Raphaël Chetrite

Fluctuation Relations for Diusions Thermally Driven by a Non-Stationary Bath

In the context of the dynamical evolution in a non-stationary thermal bath, we construct a family of uctuation relations for the entropy production that are not veried by the work performed on the system. We exhibit uctuation relations which are global versions either of the generalized Fluctuation-Dissipation Theorem around a non-equilibrium diusion or of the usual Fluctuation-Dissipation Theorem for energy resulting from a pulse of temperature.

Introduction

One important recent progress in non-equilibrium statistical physics is the discovery of various uctuation relations which can be viewed as non-perturbative extensions of the usual Fluctuation-Dissipation Theorem (FDT) [START_REF] Kubo | The uctuation-dissipation theorem[END_REF][START_REF] Bettolo Marconi | Fluctuation-dissipation: response theory in statistical physics[END_REF]. Such relations pertain either to non-stationary transient situations [START_REF] Evans | Equilibrium microstates which generate second law violating steady states[END_REF][START_REF] Jarzynski | Hamiltonian derivation of a detailed uctuation theorem[END_REF] or to stationary regimes [START_REF] Gallavotti | Dynamical ensemble in a stationary state[END_REF]. In particular, a family of uctuation relations holds for the distribution of work performed on a system [START_REF] Jarzynski | Hamiltonian derivation of a detailed uctuation theorem[END_REF][START_REF] Crooks | Path ensembles averages in systems driven far from equilibrium[END_REF][START_REF] Seifert | Stochastic thermodynamics: Principles and perspectives[END_REF][START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF] which evolves in an equilibrium bath. This developments had an important impact on the physics of nanosystems and biomolecules [START_REF] Ritort | Nonequilibrium uctuations in small systems: From physics to biology[END_REF]. Here, we examine the question of the extension of such relations to an evolution in a non-equilibrium medium. More precisely, we consider systems placed in the thermal bath that is non-stationary, a situation which can be realized experimentally by adding or extracting heat, or by modulating the pressure. Let us remark that non-stationarity of the temperature is linked via Stokes law to non-stationarity of friction and that particles with time-dependent radii have non-constant mass and friction. This last situation is an important problem in astrophysics for the formation of planets through dust aggregation [START_REF] Blum | Growth and form of planetary seedlings: results from a microgravity aggregation experiment[END_REF][START_REF] Ausloos | Brownian particles having a uctuating mass[END_REF]. There is another situation where the non-stationarity of friction can be realized with particles diusing in ferroelectric uids when external magnetic elds are controlling the intrinsic viscosity [START_REF] Gunberg | Layered magnetic structures: evidence for antiferromagnetic coupling of Fe layers across Cr interlayers[END_REF].

A famous example of a system which evolves in a non-stationary bath is the Temperature Ratchet Model [START_REF] Reimann | Brownian motors driven by temperature oscillations[END_REF] of a Brownian motor. We can also consider the evolution of a system initially in equilibrium at high temperature (i.e. in the state with the probability density ρ i = exp(-H Ti )) put in contact with a non stationary bath reaching a low temperature T f . The function T (t) is then called the cooling schedule. A particularly interesting case is the instantaneous quench of the initial system in a thermostat at temperature T f which correspond to the cooling schedule with instantaneous initial change of temperature. The age (or waiting time) of the system is then the time elapsed since the quench. Finally, at the more formal level, such non-stationary evolution rules appear naturally from a stationary dynamics after system size or coarse grained expansion [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF]. Diusion properties in a non-stationary medium have been recently studied in the case of unidimensional overdamped dynamics driven by multiplicative non stationary noise [START_REF] Denisov | Anomalous diusion for averdamped particles driven by cross-correlated white noise sources[END_REF][START_REF] Vitrenko | Exactly solvable nonlinear model with two multiplicatives Gaussian colored noises[END_REF].

The present paper consists of ve sections. Sect. 2 sets the stage and notations for the model of non-equilibrium and non-linear Langevin dynamics that we consider. In Sect. 3, we recall the main result of [START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF] and we list dierent time inversions that permit to obtain uctuation relations for entropy production. We observe that in this context the work performed on the system no longer obeys the uctuation relations but that there still exists a functional that upon averaging gives the free energy change in a non-equilibrium protocol. In Sect. 4. we explicitly construct uctuation-dissipation relations that result from the Taylor expansion of the last uctuation relation. In particular, we recover the generalized FDT [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF] around a non-equilibrium diusion and the usual FDT for the energy which results from a pulse of temperature [START_REF] Risken | The Fokker Planck Equation. 2 nd Edition[END_REF]. We discuss also the physical meaning of the eective temperature of an harmonic oscillator in a non-stationary bath. Finally, Sect. 5 presents our conclusions.

Acknowlegements. The author thanks Krzysztof Gawedzki, François Delduc and Gregory Falkovich for discussions and acknowledges support of the Koshland center for Basic Research.

2

Thermally Driven Non-Linear Langevin Dynamics

The general dynamics that we consider is described by a non-autonomous diusion process that we call Langevin dynamics (or non-linear Brownian motion) in a d-dimensional (phase-)space E.

The process obeys the stochastic dierential equation (SDE

) ẋi = -Γ ij t (x)∂ j H t (x) + Π ij t (x)∂ j H t (x) + G i t (x) + w i t (x) + η i t (x) (1)
which should be interpreted in the Stratonovich convention. Here H t (x) is the Hamiltonian of the system (the time indexation corresponds to an explicit time dependence), Γ t (x) is a family of non-negative matrices, Π t (x) is a family of antisymmetric matrices, G t (x) is an external force (or a shear) and w t (x) is an additional corrective term which will be explicited in eq. ( 3) below and which comes from the x-dependence of Γ t and Π t . Finally η t (x) is a white Gaussian vector eld with mean zero and covariance

η i t (x)η j s (y) = 2D ij t (x, y)δ(t -s) with D t (x, x) ≡ d t (x) = 1 β t Γ t (x).
(

) 2
The last equality, which contains the temperature of the bath T t = 1 βt , is called the Einstein relation. In the case when the bath is in equilibrium, this equation reects the fact that the friction and the noise are two dual eects of the interaction with the bath. In this non stationary setup, the Einstein relation is valid under the assumption of local thermal equilibrium, provided the thermalisation of the immediate environment of the Brownian particle is much faster than the characteristic variation of temperature. Let us underline that in this context the term nonlinear concerns the non-homogeneous properties of Γ t (x) and Π t (x) (which may characterize non homogeneous properties of the bath) and not the fact that equation ( 1) is non-linear. Such nonlinear properties appear naturally in many situations: non-ideal plasmas and gases [START_REF] Klimontovich | Ito, Statonovich and kinetic forms of stochastic equations[END_REF][START_REF] Klimontovich | Non Linear Brownian motion[END_REF], ultracold clusters of atoms or molecules cooled by interaction with laser radiation, active Brownian particles [START_REF] Ebeling | Klimontovich's contributions to the kinetic theory of nonlinear Brownian motion and new developments[END_REF]. Finally, the additional corrective term in (1) is given by the expression

w i t (x) = ∂ y j D ij t (x, y)| x=y - 1 β t ∂ j Π ij t (x). (3)
The presence of this term assures that in the case with stationary Hamiltonian and temperature (i.e. H t = H, β t = β) and without external force (i.e G = 0), the Gibbs density exp(-βH) is an invariant density 1 . The presence of this term w t (x) in equation ( 1) can appear as a makeshift arrangement, but it was extensively studied in the literature of non linear Brownian motion [START_REF] Klimontovich | Ito, Statonovich and kinetic forms of stochastic equations[END_REF][START_REF] Klimontovich | Non Linear Brownian motion[END_REF]. Note that w t vanishes in the case of linear Brownian motion where D t (x, y) = D t and Π t (x) = Π t . We call the deterministic part of the second member of eq. ( 1):

u t (x) = -Γ ij t (x)∂ j H t (x) + Π ij t (x)∂ j H t (x) + G i t (x) + w i t (x) (4)
1 In fact, the Gibbs density is then an equilibrium density: an invariant density with vanishing modied probability current [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF]. the drift term. An elementary case of non-linear Brownian motion is the Landau-Lifshitz-Bloch dynamics of a Brownian spin [START_REF] Garcia-Palacios | On the statics and dynamics of magneto-anisotropic nanoparticles[END_REF] in an eective magnetic eld B ef f t (x) = -∇H t (which can incorporate interaction with other spins). It follows the dynamics

ẋ = -x × ∇H t + λ t x × (x × ∇H t ) + G t (x) + x × ζ t with ζ i t ζ j s = 2λ t β t δ ij δ(t -s). (5)
The rst term on the right hand side is the precession term, the second one is the damping term and the third one an external torque. The noise (and the damping term) accounts for the eect of the interaction with the microscopic degrees of freedom (phonons, conducting electrons, nuclear spins...). This dynamics is a particular case of (1) with Γ ij t (x) = λ t δ ij x 2 -x i x j and Π ij t (x) = ε ijk x k with ε ijk the totally antisymmetric tensor. In this example one sees the need for the term Π ij t (x)∂ j H t (x) in eq. ( 1) corresponding to the Hamiltonian vector eld which also permits to describe systems of non-overdamped Brownian particles in the phase-space with coordinates q p , non-stationary mass, in an external potential V t , subjected to a non-conservative force f t and in a non-stationary bath giving rise to the noise and the non-homogeneous, non-linear but isotropic drag γ t (q, p). The Stratonovich SDE which governs this model is then q = p mt ṗ = -γt(q,p) mt p -∇V t (q) + f t (q) + ∇pγt(q,p) 2βt

+ 2γt βt η with η i t η j s = δ ij δ(t -s) (6)
which is again a sub-case of (1) with

Γ t = 0 0 0 γ t , Π = 0 Id -Id 0 , H t = p 2 2m t + V t (q) and G t = 0 f t (q)
. The Kramers case corresponds to the Stokes law of friction γ t (q, p) = γ t (q).

Another example of friction with γ t (q, p) = γ(p 2 -p 2 0 ) appears in the Rayleigh-Helmholtz theory of sound [START_REF] Rayleigh | The Theory of Sound[END_REF]. We start by collecting the elementary properties of diusion processes that we shall need [START_REF] Stroock | Multidimensional Diusion Processes[END_REF].

The Markovian generator L t of the process x t satisfying the SDE (1) is dened by the relation

∂ t f (x t ) = (L t f )(x t ) , where L t = u t • ∇ + ∇ • Γ t βt ∇, (7)
where the modied drift u t is dened in term of the drift (4):

u i t (x) = u i t (x) -∂ y j D ij t (x, y)| x=y . (8)
The time evolution of the instantaneous probability density function of the process ρ t

(x) = δ (x t -x) is governed by the formal adjoint L † t of the generator L t ∂ t ρ t = L † t ρ t (9)
which can be rewritten as a continuity equation (resp. an hydrodynamic advection equation) by dening the probability current j t (resp. the mean local velocity v t :)

∂ t ρ t = -∇ • j t = -∇ • (ρ t v t ) where j i t ≡ ( u i t -β -1 t Γ ij t ∇ j )ρ t and v i t ≡ j i t ρ t . (10)
As was explained in [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF], it is convenient to use the freedom to add a divergenceless term in the denition of the probability current to obtain the modied current and the modied local velocity

j i t ≡ j i t + β -1 t ∇ j Π ij ρ t and v i t ≡ j i t ρ t (11)
which verify also the continuity equation ( 10) but vanish in the case of stationary Hamiltonian and temperature (i.e. H t = H , β t = β) for vanishing external force G t = 0.

Fluctuation relations and time inversion

In [START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF], various uctuation relations were discussed for arbitrary diusion processes. We recall here the main result in the context of systems with dynamics of type [START_REF] Ausloos | Brownian particles having a uctuating mass[END_REF], see also Sect. 3 of [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF]. With the use of combined Girsanov and Feynman-Kac formulae, one obtains the detailed uctuation relation (DFR) µ 0 (dx) P T (x; dy, dW ) e -W = µ r 0 (dy

* ) P r T (y * ; dx * , d(-W )) , (12)
where 1. µ 0 (dx) = 0 (x) dx is the initial distribution of the original forward process (1), 2. µ r 0 (dx) = r 0 (x) dx is the initial distribution of the backward process obtained from the forward process by applying a time inversion (see below),

3. P T (x; dy, dW ) is the joint probability distribution of the time T position x T of the forward process starting at time zero at x and of a functional W T (linked to the entropy production) of the same process on the interval [0, T ] (described later), 4. P r T (x; dy, dW ) is the similar joint probability distribution for the backward process.

The time inversion acts on time and space by an involution

(t, x) → (t * = T -t, x * ) . (13)
Such an involution induces the action x → x on trajectories by the formula x t = x * T -t and, further, the action on functionals of trajectories F → F by setting

F [x] = F [ x].
To recover various uctuation relations discussed in the literature [START_REF] Kurchan | Fluctuation theorem for stochastic dynamics[END_REF][START_REF] Lebowitz | A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics[END_REF][START_REF] Crooks | Path ensembles averages in systems driven far from equilibrium[END_REF][START_REF] Jarzynski | Hamiltonian derivation of a detailed uctuation theorem[END_REF][START_REF] Speck | Integral uctuation theorem for the housekeeping heat[END_REF][START_REF] Chernyak | Path-integral analysis of uctuation theorems for general Langevin processes[END_REF], one divides the drift part ( 4) of ( 1) into two parts, u = u + + u -, with u + transforming as a vector eld under the space-time involution (13) and u -as a pseudo-vector eld:

u r i T -t,± (x * ) = ±(∂ k x * i )(x) u k t,± (x) , u r = u r + + u r -. (14)
The random eld η t (x) may be transformed with either rule. By denition, the backward process satises then the Stratonovich SDE

ẋ = u r t (x) + η r t (x) (15)
and, in general, diers from the naive time inversion x t of the forward process. The functional W T is given by the expression

W T = ln 0 (x 0 ) -ln( r 0 (x * T )σ(x T )) + T 0 J t dt (16)
where σ(x) = det ∂x * ∂x is the Jacobian of the spatial involution. The intensive functional J t has the interpretation of the rate of entropy production in the environment and is given by the expression

J t = β t u t,+ (x t ) • Γ -1 t (x t ) ẋt -u t,-(x t ) -(∇ • u t,-)(x t ) , (17)
The time integral in eq. ( 16) is taken in the Stratonovich sense. When µ r 0 (dx * ) = µ T (dx) then the boundary contribution [ln 0 (x 0 ) -ln( r 0 (x * T )σ(x T ) )] to W T gives the change in the instantaneous entropy of the process. In this case, the functional W T becomes equal to the overall entropy production. Moreover, with the interpretation of J t as the entropy production in the environment, the First Principle gives us that the work T T performed on the system can be expressed in term of J t :

T T = H T (x T ) -H 0 (x 0 ) + T 0 J t β t dt. ( 18
)
We can underline that in this setup, and contrary to the case of a the stationary bath [START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF], the work T T cannot be identied with the functional W T for an appropriate choice of initial densities of the forward and backward processes. This means that the work does not verify the DFR.

The DFR [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF] holds even if the measures µ 0 and µ r 0 are not normalized, or even not normalizable. When they are normalized, let us denote byand by -r the expectations of functionals of, respectively, the forward and the backward process on the time interval [0, T ], with initial distributions µ 0 and µ r 0 . One of the immediate consequences of the DFR equation ( 12) is the (generalized) Jarzynski equality [START_REF] Jarzynski | Hamiltonian derivation of a detailed uctuation theorem[END_REF] e -W T = 1 [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF] obtained by the integration of the both sides of eq. ( 12). It implies the inequality W T ≥ 0 that has the form of the Second Law of Thermodynamics stating the positivity of the average entropy production. With a little more work [START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF], the DFR ( 12) may be cast into a form of the (generalized) Crooks relation [START_REF] Crooks | Path ensembles averages in systems driven far from equilibrium[END_REF]:

F exp(-W T ) = F r . ( 20
)
We will now restrict ourselves to the class of time inversions [START_REF] Evans | Equilibrium microstates which generate second law violating steady states[END_REF] such that there exists a non-stationary density f t such that

u +,t = β -1 t Γ t ∇ ln f t and then u -,t = -Γ t ∇(H t + β -1 t ln f t ) + Π t ∇H t - 1 β t ∇ • Π T t + G t . (21)
After a straightforward calculation, the rate of entropy production in the environment may be expressed as

J t = ẋt • (∇ ln f t )(x t ) + ((f t ) -1 L † t f t )(x t ). (22)
With the choice µ 0 (dx) = f 0 (x)dx et µ r 0 (dx * ) = f T (x)dx the functional W T takes then the simple form

W T = T 0 (f t ) -1 L † t f t -∂ t ln(f t ) (x t ) dt. ( 23
)
We shall see that the uctuation relations associated to this peculiar family of inversions are the natural generalizations of the FDT. We shall now describe particular cases in this family of time inversions.

Complete reversal [5]

As the function f t in [START_REF] Gomez-Solano | Experimental verication of a modied uctuation-dissipation relation for a micron-sized particle in a non-equilibrium steady state[END_REF] we take the instantaneous density function (i.e. f t = ρ t ) of the forward process (1) distributed with initial condition f 0 . Here, the functional (23) trivially vanishes W T = 0 and the DFR [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF] takes the form of the generalized detailed balance µ 0 (dx) P T (x; dy) = µ T (dy) P r T (y * ; dx * ) .

(

) 24
One may show that ρ r t (x) ≡ ρ t * (x * ) is the instantaneous density of the backward process and that the corresponding probability current satises the relation

j i,r t (x) = -(∂ k x * i )(x) j k t * (x * ) . ( 25
)
This inversion is employed in many articles in probability theory [START_REF] Azema | Théorie générale des processus et retournement du temps[END_REF][START_REF] Millet | Integration by parts and time reversal for diusion process[END_REF][START_REF] Follmer | An entropy approach to the time reversal of diusion process[END_REF][START_REF] Petit | Time reversal and reected diusions[END_REF][START_REF] Nelson | Dynamical Theories of Brownian Motion[END_REF]. It corresponds to the vanishing overall entropy production.

3.2 Current reversal [START_REF] Chernyak | Path-integral analysis of uctuation theorems for general Langevin processes[END_REF][START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF] Another useful choice of time inversion, called the current reversal is based on the choice f t = π t where π t satises L † t π t = -∇• j t = 0. In the case where G t = 0, we have π t = exp(-β t (H t -F t )) with F t the free energy (i.e. exp(-

β t F t ) = exp(-β t H t ) ). One can show [5] that π r t (x) ≡ π t * (x *
) is the density for the backward process which correspond to the conserved current ∇• j r t = 0 and that eq. ( 25) still holds. The functional [START_REF] Follmer | An entropy approach to the time reversal of diusion process[END_REF] takes now the form

W ex T = - T 0 (∂ t ln π t )(x t ) dt, (26)
where the index ex stands for excess [START_REF] Seifert | Stochastic thermodynamics: Principles and perspectives[END_REF][START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF]. For the backward process, the functional W ex,r T is given by the same expression with π t replaced by π r t . The Jarzynski equality [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF] for this case was rst proven in one dimension in [START_REF] Hatano | Steady-state thermodynamics of Langevin systems[END_REF] and in the general case in [START_REF] Ge | Generalized Jarzynski's equality of inhomogeneous multidimensionnal diusion processes[END_REF][START_REF] Liu | A generalized integral uctuation theorem for diusion processes[END_REF][START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF].

Canonical inversion [5]

A natural choice for the system (1) is to take f t to be the Gibbs density exp(-β t (H t -F t )) where F t is the free energy (i.e. exp(-β t F t ) = exp(-β t H t ) ) if the Gibbs density is normalizable and zero otherwise. This corresponds to the choice u +,t = -Γ t ∇H t in [START_REF] Gomez-Solano | Experimental verication of a modied uctuation-dissipation relation for a micron-sized particle in a non-equilibrium steady state[END_REF]. The functional [START_REF] Follmer | An entropy approach to the time reversal of diusion process[END_REF] becomes

W ci T = -(β T F T -β 0 F 0 ) + T 0 [∂ t (β t H t ) + β t G t • ∇H t -∇ • G t ] (x t ) dt (27)
where the index ci means canonical inversion. The generalized Jarzynski equality ( 19) can be rewritten in the form:

exp - T 0 [∂ t (β t H t ) + β t G t • ∇H t -∇ • G t ] (x t ) dt = exp [-(β T F T -β 0 F 0 )] (28)
which permits to extract the dierence of free energy out of a non-equilibrium experiment in a non-stationary bath (but the connexion with the work performed is lost). For example, for the Brownian particle [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF], this functional takes the form:

W ci T = -(β T F T -β 0 F 0 )+ T 0 βt p 2 t 2mt +V t (q t ) +β t - p t ṁt m 2 t +(∂ t V t )(q t )+(f t .∇V t )(q t ) dt (29)
which, as compared to the functional which appears in the usual Jarzynski equality [START_REF] Jarzynski | Hamiltonian derivation of a detailed uctuation theorem[END_REF][START_REF] Kurchan | Fluctuation theorem for stochastic dynamics[END_REF],

contains new terms proportional to the variation of temperature ∂ t β t and of mass ∂ t m t . From the Jarzynski equality [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF], one can deduce the inequality which constrains the evolution in a non-stationary bath

W ci T ≥ 0 .
In the case with a stationary Hamiltonian (i.e. H t = H ) and without external force (i.e. G t = 0)

this constraint reads T 0 (∂ t β t ) H(x t ) dt ≥ (β T F T -β i F i ).
Let us consider now the overdamped particle in a 3-dimensional time-dependent harmonic potential. Such example admits an analytical computation of the distribution of the functional W ci T . We consider a non stationary bath (γ t , β t ) and the harmonic potential U t (x) = kt 2 (x -a t ) 2 where k t is the stiness coecient and a t is the instantaneous center of the potential. The particle is initially distributed with the Gibbs density ρ 0 (x) = exp(-

β 0 (U 0 (x) -F 0 )) = C 2π 3 2 exp -C 2 (x -a 0 ) 2 .
Further, we will restrict our study to the particular case, not necessarily physical, where the temperature of the bath and the stiness coecient are such that their product is stationary: k t β t = C. This setup generalizes the unidimensional stationary case (k t = k , β t = β, γ t = γ and a t = ut) considered in [START_REF] Van Zon | Theory of the Transient and Stationary Fluctuation Theorems for a Dragged Brownian Particle in a Fluid[END_REF][START_REF] Mazonka | Exactly solvable model illustrating far-from-equilibrium predictions[END_REF]. The system satises the linear SDE

ẋ = - k t γ t (x -a t ) + η t with η i s η j t = 2 β t γ t δ(t -s)δ ij . ( 30
)
The functional W ci T = W ex T takes here the form

W ci T = -C T 0 ȧt • (x t -a t ) .
The distribution of W ci T for a process with the initial Gaussian density ρ 0 is Gaussian due to the linearity of [START_REF] Landauer | Distribution function peaks generated by noise[END_REF]. A straightforward calculation gives the mean

W ci T = C T 0 dt t 0 ȧt • ȧs exp - t s k u γ u du ds (31)
and the variance of this Gaussian (we assume that the integrals exist):

V T ≡ W ci T -W ci T 2 = 2C T 0 dt t 0 ȧt • ȧs exp - t s k u γ u du ds. ( 32
)
The distribution of W ci T is then

P T (W ) = 1 √ 2πV T exp - W -W ci T 2 2V T (33)
and an elementary calculus shows that the Jarzynski equality ( 19) is equivalent to the fact that

V T = 2 W ci T
which is evident from the comparison of ( 31) and (32).

New inversion

We choose for the function f t the mean instantaneous density ρ t of another Langevin dynamics (1) which possesses the same parameters Γ t , Π t , G t but with another non autonomous Hamiltonian H t and another bath temperature β t . We note L t (resp. L t ) the Markovian generators of the process with the Hamiltonian H t and the bath temperature β t (resp. H t and β t ). The functional [START_REF] Follmer | An entropy approach to the time reversal of diusion process[END_REF] takes now the form

W T = T 0 (ρ t ) -1 (L t -L t ) † ρ t (x t ) dt. (34)
This new inversion will permit to recover new generalizations of the FDT around non-stationary non-equilibrium diusions, see also [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF][START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF][START_REF] Baiesi | Fluctuations and response of nonequilibrium states[END_REF].

Generalizations of the Fluctuation-Dissipation Theorem

As noted in [START_REF] Gallavotti | Extension of Onsager's reciprocity to large elds and the chaotic hypothesis[END_REF][START_REF] Lebowitz | A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics[END_REF], the uctuation relations may be viewed as extensions to the non-perturbative regime of the Green-Kubo and Onsager relations for the non-equilibrium transport coecients valid within the linear response description of the vicinity of the equilibrium. Ref. [START_REF] Chetrite | Fluctuation relations for diusion processes[END_REF] contains a detailed argument showing that if in a stationary bath one perturbs an equilibrium system by introducing a weakly time dependent Hamiltonian H t (x) = H(x) -g a,t O a (x) then the Jarzynski equality associated to [START_REF] Klimontovich | Ito, Statonovich and kinetic forms of stochastic equations[END_REF] or [START_REF] Klimontovich | Non Linear Brownian motion[END_REF] gives in the second order of the Taylor expansion in g the usual FDT. Ref. [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF] showed that similar correspondence still holds around non-equilibrium steady states for a stationary dynamics with an external force (i.e. G = 0). In this case, it is the Crooks relation ( 20) associated to the functional [START_REF] Klimontovich | Ito, Statonovich and kinetic forms of stochastic equations[END_REF] which gives the modied Fluctuation-Dissipation Theorem [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF][START_REF] Gomez-Solano | Experimental verication of a modied uctuation-dissipation relation for a micron-sized particle in a non-equilibrium steady state[END_REF] after the rst order Taylor expansion in g. The second order expansion of Jarzynski equality ( 19) associated to the functional [START_REF] Klimontovich | Ito, Statonovich and kinetic forms of stochastic equations[END_REF] gives in such a situation only a special case of this theorem. We shall now investigate which type of uctuation-dissipation identities may be deduced by Taylor expanding the uctuation relation corresponding to the time inversion of Sect 3.4.

FDT around non-stationary diusions

We consider the system (1) with the Hamiltonian H t (x) = H 0 t (x)-g a,t O a (x). Following Sect. 3.4, we choose f t as the mean instantaneous density ρ 0 t of the unperturbed system with g = 0. The functional [START_REF] Mazonka | Exactly solvable model illustrating far-from-equilibrium predictions[END_REF] becomes

W T = T 0 g a,s (ρ 0 s ) -1 M a † s ρ 0 s s ds with M a s = (Γ s ∇O a -Π s ∇O a ) • ∇, (35)
where the subscript s on (ρ 0 s ) -1 M b † s ρ 0 s signals that the latter function should be taken at the point x s , Let us now write a particular case of Crooks relation [START_REF] Ge | Generalized Jarzynski's equality of inhomogeneous multidimensionnal diusion processes[END_REF], where the average is in the system (1) with the Hamiltonian H t , associated to a single time functional

F [x] = O a (x t ) ≡ O a t ( 0 < t < T ): O a t e -W T = O a T -t r . ( 36
)
We shall denote by . 0 the average of the process the dynamics driven by H 0 t and by L 0 , v 0 and v 0 , respectively, its Markovian generator, its mean local velocity and its modied mean local velocity. The rst order Taylor expansion

exp(-W T ) = 1 + T 0 g b,s (ρ 0 s ) -1 M b † s ρ 0 s s ds + O(g 2 ) ( 37
)
in [START_REF] Bettolo Marconi | Fluctuation-dissipation: response theory in statistical physics[END_REF] gives the relation

O a t 0 + g b,s δ δg b,s g=0 O a t ds - T 0 g b,s O a t ρ 0 s -1 M b † ρ 0 s s 0 ds + O(g 2 ) = O T -t r . ( 38
)
The right hand side has a functional dependence only on {g u , u > t}, so if we apply δ δg b,s | g=0 for 0 < s ≤ t to the last identity, we obtain:

δ δg b,s g=0 O a t = (ρ 0 s ) -1 M b † ρ 0 s s O a t 0 . (39)
A short calculation gives:

(ρ 0 s ) -1 M b † s ρ 0 s = β s v 0 s • ∇O b -β s L 0 s O b s + Π ij s (∂ j O b )∂ i ln(ρ 0 s ) + (∂ i Π ij s )∂ j O b (40) = β s 2 v 0 s • ∇ -L 0 s O b -β s v 0 s • ∇O b + Π ij s (∂ j O b )∂ i ln(ρ s ) + (∂ i Π ij s )∂ j O b = β s 2 v 0 s • ∇ -L 0 s O b -β s v 0 s • ∇O b .
Moreover, we have the sum rule (for s ≤ t)

∂ s O b s O a t 0 = (2 v 0 s ∇ -L 0 s )O b s O a t 0 . (41)
With ( 39), ( 41) and( 40), we obtain the Modied Fluctuation-Dissipation Theorem :

∂ s O b s O a t 0 = 1 β s δ δg b,s g=0 O a t + (v 0 s • ∇O b ) s O a t 0 . (42)
This is a generalization of the FDT around a non-equilibrium diusion process in a stationary bath (β t = β) of refs. [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF][START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF][START_REF] Baiesi | Fluctuations and response of nonequilibrium states[END_REF][START_REF] Cugliandolo | O equilibrium dynamics and aging in unfrustrated systems[END_REF] and of FDT around non-stationary Langevin equation of ref. [START_REF] Zimmer | Fluctuations in Nonequilibrium Systems and Broken Supersymmetry[END_REF]. It may be also proven as in [START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF] using the fact that the diusion process becomes an equilibrium one in the Lagrangian frame of its modied mean local velocity v 0 , verifying in that frame the usual FDT. The transformation of the latter back to the Eulerian (i.e. laboratory) frame leads to [START_REF] Risken | The Fokker Planck Equation. 2 nd Edition[END_REF]. Let us remark that here, similarly as in the stationary case discussed in [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF],

the Jarzynski equality [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF] for the functional (34) leads upon the second order expansion in g to a particular case of the MFDT where the observable O a t is replaced by

A a = (ρ 0 t ) -1 M a † t ρ 0 t which is a (time-dependent) functional of O a .
The violation of the usual FDT can be parametrized by using [START_REF] Risken | The Fokker Planck Equation. 2 nd Edition[END_REF] via the introduction of the so-called eective temperature [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF][START_REF] Crisanti | Violation of the uctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence[END_REF] dened by

T ef f (s, t, O a ) ≡ ∂ s O a s O a t 0 δ δga,s g=0 O a t = 1 β s + (v 0 s • ∇O a ) s O a t 0 δ δga,s g=0 O a t . (43)
We shall consider now the case where this eective temperature may be computed analytically in order to verify its physical consistency.

Unidimensional harmonic oscillator in a non-stationary bath. The SDE which governs this systems is

ẋ = - k γ x + η , with η t η s = 2T t γ δ(t -s). (44)
We take the cooling schedule T t such that the bath passes from an initial temperature (T 0 = T i ) to a lower nal temperature (T τ = T f < T i ) during a time τ. The system is initially in equilibrium with the bath and its initial density is ρ 0 (x) = exp(-k 2Ti x 2 )/Z i . We consider two particular examples of cooling schedules:

• Instantaneous quench

T t = T i if t = 0 T f if t > 0 . (45)
• Linear decrease of temperature:

T t = T i + t τ (T f -T i ) if t ≤ τ T f if t ≥ τ . (46)
Due to the linearity of eq. ( 44), one can compute explicitly the response of the position to an

external perturbation V (x) = k 2 x 2 → V t (x) = V (x) -g t x with g 0 = 0: δ δg s g=0 x t = 1 γ exp - k γ (t -s) . (47)
It has a stationary form and is independent on the cooling schedule. In a similar way, we obtain also an explicit expression for the dynamical 2-time correlation function of the position in the unperturbed system in the two cooling schedules. For s ≤ t, we obtain for the instantaneous quench:

x s x t 0 = T i -T f k exp - k γ (s + t) + T f k exp - k γ (t -s) (48)
and for the linear decrease of temperature schedule:

x s x t 0 =    Ti k + T f -Ti kτ s -γ 2k exp -k γ (t -s) + γ(T f -Ti) 2k 2 τ exp(-k γ (s + t)) if s ≤ τ, s ≤ t T f k exp -k γ (t -s) - γ(T f -Ti) 2k 2 τ exp( 2k γ τ ) -1 exp -k γ (s + t) if τ ≤ s ≤ t.
(49) We see in these two formulae that the characteristic time of convergence toward the Gibbs density exp(-k 2T f x 2 )/Z f is γ k for the instantaneous quench and τ + γ k for the linear decrease schedule. Note the relation

x s x t 0 = x 2 s 0 exp - k γ (t -s) = x 2 s 0 γ δ δg s g=0 x t (50)
holding for both cooling schedules. It shows that at very large times (i.e. t > s τ, γ k ) the correlation functions ( 48) and ( 49) take a stationary form depending on t -s. The instantaneous mean density of the process is Gaussian at all times. It follows that the mean local velocity has the form

v 0 s (x) = - k γ + T s γ x 2 s 0 x (51)
and the corrective term in the FDT ( 42) is

(v 0 s • ∇x) s x t 0 = (v 0 s ) s x t 0 = - k γ + T s γ x 2 s 0
x s x t 0 [START_REF] Zimmer | Fluctuations in Nonequilibrium Systems and Broken Supersymmetry[END_REF] Using the relations ( 50) and ( 52), the FDT (42) may be rewritten in this case as the identity

∂ s + k γ x s x t 0 = 2T s δ δg s g=0 x t (53)
which is easy to check directly.

The eective temperatures for the instantaneous quench T ef f Q and for the linear decrease of temperature schedule T ef f LD are:

T ef f Q (s, t, x) = T f + (T f -T i ) exp(- 2k γ s) if 0 < s ≤ t (54)
and

T ef f LD (s, t, x) =    T i + 1 τ (T f -T i ) s + γ 2k 1 -exp(-2k γ s) if 0 < s ≤ τ, s ≤ t T f + (T f -T i ) exp -2k γ s exp( 2k γ τ )-1 2k γ τ if τ ≤ s ≤ t. (55)
The two eective temperatures have an expected behavior for large time s converging toward T f . However, we may see in this system the problems with the physical interpretation of the eective temperature [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF][START_REF] Crisanti | Violation of the uctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence[END_REF]. For example, lim s→0 T ef f Q = 2T f -T i = T i and this expression can be negative if T f < Ti 2 . The possibility to nd negative eective temperature has been observed also in [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF] and, for the kinetically constrained model, in [START_REF] Mayer | Activated Aging Dynamics and Negative Fluctuation-Dissipation Ratios[END_REF]. Moreover, the eective temperature grows toward its limit T f , which does not correspond to the physical intuition for the temperature of a cooled system. The investigation of the linear-decrease cooling schedule is instructive for the understanding of these two problems. For this schedule, lim s→0 T ef f LD = T i and the eective temperature decreases from this value until time τ when it reaches

T ef f LD (τ ) = T f + (T f -T i ) 1-exp(-2k γ τ ) 2k γ τ
< T f . So, the problem with the initial time limit of T ef f Q was due to the instantaneous modeling of the quench. On the other hand, the second part of the evolution for the linear cooling schedule (i.e. for s ≥ τ ) begins with an eective temperature below T f that may be even negative. The last features are not really physically satisfying but the rst one explains why the eective temperature T ef f Q converges toward T f by growing, the fact which is conrmed by (55). We represent below in Figure 1 typical joint evolution for the linear cooling schedule of the bath temperature T (the crosses) and the eective temperature T ef f LD (the solid lines) in the case where T i = 300K, T f = 200K and 2k γ = 1s -1 for τ = 0.1s (red) τ = 1s(blue) and τ = 10s (black). The singularity of the limit τ → 0 is evident on this graph.

Response of a diusion to a pulse of bath temperature

Let us consider the system whose dynamics is governed by eq. ( 1) with the variable bath temperature β -1 t = (1 + g t )β -1 0 . We choose as the function f t the mean instantaneous density ρ 0 t of the similar system with the constant bath temperature β -1 0 . The functional [START_REF] Mazonka | Exactly solvable model illustrating far-from-equilibrium predictions[END_REF] becomes With the same reasoning as in Sect. 4.1, we nd the link between the response to a pulse of temperature at time s and the dynamical correlation function in the system with stationary inverse temperature β 0 :

W T = T 0 g s (ρ 0 s ) -1 M † s ρ 0 s s ds with M s = (β 0 ) -1 -∇ j Π ij s + ∇ j Γ ij s ∇ i + Γ ij s ∇ i ∇ j . ( 56
)
δ δg s g=0 O t = (ρ 0 s ) -1 M † ρ 0 s s O t (x t ) 0 . (57)
Here, there does not seem to exist a simplication of this relation in the spirit of (42) and we cannot say more except for the case when the system with the bath temperature β 0 is an equilibrium one.

Temperature pulse around equilibrium.

In the case where the system with the bath temperature β 0 is in equilibrium (i.e. without external force G t = 0 and with a stationary Hamiltonian H t = H and the Gibbsian instantaneous density), the functional (27) takes the form:

W ci T = W ex T = -(β T F T -β 0 F 0 ) + T 0 βt H(x t ) dt. (58)
We want to prove that the Taylor expansion in the second order of the Jarzynski equality [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF] associated to this functional gives the usual Fluctuation-Dissipation Theorem for the energy [START_REF] Risken | The Fokker Planck Equation. 2 nd Edition[END_REF] ∂

s H s H t 0 = 1 β 0 δ δg s g=0 H t . (59)
The equality [START_REF] Gardiner | Handbook of Stochastic Methods for Physics[END_REF] takes now the form exp -

T 0 βt H(x t ) dt = exp (-β T H(x)) dx exp (-β 0 H(x)) dx . (60)
We develop the left member in second order in g t or h t = g t -g 2 t assuming that g t vanishes for t ≤ 0:

exp - T 0 βt H(x t ) dt = exp β 0 T 0 ḣt H(x t ) dt + O(h 3 ) = 1 + β 0 T 0 ḣt H(x t ) dt + β 2 0 2 T 0 dt T 0 ḣt ḣs H(x t )H(x s ) ds + O(h 3 ) = 1 + β 0 T 0 ḣt H(x t ) 0 dt + β 0 T 0 dt t 0 ḣt h u δ δg u g=0 H t du + β 2

Conclusions

We have discussed uctuation relations for diusion processes (1) in a non-stationary thermal bath. Those included the uctuation relations for the entropy production [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF]. The work performed on the system no longer veries such uctuation relations, but that there still exists a relation [START_REF] Kurchan | Fluctuation theorem for stochastic dynamics[END_REF] which permits to extract the free energy dierence in a non-equilibrium experiment. We proved that the uctuation relations involving the functional [START_REF] Mazonka | Exactly solvable model illustrating far-from-equilibrium predictions[END_REF] are global versions of the Modied Fluctuation-Dissipation Theorem (MFDT) (42) around a non-equilibrium diusion extending the MFDT obtained before in [START_REF] Chetrite | Fluctuation relations in simple examples of non-equilibrium steady states[END_REF][START_REF] Chetrite | Eulerian and Lagrangian pictures of non-equilibrium diusions[END_REF] and of the usual FDT for energy (59) [START_REF] Risken | The Fokker Planck Equation. 2 nd Edition[END_REF] resulting from a pulse of temperature. On the way, in Sec 4.1 we illustrated the extended MFDT on a simple example of a harmonic oscillator in a thermal bath with variable temperature and we investigated the physical meaning of the eective temperature introduced in [START_REF] Cugliandolo | Energy ow, partial equilibration, and eective temperatures in systems with slow dynamics[END_REF][START_REF] Crisanti | Violation of the uctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence[END_REF] for such a system. One should underline that the interaction with a non-stationary bath is one among many ways to thermally drive a system. For example the thermodiusion eect (or Sorret eect) which appears in a bath with non-uniform temperature (i.e β(x) = 1 T (x) ) has been explained in [START_REF] Van Kampen | Relative stability in nonuniform temperature[END_REF], for the unidimensional case, using a stationary non-equilibrium microscopic model of the type (1) with G = 0 but with the thermophoretic force -Γ dT dx added to the drift (4). In the same spirit, many years ago, Landauer [START_REF] Landauer | Distribution function peaks generated by noise[END_REF] proposed a model with the wall temperature varying along a very narrow pipe lled with the Knudsen gas described by a stationary non-equilibrium microscopic model of the type (1) but with the thermophoretic force -Γ dT dx and the chemical force -T dΓ dx added to the drift. Finally, other way to drive a system is to consider uctuating coecients in (1), for example [START_REF] Ausloos | Brownian particles having a uctuating mass[END_REF] considered a uctuating mass and [START_REF] Rozenfeld | Brownian motion in a uctuating medium[END_REF][START_REF] Luczka | Diusion of Brownian particles governed by uctuating friction[END_REF][START_REF] Talkner | Brownian motion in a d-Dimensionnal Space with Fluctuating Friction[END_REF] a stochastic friction. It would be interesting to describe the uctuation relations in those setups.

Figure 1 :

 1 Figure 1: Red solid: T ef f LD for τ = 0.1s. Red crosses: T for τ = 0.1s. Blue solid: T ef f LD for τ = 1s. Blue crosses: T for τ = 1s. Black solid: T ef f LD for τ = 10s. Black crosses: T for τ = 10s.

  H(x t )H(x s ) 0 ds + O(h 3 ) = 1 + β 0 T 0 ḣt H(x t ) 0 dt + β 0 H(x t )H(x s ) 0 ds + O(h 3 ),where the last equality was obtained by expressing h u = u 0 ḣs ds in the second term and changing the order of integration over s and u. Expansion of the right member of eq. (60) gives in turn exp (-β T H(x)) dx exp (-β 0 H(x)) dx = 1 + β 0 h T H 0

  The comparison of the terms quadratic in ḣ leads to the identity s ≤ t which gives the relation (59) by the derivation with respect to s. Once again the Jarzynski equality appears as a global version of the FDT.

	1 β 0	s	t	du	δ δg u g=0	H t = H 2 t 0 -H s H t 0	(61)

for


Sandra Morgen

Contestations, appropriations et productions de savoirs par les femmes. Réflexions sur les catégories d'analyse d'un champ de recherche émergent

par Émeline Fourment (Université de Rouen, CUREJ) et Anne Kwaschik (Université de Constance)

Longtemps négligés par la recherche sur le militantisme 1 , les mouvements sociaux pour la santé commencent à intéresser les études de l'action collective dans les années 1990, notamment via les travaux consacrés aux luttes environnementales et contre le VIH 2 . Le mouvement états-unien pour la santé des femmes devient alors un objet d'étude de plus en plus considéré, ce qui se traduit dans les années 2000-2010 par la publication de cinq monographies 3 et de nombreux articles. Cet intérêt se retrouve également dans d'autres champs de recherche, notamment en histoire et en sociologie de la médicine, avec des analyses qui portent autant sur l'évolution de la relation patient/médecin que sur l'implication des profanes dans la médecine. Le fait que ces littératures se concentrent sur l'engagement de femmes pour la santé sexuelle et reproductive renvoie à une réalité genrée maintes fois constatée : les femmes, ainsi que les hommes qui ne correspondent pas aux normes de la masculinité (les homosexuels notamment), sont majoritaires dans ces mobilisations 4 . Cela s'explique non seulement par la division genrée du travail qui assigne les femmes aux tâches de soin dans l'espace domestique 5 , mais aussi par l'histoire de la domination médicale, qui s'est notamment établie à travers un processus de médicalisation du corps et de la sexualité des femmes et des déviant•es du sexe, du genre et de la sexualité 6 .

En France comme en Allemagne, un champ de recherche interdisciplinaire émerge sur les questions de santé reproductive et sexuelle, approchées depuis une perspective genrée et articulées avec un questionnement sur les mobilisations de patientes 7 . Ce dossier participe de cet essor en croisant des réflexions issues de travaux sur la construction des frontières de la médecine instituée avec des approches propres à l'étude des mouvements sociaux 8 . Ce faisant, les articles interrogent la place que prend l'engagement des femmes pour leur santé dans les conflits de légitimité entre savoirs : contestent-elles les savoirs institués, essentiellement médicaux, ou au contraire se les approprient-elles ? Comment ces contestations ou appropriations se traduisent-elles dans les pratiques militantes ? Ces engagements mènent-t-ils à la production de nouveaux savoirs sur la santé des femmes et, si oui, celle-ci se fait-elle en alors peu critiques de la domination médicale 12 . Kathy Davis a par ailleurs étudié les différentes traductions et adaptations de Our Bodies, Ourselves à travers le monde, en mettant en avant la capacité d'agir épistémique (epistemic agency) des traductrices, qui leur a permis de créer de nouvelles articulations entre les savoirs médicaux et leurs savoirs expérientiels sur le corps des femmes 13 . K. Davis montre combien le manuel fait l'objet d'appropriations différenciées selon les contextes -en lien avec la place accordée à la religion, à l'avortement à la sexualité lesbienneet selon le type d'actrices impliquées dans sa traductiongroupes de conscience ou ONG 14 . Our Bodies, Ourselves a justement inspiré Notre corps, notre santé. La santé et la sexualité des femmes en Afrique subsaharienne, que nous présentons ici dans le cadre d'un entretien avec sa co-éditrice, Fatou Sow. À rebours des lectures qui situent ce manuel dans la continuité de la publication états-unienne 15 , la sociologue sénégalaise montre combien cet ouvrage s'en distingue du fait de son contexte de production.

Ce dossier poursuit donc le travail amorcé par les recherches portant sur d'autres cas que les États-Unis, et qui mettent en lumière la diversité des formes prises selon les contextes par les engagements des femmes pour leur santé et leur rapport aux savoirs institués. Se décentrer des États-Unis amène alors à interroger les catégories de pensée usuelles, qui relèvent en grande partie des recherches sur les militantismes occidentaux et de l'histoire de la médicine occidentale. À cet égard, la direction de ce dossier a été riche d'enseignements. En effet, c'est moins l'absence de travaux sur des terrains non occidentaux que la manière d'approcher ces questions qui semble différer et complexifier le dialogue entre les champs de recherche établis. Les travaux sur les politiques reproductives en Afrique ou en Amérique latine sont très développés, mais leurs auteurs et autrices n'y adoptent pas une approche en termes de « mobilisation » ou d'« engagement » pour la santé : ils et elles s'intéressent plus aux politiques publiques et aux patientes ordinaires qu'aux femmes engagées dans des associations ou des ONG pour la santé 16 . Il est également probable que le caractère occidentalo-centré des catégories de pensée de l'étude des mouvements sociaux 17 contribue à son manque d'écho dans la recherche sur des pays non occidentaux : ces catégories supposent bien souvent l'existence d'un régime démocratique et libéral, une liberté de la presse et un droit d'association. La mise en regard de plusieurs cas d'étude et de plusieurs périodes permet d'amorcer une réflexion sur les catégories à partir desquelles sont pensés les engagements des femmes pour la santé.

___________________

Contextualiser le rapport des femmes engagées aux savoirs institués

Varier les contextes permet de mettre au jour une diversité de rapports aux savoirs institués des femmes engagées pour leur santé. L'idée selon laquelle la critique de ces savoirs serait caractéristique de ce type d'engagement est ainsi remise en question au profit d'une analyse des effets des situations politiques et du statut social des femmes engagées sur la place que ces dernières prennent dans les conflits de légitimité entre savoirs sur la santé.

Si toutes les actrices étudiées appartiennent à la classe moyenne éduquée de leur pays, les contributions réunies ici mettent en évidence des différences dans leurs modes d'organisation et leur positionnement vis-à-vis des institutions. Les Européennes se ressemblent d'abord beaucoup, et ce, malgré les quarante ans qui séparent leurs mobilisations respectives (années 1970 pour les Allemandes, 2010 pour les francophones) : elles participent à des collectifs et réseaux militants largement informels et organisent des ateliers de self-help à l'arrière d'une librairie (Europe francophone) ou dans un centre de femmes (Allemagne). Elles ne sont que peu en contact avec les institutions, voire leur sont largement hostiles, pour ce qui concerne les Allemandes. Leurs similitudes témoignent d'une inspiration commune (le mouvement états-unien pour la santé des femmes) mais aussi du travail mémoriel accompli par les militantes étudiées par L. Quéré, qui revendiquent la continuité avec les années 1970 18 . Ces points communs ne doivent néanmoins pas faire oublier des différences importantes : dans les problème public intime et sensible ? : dilemnes et inconforts des acteurs autour de l'avortement au Burkina Faso », Anthropologie et développement, Hors-Série, 2021, p.291-301). 17 Johanna Siméant-Germanos, « Protester/mobiliser/ne pas consentir. Sur quelques avatars de la sociologie des mobilisations appliquée au continent africain », Revue internationale de politique comparée, 20 (2), 2013, p. 125-143 ; Emmanuelle Bouilly, Ophélie Rillon, Hannah Cross, « African Women's Struggles in a Gender Perspective », Review of African Political Economy, 43 (149), 2016, p. 338-349. 18 Lucile Quéré, « Feminist Collective Memory and Nostalgia in Gynaecological Self-Help in Contemporary Europe », European Journal of Women's Studies, 28 (3), 2021, p. 337-352. années 1971-1976 Ces différences se retrouvent dans le rapport aux savoirs institués qu'entretiennent les femmes étudiées. À nouveau, les Européennes partagent une ambivalence vis-à-vis de la médecine : elles la critiquent et cherchent à développer des alternatives à la médicalisation, tout en 19 L. Ruault, Le spéculum, la canule et le miroir. Pratiques des MLAC, entre appropriations féministe et médicale de l'avortement (1972-1984), Paris, ENS Éditions, à paraître en 2023. 20 Lee Ann Banaszak, The Women's Movement Inside and Outside the State, Cambridge, Cambridge University Press, 2010 ; Anne Revillard, La cause des femmes dans l'État. Une comparaison France-Québec, Grenoble, Presses universitaires de Grenoble, 2016 ; Laure Bereni, « Penser la transversalité des mobilisations féministes : l'espace de la cause des femmes », dans Christine Bard (dir.), Les féministes de la deuxième vague, Rennes, Presses universitaires de Rennes, 2012, p. 27-41. 21 Nathalie Bernard-Maugiron, Marième N'Diaye, « Les femmes au coeur des processus de réformes juridiques dans les Afriques musulmanes », Cahiers d 'études africaines, 242, 2021, p. 265-285. 22 O. Rillon, « Mobilisations féminines en contexte autoritaire. La "dépolitisation" comme outil d'émancipation dans le Mali des années 1970 », dans Kadya Tall, Marie-Emmanuelle Pommerolle, Michel Cahen (eds), Collective Mobilisations in Africa/Mobilisations collectives en Afrique: Enough is Enough!/Ça suffit !, Leyde, Brill, 2015, p. 164-182.

  débats souvent « présentistes » sur le sujet 9 . Nous détaillons ici l'apport de ce dialogue pour notre questionnement sur le rapport aux savoirs institués des femmes engagées pour leur santé.__________________Faire dialoguer des travaux sur l'Europe occidentale et l'Afrique subsaharienne Travailler sur l'engagement des femmes pour la santé amène nécessairement à se pencher sur la littérature consacrée au mouvement états-unien pour la santé des femmes des années 1970, qui semble, à première vue, dominer ce champ d'étude. Au-delà de la diversité des questions qu'ils posent, ces travaux semblent s'accorder sur un point : ce mouvement se caractérise par la remise en cause, à l'aune de l'expérience que les femmes font de leur corps, des savoirs institués sur la santé ; sa critique vise avant tout les savoirs médicaux, et se traduit par le développement d'alternatives à la médicalisation du corps des femmes. C'est dans ce cadre qu'il faut comprendre deux des piliers du mouvement : la pratique du self-help (auto-santé) et la production de manuels pour la santé. Le self-help renvoie à des pratiques d'autoauscultations gynécologiques, d'échanges d'expériences et/ou de recettes dites « de grandmère ». L'auto-auscultation gynécologique vise notamment à permettre aux femmes de contrôler leur fertilité à partir de l'observation de leur col de l'utérus et des glaires cervicales, qui changent d'aspect et de texture les périodes du cycle menstruel. Quant aux manuels de santé, ils rassemblent des savoirs médicaux vulgarisés et des témoignages de femmes sur leur corps et leurs maux. Le plus connu d'entre eux, Our Bodies, Ourselves, est publié pour la première fois sous le titre Women and Their Bodies par le Boston Women's Health Book Collective en 1970, puis réédité et actualisé à neuf reprises (la dernière version date de 2011). Si la littérature sur les États-Unis domine le champ, les recherches portant sur les contextes européens ainsi que sur la circulation des savoirs du mouvement états-unien commencent à se développer (articles de Lucile Quéré et Émeline Fourment). Auparavant, d'autres recherches se sont penchées sur les voyages du self-help en Australie 10 , en Grande Bretagne 11 , ainsi qu'en France, où cette pratique a rencontré peu de succès, la majorité des féministes étant 9 François Hartog, Régimes d'historicité : présentisme et expérience du temps, Paris, Le Seuil, 2003. 10 Gwendolyn Gray Jamieson, Reaching for Health: The Australian Women's Health Movement and Public Policy, Canberra, Australian National University (ANU) Press, 2012. 11 Tess Little, The Women's Liberation Movement in Transnational Perspective: The US,UK, and France, 1967-79, Oxford, Université of Oxford, 2020.

  figure de la militante grassroot.

  lien, ou non, avec le corps médical ? Pour répondre à ces questions, il nous a paru essentiel de varier les contextes investigués en rassemblant des articles et un entretien portant sur l'Europe francophone des années 2010, l'Allemagne de l'Ouest et le Mali des années 1970 ainsi que le Sénégal de ces vingt dernières années. En effet, le dialogue entre les travaux menés sur les contextes occidentaux, d'une part, Pour la France, nous pensons aux travaux portés par le laboratoire junior Contraception&Genre (https://contraceptiongenre.wordpress.com/) ainsi qu'au colloque « Genre et gynécologie. Savoirs, pratiques et mobilisations » qui a eu lieu du 18 au 20 avril 2023 à la MSH Paris-Nord. Pour l'Allemagne, nous pensons au colloque « Women's Bodies, Women's Rights. Health Feminism, Reproductive Knowledge and Women's Activism across Europe in the long 20th Century » qui aura lieu du 19 au 21 octobre 2023 à l'Université de Constance, ainsi qu'au projet de Karen Nolte et Ulrike Klöppel « Frauen in verrückten Lebenswelten » à l'Université d'Heidelberg. 8 Donatella Della Porta, « Making the Polis: Social Forums and Democracy in the Global Justice

	non occidentaux, d'autre part, nous paraît encore trop peu développé. Par ailleurs, il nous
	semble important d'étendre le champ d'étude au-delà des très étudiées années 1970 de façon
	non seulement à interroger les continuités et ruptures temporelles, mais aussi à enrichir les
	en santé publique), 2001, p. 303-328 ; Anne-Sophie Cousteaux, « Santé », dans Juliette Rennes (dir.),
	Encyclopédie critique du genre. Corps, sexualité, rapports sociaux, Paris, La Découverte, 2021,
	p. 679-691.
	6 Anne Fausto-Sterling, Sexing the Body: Gender Politics and the Construction of Sexuality, New
	York, Basic Books, 2000 ; Delphine Gardey, Histoire politique du clitoris, Paris, Textuel, 2021 ;
	Patrick Pognant, « Les interdits hors la loi : la répression institutionnelle et médicale de la sexualité
	(1850-1930) », Droit et cultures. Revue internationale interdisciplinaire, 57 (5), 2009, p. 129-142.

7 Movement », Mobilization: An International Quarterly, 10 (1), 2005, p. 73-94 ; Laurence Cox, Cristina Flesher Fominaya, « Movement Knowlegde: What Do We Know, How Do We Create Knowledge and What Do We Do with It », Interface: A Journal for and about Social Movements, 1 (1), 2009, p. 1-20 ; Ron Eyerman, Andrew Jamison, Social Movements: A Cognitive Approach, Cambridge, Polity Press, 1991.

Phil Brown, Stephen Zavestoski, « Social Movements in Health: An Introduction », Sociology of Health & Illness,

(6), 2004, p. 679-694 ; Danièle Carricaburu, Marie Ménoret, Sociologie de la santé. Institutions, professions et maladies, Paris, Armand Colin, 2004. 2 Janine Barbot, Les malades en mouvements. La médecine et la science à l'épreuve du sida, Paris, Balland, 2002 ; Marcel Calvez, « Les signalements profanes de clusters de cancers : épidémiologie populaire et expertise en santé environnementale », Sciences sociales et santé,

(2), 2009, p. 79-106. 3 Ayse Dayi, « The Empowerment of Women in Reproductive Services: A Postructural Feminist Case Study of Two Women's Health Centers », thèse de doctorat en « Human Development and Family Studies », The Pennsylvania State University, 2005 ; Wendy Kline, Bodies of Knowledge: Sexuality, Reproduction, and Women's Health in the Second Wave, Chicago, The University of Chicago Press,

Geneviève Cresson, « Les soins profanes et la division du travail entre hommes et femmes », dans Pierre Aïach, Dominique Cèbe, Geneviève Cresson, Claudine Philippe (dir.), Femmes et hommes dans le champ de la santé. Approches sociologiques, Rennes, Presses de l'EHESP (École des hautes études

mais en affirmant leur foi. En parallèle, elles s'appuient sur les savoirs médicaux, autres savoirs d'autorité, pour défendre la santé comme un droit des femmes.On voit ici combien la diversification des terrains d'étude permet de réviser les catégories analytiques initialement forgées pour le contexte états-unien, et tout ce qu'elle apporte à l'analyse des rapports aux savoirs institués des femmes engagées pour leur santé. En cela, ce dossier n'est que la première pierre d'un édifice qui reste à construire, tant il appelle à de nouvelles recherches sur les contestations, appropriations et productions de savoirs par les femmes sur leur propre santé 26 .26 Les directrices du dossier remercient l'ensemble de l'équipe de Critique internationale ainsi que les évaluatrices et évaluateurs externes pour leur implication et leurs précieux commentaires.


Brice Goglin

Loïc Prylli

Design and Implementation of ORFA

ORFA is a user-level protocol that aims at providing an efficient remote file system access. It uses high bandwidth local networks and their specific API to directly connect user programs to data in the remote server. The ORFA model and its first implementation are described in this paper. The client is an automatically preloaded shared library that transparently overrides GLIBC I/O calls and redirect them to the server if concerned files are remote. The server may be a user program implementing a custom memory file system, or accessing native file systems.

Introduction

Achieving high performance for remote file access in a cluster environment require a high bandwidth data access to satisfy massive parallel applications running on MPI-IO, low latency for metadata access, and a usually strong coherency.

Existing projects generally focus on one of these important points by using stripping and parallezing or complex caching protocols. Optimized Remote File System Access is a new protocol that was designed for high performance general-purpose file access, making the most out of the cluster architecture, that is a high bandwidth and low latency local area network.

In this paper, we present ORFA design and describe its first implementation that is based on an automatically preloaded shared library client and a user-level server. We focus on ORFA protocol in Section 1, while the client architecture is discussed in Section 2 and the server in Section 3. ORFA was designed for LINUX in a homogeneous cluster environment. Thus no XDR-like layer [START_REF]XDR: eXternal Data Representation Standard[END_REF] is provided to ensure compatibility among different architectures and operating systems. Nevertheless a 32/64 bits conversion is provided by the protocol, which is fully based on 64 bits data structures. ORFA remains on the user-level file access API that is based on file paths, file descriptors and offsets. ORFA protocol was also built on main POSIX I/O calls. This model is not supposed to be used with caching clients. No flush or revalidate operation is available.

Contents

API Description

ORFA protocol is not stateless as NFS is [START_REF]NFS Version 3 Protocol Specification[END_REF]. Opening a remote file or directory in the client links a virtual descriptor on the client to the real descriptor in the server through an identifier. Following calls will use this identifier as the target object (see Table 1).

Opening a file as a file descriptor with open (or as a FILE* through fopen) generates the use of a FD identifier on the ORFA protocol for following calls. However directory reading is handled in a special way, the opendir call creates a DIR* identifier.

This state-oriented model implies that the client may not handle a server failure properly. Actually on network or server error the client receives an error (usually EIO) and is supposed to discard its virtual descriptors.

ORFA clients can only handle seeking locally. All other I/O calls are supposed to be forwarded to the server, even descriptor manipulations such as changing open flags with fcntl or resetting a directory with rewinddir.

Any time an I/O call passes a user buffer as an argument, the corresponding ORFA request uses it as an input or output data for network requests. Table 1 summarizes requests need for such data.

Network Layer

ORFA was designed to use high bandwidth local network, especially Myrinet [BCF • 95] (with GM [START_REF] Myricom | The GM Message Passing System[END_REF] or BIP software layer [START_REF] Prylli | BIP Messages User Manual[END_REF]). ORFA also provides a full TCP support through the socket layer. A ORFA request is composed of a header containing request parameters that may be followed by request data containing for example data to write or the file path to access. The reply is made of a reply header, eventually followed by a message containing for example read data or attributes.

Protocol Internals

Sending a memory buffer containing both request and associated data is faster than sending two distinct messages. However the input (or output) data buffer is given by the user program and is thus not contiguous to the ORFA header. Avoiding memory copy is important to maximize data transfer, especially for large buffers. That is the reason ORFA uses two distinct buffers instead of trying to merge them using memory copies. A support for vectorized message in GM or BIP would solve this issue.

Thus the process is summarized in Figure 2. The client sends a first message, describing the request type, the target file, the context (user's uid/gid for example) and the size of the data if it exists. Then the data buffer is sent. The client receives from the server a reply composed of the result of the request (error code or bytes read for example) and the amount of data if it exists. Then the data buffer is received. The asynchronous network APIs we use (GM and BIP) need a receive buffer to be posted before a message arrives. As the server cannot predict what kind of request it will receive, it cannot predict how many receive buffers need to be pre-posted and what size they need to be.

Pre-provided Buffers

ORFA Server

Pre-Posted

Client

Thus ORFA provides receive buffers in advance. The protocol sets a maximum size (generally 64 kB) and the server is supposed to pre-post enough header buffers or data buffers of this size for all messages it may receive (either headers or data).

If the client needs to send a smaller data, it can be sure a corresponding receive buffer has been posted on the server side and thus it can send safely (see Figure 3(a)).

If it needs to send a larger data, it waits for a confirmation message from the server. When the server receives the header, it notices that the amount of data to be received is higher than the size set by the protocol, and thus posts a larger buffer, and then sends a small confirmation message to the client (see Figure 3(b)).

This method maximizes the network utilization for limited data buffers. Only large buffers need to wait for a small message to be received, that is less than 10 s, which is negligible compared to the buffer sending time.

It is important to see that letting clients try to send their header and input data buffer without any control will generate many missing receive buffers. This implies the lost of a large part of the throughput, that means poor performance.

Client Implementation

ORFA goals are similar to those of DAFS [START_REF] Magoutis | Design and Implementation of a Direct Access File system (DAFS) Kernel Server for FreeBSD[END_REF], that is use the fastest way from the client to remote files, especially through the underlying high performance network. However the design is different, particularly on the client side. While DAFS defines its own specific API, which is fully asynchronous and also requires programs to be rewritten and recompiled, ORFA client tries to keep existing programs and add a transparent access to remote file systems.

Overview

The ORFA client is based on a shared library that intercepts I/O calls and redirects them to local files or remote servers. Assuming that physical networks available in clusters provide a very low latency (less than 10 s), it has been designed as a lightweight client without intelligence. Each I/O requests is forwarded to the server and no caching is available. The client architecture is described on Figure 4 and is explained in following sections.

This drastically increases the number of small requests the ORFA client needs. For example, reading a directory usually requires a readdir request and a stat for each entry. This is huge compared to NFS, which gets lots of entries and their attributes through only one READDIRPLUS request.

To measure an example of metadata processing overhead, the ORFA client may use read-ahead when getting directory entries. Following readdir calls from the user application will be handled locally. This is the only case where the client shows intelligence. All other intercepted calls concerning remote files imply a remote call, that is a network request to the ORFA server.

Intercepting I/O Calls

Intercepting a function of the usual GLIBC library is done by defining a same name function in our library, and force its preloading. Setting LD_PRELOAD environment variable to the ORFA shared library client make it be loaded before all standard libraries. Dynamic linking then uses ORFA symbols instead of the GLIBC ones.

The ORFA shared client is organized around a main wrapper.c file, which contains redefinitions of I/O symbols (see the list in Appendix B). These symbols check whether the file is a local or a remote one and calls the corresponding sub-routine. Local/remote check is done through is_remote_* functions. is_remote_pathname scans the full pathname, adding the current working directory, removing . and .., and checks for a /REMOTE@server/ beginning (which is currently used to mean remote file on server server). This function copies the full-simplified path into a newly allocated buffer to reduce following parsing work.

is_remote_fd and is_remote_dir functions check whether the given descriptor is a virtual descriptor associated with a remote file (see Section 2.5).

Finding Real Calls with dlfcn

When the target object is a local one, the wrapper function calls the real function of the GLIBC. Some previous work (such as smbsh or user-level PVFS implementation [START_REF] Philip | PVFS: A Parallel File System for Linux Clusters[END_REF]) used a copy of the internal GLIBC code (generally just a syscall wrapper) to reproduce real calls behavior for local files. This prevents from supporting the evolution of the GLIBC code, and several special cases where another library (for example libpthread) is loaded before GLIBC to fix its behavior (see Figure 2(a)).

ORFA solves this issue by calling the real GLIBC function, which has been overridden. This is done through the dlfcn library, which provides a way to look for another occurrence of a symbol in the process address space. real_fsync = dlsym(RTLD_NEXT, "fsync");

Remote Calls

If the wrapper recognizes a remote file, it calls the dedicated routine, which prepares ORFA requests and then asks the network layer for data transfer.

.] }

As described in Section 1.1, the ORFA header is filled with its type and parameters. For file or directory descriptor oriented requests, the descriptor's identifier is copied into the header structure. For path-oriented requests, the application path buffer is given as an input data buffer to the network layer. The same mechanism is used for other kinds of input and output data buffers.

req.in_buf = in_buffer; req.out_buf = out_buffer;

Virtual File Descriptors

When accessing a file through its pathname, the localization of this remote file is described in this path (/REMOTE@server/ references the root on the remote server server). However when using a file or directory descriptor, the ORFA client must be able to get its localization back. An efficient method to support all these special calls has been implemented by storing this structure into a memory-mapped file, whose file descriptor is given to the user application. Storing directory descriptors also uses some of these aspects.

This implementation provides a full and transparent support for manipulation of remote file descriptors through fork, dup or exec. Multithreading support is also available through protection of virtual descriptors against concurrent modifications.

Memory Registration

Any asynchronous network API requires buffers to be registered to the network interface and pinned in physical memory until the data transfer really occurs. This registration operation can be long on VM-oriented API such as GM. Thus memory registration needs to be used only when necessary.

ORFA client must use registered zones for headers and input/output data buffers. Instead of registering and deregistering each of these, for each I/O call, all small zones are copied into static zones that are registered during initialization. This additional memory copy has an overhead that is largely negligible against the registration it avoids.

Nevertheless a registration cache, which is generally more efficient, can replace this optimization. This cache consists of delaying deregistration until too much pages have been registered. This drastically reduces registration overhead when concerned pages are redundant. This argument is especially true for I/O applications because they generally use the same buffers for several consecutive data accesses. These buffers are registered only once.

Variant of ORFA Implementation on Client side using FUSE

Actually ORFA also provides the possibility to mount a remote file system in the kernel rather than using its LD_PRELOAD based shared library client (for cases that LD_PRELOAD cannot handle, for example, statically linked programs). Instead of porting the entire ORFA client as a kernel module, we based our kernel support on the FUSE project (File system in USEr-space) that allows connecting a file system implemented in a user program to the VFS layer. Mapping a remote file into a user program memory was the only function that was impossible to support with ORFA shared client, while mounting a remote file system with FUSE gives automatic kernel support for all Unix functionalities. This implementation has the drawback of requiring data to go from the user program into the FUSE kernel module, and then back to user-space in the FUSE-ORFA client that communicates to the ORFA server. This implies an important latency and limits the bandwidth in the client because of the two memory copies (see Figure 5). These added costs have an impact only on the client side, so it is still useful for situations where the shared library approach is not suitable, but where we want to increase the scalability of the server side.

FUSE-ORFA Client Application

Network Request

Server Implementation

Getting the most out of the underlying network layer requires ORFA to use the fastest way from clients to storage space. This implies removing layer redundancies and memory copies as often as possible. An interesting solution would have been to use a kernel server so that going from user-level to kernel would have been done during the network traversal.

We began our work on the server by using a user-level implementation which is simpler than a kernel one. We will mention in 3.1 several advanced subsystems on which this implementation relies. We also implemented two variants of this server. The first is based on a local file system on the server. The second is based on a custom memory file system (by removing the overhead of using syscalls and going through the VFS layer, this later variant allow to test the ORFA protocol independently of the storage subsystem of the server).

The ORFA server has to deal with all network requests coming from ORFA clients, process these requests, and send all replies. It has been designed to provide an experimental platform for testing several methods of handling all this work.

Events Handling

When using TCP network layer the ORFA server may have to deal with hundreds of connections, that means being able to get events from them in an efficient way. As the usual poll/select strategy is known to be not scalable to hundreds of clients, we also added support for the epoll and LINUX AIO API that provide much better performance.

Whatever network layer is used, the server then has to process many requests. The sequential architecture consists of receiving all pending requests, processing them all and then sending all answers back to clients. The ORFA server also provides a queue-thread architecture that uses one thread for receiving, one for processing and another one for sending. Moreover, it is possible a connection-thread architecture, that is to use a thread for each TCP connection.

Finally the server tries to make the most out of asynchronous API such as GM and BIP for network access, and also LINUX AIO for data access. This provides an efficient event-based architecture, even if full kernel support for disk AIO is still in development.

All these techniques provide a way to test the efficiency of several server architectures and find out which one is the most adapted to the workload the ORFA has to deal with.

Storage Systems

Native File System

The main ORFA server converts incoming requests into I/O calls to the underlying native subsystems through syscalls. Data is stored as local files on the server host.

As described in the previous section, LINUX AIO is also used to make data access asynchronous, and thus increase the server reactivity. This also shares workload across different processors on the host (without using user-level threads)

This mechanism suffers from memory copies between user and kernel space. Thus sendfile is used to reduce per byte overhead on the server, when using TCP or BIP (see Figure 6(a)). As sendfile was developed to increase web-server efficiency, no recvfile implementation is currently available (we developed an experimental one in BIP but we do not use it). This is an important limitation concerning the memory copy avoidance (see Figure 6(b)).

User-Level Memory File System

Using native file systems require our user-level server to use a syscall and go across the VFS layer for each request. This implies a significant overhead compared to the total request processing time.

It is noticeable that data that is exported by the ORFA server do not necessarily require being accessible as local files on the server. We may imagine a physical storage space that would be dedicated to the ORFA server and not accessible in other way. This has the advantage of removing VFS overhead (used to protect data from concurrent modifications and provide support for several specific features, what is useless here).

Instead of implementing a new physical storage system in a dedicated partition, we simply developed a memory file system in the ORFA server. This explicitly removes the copy between user and kernel space, syscall overhead, and several other constraints. This memory file system is implemented with a usual API while its storage organization is adapted to maximize ORFA protocol performance.

Conclusion

We described in this paper the protocol of ORFA, the client implementation as an automatically preloaded shared library and the user-level server. The lightweight client has been designed to use the fastest way to transfer data on the server side, while the absence of caching may imply lots of small network requests. The ORFA server wants to be as efficient as possible, using several techniques to reduce its CPU workload and increase its reactivity. Finally the implementation tries to make the most of the underlying high bandwidth local network.

This aims at providing a high performance remote file access, that means efficiency for parallel computing applications. However the full support the POSIX API on remote files and the client transparency make it available for any kind of interactive usage, or shell scripts. This contributes to the general-purpose file access ability of ORFA protocol.

The first version of the ORFA project produced user-level implementation of the client and server. We are now working on a second version where the client is implemented in the kernel directly under the VFS layer of the operating system. The server will still be user-level and the protocol is being changed to take advantage of the VFS layer properties. fcntl, _fcntl, __fcntl, __libc_fcntl flock, lockf utime, utimes fsync, _fsync, __fsync, __libc_fsync fdatasync sendfile, sendfile64 getdents, getdents64 opendir, closedir readdir, readdir_r, readdir64, readdir64_r seekdir, telldir, rewinddir To ensure that the server acts on its files in the exact same way the client does, several functions have to be intercepted to in order to transmit the I/O context to the server. umask setuid, seteuid, setreuid, setresuid setgid, setegid, setregid, setresgid Allowing the use of a remote directory as working directory requires overriding usual behavior.

fchdir, chdir, getcwd

Keeping the POSIX behavior of file descriptors for remote files needs interception of several calls. dup and fork calls just have to increase usage counters. exec has to keep remote descriptor details during process reset and deal with close-on-exec flags.

fork, _fork, __fork, __libc_fork dup, _dup, __dup, dup2, _dup2, __dup2 vfork, __clone execve, fexecve, execv, execvp, execle, execl, execlp Memory mapping explicitly requires to deal with the buffer cache. This is the reason ORFA shared client cannot support mapping of remote files (even if a read-only mapping support may be imagined when non-concurrent modification is ensured). Nevertheless mmap-like functions are intercepted to update register caching when the process address space is modified. mmap, mmap64, mremap, munmap msync, _msync, __msync, __libc_sync

The usual GLIBC malloc library uses __mmap-like functions to manipulate huge free memory zones. These functions are not exported by the GLIBC and thus cannot be intercepted. Therefore it was required to intercept memory allocating functions. This was done by including an entire memory allocation library in ORFA shared client. sbrk malloc, calloc, realloc, free

  Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Network Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Protocol Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Pre-provided Buffers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Client Implementation 6 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Intercepting I/O Calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Finding Real Calls with dlfcn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Remote Calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Virtual File Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Memory Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 Variant of ORFA Implementation on Client side using FUSE . . . . . . . . . . . . . . 10 3 Server Implementation 11 3.1 Events Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: ORFA User-Level Remote File Access Protocol Model.

Figure 2 :

 2 Figure 2: Message Passing Model used in ORFA

Figure 3 :

 3 Figure 3: Maximal Size Buffer Pre-Posting Model

Figure 4 :

 4 Figure 4: Architecture of the ORFA client.

  Dynamic linking with dlfcn Table 2: Real Symbol Finding Models This ensures a full transparency for local file access because the wrapper completely remains on usual GLIBC I/O calls (see Figure 2(b)).

Figure 5 :

 5 Figure 5: Kernel Mounting with ORFA and FUSE.

Figure 6 :

 6 Figure 6: Path of data in the server during read (with sendfile) and write request.

  ORFA is a user-level remote file access protocol. A user program accesses remote files through a user-level client that converts I/O calls into network requests, as described on Figure1.

	1 Remote File Access Protocol
	I/O Call	Application
		ORFA Client	Network	ORFA Server	I/O Syscall
	User		Request
	Kernel
				VFS
	Conclusion				13
	References				13
	A Usage				14
	B Supported I/O Calls				14

. . . . . 11 3.2.1 Native File System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.2 User-Level Memory File System . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Table 1 :

 1 ORFA Protocol Description. The first table presents targets of each kind of ORFA requests. Following tables shows whether a request requires an input and/or output buffer.

	ORFA Request Type

  That is the reason each remote open file or directory is described by a local structure, a virtual descriptor.Moreover this descriptor stores several details to reproduce the usual behavior during lseek or several special calls such as dup or fork.

	struct file_descr {
	/* identifier */
	server_t server;
	fd_id_t fd_id;
	[...]
	};
	/* usual file descriptor fields */
	off64_t size;
	off64_t seek;
	/* use count */
	int use_count;

A Usage

The ORFA implementation that is described in this paper is available as a tarball on http:// www.ens-lyon.fr/~bgoglin/work.html. Source files are organized in client and server directories and their respective subdirectories, while common headers are centralized in include. When compiled, binaries are stored in the bin-${ARCH} directory.

The shared library client needs to be placed in the LD_PRELOAD environment variable. This enables its automatic preloading for all programs that will be run later. Files exported on server named server are then available under /REMOTE@server/. Setting this environment variable in a shell does not enable remote files for shell specific features (such as completion). Launching a new shell is necessary.

The main directory also contains a patch to be applied to FUSE 0.95 to convert its API to fit with ORFA protocol (see Section 2.7).

B Supported I/O Calls

Following functions, when available in the host GLIBC, are supported for remote files. Several functions support is provided through the interception of underlying functions that their GLIBC implementation is based on. open, _open, __open, __libc_open open64, _open64, __open64, __libc_open64 creat, creat64 close, _close, __close, __libc_close read, _read, __read, __libc_read pread, _pread, __pread, __libc_pread pread64, _pread64, __pread64, __libc_pread64 write, _write, __write, __libc_write pwrite, _pwrite, __pwrite, __libc_pwrite pwrite64, _pwrite64, __pwrite64, __libc_pwrite64 lseek, _lseek, __lseek, __libc_lseek lseek64, _lseek64, __lseek64, __libc_lseek64 llseek, _IO_file_seek fopen, fdopen, fclose, fread, fwrite, fseek, feof truncate, ftruncate, truncate64, ftruncate64 stat, __xstat, _stat, __stat lstat, __lxstat, _lstat, __lstat fstat, __fxstat, _fstat, __fstat stat64, __xstat64, _stat64, __stat64 lstat64, __lxstat64, _lstat64, __lstat64 fstat64, __fxstat64, _fstat64, __fstat64 statfs, fstatfs, statfs64, fstatfs64 mknod, __xmknod, _mknod, __mknod mkdir, rmdir, unlink, rename, link symlink, readlink access, euidaccess chmod, fchmod, chown, lchown, fchown