Gilles Montambaux
email: gilles.montambaux@u-psud.fr

Generalized Stefan-Boltzmann law

Keywords:

come

Introduction

In 1879, J. Stefan found experimentally that the total power emitted by a black body scales as the fourth power of the temperature. [START_REF] Stefan | Über die beziehung zwischen der wärmestrahlung und der temperatur[END_REF] This law was derived theoretically by L. Boltzmann in 1884, [START_REF] Boltzmann | Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie[END_REF] without any knowledge on the spectral repartition of the black body emission which was discovered later by M. Planck. [START_REF] Planck | Über das gesetz der energieverteilung im normalspectrum[END_REF] Nowadays, the now well-established expression of the Stefan-Boltzmann law with its correct prefactor [START_REF]Stefan-Boltzmann law is the expression for the emitted power per unit area P(T ) = c 4 u(T ) ≡ σT 4 where σ is named the Stefan-Boltzmann constant[END_REF]

u(T ) = π 2 k 4 B 15 3 c 3 T 4 , (1)
where u = U/V is the energy per volume unit, is simply deduced in most textbooks from the Planck law integrated over frequencies. It is however interesting to recall that the total emission can be derived from purely classical and thermodynamical grounds, as Boltzmann did. The main two ingredients for this derivation are the second law of thermodynamics and the result obtained by J.C. Maxwell[5] that the radiation exerts a pressure p(T ) proportional to the total energy density, p(T ) = 1 3 u(T ) (This pressure is now understood as the pressure of massless particles which are the photons). The 1/3 factor, obtained by Boltzmann, results from angular averaging in 3D and is indeed the inverse of the space dimensionality.

In this paper, we show how the Stefan-Boltzmann (SB) law can be generalized to any gas of free particles, and we stress the importance of a third ingredient for its derivation, a zero chemical potential, as it was implicit in the Boltzmann argument. In particular, we show how the simple relation between pressure and internal energy for a gas of massive particles p(T ) = 2 3 u(T ) leads directly, on purely classical grounds, to the temperature dependence of the total energy of a 3D saturated Bose gas, that is an ideal gas below the Bose-Einstein condensation temperature.

More generally, we consider free gases of particles with general, including hybrid dispersion relations, and show that the classical argument applies not only to the saturated Bose gas in various geometries (box or harmonic potentials), but also to gases of Fermions in various situations under the condition that the Fermi energy is temperature independent, which is the case for halffilled band systems. Such systems will be called "compensated Fermi gases". We regroup in the same framework the thermodynamic properties of many different bosonic or fermionic free gases, which until now were described in completely different contexts. The outline of the paper is the following. In next section, we recall the main elements of the Boltzmann derivation, including homogeneity arguments. In section 3, we generalize it to massive particles and show that this classical derivation accounts for the thermodynamics of the saturated Bose gas. More general spectra are considered in section 4, where we discuss specifically the case of the saturated Bose gas in various geometries including boxed or harmonic potentials. In section 5, we consider gases of particles with hybrid spectrum, that is massive in some directions, massless in the other directions, and we present several examples including bosons in mixed boxed-harmonic potential or fermions with exotic dispersion relation (section 6). For particles which are massive in d m directions and massless in d c directions, the generalized Stefan-Bolzmann law may be summarized in the form

u(T ) ∝ (mc 2 ) d-κ (hc) d (k B T ) κ+1 , (2)
where

d = d c + d m and κ = d c + d m /2.
We conclude by the presentation of a table with many different physical situations described by this generalized Stefan-Boltzmann law.

Stefan-Boltzmann law

We first briefly recall the derivation of this law as first theoretically derived by Boltzmann. [START_REF] Boltzmann | Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie[END_REF] We start by considering the entropy S(U, V ). The variation of entropy for an infinitesimal transformation is

T dS = dU + pdV = ∂U ∂T dT + p + ∂U ∂V dV . (3)
Since the entropy is a total differential, the equality between cross derivatives leads to:

∂p ∂T = p + u T . (4)
Inserting the Maxwell-Boltzmann relation p(T ) = u(T )/3 in Eq.( 4) gives immediately

1 4 du u = dT T , (5)
leading to the Stefan law u(T ) = K 1 T 4 . Implicitly, in Eq.( 3) the chemical potential is set to zero, which is essential for the demonstration here. Of course the correct numerical factor can only be obtained from integration over frequencies of the spectral function u(ν, T ) and requires the famous form of the Planck law discovered later in 1900. [START_REF] Planck | Über das gesetz der energieverteilung im normalspectrum[END_REF] However, the combination of universal constants could be obtained simply by dimensional arguments, where a quantity having the dimension of an action is needed. The Planck constant was not introduced at that time, but the existence of a natural unit for action does not require quantization hypothesis. Indeed such natural unit was already introduced in 1899 by Planck [START_REF] Planck | Über irreversible strahlungsvorgänge[END_REF](see Appendix A), so that Eq.( 1) without the numerical factor can be obtained from purely classical arguments, h being a natural unit of action obtained experimentally [START_REF] Planck | Über irreversible strahlungsvorgänge[END_REF](see also [7]).

We now turn to the generalization of the SB law to a gas of massive particles with zero chemical potential.

Massive particles

Since the unique assumptions for deriving SB law are the thermodynamic principles and the relation between internal energy and pressure, it is interesting to reconsider now the Boltzmann derivation in the case of massive particles. However an important remark is in order. Contrary to photons whose number is not fixed, the number of massive particles is in general fixed (in average) and controlled by the chemical potential µ. Eq.(3) should indeed be written as

T dS = dU + pdV -µdN . (6)
Therefore, the use of Eq.( 3) implies that the chemical potential is zero, which is actually the case for a gas of photons, leading to the SB law. If one now considers a gas of massive particles, we know from Maxwell-Boltzmann kinetic theory of gases that p and u are related by p = 2 3 u. The Boltzmann derivation still holds under the condition that µ = 0. In this case, from (4) one gets, instead of (5): 2 5

du u = dT T (7)
so that u(T ) = K 2 T 5/2 . A dimensional argument which necessarily involves an action, the Boltzmann constant and a mass (instead of a velocity as before) leads to

u(T ) ∝ m 3/2 h 3 (k B T ) 5/2 (8)
This behavior is actually well-known: this is the internal energy of the 3D saturated Bose gas, that is an ideal gas below the Bose condensation temperature, or of a gas magnon excitations in a 3D ferromagnet. We stress that this expression does not require the knowledge of the thermal repartition of energy levels and no quantum mechanical input: the introduction of h as a unit of action in the prefactor does not imply any quantization. It is therefore quite interesting that the thermodynamic of the saturated gas can be described in the framework of classical physics, with the unique hypothesis that µ = 0. More generally, the Maxwell-Boltzmann kinetic theory of gases can be used for any dispersion relation between energy and momentum of the form ∝ p ν in dimension d, and leads to the general relation u(T ) = κp(T ) with κ = d/ν (ν = 1 for photons and ν = 2 for massive particles). The thermodynamics of such gases is described below.

General spectrum

Consider now any gas of particles at zero chemical potential whose relation between pressure and energy is given a linear relation u(T ) = κp(T ). Then Eqs. [START_REF] Maxwell | A treatise on electricity and magnetism[END_REF]7) have to be replaced by [START_REF] De Lima | Such a generalization has also been discussed[END_REF]

1 κ + 1 du u = dT T . (9)
Integration of this relation together with dimensionality arguments leads to the connection between energy-pressure relation and the generalized Stefan-Boltzmann law:

u(T ) = κp(T ) =⇒ u(T ) ∝ (mc 2 ) d-κ (hc) d (k B T ) κ+1 (10)
which generalizes Eqs. [START_REF] Stefan | Über die beziehung zwischen der wärmestrahlung und der temperatur[END_REF][START_REF] De Lima | Such a generalization has also been discussed[END_REF].

We propose here a quicker, simpler and more general derivation : instead of pressure, which may not be well defined in certain situations that we will consider below, we consider the grand potential Ω, and suppose a relation of the form U = -κΩ which generalizes the relation u = κp. Then the thermodynamic relation between the grand potential and the energy (this is an interesting exercise to show that the following relation is equivalent to the thermodynamic relation Ω = U -T S -µN , see Appendix B):

U = ∂βΩ ∂β α (11)
where α = βµ = µ/(k B T ), together with U = -κΩ, leads immediately to

∂βU βU α = -κ ∂β β . (12)
Integration of this relation, combined with dimensional arguments, gives

U (T ) = F (α) L hc d (mc 2 ) d-κ (k B T ) κ+1 , (13)
where F (α) is a multiplicative integration function. Here no assumption is made on the structure of F (α) (which is actually an integral of Bose or Fermi factors [START_REF](α) = g 1+κ (e α ) = Li 1+κ (e α ) for Bosons and F (α) = f 1+κ (e α ) = -Li 1+κ (-e α ) for Fermions, where Liν (z) is the polylogarithm function[END_REF]). The generalized Boltzmann law [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF] follows immediately when the chemical potential vanishes. Finally since the number of particles N (T ) is given by N (T ) = -∂βΩ ∂α β

, we find quite generally that

U (T ) = κ F (α) F (α) N (T )k B T . (14)
In the following, we introduce the thermal de Broglie wavelength λ dB = h/(2πmk B T ) 1/2 and another characteristic wavelength for a massless spectrum λ W = hc/k B T , that we may call "Wien wavelength" since it separates classical and quantum behaviors. Then, the total number of particles can be rewritten as (within a multiplicative numerical constant):

N (T ) = F (α) V λ 2(d-κ) dB λ 2κ-d W ( 15
)
and the total energy is given by ( 14) which are forms of the generalized SB law [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF]. The relations [START_REF] Montambaux | A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal[END_REF][START_REF] Banerjee | Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point[END_REF] are general, for any α. Then two cases are of interest:

• For the classical ideal gas where the particle number N is fixed and U (T ) = κN k B T , one deduces that F (α) = e α , and from (15) one finds the relation for the chemical potential

µ = -k B T ln V N λ 2(d-κ) dB λ 2κ-d W ( 16
)
which generalizes the case of massive particles (κ = d/2).

• When α = 0, for example for the saturated ideal gas where the chemical potential is fixed to 0, we get the generalized Stefan-Boltzmann law. Then the number of particles above the condensate N (T ) varies with temperature, and we have the relations

U (T ) = κN (T )k B T , S(T ) = (κ + 1)N (T )k B . (17)
For more complex dispersion relations as hybrid dispersion relations discussed later in sections 5, 6, the calculation of pressure from kinetic theory may be difficult to achieve because of complex angular averages. The pressure may even not be properly defined. Then we may obtain the coefficient κ and deduce the SB law directly from the energy dependence of the density of states ρ( ), when it is a power law. Here we rather consider the integrated density of states (also called counting function) denoted as N < ( ). We start with an integrated density of states of the form N < ( ) = B κ , where B has necessarily the dimensions

B ∝ L d (mc 2 ) d-κ (hc) d . ( 18
)
When α = 0, the total energy U (T ) has the form

U (T ) = ρ( )f (β )d (19)
where no assumption is needed on the form on the function f except that it is a function of the single combination β (which is Wien's displacement law for light). This immediately leads to a power law T κ+1 for the total energy and to the relation U (T ) = -κΩ(T ) between energy and grand potential. Therefore, we have the following correspondence

ρ( ) ∝ κ-1 ⇐⇒ U = -κΩ ⇐⇒ U ∝ T κ+1 . ( 20
)
This is the main result of this paper. It may appear quite natural since the grand potential is actually a double integral of the density of states. It has however a profound origin which is Wien's displacement law which is valid for any gas of zero chemical potential. Since it is obtained from purely classical arguments, no assumption is made on the nature of the particles. The argument applies as well for Bosons as for Fermions under the condition that µ = 0. In the following, we review many different physical systems whose thermodynamics is unified in this unique framework.

Until now, we have only considered a free gas in a box of volume V . Particularly important in the context of Bose-Einstein condensation (BEC), is the case of a Bose gas trapped in a harmonic potential. To avoid any detailed calculations, it is useful to remark that the density of states of massive particles in a harmonic trap is the same as the one of free massless particles. This is of course reminiscent of the wave-particle duality that the radiation spectrum can be described either as an ensemble of resonators or as a gas of massless particles. The substitution to be done is

πc L → ω or L λ W → k B T 2 ω (21)
and we have κ = d. Therefore the integrated density of states has the form, for an isotropic harmonic well

N < ( ) ∝ ω d . (22)
Again has been introduced from homogeneity, but the scaling dependence of the density of states does not need quantization. The above argument leads to

N (T ) ∝ k B T ω d , U (T ) ∝ N (T )k B T (23)
Notice that in this case Maxwell-Boltzmann argument cannot be used directly since volume and pressure are obviously not defined. However, Ref. [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF] defines a "harmonic volume" V = 1/ω 3 , consistent with the above substitution (21), and the conjugated "harmonic pressure" P = -∂Ω/∂V. [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF] 5 Hybrid spectrum Up to now, we have only considered situations of either massless or massive particles, so that obviously a mass and a velocity cannot enter both in [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF][START_REF] Wunsch | Dirac-point engineering and topological phase transitions in honeycomb optical lattices[END_REF][START_REF] Banerjee | Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point[END_REF]. One has respectively either κ = d or κ = d/2, so that

N (T ) = V λ d W or N (T ) = V λ d dB . ( 24
)
Interesting is the more general situation of a hybrid spectrum, massive in d m directions and massless in d c directions. In this case, the Maxwell relations u = 3p for massless particles or u = 3p/2 for massive particles can be generalized to

u = d c + d m 2 p . (25)
Quite generally, assume a dispersion relation written in the short dimensionless notation

∼ j A j n νj j (26)
with ν j = 2 along d m massive directions and ν j = 1 along d c massless directions. A j have the dimensions of an energy and n j are positive integer numbers. The integrated density of states is still a power

N < ( ) = B κ with κ = j 1 ν j = d c + d m 2 ( 27
)
and B is given by Eq.( 18). Therefore the number of particles has the form

N (T ) ∝ V λ dm dB λ dc W (28)
and the expressions for energy and pressure result immediately.

For completeness, we consider the hybrid situation of a saturated Bose gas in a potential with a box (or size L) profile in d m directions and an harmonic profile in d c directions. The substitution (21) in Eq.(28) gives

N (T ) = L λ dB dm k B T ω dc . ( 29
)
6 Compensated Fermi gas

Another situation where the Boltzmann derivation can be applied is the one of a Fermi gas, under the condition that µ = 0. This peculiar situation occurs when the Fermi level is pinned at a neutrality point with electron-hole symmetry. This is for example the case for Dirac massless fermions, extensively studied to describe electronic states of graphene at half-filling, where the Fermi energy lies in the middle of the energy spectrum and the chemical potential is independent of temperature. We denote this system as a "compensated Fermi gas". The T = 0K filled lower band plays the role of the condensate, and the finite T excited electrons play the role of the saturated gas (Fig. 1). Consider first the case of 2D Dirac fermions like in graphene, for which d = 2, ν = 1, so that κ = 2 (the density of states is linear). The general relation [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF] gives immediately

u(T ) ∝ (k B T ) 3 (hc) 2 (30)
leading to the well-known specific heat ∝ T 2 of graphene for the half-filled band. [START_REF] Srivastava | Thermodynamic Equations of State for Dirac and Majorana Fermions[END_REF] Similarly, in bilayer graphene, the spectrum is quadratic so that ν = 2 and κ = 1, and from [START_REF] Romero-Rochín | Equation of state of an interacting Bose gas confined by a harmonic trap: the role of the "harmonic" pressure[END_REF] we have

u(T ) ∝ m h 2 (k B T ) 2 . ( 31
)
The above arguments can be applied to the now recently extensively studied Weyl fermions in semi-metals. [START_REF] Rao | Weyl semi-metals: a short review[END_REF] Consider Weyl Fermions in d dimensions. The energy spectrum being linear, one has ν = 1, and κ = d, so that

u(T ) ∝ (k B T ) d+1 (hc) d (32)
The thermodynamics of Weyl Fermions is therefore the same as the one of the black body. Finally, we remind that for an intrinsic compensated semiconductor in 3D, in the limit of a vanishing gap, the number of elecrons in the conduction band varies as

N (T ) ∝ m 3/2 h 3 (k B T ) 3/2 (33)
which is also a form of the Stefan-Boltzmann law. It has been shown that under appropriate lattice distortion, the two Dirac points characteristic of the graphene electronic spectrum can merge into a single point, where the spectrum stays linear in one direction but becomes quadratic along the other direction. [START_REF] Wunsch | Dirac-point engineering and topological phase transitions in honeycomb optical lattices[END_REF][START_REF] Montambaux | A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal[END_REF] This hybrid point has been called a "semi-Dirac" point. [START_REF] Banerjee | Tight-Binding Modeling and Low-Energy Behavior of the Semi-Dirac Point[END_REF] This is an interesting example of an hybrid spectrum characterized by both a mass and a velocity (Fig. 2).

= ± p 2 x 2m 2 + c 2 p 2 y ( 34
)
whose thermodynamic properties have been studied. [START_REF] Wunsch | Dirac-point engineering and topological phase transitions in honeycomb optical lattices[END_REF][START_REF] Montambaux | A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal[END_REF] This is special case of a general dispersion relation of the form

∼ ± j A j n 2νj j (35)
which leads to the same energy dependence of the density of states as a spectrum of the form (26) (see Appendix C), and the energy density if given by Eq.( 10) where κ = d c + d m /2 = 3/2. The thermodynamic properties of a semi-Dirac Fermi spectrum are therefore the same as the saturated Bose gas in a mixed boxed-harmonic potential. More generally a gas of Weyl fermions close to a merging point, the spectrum can transform from a full linear dispersion relation to a linear-linear-massive, or a linear-massive-massive, to a full massive spectrum. The corresponding thermodynamics is summarized in Table 1.

Fig. 3 Semi-Dirac fermions in 2D (let), a Bose gas in a 2D harmonic-boxed potentiel (right ), a 3D Bose gas in a boxed potential, or a compensated semiconductor in 3D (limit of small gap) obey the same Stefan-Boltzmann law.

Conclusion

We have generalized the Boltzmann derivation of the Stefan-Boltzmann law to ideal gases of any kind of particles, under the condition that the chemical potential stays zero. From the Maxwell-Boltzmann relation between energy and pressure, one simply recovers the thermodynamic properties of a saturated gas of massive particle below the Bose-Einstein condensation temperature. Then we generalize the derivation to the case of hybrid dispersion relations, massive in some directions, massless in other directions. This situation is achieved in Bose gases confined in hybrid boxed-harmonic potentials. We have also shown that fermionic gases with particle-hole symmmetry behave indeed as saturated gases with fixed chemical potential. This remark leads to draw similarities between quite different problems like a semi-Dirac Fermi gas and a 2D Bose gas in a mixed boxed-harmonic potential. The recently studied physics of Weyl points in 3D is even richer since the dispersion relation can be linear in two directions and massive in a third direction, a situation thermodynamically similar to a 3D Bose gas confined in a harmonic trap in two directions and a boxed potential in the third direction. The unifying picture is summarized in Table 1.

κ = 1 κ = 3/2 κ = 2 κ = 5/2 κ = 3 d=2 black body
massive fermions semi-Dirac fermions massless fermions

N (T ) ∝ L λ 2 dB N (T ) ∝ S λ dB λ W N (T ) ∝ S λ 2 W d=2 BEC + + BEC + + N (T ) ∝ L λ dB k B T ω N (T ) ∝ k B T ω 2 d=3
black body F-magnons phonons, AF-magnons massive fermions hybrid MML fermions hybrid MLL fermions massless fermions

N (T ) ∝ V λ 3 dB N (T ) ∝ V λ 2 dB λ W N (T ) ∝ V λ dB λ 2 W N (T ) ∝ V λ 3 W d=3 BEC + + + BEC + + + BEC + + + BEC + + + N (T ) ∝ V λ 3 dB N (T ) ∝ S λ 2 dB k B T ω N (T ) ∝ L λ dB k B T ω 2 N (T ) ∝ k B T ω 3 u(T ) T 2 T 5/2 T 3 T 7/2 T 4
Table 1 This table summarizes the thermodynamic properties of various massive, massless or hybrid saturated (µ = 0) gases. The properties depend both on the space dimensionality d and the proportionality coefficient κ between internal energy u(T ) and pressure p(T ). The symbols and denote respectively a box potential or a harmonic potential in a given direction. The examples of bosons are shown in normal characters, and the example of fermions are denoted in italics. MML denotes a massive-massive-massless spectrum, MLL a massive-linear-linear spectrum. One has N (T ) ∝ T κ and U (T ) ∝ T κ+1 .

A Planck's natural unit of action

In his 1899 paper, [START_REF] Planck | Über irreversible strahlungsvorgänge[END_REF] more than one year before his discovery of the black body law with quantization of energy exchange, M. Planck found a natural unit of action from his analysis of the Stefan-Boltzmann law and Wien's law:

u(ν, T ) = 8πb c 3 ν 3 e -aν/T (36)
where a and b are parameters to be fixed from the experiment. Comparing with experimental estimate of the total energy

u(T ) = ∞ 0 u(ν, T )dν = 48πb a 4 c 3 T 4 (37)
and Wien's displacement law, Planck could extract the parameters b/a 4 and a, and obtain the values b = 6.885 10 -34 J.s and a = 0.4818 10 -10 K.s. This was the first estimate of a unit of action, which was not yet related to any idea of quantization, and which will be denoted later as h in the 1900 discovery of the black body law. Moreover, in our modern notations, a = h/k B so that the obtained value of a leads to an estimate of k B = 1.43 10 -23 J/K (that Planck will introduce one year later in his seminal paper).

B Thermodynamic relations

Here we show the equivalence between these to relations

U = ∂βΩ ∂β α ⇐⇒ Ω = U -T S -µN (38)
One first explicitly write the derivative as

U = β ∂Ω ∂β α + Ω = -T ∂Ω ∂T α + Ω (39)
The next step is to rewrite ∂Ω ∂T α

. We first write the grand potential Ω as (k B = 1 in this section)

Ω α = µ T , T . (40)
Consider the derivatives

∂Ω ∂µ T = 1 T ∂Ω ∂α T (41) ∂Ω ∂T µ = - µ T 2 ∂Ω ∂α T + ∂Ω ∂T α (42)
Inserting the first equation into the second one in order to eliminate α, we obtain

T ∂Ω ∂T α = µ ∂Ω ∂µ T + T ∂Ω ∂T µ = -µN -T S (43)
Inserting this last relation in eq. ( 39), we find the expected result.

C Exact thermodynamic results

We give here for completeness the exact expressions for the density of states and thermodynamic quantities of gases of particles in various dimensions and geometries.

C.1 Integrated density of states

Here we give complete expressions fot the integrated density of states for particles with various dispersion relations.

• Free massive particles, = p 2 2m :

N<( ) = C d L 2π d 2m 2 d/2 . ( 44
)
where

C d = π d/2 /Γ (1 + d/2
) is the unit volume in d dimensions.

• Free massless particles, = |p|c :

N<( ) = C d L 2π d c d . (45)
• Particles in a isotropic harmonic potential, = (nx + ny + nz) ω :

N<( ) = 1 d! ω d . (46)
• 2D semi-Dirac fermions = with κ = dc +dm/2. The previous equations (44,45,46,48) may be obtained from this formula with ν j = 1, A j = πc j /L j or A j = ω j ν j = 2, A j = ( 2 π 2 /(2m j L 2 j ), and the integrated density of states may be summarized in the form:

N<( ) = κ Γ (1 + κ) dc j=1 L j π c j dm j=1 2πm h 2 L j ( 50
)
for a massless spectrum in dc directions and in the form:

N<( ) = κ Γ (1 + κ) dc j=1 1 ω j dm j=1 2πm h 2 L j . ( 51
)
for a harmonic spectrum in dc directions.

• Particles with dispersion relation = j (A j n j ) 2ν j , n j > 0:

C.2 Bosons and Fermions with zero µ

Here we collect the exact thermodynamic quantities for the saturated Bose gas and the compensated Fermi gas (µ = 0). We start from a power law for the integrated density of states , N<( ) = B κ . The total energy is

U (T ) = Bκ ∞ 0 κ e ν -α ± 1 d . ( 54
)
For the saturated Bose gas, the number of particles and the total energy read:

Fig. 1

 1 Fig.1Left : a Bose gas having N 0 particles in the ground state. At finite temperature, N (T ) particules are excited and form the saturated Bose gas, keeping the chemical potential µ = 0, until the Bose-Einstein temperature Tc is reached when N (Tc) = N 0 . Right: An electron compensated gas transfers N (T ) electrons from the valence band to the conduction band, keeping µ = 0 when the spectrum is symmetric. There is no transition in this case since the reservoir is infinite.

Fig. 2

 2 Fig. 2 Stefan-Boltzmann law for compensated Fermions, massive, massless and hybrid particles

•

  Particles with dispersion relation = j A j n ν j j , n j > 0 :

  expressions of A j , the integrated density of states can be summarized as: same as for the dispersion = j A j n ν j j , within different numerical factors.

[ 16 ]Uacknowledgments

 16 N (T ) = BΓ (1 + κ)ζ(κ) T κ U (T ) = κ ζ(κ + 1) ζ(κ) N (T )k B T .(55)For the compensated Fermi gas, we haveN (T ) = BΓ (1 + κ)(1 -2 1-κ )ζ(κ) T κThe author acknowledges useful comments form J. Dalibard, J.-N. Fuchs and M.-O. Goerbig.


3D reconstruction of poral network at nanoscale from 2D slices.

Q. CORLAY a , P. COLLON b , F. GOLFIER c , K. KALO d , A-J. TINET e a. Université de Lorraine, CNRS, GeoRessources, F -54000 Nancy, France b. Université de Lorraine, CNRS, GeoRessources, F -54000 Nancy, France pauline.collon@univ-lorraine.fr c. Université de Lorraine, CNRS, GeoRessources, F -54000 Nancy, France fabrice.golfier@univ-lorraine.fr d. Université de Lorraine, CNRS, GeoRessources, F -54000 Nancy, France kassem.kalo@univ-lorraine.fr e. Université de Lorraine, CNRS, GeoRessources, F -54000 Nancy, France anne-julie.tinet@univ-lorraine.fr

Résumé :

En France, les formations d'argilites sont des candidates privilégiées pour le stockage géologique de déchets haute-teneur du fait, en particulier, de leur faible perméabilité. Cependant, de façon à mieux prédire le comportement de tels milieux, la caractérisation du réseau poral est essentielle. De tels matériaux à faible perméabilité peuvent avoir une forte porosité totale mais la majorité de celle-ci est constituée de pores de

Abstract :

In France, claystone geological formations are privileged candidates for deep underground nuclear waste storage, due, in particular, to their low permeability ability. However, in order to better predict the behavior of such media, the characterization of the pore network is essential. Such low permeability materials may have a high overall porosity but the majority of this porosity is constituted by pores smaller than 100 nm and whose distribution is still poorly known. The connectivity and the topology of these pores influence the porous media properties, for instance in regards to gas transport. In order to better understand the pore network structure, imaging techniques have been used to provide 2D and 3D images, such as microtomography or FIB-SEM. The latter allows a nanometric resolution (5 to 20nm) at the expense of the total imaged volume (usually a few µm3). Such a sample size is usually below the REV (Representative elementary volume), especially in regards to transport properties. In this project, we explore the possibility to distance the imaged 2D slices and use a numeric method to reconstitute the 3D volume at the REV scale. To this end, we study a stack of 180 parallel FIB-SEM slices of a synthetic clay of nanometric porosity. We compare two methods, based on multiple-point statistic algorithm (MPS), to reconstruct a coherent 3D configuration of its porous structure from 2D sections with different sampling spaces. The results of the reconstruction based on sparse data are then compared to the initial volume in regards to global characteristicssuch as porosity -, morphological parameterssuch as pore size distribution and connectivity index -as well as Lattice Boltzmann flow simulations to assess the capacity of the reconstruction method to provide satisfactory results. The first method, using a slice sequential reconstitution, produces results with low sensitivity to conditioning data but induces planar effects. The second method, aggregating values of three orthogonal directions allows a better reconstruction of the pore structure. This method is, however, more sensitive to conditioning data. The permeability values resulting from flow simulations on the second method are similar to those obtained on the total image for a distance between conditioning slices up to 11 cells. Thus, by reducing the number of direct images to acquire, this method opens the possibility of exploring volumes 5 to 10 times larger than those currently analyzed, therefore leading to a better representativity of the porous medium.

Mots clés: milieu nanoporeux, reconstruction, morphologie, échelle du pore

Introduction

The study of fluid transport in nanoporous geomaterials is essential for various applications such as gas storage (CO2, H2), geological storage of radioactive water or non-conventional reservoirs exploitation. In order to assess the properties of a nanoporous material, a DRP (Digital Rock Physics) approach may be undertaken. Such an approach requires the acquisition of a 3D image, subsequent image treatment so as to perform a numerical upscaling of the physical properties of the medium (permeability, dispersivity, etc.). For geomaterials, a commonly used image acquisition technology is the microtomography. However, the resolution of microtomography is usually of around 1 µm, recent techniques allowing resolution of 0.3 µm. In nanoporous materials such as clayrock or shales, pore size distribution range is often below such resolution. Recently, SEM (Scanning Electron Microscopy), which allows nanometric resolution of 2D slices, has been coupled with FIB (Focused Ion Beams) to allow a 3D description at a few nanometers resolution [START_REF] Holzer | 3Dmicrostructure analysis of hydrated bentonite with cryo-stabilized pore water[END_REF]- [START_REF] Song | Multi-scale pore structure of COx claystone: Towards the prediction of fluid transport[END_REF]. This technique, however, remains rather difficult and is constrained by the total size of the image (usually a few µm 3 ), leading to questioning in regards to the representativity of the imaged pore structure, especially when considering more complex behavior such a multi-phase flow or reactive transport. We therefore evaluate the capacity of spacing the imaged slices and numerically reconstructing the 3D volume at a larger scale to accurately recover the desired properties of the material. To this end, we will test two different reconstruction methods ( [START_REF] Okabe | Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[END_REF], [START_REF] Comunian | 3D multiple-point statistics simulation using 2D training images[END_REF]) on a fully-known set of FIB-SEM images of synthetic clay (from [START_REF] Gaboreau | Optimization of pore-network characterization of a compacted clay material by TEM and FIB/SEM imaging[END_REF]). In order to assess the performance of the approaches, the reconstructed results will be compared through a variety of morphological descriptors and effective properties.

Materials and Method

Initial data

The sample studied is a synthetic clay, i.e. compacted illite powder, with an overall porosity of 32%.

The sample has been studied and imaged by [START_REF] Gaboreau | Optimization of pore-network characterization of a compacted clay material by TEM and FIB/SEM imaging[END_REF]. The image obtained from the FIB-SEM has then been segmented by a specific method as described in [START_REF] Gaboreau | Optimization of pore-network characterization of a compacted clay material by TEM and FIB/SEM imaging[END_REF]. For the purpose of the present study, we used the segmented image of size 1096x1095x180 voxels with a resolution of 5nm provided by [START_REF] Gaboreau | Optimization of pore-network characterization of a compacted clay material by TEM and FIB/SEM imaging[END_REF] as reference data. Considering the size of this domain and the subsequent computing time, we only considered a set of 180x180x180 voxels for development purpose within this study.

Reconstruction methods

Multiple reconstruction approaches exist and have been applied in geological context [START_REF] Okabe | Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[END_REF], [START_REF] Comunian | 3D multiple-point statistics simulation using 2D training images[END_REF], [START_REF] Wu | Reconstruction of 3D porous media using multiple-point statistics based on a 3D training image[END_REF]- [START_REF] Chen | Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity[END_REF]. Most of these works were tested on microtomography images such as ones of sandstone. In the present work, we will focus on two Multiple Point Statistics (MPS) approaches based on the work of [START_REF] Okabe | Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[END_REF] (the 3DA-MPS algorithm) and [START_REF] Comunian | 3D multiple-point statistics simulation using 2D training images[END_REF] (the s2Dcd algorithm). Both approaches were tested using the SKUA-Gocad software. A first step in reconstruction methods is the acquisition of training images (TI). A training image is essentially a conceptual model that should try to include all pore structure patterns for porous media.

The training image can be 2D or 3D and may be obtained through imaging techniques such as SEM and FIB-SEM. For MPS approaches, a template should be defined to capture multi-point statistics.

The template is used to scan the TI and collect patterns. Following [START_REF] Okabe | Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[END_REF], a 9x9 template is used in the present study. The s2Dcd (sequential 2D conditioning data) algorithm is based on a sequential 2D-MPS simulation along a given surface (among the three possible directions) using the training image that describes heterogeneity along the direction of this surface and conditioned by the points already simulated in that surface or given as conditioning data at the beginning of the simulation. The 3DA-MPS (3D Aggregation -Multi-Point Statistics) algorithm reconstructs the 3D volume node by node and constrained by the porosity. At first, the probability of a given node to be associated to void space is equal to porosity, then, when sufficient conditioning data becomes available, measured statistics from the TI are being used. For each node randomly picked in the domain, a data event is extracted in each direction. In each direction, a probability value is computed using a classic direct sampling.

In the present work, the reconstruction will be based on training images from the three directions and conditioning data will be selected from the full 180x180x180 images with different spacing in one direction (every 5, 11, 22 images). For each simulation method and set of parameters, 5 reconstructed samples were generated to assess the variability of the results. Computing time was around 20 minutes for s2Dcd and 40 minutes for 3DA-MPS method, though could be improved using high-performance computing.

Morphological characterization

In order to assess the performance of the reconstruction methods, a set of morphological descriptors will be used. The descriptors were chosen as they are usually considered relevant for key physical processes such as flow, multiphase flow, transport (including reactive transport), etc.

• Porosity • Pore Size Distribution (PSD) was computed using the XLib plug-in of ImageJ [START_REF] Münch | Contradicting geometrical concepts in pore size analysis attained with electron microscopy and mercury intrusion[END_REF] using both the continuous PSD and the discrete PSD approaches. • Euler characteristic, which provides a first idea of the connectivity of the medium, was computed using the MorphoLibJ plug-in of ImageJ [START_REF] Legland | MorphoLibJ: Integrated library and plugins for mathematical morphology with ImageJ[END_REF] • Isolated pores characteristics such as volume and specific surfaces were also obtained using the MorphoLibJ package • Percolating pores, i.e. pores connecting two opposite faces of the reconstructed images, were obtained using the Object Counter process of ImageJ • A connectivity index is calculated as the ratio of the larger pore volume to the total pore volume.

On top of that, permeability of the reconstructed samples was computed using the LBM software Sailfish [START_REF] Januszewski | Sailfish: A flexible multi-GPU implementation of the lattice Boltzmann method[END_REF].

Results and discussion

The morphological descriptors representing D10, D50 and D90 (as obtained from the PSD), I (connectivity index) as well as the porosity are summarized as ratio to the reference values in the below Table. For both reconstruction methods, the result is degraded with the distance of the conditioning data for most of the descriptors. For porosity and connectivity index, this degradation is more significant for the 3DA-MPS approach than for the s2Dcd algorithm. On the contrary, the PSD for 3DA-MPS is closer to the reference PSD than s2Dcd. When the spacing between conditioning slices is larger than the template size (9x9), some larger pores are no longer recovered.

Algorithm

The ratio of permeability of the reconstructed medium to the permeability of the real medium is studied for the 3DA-MPS data. For a spacing between conditional data of 5 voxels, this ratio is close to one in each flow directions. At a spacing of 11 cells, this ratio decreases to an average of 0.6, the reconstructed permeability being lower than the real permeability. The result is even more degraded for higher distances between conditioning data. Indeed, the pores tend to be disconnected under such conditions, leading to a significantly lower permeability. These results therefore validate the 3DA-MPS method of reconstructing the porous medium for well-conditioned simulations: distances between conditioning slices being less than the template simulation size, here set at 9 cells.