P Royer

F Cherblanc

Homogenisation of advective-diffusive transport in poroelastic media

Advective-diffusive solute transport in a poroelastic medium is usually modelled at the macroscopic scale by considering a poro-elastic formulation associated with the classical advective-diffusive solute transport equation. The purpose of the present work is to rigorously determine the macroscopic model by starting with the pore-scale equations that describe transient advective/diffusive solute transport in a saturated deformable elastic porous medium. The macroscopic model is derived by using an homogenisation method. It includes Biot's model of consolidation and a transport equation of advective-diffusive type. This latter contains a coupling term which reveals a mechanically-induced solute transport mechanism.

Introduction

When solute transport occurs in a deformable porous media, some coupling effects are observed which are generally reported as "mechanically-induced solute transport" or "activated solute transport". Indeed, the consolidation of a saturated porous medium produces fluid flow which participates to the solute convective transport. These issues have been theoretically examined in the context of soil contaminant migration and particularly in capped waste deposits (Brannon and Poitdexter-Rollings, 1990;[START_REF] Alshawabkeh | A model fo contaminant mass flux in capped sediment under consolidation[END_REF][START_REF] Arega | Coupled consolidation and contaminant transport model for simulating migration of contaminants through the sediment and a cap[END_REF]. Various consolidation and contaminant transport models have been developped to capture some non-linear effects resulting from a large strain formulation [START_REF] Smith | One-dimensional contaminant transport through a deforming porous medium: theory and a solution for a quasi-steady-state problem[END_REF][START_REF] Peters | Solute transport through a deforming porous medium[END_REF][START_REF] Fox | Coupled large strain consolidation and solute transport. I: Model development[END_REF].

To get further insight into mechanically-induced solute transport, cyclic loading is applied on saturated deformable porous media as it is known to enhance solute transport mechanism [START_REF] Tate | Mixing Mechanisms and Net Solute Transport in Bone[END_REF][START_REF] Mauck | Modelling of neutral solute transport in a dynamically loaded porous permeable gel: implications for articular cartilage biosynthesis and tissue engineering[END_REF]. This coupling effect is of major interest when dealing for instance with the design of drug delivery system [START_REF] Urciuolo | Effect of Dynamic Loading on Solute Transport in Soft Gels Implication for Drug Delivery[END_REF]. This transport enhancement mechanism also participates in the cell nutrients transfer through avascular biological tissue such as articular cartilage or intervertebral disk [START_REF] O'hara | Influence of cyclic loading on the nutrition of articular cartilage[END_REF][START_REF] Evans | Dynamic Compression Augments Interstitial Transport of a Glucose-Like Solute in Articular Cartilage[END_REF]. Some attempts to develop a theroretical solute transport model a the scale of a canaliculi have been proposed [START_REF] Wang | Modeling Tracer Transport in an Osteon under Cyclic Loading[END_REF][START_REF] Goldsztein | Solute transport during cyclic flow in saturated porous media[END_REF]).

Several experimental and numerical works have focused on the identification of the main parameters (frequency, amplitude, ...) that govern this mechanism [START_REF] Sengers | An integrated finite-element approach to mechanics, transport and biosynthesis in tissue engineering[END_REF][START_REF] Urciuolo | Effect of Dynamic Loading on Solute Transport in Soft Gels Implication for Drug Delivery[END_REF]. These models are usually based on a poro-elastic formulation associated with the classical advective-diffusive transport equation. Both models are written directly at the macroscopic scale without wondering if coupling effects between solute transport and mechanical strains could affect these classical macroscopic equations.

In the present work, we propose to revisit this issue by means of a theoretical upscaling procedure to rigorously determine the macroscopic model by starting with the pore-scale equations that describe transient advective/diffusive solute transport in a saturated deformable elastic porous medium. To upscale the pore-scale description, we use the homogenisation method for periodic structures, also called homogenisation method of multiple scales asymptotic expansions (Sanchez-Palencia, 1980) and the methodology introduced in Auriault (1991) is applied. The macroscopic behaviour which is thus derived includes Biot's model of consolidation and an advective-diffusive transport equation. This latter contains a coupling term which is emerging from the scale transition and characterizes a mechanically-induced solute transport mechanism.

Homogenisation method

Medium under consideration

The method of multiple scales is based upon the fundamental assumption of separation of scales. The condition of periodicity is also required, which has no impact upon the form of the macroscopic description. We thus consider a periodic porous medium, of macroscopic characteristic size L, and made of a solid matrix and of a fluid-saturated pore space. We further denote the periodic cell by Ω, its characteristic length by l, and we formulate the condition of separation of scales by ε = l/L 1. Within the periodic cell, we denote by Ω f the fluid-saturated pore domain, by Ω s the solid matrix, and by Γ their common interface, as depicted in Fig. 1. Using the two characteristic length, l and L, and the physical space variable, X, we define two dimensionless space variables: y = X/l, x = X/L. Since the condition of separation of scales (ε 1) is supposed to be satisfied, then variables y and x can be considered as two independent space variables: y describes the local scale, while x is the macroscopic space variable. As a consequence, the unknown fields (stress tensors, displacements, fluid velocity and pressure, . . . ) are, a priori, functions of both space variables y and x. Invoking the differentiation rule for multiple variables, the gradient operator with respect to the physical variable, X, is written as

∇ X = 1 l ∇ y + 1 L ∇ x .
(1)

Dimensionless pore-scale description

The methodology introduced in Auriault (1991) firstly consists in writing the dimensionless pore-scale description, which is the set of dimensionless equations that describe the phenomena being considered within the periodic unit cell depicted in Fig. 1. Each quantity in a dimensionless equation is the ratio of its physical counterpart to its characteristic value (indexed by * ). This writing gives rise to dimensionless numbers, which are defined by means of characteristic values. We consider the elasticity equations in the solid domain Ω s , the equations of flow for a Stokesian fluid and the convection-diffusion equation for the solute in the fluid-saturated pore domain Ω f , together with the boundary conditions over the interface Γ. The dimensionless writing of the equations requires the choice of a characteristic length for the dimensionless writing of space derivatives. We arbitrarily choose L as the reference characteristic length. The dimensionless gradient operator is thus given by

∇ = L∇ X = ε -1 ∇ y + ∇ x . (2)
The dimensionless pore-scale description being considered is given by the equations presented below. Equations in the solid domain Ω s ∇ • σ s = 0,

(3) σ s = a : e ( u s ) .

(4)

Equations of fluid motion in Ω f

F L µ∆ v f -∇p f = 0, (5) σ f = -p f + F L 2µD( v f ), (6)
∇ • v f = 0, (7)
where F L is a dimensionless number, defined by

F L = µ * v * f p * f L . Advective-diffusive solute transport in Ω f N L ∂c ∂t + Pe L ∇ • ( v f c) = D∆ c, (8)
where the dimensionless numbers are defined by

N L = L 2 D * t and Pe L = v * f L D * . Conditions over the interface Γ σ f • n = σ s • n, (9)
v f = ∂ u s ∂t , ( 10
) ∇c • n = 0. ( 11
)

Orders-of-magnitude of dimensionless numbers

The methodology introduced in [START_REF] Auriault | Heterogeneous medium: is an equivalent macroscopic description possible ?[END_REF] consists in estimating the dimensionless numbers with respect to ε. As we are interested in the case which leads to a diphasic macroscopic behaviour for the fluid/solid system, the case where fluid motion is generated must be considered, which corresponds to the order [START_REF] Auriault | Heterogeneous medium: is an equivalent macroscopic description possible ?[END_REF]) F L = O ε 2 . It has been shown in [START_REF] Auriault | Taylor Dispersion in porous media: analysis by multiple scale expansions[END_REF] that in rigid porous media, a transient advective-diffusive macroscopic solute transport is obtained when N L = O ε 0 and Pe L = O ε 0 .

Homogenisation procedure

The homogenisation method being used is an asymptotic approach. It is therefore based upon the fundamental statement that the unknown fields can be written in the form of asymptotic expansions in powers of ε

ψ = ψ (0) ( y, x) + εψ (1) ( y, x) + ε 2 ψ (2) ( y, x) + ... ( 12
)
in which functions ψ (i) are Ω-periodic in variable y, with

ψ = σ s , σ f , p f , c, v f , u s .
The method consists in incorporating the asymptotic expansions in the dimensionless local description (Eqs. 3-11), while replacing the dimensionless numbers by their orders of magnitude in power of ε and taking into account the expression of the dimensionless gradient operator (Eq. 2). This leads to approximated governing equations and boundary conditions at the successive orders, which together with the condition of periodicity define well posed boundary-value problems in the periodic unit cell. Existence of solutions requires that volume averaged equations be satisfied. These latter actually describe the macroscopic behaviour at successive orders.

Macroscopic poroelastic model

Homogenisation of the fluid/solid equations (Eqs. 3-7, 9 and 10) has been performed by [START_REF] Auriault | Etude du comportement macroscopique d'un milieu poreux saturé déformable[END_REF]. It leads to the following macroscopic model, which is identical to Biot's model of poroelasticity

∇• < σ (0) > = 0, ( 13
) < σ (0) > = C : e u (0) s -α p (0) f , (14)
∇ x • < v (0) f > f -φ ∂ u (0) s ∂t = α ⊗ e u (0) s -β ∂p (0) f ∂t , ( 15
) < v (0) f > f -φ ∂ u (0) s ∂t = - K µ ∇p (0) f . ( 16
)
In the above equations < σ (0) > is the first-order averaged total stress tensor, C is the effective elastic tensor, while α represents the elasticity coefficient of Biot, β is the medium compressibility and K denotes the permeability tensor. The term < v

(0) f > f -φ ∂ u (0)
s /∂t where u

(0) s = u (0)
s ( x, t) represents the fluid relative velocity and the porosity φ is defined by φ

= | Ω f |/| Ω |.
The distinct averages being introduced are defined by:

< ψ >=< ψ > s + < ψ > f , (17)
where

< ψ > s = 1 | Ω | Ωs ψ dΩ, < ψ > f = 1 | Ω | Ω f ψ dΩ. (18)
Details on the derivation of the above model can be found in [START_REF] Auriault | Etude du comportement macroscopique d'un milieu poreux saturé déformable[END_REF].

Homogenisation of transport equations

Determination of c (0) and c (1) At the first two orders, we get boundary-value problems which are identical to those obtained in a rigid porous medium [START_REF] Auriault | Taylor Dispersion in porous media: analysis by multiple scale expansions[END_REF]. They lead to

c (0) = c (0) ( x, t) , (19)
c (1) = χ • ∇ x c (0) + c(1) ( x, t) , (20)
where c(1) is an arbitrary function and where χ is defined by the following local problem to be solved on the unit cell:

       ∇ y • (D(∇ y χ + I)) = 0 in Ω f , (∇ y χ + I) • n = 0 on Γ, < χ > f = 0, and χ is Ω-periodic. ( 21
)
Determination of the first-order macroscopic description At the third-order, transport equations lead to the following system:

∂c (0) ∂t -∇ y • D(∇ y c (2) + ∇ x c (1) ) -c (0) v (1) f -c (1) v (0) f - -∇ x • D(∇ y c (1) + ∇ x c (0) ) -c (0) v (0) f = 0 in Ω f , (22)
∇ y c (2) + ∇ x c (1) • n = 0 on Γ, (23)
v (0) f = ∂ u (0) s ∂t ( x, t) on Γ. ( 24
)
Integrating equation (Eq. 22) over the period, while taking the associated boundary-conditions (Eqs. 23 and 24) into account together with the condition of periodicity yields the first-order macroscopic transport equation. We firstly note that from the second-order expansion of (Eq. 7) and from (Eq. 19) , we have:

∇ y • c (0) v (1) f = c (0) ∇ y • v (1) f = -c (0) ∇ x • v (0) f . ( 25
)
Integrating the thus transformed transport equation (Eq. 22), we obtain:

φ ∂c (0) ∂t -∇ x • D eff ∇ x c (0) + ∇ x c (0) • < v (0) f > f -φ ∂ u (0) s ∂t = -φ∇ x c (0) • ∂ u (0) s ∂t - 1 | Ω | Ω f ∇ y • c (1) ∂ u (0) s ∂t dΩ, (26)
in which D eff represents the effective tensor of diffusion, and is defined by

D eff = DT, with T = 1 | Ω | Ω f (∇ y χ + I) dΩ, (27)
where T denotes the tortuosity tensor [START_REF] Quintard | Diffusion in isotropic and anisotropic porous systems: Threedimensional calculations[END_REF][START_REF] Dormieux | Modélisation macroscopique du transport diffusif[END_REF]. Now, using the expression of c (1) (Eq. 20), we get

-φ∇ x c (0) • ∂ u (0) s ∂t - 1 | Ω | Ω f ∇ y • c (1) ∂ u (0) s ∂t dΩ = ∂ u (0) s ∂t • < ∇ y χ + I > f ∇ x c (0) = ∂ u (0) s ∂t • T ∇ x c (0) . (28)
The first-order macroscopic transport equation finally reads

φ ∂c (0) ∂t -∇ x • D eff ∇ x c (0) + < v (0) f > f -φ ∂ u (0) s ∂t • ∇ x c (0) + ∂ u (0) s ∂t • T ∇ x c (0) = 0.
(29)

Conclusions and perspectives

Homegenisation of the pore-scale equations that describe advective-diffusive solute transport in an elastic porous medium leads to Biot's model of consolidation (Eqs. 13-16) and to transport equation (Eq. 29). This latter can be expressed in a simpler manner as follows

φ ∂c ∂t + φ V f -V s • ∇c + V s • T ∇c = ∇ • (D eff ∇c) , (30)
where V f and V s denote the fluid and solid intrinsic macroscopic velocities, respectively.

Besides the usual accumulation, advection and effective diffusion terms, the coupling term V s • T ∇c is emerging from the scale transition. This term is associated with the no-slip boundary condition at the fluid-solid interface and characterizes a mechanically-induced solute transport mechanism. It can be seen as a convective transport in the boundary layer moving at the solid phase velocity. It is weighted by the tortuosity tensor T which represents a geometrical property of the unit cell micro-structure; for isotropic porous media it comes down to a scalar ranging from 0 to 1. An important feature of this coupling term is that it occurs only whenever there exits macroscopic advection: it does not appear in the case of a purely diffusive solute transport.

The influence of this additional term should be negligible in classical cases of consolidating porous media. Nevertheless, when focusing on solute transport in porous media submitted to cyclic loading, the cumulative effect of this mechanically-induced transport mechanism may become predominant. Experimental evidence of this term may be tricky to bear out. However, a numerical approach would bring some interesting features. The above developments have been done with specific orders of the dimensionless numbers F L , N L and Pe L . Future works will include the analysis of different cases and particularly for higher Péclet number where dispersive effects come up.

Figure 1 :

 1 Figure 1: Porous medium : (a) Macroscopic sample ; (b) Periodic unit cell.


Pascale Royer

Fabien Cherblanc

]

Introduction

At the pore scale, i.e. within the periodic cell, the equations that govern solute transport by convection and diffusion in a deforming medium are the following:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂c ∂t -∇ • q = 0 within ˝f, (1)
q • n = 0 over , (2)
v f = ∂ u s ∂t over , (3)
where the flux q is given by

q = -D ∇c + c v f . (4)
Therefore, Eq. ( 2) yields (D ∇c -c v f ) • n = 0 over .

(5)

Consequently, Eq. (11) in the original paper [START_REF] Royer | Homogenisation of advective-diffusive transport in poroelastic media[END_REF] should be replaced by Eq. (5).

Homogenisation of the transport equations

The dimensionless local scale description to be homogenised is therefore the following:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂c ∂t -∇ • D ∇c -c v f = 0 within ˝f, (6)
(D ∇c -c v f ) • n = 0 over , (7)
v f = ∂ u s ∂t over . (8)
Let us now analyse the boundary-value problems that arise at the successive orders of ε, once the asymptotic expansions for c, v f and u s and the expression of the dimensionless gradient operator (Eq.

(2) in the original paper) have been incorporated.

Determination of c (0)

The boundary-value problem for c (0) is unmodified. It remains identical to that obtained for a rigid porous medium, which leads to:

c (0) = c (0) ( x, t). (9)
2.2. Determination of c (1) Noticing that 1 ∂v (0)

f i ∂y i = 0, (10)
we get the following boundary-value problem for c (1) :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂y i D ∂c (1) ∂y i + ∂c (0) ∂x i = 0 within ˝f, (11)
D ∂c (1) ∂y i

+ ∂c (0) ∂x i n i -c (0) ∂u (0) s i ∂t n i = 0 over . ( 12
)
The solution c (1) depends on two forcing terms: ∇ x c (0) ( x, t) and 2 c (0) (∂u (0)

s i /∂t)( x, t)
. By virtue of linearity, the solution for c (1) is a linear combination of both:

c (1) = j ( y) ∂c (0) ∂x j -j ( y)c (0) ∂u 0 s j ∂t + c(1) ( x, t), (13)
where c( 1) is an arbitrary function. The definition of which is given by Eqs. (21) in the original paper remains valid, while the vector is defined by the following local problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ y • (D(∇ y )) = 0 in ˝f, D∇ y • n = 0 on , f = 0, : ˝-periodic. (14)

Derivation of the first-order macroscopic description

At the third order, the transport equations lead to the following system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂c (0) ∂t -∇ y • D(∇ y c (2) + ∇ x c (1) ) +∇ y • c (0) v (1) f + c (1) v (0) f -∇ x • D(∇ y c (1) + ∇ x c (0) +∇ x • c (0) v (0) f = 0 within ˝f, (15)
D ∇ y c (2) + ∇ x c (1) • n -c (0) v (1) f + c (1) v (0) f • n = 0 over . ( 16
)
The first-order macroscopic description is obtained by integrating the balance equation (Eq. ( 15)) over the periodic cell:

∂c (0) ∂t - 1 | ˝| ˝f ∂ ∂y i D ∂c (2) ∂y i + ∂c (1) ∂x i -c (0) v (1) f i -c (1) v (0) f i d˝- 1 | ˝| ˝f ∂ ∂x i D ∂c (1) ∂y i + ∂c (0) ∂x i -c (0) v 0 f i d˝= 0, (17)
1 First-order expansion of the fluid continuity equation:

∇ • v f = 0.
2 It has been shown earlier that u

(0) s i = u (0) s i ( x, t).
where

= | ˝f | | ˝| (18)
denotes the porosity. Using Gauss' theorem and boundarycondition (Eq. ( 16)), while keeping in mind the condition of periodicity, the second term of (Eq. ( 17)) vanishes and it thus reduces to:

∂c (0) ∂t - ∂ ∂x i D ∂c (1) ∂y i + ∂c (0) ∂x i f + ∂ ∂x i c (0) v (0) f i f = 0. ( 19
)
Using the expression obtained for c (1) (Eq. ( 13)), we deduce that:

D ∂c (1) ∂y i + ∂c (0) ∂x i f = D eff ij ∂c (0) ∂x j -D ∂ j ∂y i f c (0) ∂u 0 s j ∂t , ( 20
)
where D eff represents the effective tensor of diffusion, and is defined by

D eff = DT, (21)
with

T = 1 | ˝f | ˝f ∇ y + I d˝, ( 22
)
where T denotes the tortuosity tensor.

Eq. ( 19) now reads:

∂c (0) ∂t - ∂ ∂x i D eff ij ∂c (0) ∂x j + ∂ ∂x i (c (0) v (0) f i f ) = - ∂ ∂x i D ∂ j ∂y i f c 0 ∂u 0 s j ∂t . ( 23
)
Noticing that the macroscopic relative fluid velocity 3 is:

v (0) f i f - ∂u (0) s i ∂t ,
Eq. ( 23) can be rewritten as follows:

∂c (0) ∂t accumulation term -∇ x • D eff ∇ x c (0) effective diffusion + v (0) f f - ∂ u (0) s ∂t • ∇ x c (0) advection + A∇ x • c (0) ∂ u (0) s ∂t mechanically induced solute transport = 0, (24)
where

A = D∇ y + I f . (25)
Tensor A characterises a coupling between transport and deformation mechanisms, which is induced by the fluid convection.

Properties of the coupling tensor A

By definition, the variational formulation of the boundary-value problem (Eqs. ( 11) and ( 12)) reads: ˝f ˛∂ ∂y i D ∂c (1)

∂y i + ∂c (0) ∂x i d˝= 0, (26)
3 See the macroscopic poroelastic model in the original paper.

and can be transformed as follows:

˝f ∂∂ y i D ∂c (1) ∂y i d˝= -

˝f ∂∂ y i D ∂c (0) ∂x i d+ ˝f ∂∂ y i c (0) ∂u (0) s i ∂t d˝. (27)
By taking

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c (1) = j , ∂c (0) ∂x i = I ij , (28)
˛= j , c (0) ∂u (0)

s i ∂t = 0, (29)
into Eq. ( 27), we get:

˝f ∂ j ∂y i D ∂ j ∂y i d˝= - ˝f D ∂ j ∂y i d˝. (30)
Now, by considering

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c (1) = j , c (0) ∂u (0) s i ∂t = -I ij , (31)
˛= j , ∂c

) ∂x i = 0, (0
into Eq. ( 27), we obtain:

˝f ∂ j ∂y i D ∂ j ∂y i d˝= - ˝f ∂ j ∂y i d˝. (33)
From Eqs. ( 30) and (33), we deduce:

˝f D ∂ j ∂y i d˝= ˝f ∂ j ∂y i d˝. (34)
Consequently, we have:

A = D∇ y + I f = ∇ y + I f = T, (35)
where T is the tensor of tortuosity, defined by (Eq. ( 22)).

Conclusion

The macroscopic transport equation is written as:

∂c ∂t

+ V f -V s • ∇c + T∇ • c V s = ∇ • D eff ∇c ,
where V f and V s denote the fluid and solid intrinsic macroscopic velocities, respectively. Therefore, the only difference with the original paper is that the coupling term emerging from the scale transition is T∇ • (c V s ) instead of V s • T ∇c. All the other conclusions mentioned in the original paper remain valid.


Matthias Kr Üger

David S Dean

A Gaussian theory for fluctuations in simple liquids

Assuming an effective quadratic Hamiltonian, we derive an approximate, linear stochastic equation of motion for the density-fluctuations in liquids, composed of overdamped Brownian particles. From this approach, time dependent two point correlation functions (such as the intermediate scattering function) are derived. We show that this correlation function is exact at short times, for any interaction and, in particular, for arbitrary external potentials so that it applies to confined systems. Furthermore, we discuss the relation of this approach to previous ones, such as dynamical density functional theory as well as the formally exact treatment. This approach, inspired by the well known Landau-Ginzburg Hamiltonians, and the corresponding "Model B" equation of motion, may be seen as its microscopic version, containing information about the details on the particle level.

I. INTRODUCTION

The statics and dynamics of simple liquids is of great importance both in fundamental research, [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Evans | [END_REF][START_REF] Evans | Fundamentals of Inhomogeneous Fluids[END_REF][START_REF] Götze | Complex Dynamics of Glass-forming Liquids: A Mode-Coupling Theory[END_REF] but also in industry, technology, and biology. The statics have been investigated for many years and are well understood, for instance, via the framework of classical density functional theory. [START_REF] Evans | [END_REF][START_REF] Evans | Fundamentals of Inhomogeneous Fluids[END_REF][START_REF] Roth | [END_REF] Studies of the dynamical properties of fluids, such as the viscosity, have a long history, 6 and the field is still very active. [START_REF] Viswanath | Viscosity of Liquids[END_REF][START_REF] Elliott | [END_REF] Linear response theory 9,10 connects transport coefficients to time dependent correlation functions measured in thermal equilibrium, and the time dependent correlation functions [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Dhont | An Introduction to Dynamics of Colloids[END_REF][START_REF] Risken | The Fokker-Planck Equation[END_REF] studied here are thus of particular importance.

Time dependent correlation functions can be computed from various fundamental equations, such as the Liouville [START_REF] Hansen | Theory of Simple Liquids[END_REF] or Fokker-Planck equations. [START_REF] Dhont | An Introduction to Dynamics of Colloids[END_REF][START_REF] Risken | The Fokker-Planck Equation[END_REF] Dilute systems have been examined using exact dynamical formulations, for instance, via the Boltzmann equation [START_REF] Kreuzer | Nonequilibrium Thermodynamics and its Statistical Foundations[END_REF][START_REF] Da | [END_REF] or using the Fokker-Planck equation. [START_REF] Dhont | An Introduction to Dynamics of Colloids[END_REF][START_REF] Risken | The Fokker-Planck Equation[END_REF][START_REF] Pusey | Dynamic Light Scattering[END_REF] In dense systems, approximate dynamical formulations have been used: Here, Mode Coupling Theory (MCT) [START_REF] Götze | Complex Dynamics of Glass-forming Liquids: A Mode-Coupling Theory[END_REF] is useful for the computation of time correlation functions in bulk systems and has recently also been applied in confinement. [START_REF] Lang | [END_REF][17][18][19] Classical density functional theory finds static equilibrium quantities, [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Evans | Fundamentals of Inhomogeneous Fluids[END_REF][START_REF] Roth | [END_REF] while Dynamical Density Functional Theory (DDFT) [20][21][22] is powerful for describing out of equilibrium situations. In addition to the evolution in time dependent potentials, DDFT has also been used to study driven suspensions with spherical obstacles 23,24 or with constrictions, 25 driven liquid crystals, 26 suspensions under shear, [27][28][29][30][31] and for microswimmers. 32 Such research directions have also benefitted from formal improvements within power functional theory. 33,34 Despite these many applications, DDFT provides no immediate access to the time dependent equilibrium correlation functions; however, the test particle trick has been used to derive the van Hove function from it, [35][36][37] even for dense suspensions. 38 On the experimental side, the intermediate scattering function is an important quantity characterizing the dynamics of liquids, e.g., as regards the glass transition, 39 and can also be measured in confinement. 40 More generally, the dynamics of fluids in confinement have received a lot of recent attention, [41][42][43] among other reasons due to improved experimental precision on small scales, [44][45][46][47] and in microfluidic devices 48,49 or blood flow in capillaries. 50,51 Previous approaches that discuss the stochastic dynamics of particle densities, including noise, have been presented in Refs. 52 and 53, see also Ref. 54. We will discuss their relation with the approach developed here.

In this manuscript, we propose a description of fluctuations of fluids near equilibrium by use of a Gaussian field theory, corresponding to an effective quadratic Hamiltonian for the density fluctuation field in conjunction with a corresponding Langevin equation. The Hamiltonian is constructed to yield the correct static equilibrium averages. The corresponding Langevin equation is constructed to yield the dynamics of overdamped particles. Within this theory, we derive a closed, approximate expression for the time dependent equilibrium correlation function, which agrees with the exact result (found from the Smoluchowski equation) for short times. It is thus expected to describe well the dynamics at not too high densities, and might, in particular, provide insight into dynamics in confined systems. We also demonstrate the connection between the derived dynamics and the dynamics following from dynamical density functional theory, as well as the exact stochastic equation for the density operator.

The manuscript is structured as follows. In Section II, we lay out the theoretical framework, starting with the system considered in Section II A. We define the physical observables of interest in Sec. II B. The quadratic effective Hamiltonian is introduced in Section II C, and the stochastic equation of motion is introduced in Section II E. The time dependent correlation function from this framework is computed in Section III, and we demonstrate in Section III B that it is exact for short times. In Section IV, we show that the equation of motion, proposed and derived herein, is in close correspondence with DDFT. Sec. V discusses the connection to the exact stochastic equation of motion for the density field of Ref. 52. We summarize in Section VI.

II. SYSTEM, EFFECTIVE HAMILTONIAN, AND EQUATION OF MOTION

A. System

We aim to analyze time dependent correlation functions in liquids. For this, we choose a well studied and also experimentally relevant model system, which is overdamped spherical (Brownian) particles. (The question as to how well this model system describes aspects of molecular liquids has also attracted recent interest. 55 )

Regarding the ensemble, using Brownian dynamics directly implies a canonical or grand-canonical description, where the solvent acts as a bath at the given temperature. We will generally have in mind systems for which canonical and grand canonical descriptions are equivalent due to the large (infinite) particle number (such as in the semi-infinite system bound by a planar surface). Extra care has thus to be taken for closed systems, such as particles confined in a box of finite size (see Ref. 56 for an analysis of canonical systems in DFT).

The Brownian particles with positions at x i are subject to a potential Φ({x i }), including pairwise interactions (later denoted by V ) as well as an external potential (later denoted U). The thermal energy scale is denoted by k B T ≡ β -1 , with Boltzmann constant k B and the (solvent imposed) temperature T. The bare diffusivity (the diffusivity in the absence of interactions) of the Brownian particles is denoted by D. Each particle thus obeys the stochastic differential equation,

dx i dt = D βF i + √ 2Dξ i , (1)
where ξ i is white noise in the Ito Stochastic Calculus with ξ i,µ (t)ξ j,ν (t ) = δ ij δ µν δ(tt ) , and F i is the force acting on particle i, due to the potential Φ. (Throughout, i and j label particles, while Greek indices label spatial components.) In other words, in the absence of Φ, each particle performs isotropic Brownian motion.

B. Observables-mean and fluctuating

We summarize the arising observables in Table I. The density operator, ρ(x) = i δ(xx i ), [START_REF] Hansen | Theory of Simple Liquids[END_REF] is the starting point for all considerations that follow. If averaged over the equilibrium distribution, one obtains the mean equilibrium density,

ρ(x) = i δ(x -x i ) . (2)
Here, we have introduced the equilibrium average . . . , which, for the overdamped system, is exactly given by (we introduce the phase space abbreviation Γ ≡ {x i })

. . . = ∫ dΓ . . . e -βΦ(Γ) ∫ dΓe -βΦ(Γ) . ( 3
)
(x) = i δ(x -x i ) ρ(x)
Mean density in equilibrium φ(x, t)

Fluctuation of density about its equilibrium value, φ(x, t) = ρ(x) -ρ(x) φ(x, t)φ(x , t ) Time dependent correlations of density fluctuations in equilibrium, the quantity of interest of this work ρ(x, t) neq Mean density in nonequilibrium state δρ(x, t)

Average difference from equilibrium value in a perturbed system,

δρ(x, t) = ρ(x, t) neq -ρ(x)
As noted above, for systems with infinite particle number, the grand canonical average agrees with the canonical one given here. We introduce density fluctuations,

φ(x) = ρ(x) -ρ(x) . ( 4
)
This quantity will be important for this manuscript. Such fluctuations are, e.g., characterized by their correlation function, which relates two points in space and in time t (in the following we will sometimes suppress the arguments of C),

C(x, x , t -t ) = φ(x, t)φ(x , t ) . (5)
Due to the fact that we restrict to equilibrium fluctuations, C is a function of t t , [START_REF] Risken | The Fokker-Planck Equation[END_REF] but depends on both x and x in inhomogeneous systems. Computing C is the main goal of the manuscript. If transformed to reciprocal Fourier (k-space), C is the intermediate scattering function. [START_REF] Dhont | An Introduction to Dynamics of Colloids[END_REF] For completeness, we also define the mean density in an out of equilibrium situation, i.e., ρ(x, t) neq , and its average deviation from equilibrium

δ ρ(x, t) = ρ(x, t) neq -ρ(x) . ( 6
)
Note the difference with Eq. ( 4), which is for a stochastic fluctuation in an equilibrium system, while Eq. ( 6) is an average deviation from the equilibrium average density for a perturbed system.

C. Effective Hamiltonian

Aiming at the correlation function C, we start by assuring that the equal time value of C is found correctly. We thus introduce the following effective Hamiltonian, which is a functional of the fluctuating field φ,

βH = 1 2 dxdy φ(x) 1 φ(x)φ(y) φ(y), (7)
where

1 φ(x)φ(y) ≡ φ(x)φ(y) -1 (8)
is to be understood in the sense of inverse operators. In the field theory description, the equilibrium average in Eq. ( 3) is computed via the following functional integral, [START_REF] Kardar | Statistical Physics of Fields[END_REF][START_REF] Onuki | Phase Transition Dynamics[END_REF] .

. . = ∫ Dφ . . . e -βH ∫ Dφe -βH , ( 9
)
where Dφ denotes the measure of functional integration, which is most easily implemented by discretizing space or working with discrete Fourier transforms. As mentioned before, we point out that the Hamiltonian in Eq. ( 7) with Eq. ( 9) by construction correctly finds the static averages of φ, up to quadratic order. Its average, φ = 0, as required from Eq. ( 4), and its variance is indeed, using Eq. ( 7),

φ(x)φ(y) = ∫ Dφφ(x)φ(y)e -βH ∫ Dφe -βH . ( 10
)
See, e.g., Ref. 59 for useful identities regarding Gaussian functional integrals.

D. Static correlations-Theory of liquids

The variance of φ in Eq. ( 10) is a well studied object in the theory of liquids and can be expressed in terms of the so-called direct pair correction function c [START_REF] Evans | [END_REF]1 (this equation can be seen as one way of defining c [START_REF] Evans | [END_REF] ),

1 φ(x)φ(y) = 1 ρ(x) δ(x -y) -c (2) (x, y). (11)
Using this, we can make the Hamiltonian in Eq. ( 7) more explicit,

βH = 1 2 dx φ(x) 2 ρ(x) - 1 2 dxdyφ(x)c (2) (x, y)φ(y). (12)
This shows the nature of the Hamiltonian: It has a local term, corresponding to the local compressibility of an ideal gas, and a nonlocal term, which is given by the direct correlation function. It is worth mentioning that the direct correlation function is a rather featureless function, and typically zero if |x y| is larger than the interaction range of the particles. [START_REF] Hansen | Theory of Simple Liquids[END_REF] In contrast, the correlation function φ(x)φ(y) can extend to larger distances, which, mathematically, is a consequence of taking the operator inverse. Physically, it is well known that correlations may reach further than interparticle interactions. The Ornstein-Zernike equation gives a very clear interpretation of these statements, especially if regarded recursively: [START_REF] Hansen | Theory of Simple Liquids[END_REF] The total correlation between two particles is given by their direct correlation and by "indirect" correlations, obtained from direct correlations between intermediate particles.

We finish this subsection by introducing the short hand notation for the inverse of the static density correlation,

∆(x, y) ≡ k B T 1 ρ(x) δ(x -y) -c (2) (x, y) , ( 13
)
where ∆(x, y) plays the role of an effective interaction potential between densities (we multiplied by k B T to obtain units of energy). In terms of it, the Hamiltonian is finally

H = 1 2 dxdyφ(x)∆(x, y)φ(y). (14)

E. Equation of motion

While static equilibrium averages are determined via the Hamiltonian (with Eq. ( 9)), there is some freedom, or saying it differently, some lack of information, regarding the dynamics. The Hamiltonian in Eq. ( 7) describes only a subset of degrees of freedom of the system. These degrees of freedom might however not capture all relevant features of the dynamics. [START_REF] Kardar | Statistical Physics of Fields[END_REF][START_REF] Hohenberg | [END_REF] The tools of classical mechanics are thus not applicable to deduce equations of motion from Eq. (7). One possibility to overcome this problem is to resort to Langevin equations, [START_REF] Kardar | Statistical Physics of Fields[END_REF][START_REF] Hohenberg | [END_REF] which are based on deterministic (given by the explicit degrees of freedom) as well as stochastic forces (due to the integrated degrees of freedom). 61 The former may be written in terms of the driving force δH δφ , which gives the force due to deviations of H from its minimum value. For our case,

β δH δφ = φ(x) ρ(x) -dy c (2) (x, y)φ(y) (15)
≡ β∆φ.

In the second line, we used the short hand notation of Eq. ( 13), and also a short hand notation for operator products, so that the second line contains an integration over the joint coordinate. Clearly, δH δφ = 0, vanishing in equilibrium. This force transforms into changes in φ with application of the operator R = R(x, y), so that R δH δφ involves an integral over the joint coordinate. R involves, among other things, the mobility coefficient. R having no time dependence, we have already restricted to a time local description for simplicity. Dynamics which is non-local in time may also be realized in this framework. Via the operator R, one can incorporate several types of dynamics, such as dynamics conserving the density, or not conserving it. [START_REF] Kardar | Statistical Physics of Fields[END_REF][START_REF] Hohenberg | [END_REF]62 We aim at dynamics of overdamped Brownian particles given by Eq. ( 1), for which-justified a posteriori-the proper choice for R is

R = D k B T ∇ • ρ(x) ∇δ(x -y). (17)
Note that, because R is written as a divergence, local conservation of density is given. Indeed, the chosen R in Eq. ( 17) is a version of the famous "Model B." [START_REF] Kardar | Statistical Physics of Fields[END_REF][START_REF] Hohenberg | [END_REF] We thus write the following equation of motion:

∂φ ∂t = R δH δφ + ∇ • 2D ρ η(x, t) (18) = D k B T ∇ • ρ(x) ∇ δH δφ + ∇ • 2D ρ η(x, t). ( 19
)
The included stochastic force is fixed through the choice of the operator R. The field η(x, t) is a spatio-temporal vectorial white noise field with η = 0, and whose components have the correlation function

η µ (x, t)η ν (y, t ) = δ µν δ(t -t )δ(x -y). ( 20
)
The form of the last term in Eq. ( 18) and the variance in Eq. ( 20) ensure the validity of the fluctuation dissipation theorem 62 and make sure that Eq. ( 18) yields the correct variance for φ. The explicit form of Eq. ( 18) reads ∂φ ∂t = D∇ • ∇φ -φ∇ log ρρ ∇ dy c [START_REF] Evans | [END_REF] (x, y)φ(y)

+ ∇ • 2D ρ η(x, t). ( 21
)
Examining this equation for the case of an ideal gas in the absence of an external potential, for which c [START_REF] Evans | [END_REF] = 0 and ρ spatially constant, we obtain, as expected, a diffusion equation with a conservative noise term,

∂φ ∂t = D∇ 2 φ + ∇ • 2D ρ η. (22)
While the above equation contains the correct diffusion term for the ideal gas, it yields by construction Gaussian density fluctuations (because it is implied that the noise correlation in Eq. ( 20) is Gaussian, so that higher order correlations of η can be factorized). As a side note, we remark that, interestingly, even for an ideal gas, the distribution of φ is nontrivial and is in fact Poissonian. 63 This difference in underlying statistics only shows up for higher point correlation functions, so that the present theory is exact for the two point correlations in the case of ideal gas. This will be demonstrated at the end of Subsection III A. Together with Eqs. ( 15) and ( 19) gives a closed form for the dynamics of the system, which is our first main result. This dynamics is chosen to yield exact equilibrium correlation functions but also gives the exact time-correlation function for non-interacting Brownian particles. In Secs. III-V, we will investigate the properties of the dynamics proposed here in more detail.

III. TIME DEPENDENT CORRELATION FUNCTION

In this section, we finally compute and analyze the resulting approximative form of the time dependent equilibrium correlation function, as defined in Eq. ( 5), as following from Eq. (19).

A. General result

We start by writing the Langevin equation, Eq. ( 19) in a shorter form, using Eq. ( 13),

∂φ ∂t = D k B T ∇ • ρ(x) ∇ δH δφ + ∇ • 2D ρ η = R∆φ + ∇ • 2D ρ η. (23)
Eq. ( 23) can then be easily solved for the correlation function, from its general solution (see, e.g., Ref. 62),

φ(t) = φ(t 0 ) + t t 0 ds e (t-s)R∆ ∇ • 2D ρ η(s). ( 24
)
The average of φ(t)φ(t ) over the noise contains then several terms, including terms depending on the initial value at t 0 . Aiming at the equilibrium correlation function, we let t and t formally go to infinity and obtain the steady equilibrium part, which depends only on t t (recall that, as before, . . . denotes an average in equilibrium),

C = φ(x, t)φ(x , t ) = k B T ∆ e |t-t |∆R . ( 25
)
The correlation function is generally not an exponential in time, because R and ∆ are operators. Eq. ( 25) is our second main result.

We can now show that Eq. ( 25) is exact for non-interacting particles. To see this it is best to work in Fourier space where the density operator ρ takes the form, for N particles,

ρ(k) = N i=1 exp(-ik • x i ), ( 26
)
where x i obeys Eq. ( 1) with F = 0. The ensemble average in this free gas is over the trajectories of the Brownian motions ξ i . The average of ρ(k) is given by

ρ(k) = (2π) d δ(k) ρ , (27)
with here N/V = ρ the uniform bulk density. A simple computation shows that the two point correlation function of the fluctuations φ at different times is given for large N by

φ(k, t) φ(k , 0) = (2π) d δ(k + k ) ρ exp(-Dk 2 t). ( 28
)
Transforming Eq. ( 25) (derived from Eq. ( 22)) to Fourier space, the agreement with the independently obtained Eq. ( 28) can easily be verified.

B. Comparing to exact solution for short times

For small values of time t t , we expand Eq. ( 25),

φ(x, t)φ(x , t ) = k B T ∆ 1 + |t -t |∆R + . . . . (29)
The dots represent higher order terms in t t . We shall now compare this result to the exact one for Eq. ( 1). (Recall that we assume that canonical and grand canonical systems are equivalent.) For this, we use the Smoluchowski equation corresponding to the set of stochastic equations, Eq. ( 1). The Smoluchowski equation is a partial differential equation for the distribution Ψ(Γ, t), which is a function of phase space

Γ, 12 ∂ ∂t Ψ = ΩΨ. (30)
Ω = D i ∂ i • [∂ i -βF i ]
is the Smoluchowski operator. F i is, as in Eq. ( 1), the force acting on particle i. The equilibrium time correlation function for density is then written as 64 φ(x, t)φ(x , t ) = dΓφ(x)e |t-t |Ω φ(x )Ψ e (Γ).

Here, Ψ e is the equilibrium distribution. For short times, Eq. ( 31) is expanded,

φ(x, t)φ(x , t ) = φ(x)φ(x ) + |t -t | dΓφ(x)Ωφ(x )Ψ e (Γ) + . . . , (32)
where Ψ e being the Boltzmann distribution, one has ∂ i Ψ e = βF i Ψ e . [START_REF] Risken | The Fokker-Planck Equation[END_REF]64 Using this we can rewrite the second term in Eq. ( 32) by use of partial integrations (Einstein summation convention is implied),

dΓφ(x)Ωφ(x )Ψ e (Γ) = -D dΓ(∂ i φ(x))(∂ i φ(x ))Ψ e . ( 33
)
We now employ the definition of φ(x) = i δ(xx i ) -ρ(x) , noticing that the mean density vanishes when plugged into Eq. ( 33): It does not depend on phase space and ∂ i yields zero.

With ∂ i δ(x -x i ) = -∂ x δ(x -x i ), we get -D (∂ i φ(x))(∂ i φ(x )) = -D i ∇ δ(x -x i )δ(x -x i ) ← - ∇ = -D i ∇δ(x -x ) ρ (x ) ← - ∇ = k B TR(x, x ), (34)
where we have identified the operator R from Eq. ( 17). We thus have the exact solution for short times,

φ(t)φ(t ) = k B T ∆ 1 + |t -t |∆R + . . . . (35)
Comparison with Eq. ( 29) reveals that Eq. ( 25) agrees with the exact solution of the Smoluchowski equation for short times, i.e., including the term linear in time. This linear term has been discussed in terms of the "initial decay rate," 65 or in terms of a wavevector dependent diffusivity. [START_REF] Pusey | Dynamic Light Scattering[END_REF]66,67 The current formulation agrees with these.

IV. RELAXATION TO EQUILIBRIUM-AGREEMENT WITH DDFT

While in Sec. III B, we compared the stochastic equation proposed here (Eq. ( 19)) to the exact Smoluchowski equation, in this section, we aim to demonstrate another equivalence: Near equilibrium, the relaxation dynamics of Eq. ( 19) agrees exactly with the corresponding result of DDFT. This will be seen by studying the relaxation of a system which is initially out of equilibrium.

A. Mean relaxation to equilibrium from Eq. (19)

Let us assume that the system is in an initial situation out of equilibrium, so that the mean density deviates from the equilibrium one, and we define as in Table I,

ρ(x) neq = ρ(x) + δ ρ(x). (36)
If δ ρ(x) is small, we can use Eq. ( 19) to compute the relaxation of δ ρ to zero (δ ρ(x) must be small because Eq. ( 19) is linear). Therefore, we replace φ in Eq. ( 19) by δ ρ and remove the noise term, as it vanishes when taking the mean of the equation. We obtain the following equation which is linear in δ ρ:

∂δ ρ ∂t = D∇ • ∇δ ρ -δ ρ∇ log ρ -ρ ∇ dy c (2) (x, y)δ ρ(y) . (37)
In Subsection IV B, we will compute the analogous equation from DDFT and demonstrate the agreement.

B. DDFT expanded near equilibrium

Quoting the equation of motion of dynamical density functional theory for Brownian particles in an external potential U, 21 one has

∂δ ρ ∂t = D∇ • ∇( ρ + δ ρ) + β( ρ + δ ρ)∇U + β( ρ + δ ρ)∇ δF ex δ ρ , ( 38
)
where F ex is the so-called excess free energy functional. This is a well known and well studied equation, which is an approximate solution of the Smoluchowski equation, Eq. (30). It has been successfully used in many scenarios to describe the dynamics of interacting Brownian particles. 21 We now expand this equation for small values of δ ρ, as in Eq. (37). We first note that several terms cancel, as the time derivative must vanish in equilibrium. Specifically (note that even the term in the square brackets vanishes),

0 = ∇ •       ∇ ρ + β ρ ∇U + β ρ ∇ δF ex δ ρ ρ= ρ       . ( 39
)
Furthermore, for small δ ρ, we expand the last term in Eq. ( 38) in a functional Taylor series,

δF ex δ ρ(x) = δF ex δ ρ(x) ρ= ρ + dy δF ex δ ρ(x)δ ρ(y) ρ= ρ δ ρ(y) + O(δ ρ 2 ). (40)
It is now important to note that the involved Taylor coefficient equals, by definition, the direct correlation function c [START_REF] Evans | [END_REF] , 1

-β δF ex δ ρ(x)δ ρ(y) ρ= ρ = c (2) (x, y). (41)
Another useful relation is the formal exact result for the equilibrium mean density, which is given by 1

ρ = z exp       -βU - βδF ex δ ρ(x) ρ= ρ       , ( 42
)
with the (in the following irrelevant) normalization z (the fugacity). With this equation, one can write

k B T ∇ log ρ = -∇U -∇ δF ex δ ρ(x) ρ= ρ . (43)
We finally obtain for the expansion of Eq. ( 38) linear in δ ρ,

∂δ ρ ∂t = D∇ • ∇δ ρ -δ ρ∇ log ρ -ρ ∇ dy c (2) (x, y)δ ρ(y) (44)
which is identical to Eq. (37). We have thus shown that the new equation, Eq. ( 19) is in agreement with DDFT for small deviations from equilibrium. This demonstrates a connection to the framework of Ref. 53, without, however, an obvious direct equivalence.

V. COMPARISON TO THE EXACT STOCHASTIC EQUATION

Starting from the set of stochastic equations in Eq. ( 1), the following exact stochastic equation of motion is found for the density operator ρ (see Table I

), 52 ∂ ∂t ρ(x) = D∇ • ρ∇ δ βE δ ρ(x) + ∇ • 2Dρη(x, t). ( 45
)
Here, the noise η is distributed as in Eq. ( 20), and E is the energy functional,

E = k B T dxρ(x) ln(ρ(x)) + 1 2 dxdyρ(x)V (x -y)ρ(y) + dxρ(x)U(x). ( 46
)
We have as before split the potential into an interaction part V and an external part U. Note the difference of Eq. ( 46) compared to the free energy functional of DFT. [START_REF] Evans | [END_REF]52,53 Attempting to linearize Eq. ( 45) in the fluctuations φ, the first natural choice is to replace the density operator appearing in the noise term by its equilibrium average, i.e., 2Dρη(x, t) ≈ 2D ρ η(x, t). Interestingly, in order to keep detailed balance, this choice implies that also the density operator on the right hand side of Eq. ( 45) must be replaced by its mean, i.e., the resulting approximate, consistent linear equation reads

∂ ∂t φ(x) = D∇ • ρ ∇ δ βE δφ(x) + ∇ • 2D ρ η(x, t). ( 47
)
Furthermore, we note that Eq. ( 47) yields the correct result for the variance of φ in equilibrium, φ(x)φ(y) , if the functional E coincides with H of Eq. ( 7), E = H. It is thus interesting to note that after pre-averaging the noise, Eq. ( 19) appears to be the only consistent, linear equation for φ. It has been recently shown that linearizing the interaction term in Equation ( 45) about the mean bulk density, while using the mean bulk density in the noise term, leads to an analytically soluble theory in the bulk which recovers the random phase approximation for the equal time correlation functions, [68][69][70][71][72] notably this means that Debye-Hückel theory is obtained for Brownian electrolytes. The approach has been applied to a variety of driven and out of equilibrium systems. In particular, it is capable of reproducing the full Onsager theory of electrolyte conductivity, both the Ohmic linear response regime and the first Wien effect regime where the conductivity is enhanced by the electric field. 72 While the random phase approximation is valid only for weak interactions or high temperatures, the approach here should allow the study of systems with form instance hard core interactions, relevant for ionic liquids, both in the bulk and under confinement.

VI. SUMMARY

We have derived an effective field theory for simple liquids, which allows the computation of dynamical correlation functions of the density. The result for the dynamical correlation function is approximate, but exact for small times. The described dynamics also agrees exactly at all times with dynamical density functional theory. Future work will apply this theory to study the intermediate scattering function in confinement (where recent experimental findings exist 40 ). It will also be used to find the local viscosity near surfaces and compare to previous theoretical approaches for bulk 29 and confinement. 30,31 For this, an expression for the stress tensor in this theory must be derived. Then this theory may also be used to investigate out of equilibrium Casimir forces in model B as in Ref. 73, however including effects of finite particle size.

TABLE I .

 I Observables studied in this manuscript. The lower two rows, i.e., the density in nonequilibrium states, are given for comparison to dynamical density functional theory in Sec. IV.

	Symbol	Meaning
	ρ(x)	Density operator: ρ

ACKNOWLEDGMENTS

We thank C. Rohwer for helpful discussions. This research was supported by DFG Grant No. KR 3844/2-1 and the ANR project FISICS.


Johanna Choumert
email: j.choumert.nkolo@surveybe.com

Pascale Combes Motel

Guegang Charlain
email: charlainguegang@gmail.com

Djimeli

Grégoire Rota-Graziosi

Keywords: Biofuel, Bioethanol, Biodiesel, Energy, Development, Meta-regression, Computable General Equilibrium Model

While the production of biofuels has expanded in recent years, findings in the literature on its impact on growth and development remain contradictory. This paper presents a metaanalysis of computable general equilibrium studies published between 2006 and 2014. Using 26 studies, we shed light on why their results differ. We investigate factors such as biofuel type, geographic area and the characteristics of models employed. Our results indicate that the outcomes of CGE simulations are sensitive to model parameters and also suggest heterogenous effects of biofuel expansion between developed / emerging countries and Sub-Saharan African countries. Our quantitative meta-analysis complements existing narrative surveys and confirms that results are sensitive to key hypotheses on essential parameters. Simulations on longer time periods and in multi-country studies lead to results that indicate higher impacts of biofuel expansion on growth and household income. Moreover, simulations with a shock in agricultural productivity indicate positive welfare gains, unlike simulations with a shock on land expansion. Lastly, we find that biodiesels lead to higher welfare gains than biofuels.

JEL codes

Q16, O13, C68 .

Introduction

Many authors have documented the complex relationship between energy and economic development. To date, no consensus has been reached regarding this so-called energydevelopment nexus which has mainly focused on evidencing a causal relationship between energy consumption and economic growth (see e.g. the review of Ozturk, 2010). Two avenues of research are still opened. One strand of the literature focuses on the demand aspects of energy consumption while the literature on energy transition (energy ladder) describes qualitative and quantitative changes in energy consumption and analyses how households switch from traditional to modern fuels (e.g., [START_REF] Leach | An Economy-wide Analysis of Increasing Bio-Ethanol Production in Taiwan[END_REF] or van der [START_REF] Van Der Kroon | The energy ladder: Theoretical myth or empirical truth? Results from a meta-analysis[END_REF]. It also provides a framework for the implementation of enhanced energy services. Another strand of the literature focuses on supply issues. One crucial question is related to the Malthusian-like issue of the availability of fossil energies and whether they might be a growth-limiting factor [START_REF] Carbonnier | Energy and Development[END_REF]. Access to energy has also recently been targeted by global development policies. It was not a key component of the Millennium Development Goals, but in 2012 the United Nations "Sustainable Energy for All" Initiative (universal energy access, renewable energy, and energy efficiency) renewed the interest for initiatives to reduce energy poverty. In 2015, Sustainable Development Goal #7 (Ensure access to affordable, reliable, sustainable and modern energy for all) made room for energy initiatives in development policy objectives. Biofuels are serious candidates for addressing global energy objectives. They received substantial attention in climate mitigation initiatives in the transport sector, and targets were reported by the Intended Nationally Determined Contributions (INDCs) to the UNFCCC 1 Secretariat: about 40% of INDCs submitted by countries in 2015 relied on biofuels [START_REF] Gota | Intended Nationally-Determined Contributions (INDCs) Offer Opportunity for Ambitious Action on Transport and Climate Change[END_REF]. The use of biofuels might reshape several crucial issues such as energy security and climate mitigation as well as the development energy nexus, especially in developing countries. Indeed, biofuel expansion can be seen as a means of transforming national dependence on biomass into an additional source of income and job generating activities, particularly in countries where poverty pervades. In addition, biofuel production may be a key element for energy security in countries that are net importers of energy. Moreover, since biofuel cropping uses atmospheric carbon dioxide, biofuels may contribute to mitigating greenhouse gas (GHG) emissions. Further environmental benefits may also be gained if biofuel expansion pushes up land values and incentivizes land upgrading.

However, more pessimistic views on the consequences of biofuel expansion have also been put forth. First, weak land governance is a pull factor of Foreign Direct Investments (FDIs) and is therefore reputed to contribute to land rush (e.g., [START_REF] Arezki | What Drives the Global "Land Rush[END_REF]. Deleterious effects on unwritten customary land rights have been reported, especially in Africa, where biofuel production has been a key driver of the rising demand for land allocated to large-scale cultivation of biofuels [START_REF] Toulmin | Securing land and property rights in sub-Saharan Africa: The role of local institutions[END_REF]. Second, demand for biofuel cropping may induce increases in agricultural feedstock prices [START_REF] Rosegrant | Global Scenarios for Biofuels: Impacts and Implications[END_REF]. Food security may be imperiled in countries where the food expenditure share of household income is high, such as in 1 United Nations Framework Convention on Climate Change.

Sub Saharan African countries [START_REF] Chauvin | Food production and consumption trends in sub-Saharan Africa: Prospects for the transformation of the agricultural sector[END_REF]. Third, the influential paper written by [START_REF] Searchinger | Use of US croplands for biofuels increases greenhouse gases through emissions from land-use change[END_REF] questioned the positive influence of biofuels on the environment 2 and pointed out that biofuel production competes for land, which generates indirect land use changes and contributes to GHG emissions. In short, biofuel expansion is a much-debated issue that deserves further study in the literature. This paper contributes to the debate while focusing on the income generating effect of biofuel production.

Why concentrate on income generating effect of biofuel production? The main reason is that it has dramatic implications in developing countries especially those that need to jointly reach development objectives and manage environmental constraints. Biofuel policies have been endorsed by many countries. Several African countries have tried to implement attractive conditions for biofuel producers owing to their land abundance, for instance Tanzania and Mozambique [START_REF] Amigun | Biofuels and sustainability in Africa[END_REF]Arndt et al., 2011). Land abundance is a key aspect of Brazil agricultural and energy policy (e.g., [START_REF] Stattman | Governing biofuels in Brazil: A comparison of ethanol and biodiesel policies[END_REF] as well as Asian major biofuel producers [START_REF] Kumar | A review of biofuel policies in the major biofuel producing countries of ASEAN: Production, targets, policy drivers and impacts[END_REF].

How to assess financial consequences of biofuel production? A recent upsurge in biofuel production makes it difficult to evaluate ex post the consequences of biofuel expansion on national and household incomes. 3 Simulation exercises and computable general equilibrium (CGE) modeling have attractive features as policy analysis tools and allow to focus on several key aspects of biofuel expansion. First, biofuel expansion is a worldwide phenomenon and CGE modeling allows to easily take international interdependencies into account. Second, at the national level, biofuel expansion cannot be studied in isolation from other sectors of national economies [START_REF] Kretschmer | Integrating bioenergy into computable general equilibrium models-A survey[END_REF]. In other words, CGE modeling allows for modeling a variety of feedback or indirect effects induced by biofuel policies.

CGE modeling results played a crucial role in the "fuel versus food" debate (see e.g., [START_REF] Chakravorty | Fuel Versus Food[END_REF] and study results have widely differed. One crucial question, therefore, emerges of whether the results are sensitive to parameters, modeling choices, results reporting or data assumptions. The objective of this paper is to examine the impact of biofuel policies on growth and household welfare. In contrast to existing studies and surveys on the topic (e.g., [START_REF] Gasparatos | Biofuels in sub-Sahara Africa: Drivers, impacts and priority policy areas[END_REF][START_REF] Ji | A review of the ecological and socioeconomic effects of biofuel and energy policy recommendations[END_REF], we address the subject with a meta-analysis. To the best of our knowledge, ours is the first meta-analysis on the topic of biofuel and development. In addition, our study focuses on CGE results.

The remainder of the paper is organized as follows. Section 2 discusses the use of a metaanalysis. Section 3 is devoted to the construction of the database. Section 4 develops the empirical strategy and economic results. Section 5 presents the results and Section 6 concludes.

Meta-analysis

Definition of meta-analysis

A meta-analysis or meta-regression analysis (MRA) is a quantitative summary of empirical studies on a precise topic [START_REF] Paldam | Meta-Analysis in a Nutshell: Techniques and General Findings[END_REF][START_REF] Stanley | Wheat from chaff: Meta-analysis as quantitative literature review[END_REF]. The empirical literature on the impact of the expansion of biofuels on development includes dozens of studies which cannot be fully captured in a narrative survey. Therefore, conducting an MRA on the topic will complement narrative reviews and facilitate the understanding of factors that contribute to the different outcomes of existing studies. This approach also helps us to avoid the so-called "myth of the perfect study" which can be found in narrative reviews where researchers select studies based on idiosyncratic judgments [START_REF] Hunter | Methods of Meta-Analysis: Correcting Error and Bias in Research Findings[END_REF].

The usual way of checking the results obtained in any applied work is to perform robustness tests or sensitivity analyses such as running alternative estimators, using alternative measures of interest variables or varying the geographical coverage of the dataset in econometric studies. Robustness tests can also take the form of elaborating additional scenarios or changing key parameter values in CGE simulation exercises. Conducting an MRA can be considered more thorough than robustness tests, which relate only to the sets of hypotheses made in one study. MRAs are meta-robustness analyses in the sense that they are deemed to comprehensively collect the set of hypotheses used in the existing literature. While robustness analyses mainly allow for checking the sensitivity of results to a set of particular hypotheses, MRAs help to identify the hypotheses that drive the results and explain their heterogeneity (Nelson and Kennedy, 2008, p. 346).

An MRA should be conducted according to the following steps [START_REF] Nelson | The Use (and Abuse) of Meta-Analysis in Environmental and Natural Resource Economics: An Assessment[END_REF][START_REF] Stanley | Meta-analysis of economics research reporting guidelines: reporting guidelines for meta-regression analysis in economics[END_REF]: formulation of the topic, design of the protocol for the selection of studies, definition of the economic outcome (effect size) and explanatory variables (moderators), and extraction of the data and analysis.

Recent meta-analysis on energy-related topics

The contribution of our MRA is twofold. First, to the best of our knowledge, it is the first MRA on biofuels and development. Second, our primary studies rely on general equilibrium modeling, although most MRAs are based on econometric studies.

Although our MRA is original, a large number of MRAs have been conducted on energy-related topics. Several MRAs investigate the causality between energy use (production/consumption) and economic output/GDP [START_REF] Bruns | Is There Really Granger Causality between Energy Use and Output?[END_REF][START_REF] Chen | Energy consumption and economic growth-New evidence from meta analysis[END_REF][START_REF] Kalimeris | A meta-analysis investigation of the direction of the energy-GDP causal relationship: implications for the growth-degrowth dialogue[END_REF][START_REF] Menegaki | On energy consumption and GDP studies; A meta-analysis of the last two decades[END_REF]. These MRAs do not provide a clear-cut answer on the causality relationship and its direction. Other MRAs focus on energy demand, i.e., from the perspective of experimental studies [START_REF] Delmas | Information strategies and energy conservation behavior: A meta-analysis of experimental studies from 1975 to 2012[END_REF] or in terms of residential consumption [START_REF] Lampin | Meta-analysis of Energy Use in Cities[END_REF]. Still others explore fuel/gasoline consumption (price and income elasticities) [START_REF] Espey | Gasoline demand revisited: an international meta-analysis of elasticities[END_REF][START_REF] Goodwin | Elasticities of Road Traffic and Fuel Consumption with Respect to Price and Income: A Review[END_REF][START_REF] Havranek | Demand for gasoline is more price-inelastic than commonly thought[END_REF][START_REF] Stern | Interfuel Substitution: A Meta-Analysis[END_REF]. Given the recent growth of the literature on renewable energies, there are fewer MRAs on this topic and these are very recent publications. [START_REF] Sebri | Use renewables to be cleaner: Meta-analysis of the renewable energy consumption-economic growth nexus[END_REF] examines 40 primary studies on the renewable energy consumptioneconomic growth nexus and finds a significant difference of short-term versus long-term causality and differences according to the type of data and model specification. [START_REF] Soon | Willingly or grudgingly? A meta-analysis on the willingness-to-pay for renewable energy use[END_REF] analyze 18 studies estimating the willingness-to-pay for renewable energy use of households.

While most MRAs (like those on energy previously mentioned) gather information from studies using econometric estimations, we focus on studies based on general equilibrium modeling. Despite the rapid development of meta-analysis in economics, very few MRAs extract information from general equilibrium simulations. They cover different topics such as trade liberalization [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF] or climate change policies [START_REF] Barker | Costs of greenhouse gas abatement: meta-analysis of post-SRES mitigation scenarios[END_REF][START_REF] Branger | Would border carbon adjustments prevent carbon leakage and heavy industry competitiveness losses? Insights from a meta-analysis of recent economic studies[END_REF][START_REF] Rose | A Meta-Analysis of the Economic Impacts of Climate Change Policy in the United States[END_REF]. More closely related to our topic, [START_REF] Condon | Impacts of ethanol policy on corn prices: A review and meta-analysis of recent evidence[END_REF] conducted a meta-analysis of U.S. biofuel policy with a focus on the impact of corn expansion on corn prices using 29 studies (157 estimates) published since 2007.

Construction of the database

Sampling procedure and sample characteristics

This MRA is based on 26 studies (see Annex 1) published between 2006 and 2014. To conduct our bibliographic research, we entered keywords such as 'Biofuels AND Poverty', 'Biofuel AND General Equilibrium', 'Biodiesel AND Poverty', 'Bioethanol AND Development' in bibliographic databases such as Science Direct, Google Scholar, etc. Using this list of studies, we performed snowballing and obtained a pool of 150 papers. We then applied inclusion and exclusion criteria to publications, e.g., those including only general equilibrium studies, having simulations of biofuels, and having a measure of household income or growth. A collection grid was then used to codify the information from each article. Two databases were created, one for the impact of biofuel on GDP growth and one for the impact on household income. Although most variables are common to the two databases, there are variables specific to each one. The last step in the construction of these databases was the processing of the data, coding and labelling the variables, and searching and correcting errors in entries.

Effect sizes

An effect size (ES) is an outcome measure common to all primary studies. We have two different outcomes (hence two types of ESs) measuring the effect of biofuel expansion on economic development: GDP is the variation of the GDP growth rate (percentage points); INCOME is the variation of household income growth rate (percentage points). 4 Each simulation generates a welfare variation for a single or multiple regions. An ES is the welfare change for one individual country. 5 See Table 1 for summary statistics of ESs. The range of estimates is very wide and illustrates the lack of consensus in the literature (Figure 3).

In MRAs relying on econometric estimations, it is recommended to use a measure of precision for the ESs, usually standard errors of estimates or sample size [START_REF] Doucouliagos | Democracy and Economic Growth: A Meta-Analysis[END_REF][START_REF] Stanley | Meta-regression analysis in economics and business[END_REF][START_REF] Sterne | Funnel plots for detecting bias in meta-analysis: Guidelines on choice of axis[END_REF]. As our focus is CGE studies, such measures of precision are not relevant.

Moderators

To explain the variability of ESs, we use various moderators providing information on the characteristics of publications, countries, models and scenarios (Table 2).

Publication characteristics

As shown in Figure 1, studies on the topic are recent and their number has increased over time.

There are 23 primary studies for GDP and 22 for INCOME (with some overlapping, see Annex 1 and Annex 2). This notable increase in the number of publications from 2010 perhaps reflects the evolution of the production of biofuels as shown in Figure 2. (For an analysis of the determinants of biofuel production, see, for instance, [START_REF] Kere | Do political economy factors matter in explaining the increase in the production of bioenergy?[END_REF]. We can presume that this increasing interest in the topic reflects the surge in production and associated debates within the policy arena. In their "Review of Ecological and Socioeconomic Effects of Biofuel and Energy Policy Recommendations", Ji and Long (2016) selected 124 papers (published after 2004) to review, and it appears that 66% were published after 2011, corroborating this increased interest in biofuel-related research. The evolution of ESs over time is shown in Figure 3. Their distribution illustrates the contrasting results on the impact of biofuels on development and poverty found in literature reviews (e.g., [START_REF] Ji | A review of the ecological and socioeconomic effects of biofuel and energy policy recommendations[END_REF]. Not only has the number of studies increased over time (and therefore the number of ESs) but also the interval of the values taken by the ESs. This may be explained by the following hypotheses: (i) As there are more studies, the likelihood of finding more varying results increases (ii) Over time the enthusiasm for biofuels has been challenged and it has become more acceptable to publish negative results (see the Introduction for an overview of pessimistic views on biofuels). The number of ESs per paper varies from 1 to 56 with an average of 11.5 for INCOME and from 1 to 35 with an average of 20 for GDP. This is explained by the fact that authors run several scenarios (this will be addressed in Section 4).

Regarding the publication type, we differentiate between, on the one hand, published articles and, on the other hand, working papers and institutional papers. We suspect that ESs published in peer-reviewed articles will take more "conventional" values than ESs published in others formats such as working papers. This also relates to the recurrent publication bias which is faced in meta-analysis. There may be a bias in the reporting of results depending on their sign or intensity. The sources of publication bias are diverse and complex as they depend on the behavior of authors, referees and editors. See [START_REF] Stanley | Meta-regression analysis as the socioeconomics of economics research[END_REF] and [START_REF] Stanley | Meta-regression analysis in economics and business[END_REF] for discussions on the publication bias and the "file drawer" problem.

Geographical area

We include several variables to control for the geographical area and the level of economic development of countries: (i) Binary variables for the continent where each country is located (ii) An OECD variable indicating whether the country belongs to the OCED (iii) A variable for the GDP per capita of the country. The driving forces behind biofuel expansion differ between countries and may lead to different outcomes on growth and household income. Biofuel production has followed different patterns across countries, with the USA being the largest ethanol producer and the European Union being the largest biodiesel producer. Public policies, tax exemptions, subsidies and mandatory blending targets play key roles in these expansions. See [START_REF] Sorda | An overview of biofuel policies across the world[END_REF] for an overview of policies implemented in North America, South America, Europe, Asia and Australia. [START_REF] Gasparatos | Biofuels in sub-Sahara Africa: Drivers, impacts and priority policy areas[END_REF] provide an overview of the development of biofuel policies in Sub-Saharan Africa, which have been partly stimulated by FDIs.

Model characteristics

CGE models can be static, dynamic or recursive dynamic. We have chosen to construct a binary variable Dynamic =1 if the model is dynamic and =0 if the model is static or recursive dynamic. These differences reflect different assumptions of agents' expectations. In static models, economic agents are myopic and do not look forward, i.e., beyond the current period, to make decisions. In recursive dynamic settings, the model is solved following two steps. First, through a "within-period" static CGE framework and through a "between-period" model. In such models, it is assumed that agents cannot behave according to perfect foresight. In dynamic CGE models, agents are forward looking and their intertemporal behavior is taken into account. Such a framework tends to be used for simulations focusing on intertemporal resource allocation questions, such as climate change [START_REF] Babiker | Forward-looking versus recursive-dynamic modeling in climate policy analysis: A comparison[END_REF]Dixon and Jorgenson, 2013, Chap. 5;[START_REF] Scollay | Measuring the gains from APEC trade liberalisation: An overview of CGE assessments[END_REF].

In the CGE literature the main models used are the GTAP and the MIRAGE models (GTAP, MIRAGE). The GTAP (Global Trade Analysis Project)7 model is the standard CGE model. It is multiregion, multisector, with perfect competition and constant returns to scale. The MIRAGE (Modelling International Relationships in Applied General Equilibrium) 8 model is also a multiregion, multisector CGE model with a sequential dynamic set-up, imperfect competition, product differentiation by variety and by quality, and foreign direct investment [START_REF] Bchir | MIRAGE, A Computable General Equilibrium Model for Trade Policy Analysis[END_REF]. Other models used are EXTER (Boccanfuso et al., 2013;[START_REF] Decaluwé | La politique économique du développement et les modèles d'équilibre général calculable: une introduction à l'application de l'analyse mésoéconomique aux pays en développement[END_REF], MEGABARE 9 (see Dixon and Jorgenson, 2013), ORANI (Dixon and Jorgenson, 2013), the TERM-BR (general equilibrium model of Brazil) model [START_REF] De Souza Ferreira Filho | The rise in global demand for ethanol and poverty in Brazil[END_REF], and the TIGER10 (The Trade Integrated Global Energy and Resources) model (See Gunatilake et al., 2014).

Most papers rely on an Armington specification allowing domestic and imported goods to be modeled as imperfect substitutes. Standard Armington elasticities in GTAP models are low compared to other models. Simulations on international trade tend to find lower welfare gains when relying on GTAP values [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF].

The biofuel-development nexus has been investigated through multiregion (multi_country) models as well as through single-region models. Single-region models seem more appropriate for studies on a specific topic in a small region. However, using single-region CGE models, may lead researchers to ignore effects and feedback from other regions of the world and may therefore cause misleading interpretation of results (Dixon and Jorgenson, 2013). "Singlecountry GEs generally do not allow for endogenous capital inflows and they often assume fixed trade balances and exchange rates. They might therefore be expected to generate lower welfare gains than multi-country GEs." [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF]. Accordingly, we should expect lower welfare gains in single-region CGE models.

We introduce the length of the model (Time_estimation) to account for long-run versus shortrun simulations. Following [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF] long-run CGE models should generate higher welfare gains than short-run ones. This reflects that long-run CGE models take into account increases in the scope of the world economy.

Scenario

Authors implement various scenarios in their simulations and extracting them was rather difficult. In the MRA, we control for whether the scenario involves an expansion of land allocated to biofuels (Land) and whether the scenario implies an increase in agricultural productivity or biofuel productivity (Productivity). In some instances, we have been able to calculate the surface allocated to biofuels in the simulation by multiplying the expansion rate of land allocated to biofuels and the surface of the country (Land_surface).

Biofuel type

There are two types of biofuels, namely biodiesel and bioethanol. Biodiesel encompasses jatropha, caster bean, palm oil, soybean, rape sunflower oil; bioethanol includes sugarcane, cassava, wheat, maize, corn, potatoes and sorghum. In the primary studies, authors carry out a simulation on biodiesel, bioethanol or both.

The production of these biofuels can have multiple impacts. [START_REF] Gasparatos | Biofuels in sub-Sahara Africa: Drivers, impacts and priority policy areas[END_REF] list (i) Economic impacts: economic development, energy security and foreign exchange savings (ii) Environmental impacts: water availability, water quality, GHG emissions and biodiversity loss (iii) Social impacts: poverty alleviation, food security and access to land. These impacts vary per the scale of production.

Household type

For the ES INCOME, we add a variable on household characteristics, indicating whether they are rural, urban, poor or better-off.

Empirical strategy

Base model

We estimate two complementary models corresponding to each type of ES, GDP and INCOME. The base model is: or is a vector of effect sizes (GPD and INCOME) that is the welfare change in a country due to a biofuel policy; is a matrix of moderators; is a vector of meta-regression coefficients. is a vector of residuals. Most MRAs using estimates from CGE models are estimated using OLS (e.g., [START_REF] Barker | Costs of greenhouse gas abatement: meta-analysis of post-SRES mitigation scenarios[END_REF][START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF]. Authors of such MRAs have stressed the need to account for multiple ESs in primary studies [START_REF] Barker | Costs of greenhouse gas abatement: meta-analysis of post-SRES mitigation scenarios[END_REF][START_REF] Branger | Would border carbon adjustments prevent carbon leakage and heavy industry competitiveness losses? Insights from a meta-analysis of recent economic studies[END_REF][START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF].

MRAs using ESs obtained using econometric estimations can be estimated in several ways, which are discussed in the literature (see [START_REF] Ringquist | Meta-Analysis for Public Management and Policy[END_REF]Stanley and Doucouliagos, 2014a, 2014b, 2012). Stanley and Doucouliagos (2012, p. 122) advise using very simple econometric models and promote the use of Weighted Least Squares (WLS), which is discussed in [START_REF] Stanley | Wheat from chaff: Meta-analysis as quantitative literature review[END_REF]Doucouliagos (2014a, 2014b). WLS allow for correcting for heteroscedasticity and excess heterogeneity using sample size as weight (or other precision measures). In this MRA, we cannot use such models as we do not have any precision measure. Consequently, we estimate robust OLS. Additionally, to capture dependence among ESs when multiple ESs are reported in studies, we estimate cluster-robust regressions, taking a study as a cluster. The cluster-robust MRA will differ from the robust OLS MRA in terms of standard errors: these will be computed considering the potential dependence within a cluster. Finally, we estimate multi-level regressions (mixed-effect model), which are an alternative way to capture dependence among ESs reported in studies, with two levels, i.e., ESs and primary studies.

Selection of moderators and robustness checks

Firstly, our moderators present high correlations between them. Therefore, we choose to present several specifications for each model. Secondly, ESs belonging to a same primary study may be correlated, which can introduce a bias in the MRA. We take this into account by using cluster robust regressions as recommended by [START_REF] Stanley | Meta-regression analysis in economics and business[END_REF] with primary studies as clusters. Robust OLS regressions are presented in Table 4 andTable 5

Results

The meta-regressions explain 30-37% and 25-48% of the variance of ESs for GDP and INCOME, respectively.

We do not find a significant impact of the year of publication (year) or its type (article). We find a positive and significant sign for time_estimation, for both GDP and INCOME, suggesting welfare gains increase with the length of the simulation. This is line with results found by [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF]. We also find a positive and significant impact of multi_country for both ESs and in all specifications. Such an effect is also found by [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF] and indicates increased welfare gains in multi-region models which account for the reallocation of international factors and comparative advantages.

The results for the Armington variable are mixed and do not allow us to derive useful results. MIRAGE and other models tend to generate higher welfare gains than GTAP models. This result is again in line with [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF]. As previously explained, standard Armington elasticities in GTAP models are lower than other models, which could explain why simulations relying on GTAP values tend to find lower welfare gains [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF].

Dynamic models lead to higher welfare gains for HH INCOME compared to static or recursive dynamic models. In dynamic models, economic agents are forward-looking and their intertemporal behavior is taken into account, which implies time-consistent behaviors.

When the scenario implies the expansion of land allocated to biofuels (Land), ESs for both GDP and INCOME models decrease. This suggests that competition for land with other agricultural crops is detrimental to welfare. Interestingly, most simulations with such scenarios were performed in Sub Saharan Africa, e.g., Mozambique, Tanzania, Mali, Ethiopia (e.g., [START_REF] Arndt | Biofuels, poverty, and growth: a computable general equilibrium analysis of Mozambique[END_REF]Arndt et al., 2011;Arndt et al., 2012;Boccanfuso et al., 2013;Ferede et al., 2013;Gebreegziabher et al., 2013;Gemechis, 2012). Several authors have put an emphasis on Sub Saharan African land endowments: uncultivated land availability in this region represents roughly half the world's endowments. However, Sub Saharan African land potential is concentrated in a few countries (Deininger and Byerlee 2011) and is subjected to highly heterogeneous agro-ecological conditions [START_REF] Wicke | The current bioenergy production potential of semi-arid and arid regions in sub-Saharan Africa[END_REF]).14 Conversely, simulations containing a scenario with an increase in agricultural productivity or biofuel productivity (productivity) consistently impact ESs positively.

Results for the geographical area are not clear-cut. They tend to indicate that OECD countries have higher gains in terms of GDP and that Africa and Asia (or countries with a lower GDP per capita) have higher welfare gains for HH INCOME. Understanding the underlying mechanisms is tricky for several reasons. First, biofuel production has more severe effects on food prices in developing countries than in developed ones (FAO, 2008). Second, tax exemptions, subsidies and mandatory blending targets influence the economic profitability of biofuel production. These policies have mainly been implemented in the USA, Europe and emerging economies (FAO, 2008;[START_REF] Sorda | An overview of biofuel policies across the world[END_REF]. With such distortions, determining the real economic profitability of biofuels is a difficult task. Third, countries produce different mixes of bioethanol and biodiesel. In addition, we find that the positive impact is higher for rural HH compared to urban HH and for a mix of poor and rich HH compared to rich HH only. This could suggest that biofuel production favors rural employment (FAO, 2008) and may, therefore, contribute to rural development.

We find that the impact of a biodiesel scenario is positive compared to bioethanol only. Recall that biodiesels encompass jatropha, caster bean, palm oil, soybean and rape sunflower oil while bioethanols include sugarcane, cassava, wheat, maize, corn, potatoes and sorghum. The production costs are much higher for rape sunflower oil (e.g., 3.29 USD per gallon in the EU) than sugarcane (0.25 USD per litre in Brazil) (FAO, 2008). More generally, biodiesels tend to be more expensive to produce, which could be what is captured in the GDP growth. Also, soybean prices have been significantly rising over the past decade, with a slowdown after 2014, therefore substantially increasing the profitability of this crop.

Discussion and conclusion

This meta-analysis is to the best of our knowledge the first to look at the impact of biofuel production on economic growth and poverty. We focus on Computable General Equilibrium Studies. CGE modeling is a particularly relevant and interesting framework to investigate the impact of biofuels on various dimensions such as growth, poverty and food prices (Dixon and Jorgenson, 2013;[START_REF] Kretschmer | Integrating bioenergy into computable general equilibrium models-A survey[END_REF]. It has indeed played a crucial role in the fuel versus food debate (see e.g., [START_REF] Chakravorty | Fuel Versus Food[END_REF]. CGE studies have provided contrasting results notably because the results are sensitive to scenario parameters, modeling choices, data assumptions, etc. Our contribution is to precisely examine the impact of biofuel policies on growth and household income by considering model parameters in a meta-analysis framework.

Our quantitative meta-analysis is complementary to existing narrative surveys and confirms that results are sensitive to key hypotheses on essential parameters. Simulations on longer time periods and simulations in multi-countries lead to higher impacts of biofuel expansion on growth and household income. Moreover, simulations with a shock on agricultural productivity lead to positive welfare gains, unlike simulations with a shock on land expansion. Lastly, biodiesels lead to higher gains compared to biofuels. Overall, according to the literature, the impact of biofuel expansion appears to be positive on both growth and household welfare.

However, CGE models do not account for non-priced effects such as externalities. To get a comprehensive overview of the impact of biofuel expansion, this meta-analysis should be combined with models that assess the environmental and social sustainability of biofuels as they both offer advantages and risks. See FAO (2008) for a thorough review of these environmental and social challenges. One way forward might consist in better integrating damage functions into CGE models. This is done, for instance, in long-term growth models that assess the social benefits of greenhouse gases abatement, following the pioneering work of [START_REF] Nordhaus | To Slow or Not to Slow: The Economics of The Greenhouse Effect[END_REF]. Crucial hypotheses pertaining to the convexity of the damage functions induce substantial variations in the social outcome (see, for example, e.g., the analysis of [START_REF] Dietz | Endogenous Growth, Convexity of Damage and Climate Risk: How Nordhaus' Framework Supports Deep Cuts in Carbon Emissions[END_REF].

Several questions were answered through this meta-analysis but other interesting research avenues remain. One crucial question that prevails involves the effect of biofuel on land allocation and land use. Assuming productivity gains in agriculture could induce a rebound effect, if agriculture is more profitable, more resources will be allocated to agriculture. The underlying mechanism is similar to a Jevons' Paradox. Therefore, more research efforts should be devoted to addressing the question of which environmental consequences of biofuel expansion are the most critical. Furthermore, on the methodological side, MRAs that use econometric studies as primary studies include a measure for the precision of estimates. In the case of primary studies relying on CGE, such measures are not available; thus, future research in meta-analysis methodology should focus on identifying alternative precision measures. Finally, as raised by [START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF], the results of CGE models are a "complex function of the many factors…[]…and of interactions between them". It is not possible to code all characteristics of publications given the enormous amount of data and computation needed to run a CGE and the fact that not all of the information needed for a CGE is provided in published papers.

Figure 1 .

 1 Figure 1. Number of primary studies over time GDP INCOME

Figure 2 .

 2 Figure 2. Evolution of biofuel production in the world (2000-2012)

Figure 3 .

 3 Figure 3. Evolution of ESs over time GDP INCOME

Table 1 .

 1 Descriptive statistics of effect sizes

	Variable	Obs	Mean	SD	Min	Max
	GDP	227	0.18	0.98	-1.62	9.82
	INCOME	255	0.57	1.22	-3.84	8.90

Table 2 .

 2 List of moderators

	Variable	Description
	Publication characteristics
	Year	Year of publication
	Article	= 1 if published article; =0 if working paper or report
	Geographical area
	Africa	= 1 if African country
	Europe	= 1 if European country
	Asia	= 1 if Asian country
	South_america	= 1 if South American country
	North_america	= 1 if North American country
	OECD	= 1 if OECD country
	GDP_cap	GDP per capita of the country (source: World Bank Data 2014)
	Model characteristics
	Time_estimation	Number of years of the simulation
	Multi_country	= 1 if multi-countries model
	Dynamic	= 1 if dynamic model; = 0 if static or recursive dynamic model
	GTAP	= 1 if GTAP model
	MIRAGE	= 1 if MIRAGE model
	Model_other	= 1 if model other than GTAP or MIRAGE
	Armington	= 1 if Armington specification
	Biofuel type
	Bioethanol	= 1 if bioethanol only
	Biodiesel	= 1 if biodiesel only
	Biodiesel_bioethanol	= 1 if biodiesel and bioethanol
	Type of households (for INCOME only)
	HH_rural_urban	= 1 if rural and urban households
	HH_rural	= 1 if rural households
	HH_urban	= 1 if urban households
	HH_poor_rich	= 1 if poor and rich households
	HH_poor	= 1 if poor households
	HH_rich	= 1 if rich households

Scenario Land = 1 if scenario with expansion of land allocated to biofuels Land_surface Expansion rate of land allocated to biofuels*surface of the country Productivity = 1 if scenario with an increase of the agricultural productivity or biofuel productivity

Table 3 .

 3 Descriptive statistics

	GPD (N=227)	INCOME (N=255)

Table 5 .

 5 Robust OLS for INCOME

	VARIABLES	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
	Year	0.178			-0.0399
		(0.117)			(0.0407)
	Article	-0.279				0.257
		(0.343)				(0.202)
	Africa								0.827*
									(0.451)
	Europe	2.661***							-0.153
		(0.738)							(0.567)
	Asia	0.462							0.887*
		(0.474)							(0.525)
	South_amer	-0.155							-0.0715
	ica
		(0.768)							(0.404)
	OECD	1.747						-0.597
		(1.736)						(0.526)
	GDP_cap	-6.56e-05*								-2.76e-05**
		(3.78e-05)								(1.15e-05)
	Time_estim	0.0927**	0.0950***	0.0975***	0.0982***	0.101***	0.0845***	0.103***	0.0870***	0.110***	0.0897**
	ation
		(0.0423)	(0.0332)	(0.0357)	(0.0346)	(0.0345)	(0.0317)	(0.0361)	(0.0334)	(0.0356)	(0.0359)
	Multi_count	3.262***					2.226***
	ry
		(0.692)					(0.396)
	Dynamic	0.967**		0.458**
		(0.424)		(0.211)
	MIRAGE	3.253***	0.446**	0.282	0.481**	0.408*	1.938***	-0.0167	-0.129	-0.282	0.487**
		(0.847)	(0.213)	(0.246)	(0.202)	(0.228)	(0.372)	(0.510)	(0.556)	(0.413)	(0.199)
	Model_othe	2.994***	1.063***	0.910***	1.090***	0.958***	2.279***	0.561	0.470	0.300	1.077***
	r
		(0.664)	(0.243)	(0.227)	(0.246)	(0.240)	(0.329)	(0.466)	(0.545)	(0.336)	(0.250)
	Armington	-0.660*		0.146

This paper has been quoted more than

3,800 times as of September 2016.3 There are several exceptions. See for instance Andrade de[START_REF] Andrade De Sá | Dynamics of indirect land-use change: Empirical evidence from Brazil[END_REF].

Note that some studies provide the two types of ESs while others provide one GPD or INCOME. See Annex 2 for a detailed list.

[START_REF] Hess | A Meta-Analysis of General and Partial Equilibrium Simulations of Trade Liberalisation under the Doha Development Agenda[END_REF] propose a similar way of depicting ESs.

http://www.eia.gov/cfapps/ipdbproject/IEDIndex3.cfm?tid=79&pid=79&aid=1(last consulted 08/07/2016).

https://www.gtap.agecon.purdue.edu/models/current.asp

http://www.mirage-model.eu/

MEGABARE model is developed by Australian Bureau of Agricultural and Resource Economics(Lee et al., 2007)

"The TIGER model is based on the World Bank's LINKAGE model, elaborated to take fuller account of energy generally and biofuels in particular"(Gunatilake et al., 2014). The LINKAGE model is a dynamic CGE based on GTAP data.

For an in-depth discussion of bioenergy expansion in Sub Saharan Africa, see, for instance,Brun et al. (2016).

Acknowledgements

We thank the ANR (project REVE, ANR-14-CE05-0008) and the French Ministry of Research for their financial support. The authors wish to thank participants of the What Works Global Summit 2016 (London 26-28 September 2016) as well as an anonymous referee from the French Association of Environmental and Resource Economists for their useful suggestions.

The usual disclaimers apply.

Appendix

Annex 1. Full references for the 26 primary studies Arndt, C., Benfica, R., Tarp, F., Thurlow, J., Uaiene, R., 2008. Biofuels, poverty, and growth: a computable general equilibrium analysis of Mozambique. IFPRI Discussion Paper 00803.