers, she decides whether their participation to the discussion group could benefit them and/or the group, or not. ix Organizing face-to-face encounters between victims and aggressors brings to mind the notion of so-called restorative justice with its four-fold purpose: to identify the damage caused to the victim and take action to repair it; to involve all interested parties; to reintegrate the author into society; and to transform the traditional relations in order to restore social peace. The DELIC team (working on a corpus computerized linguistic description under the leadership of Jean Véronis) is a previous support team (EA 3779) at the University of Provence. It is now called TALEP (referring to the automatic treatment of written and spoken language), a team created in the computer science laboratory of Marseille from the fusion of the previous CALN (automatic natural language processing) team led by Paul Sabatier from LIF and part of the DELIC team. xvi In the three years the corpus was built, no female person participated in the discussion group as possessor and user of child pornography videos.

xvii Each speaker is anonymized, though the first name gender is retained. xviii The Trameur (a software system for the statistical analysis of annotated text data) classifies the following connectors in descending order of frequency: que/parce que/that/because (780), mais/but (236), et/and (190), quand/when, (161), si/if (115), etc. xix In 1958, Perelman and Olbrechts-Tyteca proposed the following definition of argumentation: "The subject of argumentation theory is the study of the discourse techniques enabling people to create or increase others' adherence to the propositions that are presented for assent" (1958:5). xx See Figure 4 showing the distribution of occurrences of mais/but.

Figure 2 :

 2 Figure 2: Distribution of que/parce que

Figure 3 :

 3 Figure 3: Speaker-based distribution of the most frequent speech connectors

  La Défense and err my wife calls me because the police is ringing at the door and seizing our computers because they were searching for ... well there's a complaint.Thomas' story I was using marijuana a lot, in fact ... with time... I've realized that it was simply because I wanted to run away from my problems […] Now, I still bet online a lot but I don't derive as much pleasure because it's just to run away in fact.

Figure 4 :

 4 Figure 4: Semantic distribution of mais/but

  paid even more because precisely I would have hidden behind this thing ... exactly ...But well, what was bound to happen, did happen. Édouard's story I downloaded photos from the Internet, she [his wife] is right, I cheated on her. I desired other people. I masturbated while thinking about others, but hey, I still love her, and that's what matters, right?

  x The Research Center for the Study of the Conditions of Life (CREDOC) publishes its annual study "Digital barometer" undertaken for the General Council of the Economy (CGE), the French Regulatory Authority for Electronic and Postal Communications (ARCEP), and the Agency for Digital Technology. See "The 2016 Study, The use of digital technology in France", by Thomas Coëffé, 2016, December 2nd. https://www.blogdumoderateur.com/barometre-numerique-france-2016-credoc/ xi Hentaï (Japanese pornographic mangas) depict "lolicons" and "shotacons". xii In November 2013, François Perea, Associate Professor of Language Sciences at the University Paul Valéry Montpellier 3, defended his Accreditation to supervise research (HDR) on the theme of sensitive topics. xiii In September 2015, Béatrice Damian, Associate Professor of Information and Communication Sciences at the Institute of Political Studies in Rennes, defended her Accreditation to supervise research (HDR) on the theme of sexual fantasies for men through the use of women's bodies. xiv S. Branca-Rosoff, S. Fleury, F. Lefeuvre, M. Pires, Discours sur la ville. Corpus de Français Parlé Parisien des années 2000 (CFPP2000), http://ed268.univ-paris3.fr/CFPP2000/ xv

ix Example 2 Marie's story "Yes, my name is Marie, today is the third time I've come. It's the third time and every time I learn ..."

  Speaker=moderatorOf course, it's always like this, discussion groups.

They're all different. Even if you speak every time.

	[…]
	"Even today, he [the person who raped her when she was 14
	years old]

had power over me, and I ... and so, now, with all this work, the fact that I come here, that I understand, and all, well a little, the pro-the different profiles around the table, that I understand a little what goes on in people's heads, in their heads, in the heads of these people, and that I have identified what he was, what he was capable of doing … ".

Table 1 :

 1 Distribution of speakers and number of forms

	Anonymized	Number of forms
	speaker xvii
	Alain	1025
	Albert	549
	Jules	663
	Thomas	510
	Édouard	1591
	Tom	262
	Louis	830

Figure 1 : Correspondence factor analysis of speakers

Table 2 :

 2 Signifying designations

	Self-designations of the user of	Designation of the minor	Designation of the act with its
	child pornography videos		coreferents
	Pedophile(s) (38), monster (6),	Child (children) 71, girl(s)	This (513), thing (35), story
	aggressor (4), sex offender (4),	63, girl (31) youngster(s) 28,	(4), this event (7), act (7),
	abuser (2), cybercriminal (2),	[male] kid(s) 16, [female]	memory (2).
	criminal (2).	kid(s) 12, victim 11, boy(s) 9,
		minor 3, little girl(s) 2.

  The argumentative dimension of discourse raises questions among both the possessors of child pornography videos and the discussion group participants. The enunciative constructions that designate pedophiles and the argumentative constructions triggered by because and but seek to act upon the receivers by helping them understand and evaluate more or less implicit mechanisms. Questions are raised regarding constructions designed to instrumentalize naturism, refute detention conditions, confuse listeners by displaying one's vulnerable aspects, minimize one's actions through emphasizing the children's age. Do these constructions contribute to making these life narratives more acceptable? Do they enable the speakers to reach greater self-awareness and awareness from others?

	c'était 16		pas -4	28 ans	40 dire
		j'ai 14	envie -4	27 fois	28 être
	euh… 13		Elle -5	17 filles	19 faire
		c'est 11		de -5	15 peur	9 voir
		ben 9	femme -7	14 moment	8 arrêter
	pédophile 7		euh -14	14 fille	6 regarder
		mail 7		ne -14	13 amis	5 consommer
		m'a 7		ai -19	11 vidéos	5 travailler
		ouais 6		est -23	10 Internet	4 aider
					10 besoin	4 donner
		Tom and Edouard stories		Tom and Edouard stories
	item	ind-specif +	Item	ind-specif-	frequency Subst.	frequency	verbe
		C'est 24		dire -6	43 femme	30 être
	femme 13	était -7	31 enfants	26 dire
		lui 11	étais -8	25 fois	24 faire
		Elle 10		je -9	20 ans	21 voir
	qu'elle 9		ne -10	18 temps	19 avoir
		Ben 9	avais -12	18 chose	13 aller
		mais 9		est -17	17 choses	12 regarder
	enfants 7		euh -22	17 moment	6 arrêter
		et 7		ai -29	14 besoin	5 changer
					11 cannabis	5 revoir

Appendix