Valentin Gauthier

Agnès Arnould

Hakim Belhaouari

Sébastien Horna

Michel Perrin

Mathieu Poudret

Jean-Francois Rainaud

A Topological Approach for Automated Unstructured Meshing of Complex Reservoir

come

Topological based sub-surface model

Principle :

• A topological structure includes :

-Topological cells (vertices, edges, faces, volumes) -Neighbor relationship between topological cells Benefit :

• 1 rich model that contains all information : 2 geometries (the deposit domain and the actual domain), semantic labels (fault lips, surface name . . . )

• 1 model mathematically defined that the consistency can be guaranteed

Workflow

Geological Interpretation

The data collected by seismic and drilling survey are interpreted by geologist to determine the location of geological events (horizons, faults, onlap. . . ). The RESQML standard is used to store all interpretations.

Structural part

The Geosiris GeoTopoModeler prototype provides tools to specify the various contacts of the structural model (horizon/horizon and horizon/fault contacts) and to accordingly operate the necessary intersections and the removing of the parts with the "STOPS AT" process.

3D Meshing

Linking top and bottom surfaces to compute 3D mesh, from the deposit domain.

The fault position does not exist in the deposit domain, but it is necessary for the linking process.

From the actual domain, the fault is interpolated in the deposit domain by using fault lips as control points.

After the interpolation, the fault has a first geometry for the hanging wall, and a second for the footer wall. The topological model offers the possibility to keep topological relationship between both sides of the fault, with different geometry for both sides.

In the deposit space, the top and bottom are vertically linked far from faults. Near the faults, pillars extremities are delimited by faults. This extremities are interpolated to the actual domain.

2DMeshing A new quad surface is created based on structural surfaces. A regular grid is projected (in the deposit domain) on each horizon. Advantage : The 2D mesh is the same for all surfaces (excepted at fault lips).

Block Extraction

The semantics labels and geometries are used to define a dedicated simplification operation. This operation allows the individuals blocks extraction, from the resulting 3D mesh.

Results

Conclusion

The topological approach makes keeps the mesh consistency. Thus, with a single model, all information related to the mesh is stored in the same structure. It is also easy to extract information to work on these result by using the RESQML format.

Future works

To finalize the mesh, the blocks must be linked together. Each faces along faults must be splited to match faces of the oposit block.

  A Topological Approach for Automated Unstructured Meshing of Complex Reservoir V. Gauthier 1 , A. Arnould 1 , H. Belhaouari 1 , S. Horna 1 , M. Perrin 2 , M. Poudret 2 & J.F. Rainaud 2 1 University of Poitiers, XLim , 2 Geosiris A Topological Approach for Automated Unstructured Meshing of Complex Reservoir V. Gauthier 1 , A. Arnould 1 , H. Belhaouari 1 , S. Horna 1 , M. Perrin 2 , M. Poudret 2 & J.F. Rainaud 2 1 University of Poitiers, XLim , 2 Geosiris

	Objectives
	Sub-surface meshing for fluid-simulation.
	Originality :
	• Automated meshing based on geological interpretation
	• Model consistency preserved
	Image à insérer!


Fabienne Michelini
email: fabienne.michelini@im2np.fr

Adeline Crépieux

Katawoura Beltako

Entropy production in photovoltaicthermoelectric nanodevices from the non-equilibrium Green's function formalism

Keywords: radiation laws, quantum thermodynamics, quantum dots, photovoltaics, non-equilibrium green's functions

HAL is

Introduction

Nanoscale structures still reveals new insights and potential applications which alter our vision of technology. Deviceintegrated nanostructures fundamentally change the transport properties of electrons through quantum effects like tunneling, confinement or entanglement. As a result, manybody quantum methodologies are developing for practical purposes, such as for modeling electronic devices, like transistors [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF], and more recently energy conversion systems like photovoltaics [START_REF] Aeberhard | [END_REF]3], optomechanics [4] or thermoelectrics [5,6]. The formalism of non-equilibrium Green's function nanotechnology-enabled concepts accelerates the development of NEGF-based approaches [24,25]. Another possible application follows the idea of cogeneration, which combines outputted energy forms from a single sustainable energy source, like the simultaneous production of electrical and thermal energies from a single light source. At the commercial level, stacked photovoltaic-thermoelectric macroscopic modules have been created and analyzed [26,27]. Perovskites show potentials both as photovoltaics [28] and thermoelectrics [29], which suggests that a combined energy conversion with these materials could be a success [30]. Nanoscale architectures have been theoretically proposed for cooling using a photon source [31][32][33][34], and for joint cooling and electrical energy production [35]. These proposals address the idea of cogeneration inside a unique nanoscale module, which requires a deeper look at the quantum aspects of the energy conversion involving light-matter interaction. This perspective is related to the thermodynamics of light [36] which has naturally emerged in photovoltaics regarding the photon source as a thermal bath [START_REF] Würfel | Physics of Solar Cells[END_REF], but also the contact as a thermoelectric junction between the absorber and the lead [START_REF] Humphrey | [END_REF]39]. In these nanoscale proposals, reversibility remains a cornerstone as it defines the maximal efficiency, and the efficiency at maximum power achievable [40]: the entropy production rate is a crucial ingredient of device working. However, entropy cannot be represented by a Hermitian operator in quantum physics.

In this work, we derive the photon energy and particle currents in open nanosystems interacting with light using the framework of NEGFs. The Hamiltonian model is introduced in section 2, and the main steps and consequences of the derivation are given in section 3. This allows to calculate the entropy current flowing from the electron and photon reservoirs to the absorbing region of the device in section 4: we reshape and discuss the entropy production in terms of efficiencies for photovoltaic-thermoelectric nanodevices. In section 5, we finally examine a quantum-dot based architecture illuminated with a monochromatic radiation. Using a two-level model, we show that the entropy production rate is always positive at any coupling to electron reservoirs as long as one considers the same radiation properties for the spontaneous emission. However, we find that the entropy production rate can reach negative values if modifications are made inside the approach, as it is usually done for photovoltaic device simulations.

Hamiltonian model

The Hamiltonian of a quantum nanosystem in contact with electronic reservoirs and interacting with light reads

= + + + + + γ H H H H H H H , T L R 0 i nt (1)
where

† ε ∑ = H dd , n n n n 0 (2a) † ∑ ∑ = + α α α = H V c d h.c., T L R n n n n k k k , (2b) † ∑ = ζ ζ ζ H M d d A , nm nm n m q q q int , , (2c)
{ } † ε ∑ = α α α α ∈ H c c , L R n n n n k k k k , (2d) ħ † ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∑ ω = + γ ζ ζ ζ ζ
H a a 1 2 .

q q q q (2e) Subscript 0 stands for the non-interacting and isolated nanosystem, T for the transfer to the electron reservoirs, int for the interaction with light, α for the the left (L) and right (R) electron reservoirs, and γ for the photon bath. These expressions use the electron creation (annihilation) operators, † d n (d n ) in the central region and † α c nk ( α c nk ) in the electron reservoirs. On the other side, † ζ a q ( ζ a q ) and † = +

ζ ζ ζ-
A a a q q q are photon operators of the photon bath, with q the wave vector of the radiation, and ζ one of the two directions of polarization perpendicular to the propagation. The interacting central region is coupled to the electron reservoirs via parameters α V nk while it is coupled to the light radiation via parameters ζ M nm q , . Usually, calculation for optoelectronics relies on the dipole approximation: ≪ ⋅ q r 1 where r is the spatial coordinate [START_REF] Mandel | Optical Coherence and Quantum Optics[END_REF].

We introduce the notations used in this paper:

⟨ ⟩ = - β β I H Ė the energy current, ⟨ ⟩ = - β β I N ˙ with { } † = ∑ β β β ∈ N c c L R n n n k k k , and † = ∑ γ ζ ζ ζ
N a a q q q the particle current, and finally

⟨ ⟩ ⟨ ⟩ µ µ = - + = - β β β β β β β I H N I I ˙ḣ E p
for the heat current [12], where µ β is the (electro)chemical potential of reservoir β. In the case of electron reservoirs, we will also use β∈ I L R e , for the electrical current. All currents are flowing from the β γ ∈ L R , , reservoir to the central region.

Photon currents

Photon energy current

We derived the formal expression of the photon energy current

( ) ⟨ ⟩( ) = - γ γ I t H t Ė
inside an optoelectronic device following the first order Born approximation within the Keldysh formalism [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF][START_REF] Mahan | Many-Particle Physics 3rd edn[END_REF]. From the Heisenberg equation

[ ] ħ = - γ γ H H H ˙, i
, the energy current can be expressed in terms of expectation values on mixed operators which combine electron and photon operators

ħ ħ ( ) ( ) B ∑ ω = γ ζ ζ ζ ζ < I t t t M 1 Re , , E q q q q
(3)

where

( ) ⟨ ( ) ( ) ( )⟩ † B = ζ ζ < t t d t d t B t , i mn n m q q , with ( ) = ζ B t q ( ) ( ) † - ζ ζ- a t a t q q .
Here we use matrix forms to encode level and/or eventual space-discretization indices. In the framework of the Keldysh formalism, we sought the expression of the contour ordered mixed Green's function

( ) ⟨ ( ) ( ) ( )⟩ † B τ τ τ τ τ = - ′ ′ ζ ζ Td d B , i , mn t m n q q , ( 4
)
where T is the time-ordering operator. The main steps of the derivation follow the first order Born approximation [START_REF] Mahan | Many-Particle Physics 3rd edn[END_REF], which consists in switching to the interaction picture, developing the time evolution operator up to the second order in the electron-boson interaction parameter, using Wick's theorem 3 , verifying the cancellation of the disconnected graphs, including higher order contributions with self-consistency, and finally performing the Langreth's rules for the analytic continuation. We thus obtain

ħ ( ) ( ) ( ) B ∫ ∑ ω ττ τ τ τ τ τ Ξ = ′ ′ ζ ζ ζ ζ γ < M G , Tr d , , , t t q q q q 1 1 1 (5) with ħ ( ) ( ) ( ) ∑ τ τ ω ττ τ τ Ξ = ∼ γ ζ ζ ζ ζ ζ D M G M , , , . t t t q q q q q 1 1 0 1 (6)
The expression of ( )

B ζ < t t , mn q ,
is then deduced from the Langreth's rules, which finally provides the photon energy current from equation (3),

∫ Ξ Ξ = + γ γ γ < < I t t t t t t t t t t G G 1 Re Tr d , , , , . E r a 1 1 1 1 1 ħ ( ) [ ( ) ( ) ( ) ( )] (7)
These expressions use the standard Green's functions for the electrons inside the central region

( ) τ τ′ G , t , defined as ( ) ⟨ ( ) ( )⟩ † τ τ τ τ = - ′ ′ G Td d , i nm t n m
. On the other side, we introduce the photon Green's function

( ) ⟨ ( ) ( )⟩ τ τ τ τ = - ∼ ζ ζ ζ D T B A , i t q q q 0 1 1 . Function ( ) τ τ ∼ ζ D , t q 0 1 differs in a negative sign from τ τ = ζ D , t q 0 1 ( ) 〈 ( ) ( )〉 τ - ζ ζ TA t A i q q 1
which appears in the derivation of the Dyson's equation for an electron interacting with bosons [START_REF] Mahan | Many-Particle Physics 3rd edn[END_REF].

For steady-state devices, we obtain

ħ [ ] ( ) ε ε ∫ π Ξ Ξ = + γ γ γ < < I G G 1 Re Tr d 2 , E r a ( 8
)
where

∓ ∓ ∑ ω ω ω Ξ = ± + ± γ ζ ζ ζ ζ ζ ζ ζ ζ ε ε ε ⏞ N N M G G M 1 ,
q q q q q q q q absorption contribution spontaneous emission contribution

ħ ħ ħ ( ) [ ( ) ( ) ( )] ≶ ≶ ≶ (9)
and

( ) [ ( ) ( )] ( ) ( ) ∫ π Ξ Ξ Ξ Ξ Ξ = ± - + - - ′ ′ ′ ′ γ γ γ γ γ > < > < ε ε ε ε ε ε ε ε P i 1 2 2 d , r a , (10)
ζ N q is the occupation number of the radiation modes ζ q. These modes form the photon bath which is assumed to be in an equilibrium or quasi-equilibrium state of temperature γ T and chemical potential µ γ . The chemical potential can indeed be nonzero when the photon source cannot be modeled by a black body, like the radiation obtained from optical transitions in semiconducting matter [START_REF] Würfel | Physics of Solar Cells[END_REF]. The function

( ) ε Ξ γ
has the dimension of energy 2 .

We separate the three essential contributions of the electron-photon interaction in equation ( 9): the two induced processes which include the absorption (abs) and the stimulated emission (em,st), and the spontaneous emission (em,sp) which is independent of the occupation number of the photon bath and is non-zero in the vacuum state [START_REF] Mandel | Optical Coherence and Quantum Optics[END_REF]. The function Ξ γ is split as

≶ ≶ ≶ ≶ Ξ Ξ Ξ Ξ = + + γ , abs em st em sp , , (11)
with

ħ ħ ( ) ( ) ≶ ≶ ∓ ∑ ω ω Ξ = ± ζ ζ ζ ζ ζ ζ ε ε N M G M ,
abs q q q q q q (12a)

ħ ħ ( ) ( ) ≶ ≶ ∓ ∑ ω ω Ξ = ± ζ ζ ζ ζ ζ ζ ε ε N M G M ,
em st q q q q q q , (12b)

∓ ∑ ω ω Ξ = ± ζ ζ ζ ζ ζ ε ε M G M . em sp q q q q q , ħ ħ ( ) ( ) ≶ ≶ (12c)
The photon energy current thus inherits the Ξ γ splitting, and it can be cast according to different viewpoints

= + + γ I I I I E abs E em st E em sp E , , (13)
= + I I E em sp E ind , (14)
= + I I . abs E em E (15)

Energy conservation

In the case of the self-consistent Born approximation, the energy has to be conserved in the total system [START_REF] Baym | [END_REF] ⟨

⟩ ⟨ ⟩ = + + + + + = γ H H H H H H H ˙˙˙˙˙˙˙0 . L R T 0 i nt (16)
The two first terms are known from [12],

[ ] ( ) εε ε ∫ Σ Σ = - + α α α ∈ < < H G G ˙Re Tr d L R h r a , 2 with Σ α < a
, the reservoir self-energies [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF]. Energy currents related to the central region, to the transfer process and to light-matter interaction are zero for steady-state operating:

⟨ ⟩ = H ˙0 0 , ⟨ ⟩ = H ˙0 T , and ⟨ ⟩ = H ˙0 int
. We verified energy conservation starting from the photon energy current of equation [START_REF] Hughes | [END_REF] together with the expressions of ⟨ ⟩ H ˙L and ⟨ ⟩ H ˙R given in [12]. In order to eliminate the reservoir self-energies ≶ Σ L R , , we used the property

[ ] Σ Σ - = < > > < G G Tr 0 where ≶ ≶ Σ Σ = ∑ α α ∈L R , ,int
is the total self-energy [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF]. Then, we evidenced the following

quantity ħ ( ) ( ) ≶ ≶ ε ε ε ω Σ Σ - ∼ γ ζ γ q
, and performed the changes of integration variable

ħ ε ε ω = ± ′ ζ q .

Absorption and emission rates

Similarly, the photon current ( )

⟨ ⟩( ) = - γ γ I t
r a 1 1 1 1 1 ħ ( ) [ ( ) ( ) ( ) ( )] (17)
with

( ) ( ) ( ) ( ) ( ) ≶ ≶ ≶ ∑ Σ = ∼ ∼ ′ ′ ′ ′ γ ζ ζ ζ ζ t t t t t D t t t M G M , , , . q q q
q 0 (18) For steady-state operation, we obtained

ħ [ ] ( ) ε ε ∫ π Σ Σ = + ∼ ∼ γ γ γ < < I G G 1 Re Tr d 2 ,
r a (19) where

ħ ħ [ ( ) ( ) ( ) ] ≶ ≶ ≶ ∓ ∓ ∑ ω ω Σ = ± + ± ∼ γ ζ ζ ζ ζ ζ ζ ζ ε ε N N M G G M 1 , q q q q q q q (20) and still ( ) [ ( ) ( )] ( ) ( )
ε ε ε ε ε ε ε ε P ∫ π Σ Σ Σ Σ Σ = ± - + - - ∼ ∼ ∼ ∼ ∼ ′ ′ ′ ′ γ γ γ γ γ > < > < i 1 2 2 d . r a , (21)
It is worth comparing the function Σ ∼ γ with the interaction self-energy Σ γ which appears in the Dyson equation for an electron interacting with light radiation [23,[START_REF] Mahan | Many-Particle Physics 3rd edn[END_REF]

∓ ∑ ω ω Σ = + + ± γ ζ ζ ζ ζ ζ ζ ζ ε ε ε N N M G G M 1 . q q q q q q q ħ ħ ( ) [ ( ) ( ) ( ) ] ≶ ≶ ≶ (22)
Sign changes between Σ ∼ γ < and Σ γ < are intuitive: absorption(emission) means that a photon is flowing from the photon bath(central region) to the central region(photon bath). Without these sign changes, [ ] ( )

ε ε ∫ Σ Σ + = π γ γ < < G G Re Tr 0 r a d 2
(to be compared with equation ( 19)), which fulfills the condition of current conservation along the nanodevice [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF].

Similarly to the case of the photon energy current, it is meaningful to distinguish between the three radiative processes of the electron-photon interaction throughout

≶ ≶ ≶ ≶ Σ Σ Σ Σ = + + ∼ ∼ ∼ ∼ γ , abs em st em sp , induced processes , ( 23
) with ħ ( ) ( ) ≶ ≶ ∓ ∑ ω Σ = ± ∼ ζ ζ ζ ζ ζ ε ε N M G M , abs q q q q q (24a) ∓ ∑ ω Σ = ± ∼ ζ ζ ζ ζ ζ ε ε N M G M , em st q q q q q , ħ ( ) ( ) ≶ ≶ (24b) ħ ( ) ( ) ≶ ≶ ∓ ∑ ω Σ = ± ∼ ζ ζ ζ ζ ε ε M G M .
em sp q q q q , (24c)

The derivation of γ I equation ( 19) provides general expressions for the radiative rates in the stationary case. Indeed, we decompose

⟨ ⟩ = - + + γ N R R R ˙abs em st em sp ,
, and then identify (using the cycling property of the trace)

ħ ( ) ε ε T ∫ ∑ ω = ζ ζ ζ - R h N 1 d , , abs q q q (25a) ħ ( ) ε ε T ∫ ∑ ω = ζ ζ ζ + R h N 1 d , , em st q q q , (25b) ħ ( ) ε ε T ∫ ∑ ω = ζ ζ + R h 1 d , , em sp q q , ( 25c
)
where

ħ ħ ( ) [ ( ) ( )] ε ε ε T ω ω = ± ζ ζ ζ ζ ± < > M G M G , T r . q q q q
(26) Equations (25a-25c) reiterate the formula provided by Aeberhard in [45] from an analogy between the Boltzmann and Dyson equations.

Spectral photon currents

From the partition presented equation (23), it is possible to derive the three spectral photon currents using the photon density of states ħ ( )

D ω ζ u
in the large volume limit

ħ ħ ħ ħ ( ) ( ) ( ) ( ) ε ε D T ∫ ω ω ω ω = ζ ζ ζ ζ - r N h 1 d , , abs u u u u (27a) ħ ħ ħ ħ ( ) ( ) ( ) ( ) ε ε D T ∫ ω ω ω ω = - ζ ζ ζ ζ + r N h 1 d , , em st u u u u , (27b) ħ ħ ħ ( ) ( ) ( ) ε ε D T ∫ ω ω ω = - ζ ζ ζ + r h 1 d , . em sp u u u , (27c)
We have introduced the direction of light propagation / = || || u q q and abbreviated the frequency by writing ω.

The radiation is treated as a third terminal here, in contrast with other developments where the photon Green's functions are fully taken into account with their own dynamics [46]. However, the present derivations allow us to provide radiation properties from the knowledge of the matter, in terms of electron Green's functions, via the trace of

ħ ( ) ( ) ε ε ω ± ζ ζ < > M G M G u u
(see equation ( 26)). This function depends on both the electron and photon energies, and it is connected to the polarization insertion of the interaction dynamics [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF]. It is also interesting to introduce the induced spectral current = + r r r abs em st ind , given by

ħ ħ ħ ħ ( ) ( ) ( ) ( ) D A ω ω ω ω = ζ ζ ζ ζ r N , u u u u ind (28)
where

ħ ħ ( ) [ ] ( ) ε ε A T T ∫ ω ω = - ζ ζ ζ - + h 1 d , u u u (29)
is a rate of net absorption (if A > 0) or gain (if A < 0) in the optoelectronic device. Taking advantage of the equality of

ħ ħ ħ ( ) ( ) ε ε ε ε T T ∫ ∫ ω ω ω = + ζ ζ - - d , d , u u , the spectral current r ind is shaped into ħ ħ ħ ħ ħ ( ) ( ) ( ) ( ) ( ) ∫ ω ω ω ω ω = ζ ζ ζ ζ ζ + - ε ε ε D T B r h N 1 d , , , u u u u u ind 1 (30) with ħ ħ ħ ħ ħ ħ ( ) [ ( ) ( )] [ ( ) ( )] ( ) ( ) ε ε ε ε ε ε ε B T T ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ω ω ω ω ω ω = + + - = + - ζ ζ ζ ζ ζ ζ ζ < > < > - - + - M G M G M G M G , Tr Tr 1 , , 1 .
Using the dimensionless function

ħ ( ) ε B ω ζ , u
, we finally formulate the photon particle and energy currents as follows

ħ ħ ħ ħ ħ ( ) ( ) ( )[ ( ) ( ) ] ε ε ε D T B ∫ ∫ ∫ ∑ ω ω ω ω ω = Ω × - γ ζ ζ ζ ζ ζ + - I h N 1 d d d , , 1, u u
u u u 1 (32) ħ ħ ħ ħ ħ ħ ( ) ( ) ( )[ ( ) ( ) ] ε ε ε D T B ∫ ∫ ∫ ∑ ω ω ω ω ω ω = Ω × - γ ζ ζ ζ ζ ζ + - I h N 1 d d d , , 1, E u u u u u 1 (33)
where Ω d u is the elementary solid angle around the direction u along which the light propagates. It is interesting to point out the similar expressions we have for these currents: they are both written as the product of a two-dimensional spectral quantity

ħ ħ ħ ħ ( ) ( )[ ( ) ( ) ] ε ε D T B ω ω ω ω - ζ ζ ζ ζ + - N , , 1
u u u u 1
multiplied by the photon energy ħω at the power zero for the particle current, and at the power one for the energy current.

Quasi-equilibrium limits

Within NEGF formalism, the electron-photon interaction is described using the self-consistent Born approximation in terms of electron and photon Green's functions. The approach is original in the sense that it is in fact not necessary to define local thermodynamic parameters to obtain particle, energy or entropy currents which flow outside the out-of-equilibrium central region. In devices where the central region reaches the nanoscale, particles experience non-thermal states while the device is working. It is not a simple task to define local temperature, and electrochemical potential in the interacting central region [START_REF] Meair | [END_REF]49]. Indeed, all the particle statistics is encoded in NEGF formalism [START_REF] Bruus | Many-body Quantum Theory in Condensed Matter Physics[END_REF]. However, if NEGFs can be represented by quasi-equilibrium Green's functions, they will verify a Kubo-Martin-Schwinger relation [START_REF] Balzer | Nonequilibrium Green's Functions Approach to Inhomogeneous Systems[END_REF] ( )

( ) ε ε ε = µ > < - G G e k T B
, where µ and T represent the electronic chemical potential and temperature respectively. This relation generalizes the following properties of the Fermi-Dirac and Bose-Einstein functions:

( ) ( ) ε ε ε - = µ - f f 1 e k T B and ħ ħ ( ) ( ) ħ ω ω + = ω N N 1 e k T B .
More generally, let us consider the case of a semiconductor in which electrons inside the conduction band, and holes inside the valence band experience separate quasi-equilibrium states characterized by two different chemical potentials and temper atures, µ c v , and T c,v . In that case, the diagonal components of Green's functions follow local Kubo-Martin-Schwinger relations

[ ] ( ) [ ] ( ) ε ε ε = µ > < - G G e , n n k T n n B (34)
where

( ) ∈ n c v
, refers to the band index. Thanks to these relations, simplifications occur in the expression of B, equation (31).

In particular for = = T T T c v , B no longer depends on the electron energy ε, and it follows

ħ ħ ( ) ( ) B ⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ω ω µ µ = - - - - k T exp 1 , c v B 1 (35)
in which one can define T E = T and

( ) µ µ µ = - E c
v , the temper ature and chemical potential of the spontaneously emitted radiation [START_REF] Würfel | [END_REF]. Hence we obtain the full Bose-Einstein statistic function that happens in the so-called generalized Planck's law for the emission [START_REF] Würfel | [END_REF]53], that was also discussed in photovoltaic cells of quantum dot arrays using NEGFs [54], and notably used to determine the thermopower from optical measurements [55].

In the quasi-equilibrium limit, B does not depend on ε, which allows us to write the spectral emission current as

ħ ħ ħ ħ ( ) ( ) ( ) ( ) B D A ω ω ω ω = - ζ ζ ζ r . em sp u u u , (36)
Using equations ( 28) and ( 29), the two quasi-equilibrium limits of the photon particle and energy currents finally read as

∫ ∫ ∑ ω ω ω ω ω = Ω × - γ ζ ζ ζ D B A I N d d , u u u ħ ħ ħ ħ ħ ( ) ( ) [ ( ) ( )] ( ) (37)
ħ ħ ħ ħ ħ ħ ( ) ( ) [ ( ) ( )] ( ) ∫ ∫ ∑ ω ω ω ω ω ω = Ω × - γ ζ ζ ζ D B A I N d d . E u u u (38)
Equations ( 37) and ( 38) are similar to the ones obtained in [46] dealing with non-equilibrium photon Green's functions. Interestingly, our approach suggests that a generalized energy flow law would involve two-dimensional spectral functions

ħ ħ ħ ħ ( ) ( )[ ( ) ( )] ε ε ε B ω ω ω ω = - ζ ζ ζ ζ j a N ,
, ,

E u u u u (39)
with (40) in the case of a non-equilibrium nanosystem, for which charge and energy currents are given by equations ( 32) and (33).

ħ ħ ħ ħ ( ) [ ( ) ( )]/ ε ε ε T T ω ω ω ω = + - ζ ζ ζ - + a h , , , , u u u

Entropy current

The entropy current I S flowing from the central region to the reservoirs is calculated in terms of electron Green's function and self-energies using the expression of photon energy current given equation ( 8).

Spectral entropy current

The device is an open interacting nanosystem connected to three reservoirs: the two electron left and right reservoirs, and the photon bath. In this three-terminal configuration, the entropy current flowing from the central region to the three reservoirs is defined as

= + + γ γ I I T I T I T S L h L R h R h ( 41
) µ µ µ = - - - - - - γ γ γ γ H N T H N T H N T ˙˙˙L L L L R R R R 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 (42)
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ µ µ µ = + - + + - γ γ γ γ γ I T I T I T T I T I T , L E L R E R L L L R R E ( 43
)
in which we use the relation

⟨ ⟩ ⟨ ⟩ = - N N ˙L R guaranteed by charge conservation.
Implementing results of sections 3.1 and 3.2 in equation ( 41), we are hence able to derive the entropy current in terms of Green's functions from the spectral entropy current ( ) ε J S as follows

( ) ε ε ∫ = I J d , S S (44)
with

( ) [ ] ( ) S S ε ε = + < < J h G G 2 Re Tr , S r a ( 45
)
and

∑ µ µ Σ Ξ Σ = - + - ∼ α α α α γ γ γ γ < ∈ < < < ε ε ε ε ε T T 1 2 . a L R a a a , , , , , ( ) ( ) ( ) ( ) ( ) S (46)
The entropy current flowing from the central region to the reservoirs is equal to the rate of entropy production Π = I S for nanosystems maintained in out-of-equilibrium steady states [56].

Entropy production is recast in terms of efficiencies

For photovoltaic-thermoelectric converters, we define the nanodevice efficiency as the ratio of the output electrical power or useful heat current to the input power in the form of light, which is given by the heat current of the absorbed photons, I abs h . This definition contrasts with 'the maximal power conversion efficiency' defined in practice at maximal output power, and where the denominator is the incident radiant power; it thus does not depend on the processes undergone by the system [57].

In this section, we focus on three devices based on a central region interacting with light: erator on a cooling by heating process (CBH) [32], and finally a joint device which provides both cooling and electrical energy production (JCEP) [35]. For the three nanodevices, the rate of entropy production is recast in terms of efficiencies according to the device, as Whitney proposed in [58]. Indeed, all nanodevices (ND) provide the same formal rate of entropy production

[ ] η η Π = Π - , ND ND 0 rev (47)
where η ND rev is the efficiency of the reversible nanodevice, η I ND abs h rev is the output power in the reversible nanodevice, and η Π ND 0 rev is the maximum rate of entropy production achievable in the nanodevice. Ratio / Π Π 0 reflects how close to the maximum efficiency the device is working.

Table 1 summarizes the definitions and notations of the relevant efficiencies discussed for the three nanodevices. These efficiencies are named thermodynamic efficiencies as they can be manipulated following the laws of thermodynamics.

η η Π = - - η ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ I T I I 1 , abs h L em h abs h LS PV C PV rev (48)
( = T T L R , µ µ > L R ) CBH η = CBH I I R h abs h ( > T T L R , µ µ = L R ) JCEP η η η = = + - JCEP I I V I CBH P V L h L e abs h ( < T T L R , µ µ > L R ) Standard engine η rev Carnot machine (C) η = - 1 ch T T C c h ( ( ) ( ) < T T c h cold hot ) Refrigeration ( ) η = - ch T T T c h c
Heat pump (≀≀≀)

≀≀≀ η = - ch T T T h h c
Trithermal heat engine (3T) (

( ) < < T T T c i h intermediate ) η η η = × T cih i h c i 3 C
where Here, it can be worth deriving the related electroluminescent (EL) case, for which

=
= γ N 0 implies = I 0 h ind , [ ] ≀≀≀ η η Π = - - I V T T T T , L e E L E L LE EL ( 49
)
where η = - EL I P em sp h e , (with

⩾ = P I V 0 e L e
) is the efficiency of the electroluminescent device, and T E replaces γ T is the temperature of the photon bath formed by electroluminescence.

The efficiency of the reversible photovoltaic nanodevice is reduced compared to the Carnot limit: from equation ( 48), the maximum value of efficiency is

⎡ ⎣ ⎢ ⎤ ⎦ ⎥ η η = -| | 1 PV LS I I rev C em h abs h
. This maximal efficiency may be compared to the Landsberg's limit [59,[START_REF] Green | Third Generation Photovoltaics: Advanced Solar Energy Conversion[END_REF]:

⎜ ⎟ ⎡ ⎣ ⎢ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ η η = - + + γ γ γ 1 1 AS T T T T T Landsberg C 1 3 A A A 2 2
, where

A stands for ambient, and it corresponds to T L in this work.

Landsberg reconsidered the limit of the Carnot efficiency as the upper limit for photovoltaics. Starting from the model of dithermal engine, he included the energy and entropy fluxes related to the emission process. In the Landsberg's approach, the central region is a converter in a state of equilibrium, and it behaves as a black body emitting photons at temperature T C (C stands for converter). Landsberg demonstrated that the maximal efficiency the reversible device, η Landsberg , is reached when = T T C A . The Landsberg's approach ignores the details of electron properties in the converter, which is also assumed at equilibrium. However, despite these differences, the NEGF-based expression of entropy production rate, equation ( 48), provides similar conclusions to those of Landsberg: the maximum efficiency is always lesser than the Carnot limit of a heat engine producing work from electron and photon reservoirs.

4.2.2.

Cooling by heating process. We discuss the coefficient of performance of a cooling by heating process as proposed in [32] with > T T L R and V = 0 (see table 1 for the efficiency definitions),

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ η η Π = - - η I T I I 1 . abs h L em h abs h T RLS CBH 3 CBH rev (50)
From this formula, we deduce for this original cooling process

⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ η η = - - - Π T T T I I T I 1 , CBH R L R em h abs h L S L abs h C (51)
which ressembles equation (11) of [32] with the additional

reducing contribution η | | I I LE C em h abs h
to the CBH coefficient of performance. Indeed, the emission processes were not included in the approach of [START_REF] Rutten | [END_REF], which was developed in the strong optical coupling regime. Moreover, in the recent model proposed by Wang and co-authors in [34] to verify the third law of thermodynamics, the cooling regime includes a parasitic emission in the regime of weak coupling to electron reservoirs, which involves a single emission wavelength.

Joint cooling and energy production.

For a more general case, but with a specific device objective, we examine the joint cooling and energy production proposed in [35]. The joint process can be seen as a photovoltaic configuration with < T T L R , or a cooling by heating configuration with V > 0. It follows two expressions for the rate of entropy production

[ ( ) ] η η η η Π = - - -- I T T T T 1 , abs h R L L R CBH PV RL JCEP rev (52)
and

⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎤ ⎦ ⎥ ⎥ η η η η η Π = - - - I T 1 . abs h R PV CBH RL RL JCEP rev (53)
From equations ( 52) and ( 53), we deduce

η η η η < ⇒ < < 1 , RL CBH J CEP P V rev r ev rev (54) η η η η > ⇒ < < 1 . RL PV JCEP CBH rev r ev rev (55)
In terms of applications, it means that for η < 1

RL

, a joint process more efficiently converts the photon bath power than the CBH one as shown by Entin-Wohlman and co-authors in [35], and we outline here that for η > 1 RL , a joint process also more efficiently converts the photon bath power than the photovoltaic device. Additionally, writing rev , which means that the single conversion (CBH or PV) in the hybrid device is always less efficient than in the corresponding standard device.

Discussing the entropy production in a two-level system including light-matter interaction

In this section, we discuss how the second law of thermodynamics is not automatically verified depending on the model used to simulate how the nanodevice works. We focus on a minimal model of QD-based nanojunction. Such ultimate nanostructures allow us to grasp the essentials of the energy conversion at the nanoscale level using three-terminal configurations [62][63][64][65]. These kinds of configurations provide the separation between the charge and heat transport, and, at the same time, motivate innovative experimental realizations, as demonstrated recently in [66] and [67].

Basics of the modeling

We model a photovoltaic-thermoelectric junction based on quantum dots as shown figure 1. We follow a simplified methodology within the framework of NEGFs. The central region is made of a quantum dot of 1 nm 3 . The dot is described by two energy levels which interact with a resonant monochromatic radiation ħω γ through the optical coupling γ M . The upper dot level ε 2 is only connected to the left electron reservoir while the lower level ε 1 is connected to the right one, which forces the charge separation without applying an electric field. The contact self-energies are given by

( ) ( ) ( )/ ε ε ε ∓ Σ =Λ Γ i 2 L R r a L R L R , , , , [68]
. The rate of entropy production is calculated in the regime of strong coupling to reservoirs: the calculations are performed at the second order perturbation upon the optical coupling [69,70]. This approach is valid as long as the optical coupling is lower than the transfer parameter,

≪ Γ γ M L R
, . The bias voltage is symmetrically applied

/ µ = ±eV 2 L R
, between the two electron reservoirs.

We specify the basic case of a linearly polarized matic plane wave as an incident radiation of energy ħω γ and polarization ζ. Functions Ξ and Σ distinguish between the three radiative processes that are absorption, stimulated emission and spontaneous emission. For the monochromatic case, we used

∓ ω Σ = ± ∼ γ γ γ γ ε ε N M G M , abs ħ ( ) ( ) ≶ ≶ (56a) ∓ ω Σ = ± ∼ γ γ γ γ ε ε N M G M , em st , ħ ( ) ( ) ≶ ≶ (56b) ħ ( ) ( ) ≶ ≶ ∓ ω Σ = ± ∼ γ γ γ ε ε M G M , em sp , ( 56c
)
with ħ ≶ ≶ ω Ξ Σ = ∼ γ abs em st abs em st or ,
or , , and the real parts of the retarded and advanced components of the interaction self-energies are ignored. Optical coupling

γ M reads as ħ / ε ε ω = ω ζ γ ζ γ M e V P 2 nm r n m , 2 0
, where ζ P nm, is the momentum matrix element and V the volume of the interacting region.

For the two-level system, the problem is block-diagonal, and we use analytics to derive the spectral currents to the second order in = γ ωζ γ M M 12, in order to discuss the entropy production in the device.

Particle current

We focus on the spectral particle current ( )

ε J R , ( ) ε ε ∫ = I J d R R
, which is positive when an electrical power is produced. To the second order in γ M , we find

ω ω = + × - + + γ γ γ γ γ + - ε ε ε ε ε A A J h M N F N F 1 1 , R RL LR 2 1 2 ħ ħ ( ) ( ) ( ) [ ( ) ( ) ( )] (57)
with

( ) ( ) ( ( ) ) ( )/ ( ) ε ε ε ε ε ε ε A = Γ -Λ - +Γ = - G 4 2Im , R R R r 1 1 2 2 11 (58) ( ) ( ) ( ( ) ) ( )/ ( ) ε ε ε ε ε ε ε A = Γ -Λ - +Γ = - G 4 2Im , L L L r 2 2 2 2 22
(59) and

ħ ( ) ( )[ ( )] ε ε ε ω = - ± αβ α β γ ± F f f 1 , ( 60
)
for { } α ∈ L R
, and

{ } β ∈ L R , . Functions ( ) ε αβ ±
F naturally relate the photocurrent to the recent interpretation of the different contributions to the non-symmetrized noise in a quantum dot

[71]. From ( ) ( ) ε ε ε - = µ - f f 1 e k T B and ħ ħ ( ) ( ) ħ ω ω + = ω N N 1 e k T B , ( ) ε J R takes the form ( ) ( )[ ] ( ) ε ε ε = -- J J 1 e , R abs X (61) with ħ ħ ( ) ε ε ε ω µ µ ω = + - - - - γ γ γ X k T k T k T , L L R R B B B (62)
and

ω ω = + × - + γ γ γ γ ε ε ε ε ε A A J h M N f f 1 1 . abs R L 2 1 2 ħ ħ ( ) ( ) ( ) ( )( ( )) (63)
For = T T L R , the ε dependence of X drops out and ħ ( ) / ω η = - γ γ X k T eV L L C
B , where the Carnot efficiency η ch C is defined table 1. For reservoirs at the same temperature, the spectral particle current, and hence the charge current, vanishes at a voltage called open-circuit voltage in photovoltaics,

ħω η = γ γ eV L oc
C given by X = 0. This result shows an interesting analogy with the observations made by Sánchez and Büttiker in [63]. Indeed, in the three-terminal configuration they propose, the heat current is controlled from Coulomb interaction instead of the light-matter one. The authors also evidence a stall voltage, equation ( 16) of [63], that is an analogue of V oc , for which both charge and heat currents vanish. Comparing the two three-terminal configurations, this voltage is a fraction, equal to the Carnot efficiency, of the relevant energy quantum: the charging energy E C in [63] versus the photon energy ħω γ in the configuration studied here. Moreover, the heat energy current in the right reservoir is also zero at this voltage in the current configuration due to ( ) ( ) ( )

ε ε ε µ = - J J R h R R
. We moreover conclude that the heat current exchanged with the photon reservoir is also zero at V oc using this model, following

ħ ħ ω ω = = γ γ γ γ I I I h R .

Entropy production

In this nanojunction architecture, the electron current cancels in the absence of light-matter interaction: the ballistic current is null. We have analytically verified that

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ = - = -γ N N N ˙˙L R
, which means that a carrier is added to(removed from) the left reservoir when a photon is removed/ absorbed from (added/emitted to) the photon reservoir.

Assuming

> T T L R , the spectral current ( ) ε Π J is calculated using the relation ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ = - = -γ N N N ˙˙L R : ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ µ µ ω ω µ µ ω = - - - + - = + - - - - γ γ γ γ γ Π ε ε ε ε ε ε ε ε J J T T J T T J T T J T T T 1 1 1 1 . R L R R L L R R R L R L L R R ħ ħ ħ ( ) ( ) ( ) ( ) ( )
(64) This spectral entropy current is not the one directly obtained from equations ( 45) and ( 46), but the integration of Π J gives the rate of entropy production Π defined in equation [START_REF] Baym | [END_REF]. We can determine the sign of the entropy production from this integrand. Using equation ( 61) to express J R , we obtain

( ) ( ) ( )[ ] ( ) = - Π - ε ε ε J k J X 1 e . abs X B (65)
In equation ( 65): on the one hand J abs is always positive as it gives the rate of absorbed photons flowing to the active region, on the other hand the function [ ] -- X 1 e X is always positive. The rate of entropy production is hence always positive, as it results from the direct integration of ( ) ε

Π J . When = T T L R
, the entropy current vanishes for X = 0 like the particle and heat currents, which was discussed in the previous section. Moreover, the analytical model also provides that the entropy production rate is concave at the open-circuit voltage since we obtain

( ) [ ( )] ⩾ ∂ = ∂ Π I V I X V 2 0 V abs V 2 oc oc 2
. The rate of the entropy production is minimum at open circuit conditions.

We numerically calculated the charge and entropy currents for various temperature gradients between the two electron reservoirs, in the optoelectronic junction depicted figure 1. The contact self-energies are given by a unique parameter Γ in the wide band limit [68], and the other parameters are given figure 1.

/ Σ = -Γ Im i 2 L R , (Λ = 0)
Characteristics of figure 2 show that cooling the right reservoir (cathode) in this configuration enhances the open-circuit voltage V oc and the maximal output power. Additionally, the numerical results confirm that the rate of entropy production is always positive for all temper ature differences and vanishes at V oc for = T T L R , for which X does not depend on ε. Moreover, the numerical results show that the rate is minimum at V oc and that the rate curve is concave around this point.

Limits of traditional models

NEGF-based models for photovoltaics usually modify the interaction self-energies of equation ( 22) to determine the device functioning. Two post-treatments are done after the straightforward steps of derivation presented in sections 2 and 3. The Bose-Einstein function ζ N q is replaced by the number of incident photons estimated from the incoming photon flux [START_REF] Würfel | Physics of Solar Cells[END_REF], and the interaction self-energy term related to spontaneous emission is integrated over all possible photon states [72].

The number φ N is calculated from the photon flux intensity at the surface of the earth ≈ φ I 10 3 Wm -2 , which accounts for the solid angle between the sun and the earth, the volume and the refractive index, V and κ, of the absorber (here the nanoscopic dot) and the speed of light in vacuum C 0 .

Integrated emission.

In contrast with the two induced radiative processes, the self-energy term related to spontaneous emission does not explicitly depend on the properties of the incident radiation. For nanosystems strongly hybridized with electron reservoirs, the density of states broadens. It provides broad emission spectra for the set of parameters given figure 1, as illustrated figures 3(a) and (b) in the two ranges of voltage values below the open-circuit voltage and close to the QD gap, respectively. A transition occurs from black body radiation (figure 3(a)) to the luminescence peak typical of semiconductor emitters (figure 3(b)) [START_REF] Würfel | [END_REF]. It thus seems essential to integrate over all the possible transition energies of the interacting region [72]. We introduce the photon density of states ħ ħ ħ ( )

( ) / ρ ω ω κ π = V C 2 2 3 0 3 to reformulate ∓ ∫ ω ρ ω ω ω ω Σ = × ± ∼ ε ε M G M d , em sp , ħ ħ ħ ħ ħ ( ) ( ) ( ) ( ) ( ) ( ) ≶ ≶ (67a) ħ ħ ħ ħ ħ ħ ( ) ( ) ( ) ( ) ( ) ( ) ≶ ≶ ∓ ∫ ω ρ ω ω ω ω ω Ξ = × ± ε ε M G M d , em sp , ( 67b
)
where we assumed a polarization isotropy for the interacting nanosystem.

The two post-treatments can also be done in the functions Ξ of equations (12a-12c). These functions have been introduced to determine the energy and entropy currents in nanosystems interacting with light. We can thus examine the impacts of these post-treatments on the laws of thermodynamics.

5.4.3. Limits. We compare three models for the interaction self-energy. Model A is the model used before any modification: the Bose-Einstein function is used, and a photon energy equal to the one of the photon source is considered for spontaneous emission (see equations ( 57) and ( 64)). Model B: the Bose-Einstein function is replaced by the number of incident photons (see equation ( 66)), and the photon energy for spontaneous emission is identical to the one of the photon source. Model C: the Bose-Einstein function is used, and all photon energies are considered to integrate the self-energy term of spontaneous emission (see equation ( 67)).

The total energy is conserved using any of the three models in the general framework of section 3, following the procedure of section 3.2.

We numerically determined the charge and entropy currents using models A, B and C, with ) quadrant, the nanodevice produces electrical power, as expected from a photovoltaic device. Conversely, the charge current is already positive at zero voltage for the I e -V characteristics obtained using model C. The corresponding electrical power is positive: the nanodevice is consuming electrical power to produce light. That characterizes a luminescent device. The functionality of the nanodevice depends on the model choice. The I S -V curve obtained with model A exhibits a positive rate of entropy production for all voltages, as it has been proven analytically in section 5.3. Inversely, the rate of entropy production takes negative values using models B or C. Positive entropy production is no longer guaranteed using traditional post-treatments of NEGF-based simulations.

Discussion

From a fundamental viewpoint, different works show that entropy production is always positive in non-interacting nanosystems using Landauer or NEGF formalisms [11,18,58]. It seems still possible to show via a Landauerlike for mulation [73] that entropy production is positive in interacting systems for which the left and right couplings to electron reservoirs are proportional. However, such a condition is not valid in photovoltaic nanodevices, since the electron/hole selectivity requires very different couplings to the left and right reservoirs. Using NEGF formalism, light-matter interaction is included via the interaction self-energy Σ, defined equation ( 22), and its counterparts for photons Ξ and Σ ∼ , equation ( 9) and equation ( 20) respectively, to calculate particle and energy currents in these devices. Applied to a two-level model, we demonstrated that entropy production is positive when the incident and emitted radiation have the same properties which are contained in the photon density of states (energy, polarization...). We moreover examined the impacts of two traditional modifications of the interaction self-energy which are performed after deriving particule and energy currents inside the Born approximation. Usually, the Bose Einstein function is replaced by the number of incident photons, and both induced and spontaneous radiative processes are treated independently. We showed that these two modifications no longer provide a model that guarantees a positive rate of entropy production.

Model-dependent violations of the second law of thermodynamics have already been discussed in the field of photovoltaics [74]. Actually, it would be surprising that an empirical model complies with the second law of thermodynamics. In the two critical models B and C, modifications are done after the straightforward derivation of particule and energy currents starting from the hamiltonian detailed equation [START_REF] Aeberhard | [END_REF]. Models B and C can be thus regarded as empirical models. Let us reform ulate the function Ξ of equation ( 9) in a general manner that includes all models:

∓ ∓ ∓ ∑ ∑ ω ω ω ω ω ω ω ω Ξ = ± ± × ± γ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ∈ ∈ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ε ε ε ε M M C C C M G G M M G M ,
abs em st em sp q q q q q q q q q q q q q q , ,

ħ ħ ħ ħ ħ ħ ħ ħ ( ) [ ( ) ( ) ( ) ( ) ] ( ) ( )
≶ ≶ ≶ ≶ (68)
where the three functions C abs , C em,st and C em,sp account for each of the radiative processes (absorption, stimulated emission and spontaneous emission), and M and M′ represent the two ensembles of photon modes describing the properties of the light source and the spontaneous emission respectively. From the straightforward derivation that we have presented section 3.1, one obtains a generalization of model A, characterized by: (i) the two ensembles of photon modes are identical,

M M = ′, (ii) ħ ħ ( ) ( ) ω ω = = ζ ζ ζ C C N abs em st q q q ,
the Bose Einstein function, and (iii) C em,sp = 1. These results directly stem from the hamiltonian model of equation (2), and from the properties of the photon Green's function

( ) ⟨ ( ) ( )⟩ τ τ τ = - ζ ζ ζ D T A t A , i t q q q 0 1 1 [43]. Functions ≶ D
verify a Kubo-Martin-Schwinger relation [START_REF] Balzer | Nonequilibrium Green's Functions Approach to Inhomogeneous Systems[END_REF]:

ħ ħ ( ) ( ) ħ ω ω = ζ ζ ω ζ γ > < D D e k T q q q B , which gives the fluctuation dissi- pation relations ( ) = + ζ ζ ζ > D N D 1 Im r q q q and = ζ ζ ζ < D N D Im r q q q ,
and the equality ħ ħ ( ) ( )

ħ ω ω + = ζ ω ζ γ ζ N N 1 e k T q q q B
. Reciprocally, to be related to photon Green's functions, the three functions C abs , C em,st and C em,sp used in equation ( 68) shall verify:

ħ ħ ħ ( ) ( ) ( ) ħ ω ω ω + = ζ ζ ω ζ ζ γ C C C e em st
em sp k T abs q q q , , q B (69

)
for each photon mode. Actually, models B and C do not comply with such a condition. In model B, properties (i) and (iii) are verified, but . Similarly in model C, properties (ii) and (iii) are verified, but M M ≠ ′, which means that the two self-energy terms of induced processes and spontaneous emission are not calculated from the same photon modes. In that case again, equation ( 69) is not verified for each photon mode. This analysis shows that both models B and C erase the fluctuation-dissipation relations of the photon Green's functions inside the interaction self-energy, which is related to the time-reversal symmetry [75], and, hence, micro-reversibility [76].

For solar cell applications, NEGF-based simulations generally consider the whole solar spectrum. It means that the total self-energy is integrated over all photon energies. It is also possible to consider a continuous spectrum in our two-level model. The total entropy current is then calculated by integrating the analytical entropy current obtained equation ( 65) over all photon energies using a unique photon density of states (as in equation ( 67)). We thus infer that the rate of entropy production is always positive for a continuous spectrum as long as the spontaneous emission and the incident radiation are treated identically through photon properties (energy, polarization...). Nevertheless, the Bose-Einstein function shall be used. It is indeed directly related to the photon Green's function of the light source. In practice, we propose to define an equivalent temperature, T e , from the realistic number of photons entering the absorber (see equation ( 66 . This temperature depends on both the photon source and the absorber properties. However, for a monochromatic light source, model C still contains a contradiction: on the one hand, it may be crucial to integrate the selfenergy term of spontaneous emission over all photon energies in order to determine how the device is expected to work; on the other hand, this integration results in a model which does not systematically verify the second law of thermodynamics from the entropy current expression given equation [START_REF] Baym | [END_REF]. This contradiction may show that incident and emitted photons no longer form a thermal reservoir in the critical range of parameters. In that case, photon temperature and chemical potential would no longer be defined. The entropy current expression of equation ( 44) would no longer be valid, which would explain the violation of the second law of thermodynamics. In that case, coupled Dyson's equations for both the electron and photon Green's functions would form a complete and sound model for any range of parameters.

Conclusion

Following the Born approximation within the Keldysh formalism, we derived the formal expressions of photon energy and particle currents in open nanosystems interacting with light. We thus obtained generalized radiation laws in terms of non-equilibrium electron Green's functions, and we found the Planck's law at the quasi-equilibrium limit. Next, we used these expressions to formulate the spectral entropy current exchanged with the three terminals of the nanosystem, namely the two electronic reservoirs, and the photon bath. The net entropy flow was recast in the form of a difference between the efficiency of the reversible photovoltaic-thermoelectric nanodevice and its effective efficiency. We then applied these results to a two-level system illuminated with a monochromatic radiation. We showed analytically that entropy production was always positive. However, the methodology that we have presented section 3.1 is usually modified for photovoltaic applications, in order to take into account both the realistic number of photons reaching the system, and the broaden emission spectrum, notably for strong coupling to reservoirs. We found that these empirical modifications can lead to unphysical analysis since the entropy production rate can reach negative values, while energy conservation still holds. This paper provides a basic framework to discuss the ability of a quantum device to convert light energy into electrical or/and thermal energy.

4. 2 . 1 .

 21 Standard photovoltaics. For = T T L R , we can derive the photovoltaic case

Figure 1 .

 1 Figure 1. Level diagram of the junction: two discrete energy levels are available in the QD at energies ε = 0.5 1

Figure 2 .

 2 Figure 2. Charge current, output power and entropy current as a of nanojunction for different temperature gradients ∆ = -T T T R

5. 4 . 1 .

 41 Photon number. Instead of the Bose-Einstein function, realistic modelings of solar cells introduce

  models A and B are typical of photovoltaics: the charge current is negative until the voltage reaches V oc . In the (

Figure 3 .

 3 Figure 3. Spectral emission current in the QD-based nanojunction following equation (27c) for equal temperatures of electron reservoirs, varying the voltage (a) from zero to ≈ V 0.5 oc V, and (b) from V oc to 2V oc .

  Figure 4. -I V L e and I S -V curves in the QD-based nanojunction for the three models A, B and C detailed in the text. Differences between the -I V L e characteristics obtained with models A and B are not significant since we have chosen a photon number close to the value of the Bose-Einstein function at = γ k T 0.03 B

Table 1 .

 1 Efficiency notations and definitions which will be used for a photovoltaic (PV), cooling by heating (CBH) and a joint cooling and electrical energy production (JCEP) nanodevices. For PV and

	JCEP devices, device, ⩾ I 0 L h . In all cases, ⩽ I V 0 L e . For a CBH device, ⩾ I R h ⩾ I 0 abs h .	0 , and for a JCEP
	Nanodevice ( ND)	η ND
	PV	η = -PV	I V e L I abs h

Inside this derivation, the Hartree-like term is zero from the Wick's theorem using the model hamiltonian of equation (2).

Acknowledgments

We thank R Whitney, A M Daré, FGibelli, J-F Guillemoles, and LLombez for their interest in this work and for valuable discussions on different parts of this work. We acknowledge financial support from the CNRS Cellule Energie funding project 'ICARE'.


Driss Moulay

Nadir Mechaoui
email: moulaydriss.mechaoui@univ-usto.dz

Abdessamad Guetmi
email: nadir.guetmi@ensma.fr

Imine

Towards Real-Time Co-Authoring of Linked-Data on the Web

Keywords: Linked-Data, Collaborative Editing Systems, Optimistic Replication

Real-time co-authoring of Linked-Data (LD) on the Web is becoming a challenging problem in the Semantic Web area. LD consists of RDF (Resource Description Framework) graphs. We propose to apply state-of-the art collaborative editing techniques to manage shared RDF graphs and to control the concurrent modifications. In this paper, we present two concurrency control techniques. The first one is based on client-server architecture. The second one is more flexible as it enables the collaborative co-authoring to be deployed in mobile and P2P architecture and it supports dynamic groups where users can leave and join at any time.

Introduction

Recently, providing collaborative co-authoring tools in the Web Semantic is becoming more attractive as they enable semantic web data to be produced in online mode and to be available to a large public. Linked Data (LD) is recently used to replace collections of offline RDF data [START_REF] Berners-Lee | Tabulator redux: Writing into the semantic web[END_REF]. The goal of LD is to enable people to share structured data on the web as easily as they can share documents today. It uses RDF technology that (i) relies on HTTP URIs to denote things; (ii) provides useful information about a thing at that thing's URI; and (iii) includes in that information other URIs of LD. Tabulator [START_REF] Berners-Lee | Tabulator: Exploring and analyzing linked data on the semantic web[END_REF] is a LD browser, designed to provide the ability to navigate the web of linked things. In [START_REF] Berners-Lee | Tabulator redux: Writing into the semantic web[END_REF], Berners-Lee et al. raise some interesting challenges when adding collaborative co-authoring mode in Tabulator. This mode consists in collaboratively editing the LD which is represented by a RDF graph.

In this paper, we sketch two solutions that may meet to some extent the readwrite requirement in LD browser. We consider a RDF graph as a shared data which can be edited and updated by several users. To control the concurrent access to this shared data, we propose to apply state-of-the art collaborative editing techniques [START_REF] Molli | Using the transformational approach to build a safe and generic data synchronizer[END_REF][START_REF] Imine | Formal design and verification of operational transformation algorithms for copies convergence[END_REF]. The CRDT (Commutative Replicated Data Type) is a class of algorithms that is emerging for ensuring consistency of highly dynamic content on P2P networks. However, this approach incurs some overhead they do not consider directly a set as a list (or a sequence) [START_REF] Aslan | C-set : a commutative replicated data type for semantic stores[END_REF]. Also, with the continuously growing amount of structured data available on the Semantic Web there is an increasing desire to replicate such data to mobile devices. This enables services and applications to operate independently of the network [START_REF] Zander | Context-driven RDF data replication on mobile devices[END_REF][START_REF] Sacco | Fine-grained access control for RDF data on mobile devices[END_REF]. Classical replication techniques cannot be properly applied to mobile systems because they do not adopt to changing user information needs, and they do not consider the technical, environmental, and infrastructural restrictions of mobile devices. We think that Operational Transformation (OT) approach [START_REF] Ellis | Concurrency Control in Groupware Systems[END_REF][START_REF] Sun | Operational transformation in real-time group editors: issues, algorithms, and achievements[END_REF] may be a good candidate as it supports unconstrained interaction. Indeed, it allows any user to modify any shared data consistently at any time without any restrictions on users's actions.

The rest of the paper is organized as follows. Section 2 presents the ingredients of OT approach. In Section 3, we suggest two concurrency control procedures for managing the collaborative edition of RDF graphs. Section 4 discusses performance evaluation, and concludes.

Transformational Approach

Principle. Operational Transformation (OT) is an optimistic replication technique which allows many users (or sites) to concurrently update the shared data and next to synchronize their divergent replicas in order to obtain the same data [START_REF] Yi | Achieving convergence in operational transformation: Conditions, mechanisms and systems[END_REF]. The updates of each site are executed on the local replica immediately without being blocked or delayed, and then are propagated to other sites to be executed again. Accordingly, every update is processed in four steps: (i) generation on one site; (ii) broadcast to other sites; (iii) reception on one site; (iv) execution on one site.

A crucial issue when designing shared data with a replicated architecture and arbitrary messages communication between sites is the consistency maintenance (or convergence) of all replicas. To illustrate this problem, consider the following example:

Example 1. Consider the following group text editor scenario (see Figure 1.(a)): there are two users (on two sites) working on a shared document represented by a sequence of characters. These characters are addressed from 0 to the end of the document. Initially, both copies hold the string "efecte". User 1 executes operation op 1 = Ins(1, f ) to insert the character f at position 1. Concurrently, user 2 performs op 2 = Del(5) to delete the character e at position 5. When op 1 is received and executed on site 2, it produces the expected string "effect". But, when op 2 is received on site 1, it does not take into account that op 1 has been executed before it and it produces the string "effece". The result at site 1 is different from the result of site 2 and it apparently violates the intention of op 2 since the last character e, which was intended to be deleted, is still present in the final string. Consequently, we obtain a divergence between sites 1 and 2. It should be pointed out that even if a serialization protocol [START_REF] Ellis | Concurrency Control in Groupware Systems[END_REF] was used to require that all sites execute op 1 and op 2 in the same order (i.e. a global order on concurrent operations) to obtain an identical result effece, this identical result is still inconsistent with the original intention of op 2 .

Fig. 1. Serialization of concurrent updates

To maintain convergence, the OT approach has been proposed by [START_REF] Ellis | Concurrency Control in Groupware Systems[END_REF]. When User X gets an operation op that was previously executed by User Y on his replica of the shared object User X does not necessarily integrate op by executing it "as is" on his replica. He will rather execute a variant of op, denoted by op (called a transformation of op) that intuitively intends to achieve the same effect as op. This approach is based on a transformation function (or algorithm) IT that apply to couples of concurrent operations defined on the same state.

Example 2. In Figure 1.(b), we illustrate the effect of IT on the previous example. When op 2 is received on site 1, op 2 needs to be transformed according to op 1 as follows: op 2 = IT ((Del(5), Ins(1, f )) = Del [START_REF] Imine | Formal design and verification of operational transformation algorithms for copies convergence[END_REF]. The deletion position of op 2 is incremented because op 1 has inserted a character at position 1, which is before the character deleted by op 2 . Next, op 2 is executed on site 1. In the same way, when op 1 is received on site 2, it is transformed as follows:

IT (Ins(1, f ), Del(5)) = Ins(1, f ); op 1 remains the same because f is inserted before the deletion position of op 2 .
Intuitively we can write the transformation IT as follows:

IT(Ins(p1,c1),Ins(p2,c2)) = if (p1 < p2) return Ins(p1,c1) else return Ins(p1+1,c1) endif;

OT Model. Using the OT approach, each site is equipped by two main components [START_REF] Ellis | Concurrency Control in Groupware Systems[END_REF][START_REF] Ressel | An Integrating, Transformation-Oriented Approach to Concurrency Control and Undo in Group Editors[END_REF]: the integration component and the transformation component.

The integration component determines how an operation is transformed against a given operation sequence (e.g., the log buffer). It is also responsible for receiving, broadcasting and executing operations. It is rather independent of the type of the shared data. The transformation component is a set of IT algorithms which is responsible for merging two concurrent operations defined on the same state. Every IT algorithm is specific to the semantics of a given shared data.

The most known OT-based theoretical framework is established by Ressel et al. [START_REF] Ressel | An Integrating, Transformation-Oriented Approach to Concurrency Control and Undo in Group Editors[END_REF]. They define two consistency criteria:

-Causality. If one operation op 1 causally precedes another operation op 2 , then op 1 must be executed before op 2 at all sites.

-Convergence. When all sites have performed the same set of operations, the copies of the shared data must be identical.

It has been proved that any integration component can achieve convergence in the presence of arbitrary transformation paths if its IT algorithm satisfies two properties T P 1 and T P 2 [START_REF] Ressel | An Integrating, Transformation-Oriented Approach to Concurrency Control and Undo in Group Editors[END_REF]. For all op, op 1 and op 2 pairwise concurrent operations with op 1 = IT (op 1 , op 2 ) and op 2 = IT (op 2 , op 1 ):

-TP1: [op 1 ; op 2 ] ≡ [op 2 ; op 1 ].
-TP2: IT (IT (op, op 1 ), op 2 ) = IT (IT (op, op 2 ), op 1 ).

Property T P 1 defines a state identity and ensures that if op 1 and op 2 are concurrent, the effect of executing op 1 before op 2 is the same as executing op 2 before op 1 . This property is necessary but not sufficient when the number of sites is greater than two. Property T P 2 defines an update identity and ensures that transforming op along equivalent and different operation sequences will give the same operation.

Properties T P 1 and T P 2 are sufficient to ensure the convergence for any number of concurrent operations which can be executed in arbitrary order [START_REF] Ressel | An Integrating, Transformation-Oriented Approach to Concurrency Control and Undo in Group Editors[END_REF]. Accordingly, by these properties, it is not necessary to enforce a global total order between concurrent operations because data divergence can always be repaired by operational transformation. However, finding an IT algorithm that satisfies T P 1 and T P 2 is considered as a hard task, because this proof is often unmanageably complicated [START_REF] Sun | Exhaustive search of puzzles in operational transformation[END_REF]. To overcome this difficulty, we proposed in [START_REF] Imine | Formal design and verification of operational transformation algorithms for copies convergence[END_REF] a formal methodology for designing and analyzing IT algorithms by using a theorem prover.

Several OT-based integration components have been proposed in the groupware research area. These components may be categorized in two categories. The first one does not require T P 2 property: it relies on client-server architecture for enforcing a unique transformation order. We can cite in this category algorithms like SOCT4 [START_REF] Vidot | Copies convergence in a distributed real-time collaborative environment[END_REF] and TIBOT [START_REF] Li | A time interval based consistency control algorithm for interactive groupware applications[END_REF]. As for the second category, it requires T P 2 property. This constraint enables the concurrent operations to be synchronized in a decentralized way. Algorithms such as adOPTed [START_REF] Ressel | An Integrating, Transformation-Oriented Approach to Concurrency Control and Undo in Group Editors[END_REF] SOCT2,4 [START_REF] Sun | Achieving Convergence, Causality-preservation and Intention-preservation in real-time Cooperative Editing Systems[END_REF][START_REF] Vidot | Copies convergence in a distributed real-time collaborative environment[END_REF] and GOTO [START_REF] Sun | Operational transformation in real-time group editors: issues, algorithms, and achievements[END_REF] belong to this category.

Our Proposals

To manage all concurrent access for editing collaboratively a shared RDF graph, we need a concurrency control procedure. In this section, we first argue how to map a RDF graph into a sequence data structure. According to centralized and decentralized architectures, we suggest two concurrency control procedures.

RDF Graph as a sequence

When publishing LD on web, information about resources is represented using the RDF. Any expression in RDF is a collection of triples, each consisting of a subject, a predicate (also called property) and an object. The subject of a triple is the URI describing resource. The object can either be a simple literal value (e.g., a string, a number) or the URI of another resource. The predicate indicates what kind of relation exists between subject and object. The predicate is a URI too.

A set of such triples is called an RDF graph. This can be illustrated by a node and directed-arc diagram, in which each triple is represented as a node-arc-node link.

Usually a set is implemented by means of a list. It means we can use operations, such as insert and delete, to edit a shared list. Thus, we can reuse the state-ofthe-art of collaborative editing systems.

For instance, the following three english statements (this example is taken from [START_REF] Manola | Rdf primer[END_REF]):

-<http://www.example.org/index.html>has a creator whose value is John Smith -<http://www.example.org/index.html>has a creation-date whose value is August 16, 1999

-<http://www.example.org/index.html>has a language whose value is English could be represented by the RDF graph shown in Figure 2.

An RDF graph can be serialized into a sequence of triples and considered as a text where each line corresponds to a simple triple of subject, predicate and object. For example, the third statement shown in Figure 2 would be written as a text line:

<http://www.example.org/index.html> <http://purl.org/dc/elements/1.1/language> "en" .

By considering an RDF graph as a sequence, each triple is addressed simply by a position within the sequence. Therefore, we assume that the sequence of triples can be modified by the following primitive operations:

-Ins(p, t) which adds triple t at position p;

-Del(p, t) which deletes triple t at position p.

Updating a triple (e.g., by modifying the predicate URI) can be expressed by a sequence of delete (by removing the old triple) and insert (by adding the new one) operations.

Ingredients of collaboration

Each user's site has a local copy of RDF graph and a unique identity. We assume that the RDF graph is serialized in the same way on every site.

Every site generates operations sequentially and stores these operations in a stack also called a log. When a site receives a remote operation op, the integration component executes the following steps:

1. from the local log it determines the sequence seq of operations that are concurrent to op;

2. it calls the transformation component in order to get operation op that is the transformation of op according to seq;

3. it executes op on the current state;

4. it adds op to the local log.

Centralized Solution

In this section, we propose a real-time co-authoring based on client-server architecture. Indeed, users can edit collaboratively a shared RDF graph by reconciliating their divergent copies via a particular site called server. We think that SOCT4 [START_REF] Vidot | Copies convergence in a distributed real-time collaborative environment[END_REF] is most appropriate to this kind of architecture.

In SOCT4, the operations are ordered globally by using a timestamp given by the server. When an operation is generated on site s, it is immediately executed (to satisfy real-time constraint), but it is not propagated until it gets a timestamp from the server and all the operations which precede it according to the timestamp order have been received and executed on s. Moreover, this operation is transformed against all concurrent operations (operations received after its generation and preceding it in the global order) before to be propagated. To ensure convergence, SOCT4 requires only the property T P 1 to be satisfied by the IT algorithm. Example 3. Consider two users editing a shared RDF graph as described in Figure 3. Initially, each site has an empty copy. The index of each operation represents the timestamp given by the server. Two local insertion operations op 1 and op 2 have been executed by user 1 (at site 1). Concurrently, user 2 has executed another insertion operation op 3 . The added triples t 1 , t 2 and t 3 are respectively as follows (where UR1 is http://www.example.org/index.html and UR2 is http://www.example.org/staffid/85740):

<UR1> <http://www.example.org/terms/creation-date> "August 16, 1999" . <UR1> <http://purl.org/dc/elements/1.1/language> "en" . <UR1> <http://purl.org/dc/elements/1.1/creator> <UR2> .

1. At point s 1 , site 1 decides to synchronize with other sites. As there is no concurrent operation available, op 1 , op 2 are sent to site 2 (via the server) in their original forms.

2. At point s 2 , site 2 cannot send op 3 as long as it did not receive the precedent operations (according to the timestamp order). Thus the synchronization calls IT algorithm to produce the following transformations:

op 1 = IT (op 1 , op 3 ) = Ins(0, t 1 ) op 3 = IT (op 3 , op 1 ) = Ins(1, t 3 ) op 2 = IT (op 2 , op 3 ) = Ins(1, t 2 ) op 3 = T (op 3 , op 2 ) = Ins(2, t 3 )
op 1 , op 2 are executed on site 2, and op 3 is broadcast to other sites.

3. At point s 3 , site 1 decides again to synchronize. The remote operation op 3 is executed directly (without transformation) after op 1 and op 2 .

4. Note that, after point s 3 , sites 1 and 2 have the same log, namely op 1 , op 2 and op 3 . However, site 1 has performed the following sequence:

op 1 op 2 op 3 = IT (IT (op 3 , op 1 ), op 2 )
while site 2 has executed the following sequence:

op 3 op 1 = IT (op 1 , op 3 ) op 2 = IT (op 2 , op 3 )
As SOCT4 requires only T P 1 property, the above sequences are equivalent in the sense that they produce the same RDF graph. The operations are stored in the log according to the timestamp order but they may be executed in different orders at different sites.

It should be noted that SOCT4 has been used successfully in the development of a File Synchronizer [START_REF] Molli | Using the transformational approach to build a safe and generic data synchronizer[END_REF] distributed with the industrial collaborative development environment, LibreSource Community4 , proposed by Artenum Company. LibreSource is a platform for hosting virtual teams. Users can register and create channels for synchronizing shared data. On a single server, LibreSource can host several projects, several groups of users, and grant fine grain access to the resources.

Although SOCT4 ensures causality and convergence properties, it degrades the responsiveness of the system as all messages are exchanged via a server. Moreover, it does not scale because it is based on a single point of failure.

Decentralized Solution

Integration algorithms based on T P 2 property enable concurrent operations to be synchronized in a decentralized way. Thus, they avoid a single point of failure. Nevertheless, these algorithms have limited scalability with the number of users. Indeed, all proposed OT frameworks rely on a fixed number of users during collaboration sessions. This is due in the fact that they use vector timestamps to enforce causality dependency. The vector timestamps do not scale well, since each timestamp is a vector of integers with a number of entries equal to the number of users.

In [START_REF] Imine | Conception Formelle d'Algorithmes de Réplication Optimiste[END_REF], we proposed a new framework for collaborative editing to address the weakness of previous OT works. The features of our framework are as follows:

1. It supports an unconstrained collaborative editing work (without the necessity of central coordination). Using optimistic replication scheme, it provides simultaneous access to shared data.

2. Instead of vector timestamps, we use a simple technique to preserve causality dependency. Our technique is minimal because only direct dependency information between operations is used. It is independent on the number of users and it provides high concurrency in comparison with vector timestamps.

3. Using OT approach, reconciliation of divergent copies is done automatically in decentralized fashion.

4. Our framework can scale naturally thanks to our minimal causality dependency relation. In other words, it may be deployed easily in Peer-to-Peer (P2P) networks.

Example 4. Consider the scenario given in Example 3. In our framework, operations op 1 and op 2 will be related by a dependency. This is due in the fact that their added triples are adjacent (positions 0 and 1) and created by the same user. Thus, op 1 must be executed before op 2 at all sites. This dependency relation is minimal in the sense that when op 2 is broadcast to all sites it holds only the identity of op 1 as it depends on directly.

1. At site 1, op 3 is considered as concurrent. It is then transformed against op 1 and op 2 . The following sequence is executed and logged in site 1:

op 1 = Ins(0, t 1 ) op 2 = Ins(1, t 2 ) op 3 = IT (IT (op 3 , op 1 ), op 2 ) = Ins(2, t 3 )
2. At site 2, op 1 and op 2 are concurrent with respect to op 3 . They must be transformed before to be executed after op 3 according to their dependency relation. Thus, the following sequence is executed and logged in site 2:

op 3 = Ins(0, t 3 ) op 1 = IT (op 1 , op 3 ) = op 1 op 2 = op 2
Unlike the others OT-based integration algorithms, we minimize the transformation steps when integrating a remote operation depending on another operation. Indeed, at site 2, the new form of op 2 is deduced from the executed form of op 1 (without transformation as in Example 3). On the other hand, the sequences of sites 1 and 2 are not identical but equivalent.

Performance Evaluation

Our experimentation consists to compare the response time of generating and integrating a sequence of remote triples over a local ones. We use two sites (Site1 and Site2), initially the log of each sites is empty. Each site generates locally a sequence of operations; the sites communicate the generated operations to be integrated. The sizes of the sequence are varied from 50 000 to 1 000 000 triples. The percentage of insertions in the sequence and the log are variants from 50%, 80% to 100%. We implement a prototype of Optic [START_REF] Imine | Conception Formelle d'Algorithmes de Réplication Optimiste[END_REF] in java, compiled by NetBeans 6.8 with JVM heap size 1GB, and executed on a computer running Windows XP SP2 with an Intel (R) Core (TM) 2 CPU E7400 @ 2.80 GHz and 2 GB RAM. We calculate the sum of the generation time of the sequence in the Site1 with the time of integration of the same sequence in the Site2. For every generation and integrating sequence three times are executed and the average time is recorded.

The Figure 4(a) present the time of generation and integration of a varied sequences of triples over an empty RDF document. When the percentage of insertions in the sequence is 100% the performance of our algorithm increases. This is due to the minimal causality dependency between insertions operations computed during the local generation of triples. The Figure 4(b) illustrate the time of generation and integration of a varied sequences of triples over a RDF document containing 10 000 triples. The performance decreases when the percentage of deletion increases. This degradation of performance is caused by the canonizing of the log [START_REF] Imine | Conception Formelle d'Algorithmes de Réplication Optimiste[END_REF] (tidy insertion operation before deletion operations). The rate of deletion operations in the log has a direct impact on the performance of the Optic algorithm.

Conclusion

In this paper, we have dealt with the problem of the real-time co-authoring of LDW. In this respect, we have suggested two solutions based on OT approach.

In centralized and decentralized solutions we propose in this paper, the shared RDF graph is serialized into a sequence of triples that can be altered by simple operations: insertion and deletion of triples. Mapping RDF graph into sequence of triples is given in order to reuse state-of-the art collaborative editing techniques including some systems in which we participated [START_REF] Molli | Using the transformational approach to build a safe and generic data synchronizer[END_REF][START_REF] Imine | Formal design and verification of operational transformation algorithms for copies convergence[END_REF]. This mapping is simple. But, if the RDF graph must satisfy some requirements based on semantic aspects (e.g., graph connectiveness), preconditions must be added to operations. For example, we can state that the delete operation Del(p, t) is enabled iff the p exits and the object of t is not a subject of another triple. It is not sure that this delete operation will be still enabled when it is integrated in another site which has added concurrently triple t whose the subject is the object of t. Two solutions are possible: either writing another IT algorithm based on new constraints, or tolerating the violation of some requirements during some periods with the possibility to stabilize in correct state (by undoing some operations).

The question of adapting these solutions in existing semantic web browsers remains open in this paper. It will be interesting to plug these solutions in a given browser in order to evaluate the cost of mapping a RDF graph into a sequence. Using this implementation, we can also make measurements to experimentally validate the impact of OT approach on real-timeliness and scalability. On the other hand, designing a new IT algorithm for shared RDF graphs based on updates proposed in the recent version of SPARQL/Update [START_REF] Seaborne | Sparql/update: A language for updating rdf graphs[END_REF] is an exciting and challenging problem.

Fig. 2 .

 2 Fig. 2. An RDF Graph.

site 1 site 2 Fig. 3 .

 23 Fig. 3. Scenario of collaboration

(Fig. 4 .

 4 Fig. 4. Updating RDF document

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