Tony Lelièvre
email: lelievre@cermics.enpc.fr

Felix Otto
email: otto@iam.uni-bonn.de

Mathias Rousset
email: rousset@cermics.enpc.fr

Gabriel Stoltz
email: stoltz@cermics.enpc.fr

Long-time convergence of an Adaptive Biasing Force method

). Mathematically, it amounts to studying the long-time behavior of a stochastic process which satisfies a non-linear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

Introduction

In Section 1.1, we introduce the physical context of this work, namely molecular dynamics and the computation of free energy differences in the canonical statistical ensemble. In Section 1.2, we introduce the adaptive dynamics we study and the main results we prove are presented in Section 1.3.

Computations of free energy differences and metastability

Let us consider the Gibbs-Boltzmann measure dµ(q) = Z -1 exp(-βV (q))dq, [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] where q ∈ D, V : D → R, Z = D exp(-βV (q)) dq and D = {q, V (q) < ∞} is the configuration space. In the applications we consider, q represents the position of N particles so that, in the following, D is an open subset (possibly the whole) of R n , with n = 3N . All the results we prove are also satisfied if D is an open subset of T n (where T = R/Z denotes the one-dimensional torus). The function V is the energy associated with the positions of the particles and β is proportional to the inverse of the temperature. The probability measure µ represents the equilibrium measure sampled by the particles in the canonical statistical ensemble. A typical dynamics that can be used to sample this measure is

dQ t = -∇V (Q t ) dt + 2β -1 dB t , (2)
where B t is a n-dimensional standard Brownian motion. More generally, for any smooth positive function γ : D → R * + , the stochastic process Q t which satisfies

dQ t = -∇(V -β -1 ln γ)(Q t )γ(Q t ) dt + 2β -1 γ(Q t )dB t (3)
samples the measure µ.

Let us introduce a so-called reaction coordinate ξ : D → M, with M = R or M = T. For a given configuration q, ξ(q) represents a coarse-grained information, which is valuable from a physical point of view. For instance, ξ(q) may be a dihedral angle, for example to characterize the conformation of a molecule, in which case M = T, or the signed distance to an hypersurface of D (characterizing a transition state), for example to measure the evolution of a chemical reaction, in which case M = R. The function ξ is therefore related to some macroscopic information of the system. Usually, in [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF], the time-scale for the dynamics on ξ(Q t ) is larger than the time-scale for the dynamics on Q t (due to metastable states), so that ξ can also be understood as a function such that ξ(Q t ) is a slow variable compared to Q t .

In the following, we suppose that

[H1] ξ is a smooth function such that |∇ξ| > 0 on D.

Thus, the subsets Σ z = {x ∈ D, ξ(x) = z} of D are smooth submanifolds of codimension one which define a partition of D:

D = z∈M Σ z and Σ z ∩ Σ z ′ = ∅ for z = z ′ .
We denote by σ Σz the surface measure on Σ z , i.e. the Lebesgue measure on Σ z induced by the Lebesgue measure in the ambient space D ⊃ Σ z . The submanifold Σ z naturally has a (complete and locally compact) Riemannian structure induced by the Euclidean structure of the ambient space D.

The image of the measure µ by ξ is exp(-βA(z)) dz M exp(-βA(z)) dz where A is the so-called free energy defined by:

A(z) = -β -1 ln(Z Σz ) (4)
where

Z Σz = Σz |∇ξ| -1 exp(-βV )dσ Σz .
We assume henceforth that ξ and V are such that Z Σz < ∞. The free energy is actually defined up to an additive constant, the quantity exp(-βA) being then defined up to a multiplicative constant, which disappears in the normalization of the probability measure exp(-βA(z)) dz M exp(-βA(z)) dz . Many algorithms in molecular dynamics [START_REF] Chipot | Free Energy Calculations[END_REF] aim to compute the image of the measure µ by ξ, which amounts to compute free energy differences, namely quantities of the form A(z) -A(z 0 ). This is typically obtained by computing (and then integrating) the derivative A ′ (z), called the mean force. Using the co-area formula (see Appendix A), the following expression for A ′ (z) can be obtained (see [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF], or the proof of Lemma 7 below):

A ′ (z) = Z -1 Σz Σz F |∇ξ| -1 exp(-βV )dσ Σz , (5)
where F is the so-called local mean force defined by

F = ∇V • ∇ξ |∇ξ| 2 -β -1 div ∇ξ |∇ξ| 2 . ( 6
)
This can be rewritten in terms of conditional expectation as: For a random variable X with law µ, A ′ (z) = E F (X) ξ(X) = z .

In practice, free energy profiles are used for example to compare the likelihood of various conformations of a molecule, or to compute the rate of a chemical reaction. Free energy can also be useful to compute ensemble averages in the canonical ensemble using the following formula (which is a conditioning formula): For any function φ :

D → R, D φ dµ = M Σz φ dµ Σz exp(-βA(z)) dz M exp(-βA(z)) dz , (8)
where µ Σz is the probability measure µ conditioned to a fixed value z of the reaction coordinate:

dµ Σz = Z -1 Σz |∇ξ| -1 exp(-βV )dσ Σz . (9)
Notice that (5) also writes A ′ (z) = Σz F dµ Σz . Equation ( 8) may be interesting to compute averages in the canonical ensemble since, if the reaction coordinate is well chosen, it is expected that the sampling of the conditioned probability measure µ Σz is easier than the sampling of µ (the metastable features of the measure µ being mostly in the direction of the reaction coordinate ξ). The sampling of µ Σz can be done for example by projection of the gradient dynamics on Σ z (see [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]). The quantity Σz φ dµ Σz can thus be evaluated by an efficient Monte Carlo procedure, and the computation of D φ dµ through [START_REF] Evans | Measure theory and fine properties of functions[END_REF] then only requires a one-dimensional integration, and the computation of the free energy (up to an additive constant). Due to the high dimensionality of the problem (the number of particles N is usually very large), methods to compute mean forces or free energy differences are Monte Carlo methods. They typically rely on the simulation of a diffusion Markov process. The most recent methods use non-homogeneous or non-linear Markov processes. Classical examples are exponential reweighting of non-equilibrium paths (based upon the socalled Jarzynski equality, see [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF][START_REF] Lelièvre | Computation of free energy differences through nonequilibrium stochastic dynamics: The reaction coordinate case[END_REF]) or adaptive methods (see [START_REF] Darve | Calculating free energy using average forces[END_REF][START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF][START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF][START_REF] Wang | Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram[END_REF]).

We are interested here in adaptive methods to compute free energy differences, and more precisely Adaptive Biasing Force techniques (see [START_REF] Darve | Calculating free energy using average forces[END_REF][START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF]). The principle of adaptive methods is to modify the potential V during the simulation, in order to remove the metastable features of the simple dynamics (2), while approximating the free energy A. Many methods have been proposed and we refer to [START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF] for a unified presentation of these techniques, as well as a discussion of efficient parallel implementations. The aim of this paper is to propose a mathematical study of the Adaptive Biasing Force method to give a rigorous formulation and proofs of the following statements (which are the main arguments of practitioners of the field to advocate the use of adaptive methods):

[S1] The adaptive biasing force technique helps to remove the metastable features of the simple dynamics (2), and thus enables efficient exploration of the configuration space.

[S2] With the adaptive biasing force technique, the free energy A is obtained in the longtime limit, and the convergence is exponentially fast in time.

An Adaptive Biasing Force technique

The Adaptive Biasing Force (ABF) method was introduced in [START_REF] Darve | Calculating free energy using average forces[END_REF][START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF] and is recast in a general mathematical framework in [START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF]. We propose to study here one version of this method, applied to the context of Brownian (or overdamped Langevin) dynamics 1 .

The ABF dynamics we propose to study is the following non-linear stochastic differential equation:

dX t = -∇ V -A t • ξ + W • ξ -β -1 ln(|∇ξ| -2 ) (X t ) |∇ξ| -2 (X t ) dt + 2β -1 |∇ξ| -1 (X t )dB t , ( 10
)
where W is an additional well-chosen potential that we will define below and A t is the "free energy observed at time t". More precisely, the derivative of A t with respect to the reaction coordinate is defined as (compare with [START_REF] Darve | Calculating free energy using average forces[END_REF]): ∀z ∈ M,

A ′ t (z) = E F (X t ) ξ(X t ) = z , ( 11
)
where F is defined by [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]. With a slight abuse of terminology, the function A ′ t is called the biasing force. Notice that here and in the following, the notation ′ denotes a derivative with respect to the reaction coordinate values, while the notation • denotes the composition operator. Equation [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] defines A t up to an additive (time-dependent) constant, which does not modify [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF].

Compared to the simple dynamics (2), three modifications have been made to obtain (10)-( 11):

1. First and foremost, the potential V has been changed to the biasing potential V -A t • ξ. This is the bottom line of the adaptive strategy. The algorithm we study here is prototypical of many adaptive methods used in molecular dynamics (see [START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF]). In the original Adaptive Biasing Force technique as presented in [START_REF] Darve | Calculating free energy using average forces[END_REF][START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF], the conditional expectation [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] is actually "approximated" by some conditional averages over one single trajectory. The dynamics we study here is not clearly related with such a discretization, but rather with a discretization of (11) using an interacting particle system, where many replicas of the system contribute to the free energy profile (see [START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF]).

2. Second, a potential W • ξ has been added. This is actually needed only in the case when M is an unbounded domain (we recall that M is the domain where the reaction coordinate lives). In theses cases, W is chosen so that the law of ξ(X t ) converges exponentially fast to its longtime limit (more precisely, the Fisher information associated with this law converges exponentially fast to zero, see [H4] below for a more detailed statement). Besides, from a numerical point of view, such a potential is sometimes used in practice in order to separately sample some parts of the reaction coordinate space M (as in stratified sampling strategies).

3. Third, some terms depending on |∇ξ| have been introduced. This modification is made in order to obtain a simple diffusive behavior for the law of ξ(X t ) (see Proposition 1 below). It is expected that the longtime convergence of A ′ t towards A ′ still holds without this modification, by simply considering the gradient dynamics

dX t = -∇(V -A t • ξ + W • ξ)(X t ) dt + 2β -1 dB t , (12)
with the same definition [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] for A ′ t . However, we are only able to prove a weaker convergence result in this case. This is the matter of Sections 2.3 and 3.4. Notice that if |∇ξ| is constant (for example if ξ is a length), a simple change of time relates [START_REF] Lelièvre | Computation of free energy differences through nonequilibrium stochastic dynamics: The reaction coordinate case[END_REF] with [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF]. Notice also that if we take A t = W = 0 in [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF], then X t samples the original Gibbs measure µ defined by (1) (see Equation (3) above).

Remark 1 (On the computation of A ′ t (z)) From a practical point of view, with the additional terms mentioned in item 3 above, it is possible to compute the biasing force A ′ t (z) without explicitly evaluating F since (by Itô's calculus on X t that satisfies [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF], and assuming W = 0 for simplicity)

F (X t ) dt = dξ(X t ) + A ′ t (ξ(X t )) dt -2β -1 ∇ξ |∇ξ| (X t ) • dB t . ( 13
)
By a simple finite difference scheme, we thus have the following approximation

F (X tn+1 ) ≃ A ′ tn (ξ(X tn )) + ξ(X tn+1 ) -ξ(X tn ) -2β -1 ∇ξ |∇ξ| (X tn ) • (B tn+1 -B tn ) ∆t .

A PDE formulation and presentation of the main result

We would like to emphasize that our arguments are partially formal: we assume that we are given a process X t and a function A ′ t which satisfy (10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF], and such that X t has a smooth density ψ(t, •) with respect to the Lebesgue measure on D. We suppose that this density is sufficiently regular so that the computations are valid. In particular, we assume that the potential V is such that either the stochastic process X t lives in D and thus that its density ψ(t, •) decays sufficiently fast on ∂D or the stochastic process X t has some reflecting behavior on ∂D and thus that its density ψ(t, •) has zero normal derivatives on ∂D. In both cases, no boundary terms appear in the integrations by parts we perform to derive the entropy estimates. We refer for example to [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] for an appropriate functional framework in which such entropy estimates hold.

Since only the law of the process X t at a fixed time t is used in [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF], it is possible to recast the dynamics in the following nonlinear partial differential equation (PDE) on the density ψ(t, •) of X t :

           ∂ t ψ = div |∇ξ| -2 ∇(V -A t • ξ + W • ξ)ψ + β -1 ∇ψ , A ′ t (z) = Σz F |∇ξ| -1 ψ(t, •)dσ Σz Σz |∇ξ| -1 ψ(t, •)dσ Σz , ( 14
)
where F is defined by [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]. This is obtained by using the fact that if X t has law ψ(t, x) dx, then the law of ξ(X t ) is ψ ξ (t, z) dz with

ψ ξ (t, z) = Σz |∇ξ| -1 ψ(t, •)dσ Σz , (15)
and the conditional law of X t with respect to ξ(X t ) = z is µ t,z defined by

dµ t,z = ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) . ( 16
)
The probability measure ψ ξ (t, z) dz is the image of the probability measure ψ(t, x) dx by ξ. These expressions can be obtained using the co-area formula (see Appendix A). Before presenting the results, we would like to motivate the introduction of this dynamics by the following formal observation. If the potential A t and the law of X t reach a stationary state, then, from the dynamics (10) on X t (or from the partial differential equation [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF] satisfied by the distribution of X t ), we observe that this stationary law is proportional to exp(-β(V (x) -A ∞ • ξ(x) + W • ξ(x))) dx, where A ∞ denotes the stationary state for A t (this requires a uniqueness result for the law of X t , which holds for example if |∇ξ| is uniformly bounded from below by a positive constant). Then, from the definition [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] of the biasing force, we obtain that, necessarily, A ′ ∞ = A ′ (where A ′ is the mean force defined by ( 5)). This proves the uniqueness of the stationary state for this dynamics. We can thus expect that A ′ t converges to the mean force A ′ in the longtime limit.

The interest of the dynamics (10)-( 11) is actually twofold. First, as expected from the formal argument above, in the longtime limit, A ′ t converges to the mean force A ′ defined by [START_REF] Chipot | Free Energy Calculations[END_REF] (see Equation (24) below). Second, using the ABF method, the law of ξ(X t ) has a simple diffusive behavior (see Equation (20) below). The metastable feature of the simple dynamics (2) along ξ is thus corrected by the addition of the adaptive potential A t . The aim of this paper is to give a precise statement for these two assertions, which are mathematical formalizations of the two main characteristics [S1] and [S2] of adaptive techniques mentioned in Section 1.1. The proof of the longtime convergence relies on entropy techniques, and requires appropriate assumptions on the potentials V , W and the reaction coordinate ξ. We prove that under suitable assumptions, the convergence of A ′ t to A ′ is exponentially fast, with a rate of convergence limited, at the macroscopic level, by the rate of convergence of the law of ξ(X t ) to its longtime limit, and, at the microscopic level, by the rate of convergence to the equilibrium conditioned probability measures µ Σz , for all values z of the reaction coordinate.

All these results are more precisely stated in Section 2, and the proofs are given in Section 3. We would like to mention that the main arguments of the proof are given in a very simple case in Section 3.1 and that we also present a result of convergence for the dynamics ( 12)-( 11) in Section 2.3.

Precise statements of the results

In Section 2.1, we recall some well-known results on entropy and introduce the main notation used in the following to state the convergence result. Section 2.2 is devoted to the presentation of the convergence result for the dynamics ( 10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF]. Finally, we give in Section 2.3 a (weaker) convergence result for the dynamics ( 12)-(11).

Entropy and Fisher information

Let us consider ψ and A ′ t which satisfy [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF] and let introduce the long-time limit of ψ, ψ ξ (defined by ( 15)) and µ t,z (defined by ( 16)):

ψ ∞ = (ZZ ξ ) -1 exp(-β(V -A • ξ + W • ξ)), ψ ξ ∞ (z) = (Z ξ ) -1 exp(-βW (z)), dµ ∞,z = dµ Σz = Z -1 Σz exp(-βV )|∇ξ| -1 dσ Σz , where Z ξ = M exp(-βW (z)) dz.
We recall that

Z Σz = Σz |∇ξ| -1 exp(-βV )dσ Σz , Z = D exp(-βV (x)) dx.
Notice that D ψ ∞ = 1, and that the probability measure ψ ξ ∞ (z) dz is the image of the probability measure ψ ∞ (x) dx by ξ.

In order to state the results, we also need to introduce the following projection operators. For any x ∈ D, we denote by

P (x) = Id - ∇ξ ⊗ ∇ξ |∇ξ| 2 (x)
the orthogonal projection operator onto the tangent space T x Σ ξ(x) to Σ ξ(x) at point x, and by

Q(x) = ∇ξ ⊗ ∇ξ |∇ξ| 2 (x)
the orthogonal projection operator onto the normal space N x Σ ξ(x) to Σ ξ(x) at point x.

We denote by ⊗ the tensor product:

For two vectors u, v ∈ D, u ⊗ v is a n × n matrix with components (u ⊗ v) i,j = u i v j .
We measure the "distance" between ψ (respectively ψ ξ ) and ψ ∞ (respectively ψ ξ ∞ ) using the relative entropy H(ψ|ψ ∞ ) (respectively H(ψ ξ |ψ ξ ∞ )), where, for any two probability measures µ and ν such that µ is absolutely continuous with respect to ν (this property being denoted µ ≪ ν in the following),

H(µ|ν) = ln dµ dν dµ.
We recall the Csiszar-Kullback inequality:

µ -ν T V ≤ 2H(µ|ν) (17)
where

µ -ν T V = sup f, f L ∞ ≤1 { f d(µ -ν)}
denotes the total variation norm of the signed measure µν. When µ and ν both have densities with respect to the Lebesgue measure, µν T V is simply the L 1 norm of the difference between the two densities.

We denote the total entropy by

E(t) = H(ψ(t, •)|ψ ∞ ),
the macroscopic entropy by

E M (t) = H(ψ ξ (t, •)|ψ ξ ∞ ),
the "local entropy" at a fixed value z of the reaction coordinate by

e m (t, z) = H(µ t,z |µ ∞,z ) = Σz ln ψ(t, •) ψ ξ (t, z) ψ ∞ ψ ξ ∞ (z) ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) ,
and the microscopic entropy by

E m (t) = M e m (t, z)ψ ξ (t, z) dz.
It is straightforward to obtain the following result which can be seen as the extensivity of the entropy:

Lemma 1 It holds E(t) = E M (t) + E m (t).
Let us now introduce the Fisher information: For any two probability measures µ and ν such that µ ≪ ν,

I(µ|ν) = ∇ ln dµ dν 2 dµ. (18)
In the case ν is a probability measure on the (Riemannian) submanifold Σ z , ∇ actually denotes the gradient on Σ z in ( 18), namely

∇ Σz = P ∇. (19)
Therefore, for the conditional probability measures µ t,z and µ ∞,z , the Fisher information writes

I(µ t,z |µ ∞,z ) = Σz ∇ Σz ln ψ(t, •) ψ ∞ 2 ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) .
Let us finally introduce another way to compare two probability measures, namely the Wasserstein distance with quadratic cost: for two probability measures µ and ν defined on a Riemannian manifold Σ,

W (µ, ν) = inf π∈Π(µ,ν) Σ×Σ d Σ (x, y) 2 dπ(x, y).
In this expression, d Σ denotes the geodesic distance on Σ: ∀x, y ∈ Σ,

d Σ (x, y) = inf    1 0 | ẇ(t)| 2 dt w ∈ C 1 ([0, 1], Σ), w(0) = x, w(1) = y    ,
where Π(µ, ν) denotes the set of coupling probability measures, namely probability measures on Σ × Σ such that their marginals are µ and ν. We need the following definitions:

Definition 1 The probability measure ν is said to satisfy a logarithmic Sobolev inequality with constant ρ > 0 (in short: LSI(ρ)) if for all probability measures µ such that µ ≪ ν,

H(µ|ν) ≤ 1 2ρ I(µ|ν).

Definition 2

The probability measure ν is said to satisfy a Talagrand inequality with constant ρ > 0 (in short: T(ρ)) if for all probability measures µ such that µ ≪ ν,

W (µ, ν) ≤ 2 ρ H(µ|ν).
In the latter definition, we implicitly assume that the probability measures have finite moments of order 2. This will always be the case for all the probability measures we consider. We will need the following important result (see [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF]Theorem 1]).

Lemma 2 If ν satisfies LSI(ρ), then ν satisfies T(ρ).

For an introduction to logarithmic Sobolev inequalities, their properties and their relation to longtime behavior of solutions to PDEs, we refer to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF].

2.2 Convergence of the adaptive dynamics ( 10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] We are now in position to state our main results. Concerning the dynamics on the law of ξ(X t ), we have: Proposition 1 (Equation satisfied by the marginal density ψ ξ ) Let (ψ, A ′ t ) be a smooth solution to [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF] and let us assume [H1]. Then ψ ξ satisfies the following equation:

∂ t ψ ξ = ∂ z W ′ ψ ξ + β -1 ∂ z ψ ξ on M. ( 20
)
Remark 2 Notice that even if ψ ξ satisfies a closed PDE, ξ(X t ) does not satisfy a closed SDE (see Equation ( 13) above).

The fundamental assumptions we need to prove longtime convergence are the following (we recall that the local mean force F is defined by ( 6)):

[H2]

V and ξ are sufficiently differentiable functions such that

∇ξ L ∞ ≤ m < ∞ and ∇ Σz F L ∞ ≤ M < ∞, [H3]
V and ξ are such that ∃ρ > 0, for all z ∈ M, the conditional measure µ ∞,z satisfies LSI(ρ).

In Assumption [H2], the requirement on F can be seen as a boundedness condition on the coupling between the conditional measures µ ∞,z and the corresponding marginal ψ ξ ∞ , since it involves the mixed derivatives (along the tangential space and the normal space of the submanifold Σ z ) P ∇(Q∇V ) (see [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF] and Remark 11 below).

Assumption [H3] ensures that if, for a fixed value z of the reaction coordinate, the conditioned probability measure µ ∞,z were to be sampled by a simple constrained gradient dynamics (see [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]), the convergence to equilibrium would be exponential with rate ρ. We refer to ρ as the microscopic rate of convergence in the sequel.

We refer to Section 3.1 for an explicit framework where [H2] and [H3] are satisfied, and to Remark 3 below for alternative assumptions on V and ξ.

Let us now introduce the assumption we need on W .

[H4] W is such that

∃I 0 > 0, r > 0, ∀t ≥ 0, I(ψ ξ (t, •)|ψ ξ ∞ ) ≤ I 0 exp(-2β -1 r t).
Assumption [H4] is indeed an assumption on W because ψ ξ satisfies the PDE (20) where only W appears. Assumption [H4] ensures that the law of ξ(X t ) converges to equilibrium exponentially fast with rate r, which we refer to as the macroscopic rate of convergence in the sequel. We will see below (see [H4']) some sufficient explicit conditions on W for [H4] to be satisfied.

E m (t) ≤ C exp(-λt) (21)
where

C = 2 max E m (0), M β -1 |ρm -2 -r| I0 2ρ and λ = β -1 min(ρm -2 , r). ( 22
)
In the special case ρm -2 = r, E m satisfies E m (t) ≤ E m (0) + M I0 2ρ t exp(-β -1 r t). This implies that the total entropy E and thus ψ(t, •)ψ ∞ L 1 (D) both converge exponentially fast to zero with rate λ.

We thus obtain that the biasing force A ′ t converges to the mean force A ′ in the following sense: ∀t ≥ 0,

M |A ′ t -A ′ | 2 (z)ψ ξ (t, z) dz ≤ 2M 2 ρ E m (t). ( 23
)
Notice that the fact that E and ψ(t, •)ψ ∞ L 1 (D) converge exponentially fast to zero with rate λ is an immediate consequence of (21), [H4], Lemma 1 and the Csiszar-Kullback inequality [START_REF] Villani | Hypocoercivity[END_REF]. We will actually consider the two following cases for which [H4] is satisfied:

[H4']      If M = T, then W = 0. If M = R
, then W is a potential such that W ′′ is bounded from below and there exists r > 0 such that exp(-βW ) M exp(-βW ) satisfies LSI(r). Notice that in the case M = R, the assumptions stated in [H4'] on W are satisfied for an α-convex potential (namely if W ′′ ≥ α for a positive α), and then it is possible to choose r = α in [H4] (see Lemma 13 below). We refer to Remark 4 below for alternative assumptions on W . M exp(-βW ) satisfies LSI(r), then [H4] is satisfied with r = rε for any ε ∈ (0, r).

Let us now assume [H1], [H2], [H3] and [H4']. From (23), we deduce that for all compact K ⊂ M, ∃C,

t * > 0, ∀t ≥ t * , K |A ′ t -A ′ |(z)ψ ξ ∞ (z) dz ≤ C exp(-λt), ( 24
)
where λ is the rate of convergence defined by ( 22) in Theorem 1.

These results therefore show that A ′ t converges exponentially fast to A ′ (in

L 1 (ψ ξ ∞ (z) dz)-norm) at a rate λ = β -1 min(ρm -2 , r).
The limitations on the rate λ are related to the rate of convergence r at the macroscopic level, for the equation (20) satisfied by ψ ξ , and the rate of convergence at the microscopic level, which depends on the constant ρ of the logarithmic Sobolev inequalities satisfied by the conditional measures µ ∞,z . This constant of course depends on the choice of the reaction coordinate. In our framework, we could state that a "good reaction coordinate" is such that ρ is as large as possible.

The proof of these results is given in Sections 3.1, 3.2 and 3.3 below.

Remark 3 (Other possible assumptions on V and ξ) We would like to mention other possible assumptions on V and ξ than [H2]-[H3] for which the results of Theorem 1 still hold.

• First, in [H2], it is possible to change the assumption

∇ Σz F L ∞ ≤ M < ∞ to F L ∞ ≤ M < ∞.
Indeed, this simply changes the estimate (35) in Lemma 10 below to the following

|A ′ t (z) -A ′ (z)| ≤ F L ∞ µ t,z -µ ∞,z T V , ≤ M 2H(µ t,z |µ ∞,z ),
by the Csiszar-Kullback inequality [START_REF] Villani | Hypocoercivity[END_REF]. The rest of the proof remains exactly the same.

• Second, it is possible to obtain a similar result of convergence under slightly different assumptions than

M = R (or M is an unbounded domain).
In this case, it is actually also possible to satisfy [H4] by choosing W such that the dynamics is confined in a domain z∈N Σ z , where N is a bounded subset of M. This can be done by using a sufficiently confining potential W and adapting Lemma 13 below, or by adding reflexion terms to restrict ξ to N (which loosely speaking corresponds to take W zero on N and infinite on M \ N ) and adapting Lemma 12 below.

Let us make precise this latter case. Suppose for example we are interested in the values of A ′ (z) for z ∈ N = (0, 1). The dynamics is confined in the domain O = 0<z<1 Σ z . The ABF dynamics is

                 ∂ t ψ = div |∇ξ| -2 ∇(V -A t • ξ)ψ + β -1 ∇ψ , on O, ∇(V -A t • ξ)ψ + β -1 ∇ψ • ∇ξ = 0, on Σ 0 ∪ Σ 1 , A ′ t (z) = Σz F |∇ξ| -1 ψ(t, •)dσ Σz Σz |∇ξ| -1 ψ(t, •)dσ Σz , for z ∈ (0, 1),
where F is defined by [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]. From the point of view of the stochastic process X t , the boundary condition translates to a normal reflexion on the two submanifolds Σ 0 and Σ 1 . Moreover, it can be checked (using Lemma 7) that the boundary condition on ψ translates to a zero Neumann boundary condition on

ψ ξ : ∂ z ψ ξ (0) = ∂ z ψ ξ (1) = 0.
A proof similar to that of Lemma 12 then shows that I(ψ ξ |ψ ξ ∞ ) converges exponentially fast to 0, so that [H4] holds. The arguments we use to prove Theorem 1 and Corollary 1 then show that A ′ t -A ′ L 2 (0,1) goes to 0 exponentially fast.

Remark 5 (Vectorial reaction coordinate) In this work, we assume that the reaction coordinate ξ has values in T or R. The dynamics (10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] and the results of convergence presented in this section can be straightforwardly extended to the case when ξ = (ξ 1 , . . . , ξ m ) has values in T m or R m , with 2 ≤ m < n, under the orthogonality condition:

∀i = j, ∇ξ i • ∇ξ j = 0. ( 25
)
The generalization of this dynamics to non orthogonal reaction coordinates is unclear.

In this case, it is possible to resort to metadynamics (see Remark 6 below). Alternatively, the dynamics ( 12)-( 11) (and the result of convergence of Section 2.3 for this dynamics) can straightforwardly be generalized to a vectorial reaction coordinate.

Remark 6 (Metadynamics) The adaptive biasing force technique can also be used in the context of metadynamics [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF][START_REF] Bussi | Equilibrium free energies from nonequilibrium metadynamics[END_REF][START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF]. The principle of metadynamics is to introduce an additional variable z with dimension the dimension of ξ (say z ∈ R m , with 1 ≤ m < n), and an extended potential V ζ (q, z) = V (q) + ζ 2 |zξ(q)| 2 . The reaction coordinate is then chosen to be ξ meta (q, z) = z so that the associated free energy is

A ζ (z) = -β -1 ln D exp(-βV ζ (q, z)) dq,
which converges to A(z) when ζ goes to infinity. In our framework, the ABF method applied to this extended system writes:

dX t = (-∇V (X t ) + ζ(Z t -ξ(X t ))∇ξ(X t )) dt + 2β -1 dB t , dZ t = ζ ξ(X t ) -E(ξ(X t )|Z t ) dt + 2β -1 dB t ,
where B t is a m-dimensional Brownian motion, independent of B t . Notice that by construction, the orthogonality condition (25) is satisfied by ξ meta , so that the convergence results of this section apply to these kinds of models.

Remark 7 (On the initial condition) If ψ ξ (0, •) is zero at some points or is not sufficiently smooth, then A ′ 0 may be not well defined or I(ψ ξ (0, •)|ψ ξ ∞ ) may be infinite (which is in contradiction with [H4]). But since we show that ψ ξ satisfies a simple diffusion equation (see Proposition 1), these difficulties disappear as soon as t > 0. Therefore, up to considering the problem for t ≥ t * > 0, we can suppose that ψ ξ (0, •) > 0.

Remark 8 (On the choice of the entropy) In the case of linear Fokker Planck equations, it is well known that one can obtain exponential convergence to equilibrium by considering various entropies of the form h dµ dν dµ, where h is typically a strictly convex function such that h(1) = 0 (see [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] for more assumptions required on h). For example, the classical choice h(x) = 1 2 (x -1) 2 is linked to Poincaré type inequalities and leads to L 2 -convergence, while the function h(x) = x ln xx + 1 we have used here to build the entropy is linked to logarithmic Sobolev inequalities and leads to L 1 ln L 1convergence. However, for the study of the non-linear Fokker Planck equation [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF], it seems that the choice h(x) = x ln xx + 1 is necessary to derive the estimates, for example to have the extensivity property of Lemma 1.

Remark 9 (Smoother evolution in time of

A ′ t )
In practice, it may be useful to update the adaptive potential A ′ t in a smoother way in time, for example by replacing [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] by

dA ′ t (z) = 1 τ E F (X t ) ξ(X t ) = z -A ′ t (z)
dt, where F is defined by ( 6) and τ > 0 denotes a characteristic time (possibly depending on (t, z)), to be fixed. This amounts to replace A ′ t by κ τ * A ′ t in [START_REF] Iannuzzi | Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics[END_REF], where κ τ is an exponential convolution kernel. Formally, we here consider the limit case τ = 0. To prove the convergence of A ′ t towards A ′ for τ = 0 is an open problem. Remark 10 (Enhancing the macroscopic rate of convergence) Let us consider the case M = R. For an α-convex potential W , Corollary 1 states that A ′ t converges towards A ′ exponentially fast, with a rate λ = β -1 min(ρm -2 , α). This may seem surprising since for large enough α, the rate of convergence is no more limited by α. However, it is typically expected that the constant I 0 in assumption [H4] increases with growing α, which means that the constant C increases in the convergence estimate (21). Moreover, in practice, if α is very large, ψ ξ ∞ is very peaked and some parts of M are poorly sampled, so that the variance of the result is large in these areas (which can not be seen in our convergence result). Actually, a good method to enhance the rate of convergence at the macroscopic level while keeping a good sampling and thus low variance, is to use a particle systems with many replicas and a selection mechanism. We refer to [START_REF] Lelièvre | Computation of free energy profiles with adaptive parallel dynamics[END_REF] for more details.

A convergence result for the adaptive dynamics (12)-(11)

In this section, we present a weaker convergence result for another adaptive overdamped Langevin dynamics, namely ( 12)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF]. For simplicity, we only consider the case M = T and W = 0, but the results can be extended to the case M = R with a suitable W = 0, as in Section 2.2 (see [H4] and [H4']). One interest of this dynamics and this result of convergence is that they can be straightforwardly extended to the case of a multidimensional reaction coordinate (see Remark 5 above). For the sake of conciseness, we do not provide the details of the result in this case which follows exactly the same lines (see [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF] and Appendix A for formulas in the case of a multi-dimensional reaction coordinate). Let us recall the dynamics ( 12)-( 11) we consider here:

dX t = -∇ V -A t • ξ (X t ) dt + 2β -1 dB t , (26)
with the same definition as before for A t : ∀z ∈ T,

A ′ t (z) = E F (X t ) ξ(X t ) = z , ( 27
)
where F is defined by [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]. The associated non-linear Fokker Planck equation is now:

           ∂ t ψ = div ∇(V -A t • ξ)ψ + β -1 ∇ψ , A ′ t (z) = Σz F |∇ξ| -1 ψ(t, •)dσ Σz Σz |∇ξ| -1 ψ(t, •)dσ Σz . ( 28
)
The main difference with the dynamics ( 10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] considered in Theorem 1 is that the marginal distribution ψ ξ does not satisfy a closed partial differential equation. Therefore, we do not know a priori that the Fisher information I(ψ ξ |ψ ξ ∞ ) converges to 0. The strategy here is to directly estimate the derivative of the total entropy E.

We obtain a convergence result under two additional assumptions (see [H5]-[H6]).

Theorem 2 (Longtime convergence for the dynamics ( 12)-( 11)) Let (ψ, A ′ t ) be a smooth solution to (28) and let us assume [H1], [H2], [H3]. Moreover, we suppose

[H5] V and ξ are such that ∃R > 0, ψ ∞ satisfies LSI(R), (29)
and

[H6] mM β 2 √ ρ < 1.
Then the total entropy E satisfies:

E(t) ≤ E(0) exp(-λt) where λ = β -1 -1 + mMβ 2 √ ρ R is positive using [H6].
In particular, as in Theorem 1, the biasing force A ′ t converges exponentially fast to the mean force A ′ .

The proof of this result is given in Section 3.4 below.

Remark 11 (On assumption [H5])

In [START_REF] Otto | A new criterion for the logarithmic Sobolev inequality and two applications[END_REF]Theorem 2], it is shown that if µ = exp(-H(x 1 , x 2 ))dx 1 dx 2 is a probability measure on a product space X = X 1 × X 2 (where X i are Euclidean spaces), if the conditional probabilities µ(dx 2 |x 1 ) satisfy LSI(ρ 2 ) (with ρ 2 independent of x 1 ) and the marginal µ(dx 1 ) satisfies LSI(ρ 1 ), then µ satisfies LSI(ρ) provided the coupling between the two directions is bounded:

∃κ 1,2 > 0, ∀(x 1 , x 2 ) ∈ X 1 × X 2 , ∂ 2 x1,x2 H(x 1 , x 2 ) ≤ κ 1,2 .
Thus, in the simple framework of Section 3.1 for example, where the configuration space is T×R and the reaction coordinate is ξ(x, y) = x, the fact that ψ ∞ satisfies a LSI (assumption [H5]) can be deduced from the fact that the conditioned distributions µ ∞,z satisfy a LSI (which is [H3]), the marginal ψ ξ ∞ satisfy a LSI (which is related to [H4]) and the coupling is bounded (which is [H2]). Thus [H5] is not needed as an additional assumption compared to the framework of Theorem 1. The generalization of this result to the case when X is not a product does not seem to be straightforward.

Proofs

One remark to simplify the presentation of the proofs is that we can suppose β = 1 up to the following change of variable: t = β -1 t, ψ( t, x) = ψ(t, x), Ṽ (x) = βV (x) and W (x) = βW (x). Therefore, we suppose in the following that β = 1.

(30)

Proof of Proposition 1 and Theorem 1 in a simple case

In this section, we propose to prove Proposition 1 and Theorem 1 in the simple case n = 2, ξ(x, y) = x (so that we use in this section the notation x instead of z for the reaction coordinate variable) and the configuration space is D = T × R (which means that all the data are periodic with respect to the first coordinate x). In this case, we thus have ξ ∈ T (M = T) so that we choose W = 0 (see [H4']). Notice also that the local mean force F is simply given by F = ∂ x V (see ( 6)). Our aim is to introduce the main arguments in this simple case before presenting the general proof in Section 3.2.

In this simple setting, the system ( 14) writes (recall β = 1):

       ∂ t ψ = div (∇V ψ + ∇ψ) -∂ x (A ′ t ψ), A ′ t (x) = R ∂ x V (x, y)ψ(t, x, y)dy ψ ξ (t, x) , (31)
where ψ ξ (t, x) = R ψ(t, x, y)dy. Notice that in this case ψ ξ ∞ ≡ 1. It can be checked that the assumptions [H2] and [H3] are satisfied in this context for a potential V of the following form:

V (x, y) = V 0 (x, y) + V 1 (x, y) where inf T×R ∂ y,y V 0 > 0, V 1 L ∞ < ∞, ∂ x,y (V 0 + V 1 ) L ∞ < ∞. The potential V is thus a bounded perturbation of an α-convex potential, with a bounded mixed deriva- tive ∂ x,y V . Then, assumptions [H2]-[H3] are satisfied with m = 1, M = ∂ x,y V L ∞ and ρ = (inf T×R ∂ y,y V 0 ) exp(-osc V 1 ), where osc V 1 = sup T×R V 1 -inf T×R V 1 (see [2]).
Proposition 1 is simply obtained by integration of (31) with respect to y ∈ R:

Lemma 3 The density ψ ξ satisfies the following equation on T:

∂ t ψ ξ = ∂ x,x ψ ξ . ( 32
)
As stated in Corollary 1, this result already yields the exponential convergence to zero of the macroscopic Fisher information I(ψ ξ |ψ ξ ∞ ) (this is the matter of Lemma 12 below), and thus [H4] is indeed satisfied with I 0 = I(ψ ξ (0, •)|ψ ξ ∞ ) and r = 4π 2 . A fundamental lemma needed in the sequel is Lemma 4 The difference between the biasing force A ′ t and the mean force A ′ can be expressed in term of the densities as

A ′ t -A ′ = R ∂ x ln ψ ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ ψ ξ ∞ .
Proof : This is a simple computation (using the fact that ψ ξ ∞ ≡ 1):

R ∂ x ln ψ ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ ψ ξ ∞ = R ∂ x ln ψ ψ ψ ξ dy - R ∂ x ln ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ , = R ∂ x ψ ψ ξ dy + R ∂ x (V -A) ψ ψ ξ dy -∂ x ln ψ ξ , = A ′ t -A ′ .

♦

We will also use the following two estimates:

Lemma 5 Let us assume [H2]-[H3].
Then, for all t ≥ 0, for all x ∈ T,

|A ′ t (x) -A ′ (x)| ≤ ∂ x,y V L ∞ 2 ρ e m (t, x).
Proof : For any coupling measure π ∈ Π(µ t,x , µ ∞,x ), it holds:

|A ′ t (x) -A ′ (x)| = R×R (∂ x V (x, y) -∂ x V (x, y ′ )) π(dy, dy ′ ) , ≤ ∂ x,y V L ∞ R×R |y -y ′ |π(dy, dy ′ ), ≤ ∂ x,y V L ∞ R×R |y -y ′ | 2 π(dy, dy ′ ).
Taking now the infimum over all π ∈ Π(µ t,x , µ ∞,x ) and using [H3] together with Lemma 2, we obtain

|A ′ t (x) -A ′ (x)| ≤ ∂ x,y V L ∞ W (µ t,x , µ ∞,x ) ≤ ∂ x,y V L ∞ 2 ρ H(µ t,x |µ ∞,x ),
which concludes the proof. ♦ Lemma 6 Let us assume [H3]. Then for all t ≥ 0,

E m (t) ≤ 1 2ρ T×R ∂ y ln ψ ψ ∞ 2 ψ.
Proof : Using [H3], it holds:

E m = T e m ψ ξ dx, ≤ T 1 2ρ R ∂ y ln ψ ψ ξ ψ ∞ ψ ξ ∞ 2 ψ ψ ξ dy ψ ξ dx,
which yields the result since ψ ξ /ψ ξ ∞ does not depend on y. ♦

We are now in position to prove the exponential convergence of E m (t) to zero stated in Theorem 1 (see Equation ( 21)).

Equation (31) on ψ can be rewritten as:

∂ t ψ = div (ψ ∞ ∇(ψ/ψ ∞ )) + ∂ x ((A ′ -A ′ t )ψ).
Notice that the derivative dE dt can be obtained by multiplying this equation by ln ψ ψ∞ and integrating over T × R. Thus, one obtains after some integrations by parts, using a Cauchy-Schwarz inequality (to prove that (33) is non positive) and Lemma 4 (used twice):

dE m dt = dE dt - dE M dt , = - T R ∇ ln ψ ψ ∞ 2 ψ + T R (A ′ t -A ′ )∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ , = - T R ∂ y ln ψ ψ ∞ 2 ψ - T R ∂ x ln ψ ψ ∞ 2 ψ + T R ∂ x ln ψ ψ ∞ ψ dy 2 1 ψ ξ dx (33) - T R ∂ x ln ψ ξ ψ ξ ∞ ∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ , ≤ - T R ∂ y ln ψ ψ ∞ 2 ψ - T ∂ x ln ψ ξ ψ ξ ∞ ψ ξ (A ′ t -A ′ ).
We now use Lemmas 5 and 6:

dE m dt ≤ -2ρE m + T |A ′ t -A ′ | 2 ψ ξ T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ , ≤ -2ρE m + ∂ x,y V L ∞ 2 ρ E m I(ψ ξ |ψ ξ ∞ ).
Using [H4], we thus have:

d √ E m dt ≤ -ρ E m + ∂ x,y V L ∞ I 0 2ρ exp(-rt),
from which we deduce (21). Equation ( 23) is then easily obtained using Lemma 5.

Proof of Proposition 1 and Theorem 1 in the general case

We now present the proof of Proposition 1 and Theorem 1 in the more general setting of Section 2.2. The proof follows the same lines as in the simple case presented in Section 3.1, but with additional difficulties related to the geometry of the submanifolds Σ z . We need the following result

Lemma 7

The derivative of ψ ξ with respect to the reaction coordinate value reads:

∂ z ψ ξ (t, z) = Σz ∇ξ • ∇ψ(t, •) |∇ξ| 2 + div ∇ξ |∇ξ| 2 ψ(t, •) |∇ξ| -1 dσ Σz .
Proof : For any smooth test function g : M → R, we obtain (using the co-area formula (39) and an integration by parts):

M ψ ξ (t, z)g ′ (z) dz = D ψ(t, x)g ′ • ξ(x) dx, = D ψ(t, x)∇(g • ξ) • ∇ξ|∇ξ| -2 (x) dx, = - D div ψ(t, •)∇ξ |∇ξ| 2 g • ξ dx, = - M g(z) Σz ∇ξ • ∇ψ(t, •) |∇ξ| 2 + div ∇ξ |∇ξ| 2 ψ(t, •) |∇ξ| -1 dσ Σz dz,
which yields the result.

♦

Using this lemma, it can be shown that ψ ξ satisfies a simple diffusion equation, which is Proposition 1.

Lemma 8

The density ψ ξ satisfies the following diffusion equation on M:

∂ t ψ ξ = ∂ z W ′ ψ ξ + ∂ z ψ ξ . (34)
Proof : For any smooth test function g : M → R, we have (using the co-area formula (39), ( 14), an integration by parts and finally Lemma 7):

d dt M ψ ξ (t, •)g dz = d dt D ψ(t, •)g • ξ dx, = D div |∇ξ| -2 (∇(V -A t • ξ + W • ξ)ψ + ∇ψ) g • ξ dx, = - D |∇ξ| -2 (∇(V -A t • ξ + W • ξ)ψ + ∇ψ) • ∇ξ g ′ • ξ dx, = - D |∇ξ| -2 (∇V • ∇ξψ + ∇ψ • ∇ξ) g ′ • ξ dx + D A ′ t • ξg ′ • ξψ dx - D W ′ • ξg ′ • ξψ dx, = - M Σz |∇ξ| -3 (∇V • ∇ξψ + ∇ψ • ∇ξ) dσ Σz g ′ (z) dz + M A ′ t (z)g ′ (z)ψ ξ (z) dz - M W ′ (z)g ′ (z)ψ ξ (z) dz, = - M Σz |∇ξ| -3 ∇ψ • ∇ξ + div (∇ξ|∇ξ| -2 )|∇ξ| -1 ψ dσ Σz g ′ (z) dz - M W ′ (z)ψ ξ (z)g ′ (z) dz, = - M ∂ z ψ ξ (t, z) + W ′ (z)ψ ξ (z) g ′ (z) dz,
which is a weak formulation of (34). ♦

As stated in Corollary 1, this result already yields the exponential convergence to zero of the macroscopic Fisher information I(ψ ξ |ψ ξ ∞ ) under adequate assumption on W (this is the matter of [H4'] and Lemma 13 below). We suppose in the following that [H4] is indeed satisfied.

The equivalent of Lemma 4 writes

Lemma 9

The difference between the biasing force A ′ t and the mean force A ′ can be expressed in term of the densities as

A ′ t (z) -A ′ (z) = Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ ψ ψ ξ |∇ξ| -2 dσ Σz -∂ z ln ψ ξ ψ ξ ∞ .
Proof : Using Lemma 7 and the definition of A ′ t , it holds:

Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ ψ ψ ξ |∇ξ| -2 dσ Σz -∂ z ln ψ ξ ψ ξ ∞ = Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ψ ξ |∇ξ| -2 dσ Σz - Σz ∇ξ |∇ξ| • ∇ ln ψ ∞ ψ ψ ξ |∇ξ| -2 dσ Σz -∂ z ln ψ ξ + ∂ z ln ψ ξ ∞ , = 1 ψ ξ Σz ∇ξ • ∇ψ |∇ξ| |∇ξ| -2 dσ Σz + Σz ∇ξ |∇ξ| • ∇ (V -A • ξ + W • ξ) ψ ψ ξ |∇ξ| -2 dσ Σz -∂ z ln ψ ξ -W ′ (z), = ∂ z ψ ξ ψ ξ - 1 ψ ξ Σz div ∇ξ |∇ξ| 2 |∇ξ| -1 ψ dσ Σz + Σz ∇ξ • ∇V |∇ξ| 3 ψ ψ ξ dσ Σz -A ′ (z) -∂ z ln ψ ξ , = A ′ t (z) -A ′ (z).

♦

The equivalent of Lemmas 5 and 6 write:

Lemma 10 Let us assume [H2]-[H3]
. Then for all t ≥ 0, for all z ∈ M,

|A ′ t (z) -A ′ (z)| ≤ M 2 ρ e m (t, z).
Proof : For any coupling measure π ∈ Π(µ t,z , µ ∞,z ) defined on Σ z × Σ z , it holds:

|A ′ t (z) -A ′ (z)| = Σz×Σz (F (x) -F (x ′ ))π(dx, dx ′ ) , ≤ ∇ Σz F L ∞ Σz ×Σz d Σz (x, x ′ ) 2 π(dx, dx ′ ).
Taking now the infimum over all π ∈ Π(µ t,z , µ ∞,z ) and using [H2]-[H3] together with Lemma 2, we thus obtain

|A ′ t (z) -A ′ (z)| ≤ M W (µ t,z , µ ∞,z ) ≤ M 2 ρ H(µ t,z |µ ∞,z ), (35)
which concludes the proof. ♦ Lemma 11 Let us assume [H3]. Then for all t ≥ 0,

E m (t) ≤ 1 2ρ D ∇ Σz ln ψ(t, •) ψ ∞ 2 ψ.
Proof : Using [H3], it follows:

E m = M e m ψ ξ dz, ≤ M 1 2ρ Σz ∇ Σz ln ψ(t, •) ψ ∞ 2 ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) ψ ξ dz,
which yields the result, using the co-area formula (39). ♦

We are now in position to prove the exponential convergence of E m (t) to zero stated in Theorem 1 (see Equation ( 21)). Equation ( 14) on ψ can be rewritten as:

∂ t ψ = div (|∇ξ| -2 ψ ∞ ∇(ψ/ψ ∞ )) + div (|∇ξ| -2 ∇ ((A -A t ) • ξ) ψ).
Notice that the derivative dE dt can be obtained by multiplying this equation by ln ψ ψ∞ and integrating over D. Thus, one obtains after some integrations by parts, using the co-area formula (39) and Lemma 9:

dE m dt = dE dt - dE M dt , = - D ∇ ln ψ ψ ∞ 2 |∇ξ| -2 ψ + D (A ′ t -A ′ ) • ξ ∇ξ • ∇ ln ψ ψ ∞ |∇ξ| -2 ψ + M ∂ z ln ψ ξ ψ ξ ∞ 2 ψ ξ , = - D ∇ Σz ln ψ ψ ∞ 2 |∇ξ| -2 ψ - D ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ 2 |∇ξ| -2 ψ + M (A ′ t -A ′ )(z) Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ |∇ξ| -2 ψdσ Σz dz + M ∂ z ln ψ ξ ψ ξ ∞ 2 ψ ξ , = - D ∇ Σz ln ψ ψ ∞ 2 |∇ξ| -2 ψ - D ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ 2 |∇ξ| -2 ψ + M Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ |∇ξ| -2 ψdσ Σz 2 (ψ ξ ) -1 dz - M Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ |∇ξ| -2 ψdσ Σz ∂ z ln ψ ξ ψ ξ ∞ dz + M ∂ z ln ψ ξ ψ ξ ∞ 2 ψ ξ .
Using the Cauchy-Schwarz inequality:

Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ |∇ξ| -1 |∇ξ| -1 ψdσ Σz ψ ξ (z) 2 ≤ Σz ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ |∇ξ| -1 2 |∇ξ| -1 ψdσ Σz ψ ξ (z)
and Lemma 9 again, we thus obtain

dE m dt ≤ - D ∇ Σz ln ψ ψ ∞ 2 |∇ξ| -2 ψ - M ∂ z ln ψ ξ ψ ξ ∞ ψ ξ (A ′ t -A ′ ).
We now use [H2], Lemmas 10 and 11:

dE m dt ≤ -2ρm -2 E m + M |A ′ t -A ′ | 2 ψ ξ M ∂ z ln ψ ξ ψ ξ ∞ 2 ψ ξ , ≤ -2ρm -2 E m + M 2 ρ E m I(ψ ξ |ψ ξ ∞ ).
Using [H4], we thus have:

d √ E m dt ≤ -ρm -2 E m + M I 0 2ρ exp(-rt),
from which we deduce (21). Equation ( 23) is then easily obtained using Lemma 10.

Proof of Corollary 1

Convergence of the macroscopic Fisher information

Let us first show that in both cases considered in [H4'], the exponential convergence [H4] of the macroscopic Fisher information indeed holds. Let us first consider the case M = T and W = 0. We know from (20) that ψ ξ satisfies ∂ t ψ ξ = ∂ z,z ψ ξ on T, and we would like to show exponential convergence of the Fisher information I(ψ ξ (t, •)|ψ ξ ∞ ).

Lemma 12 (Convergence of the Fisher information when M = T and W = 0) Let φ be a function defined for t ≥ 0 and x ∈ T which satisfies

∂ t φ = ∂ x,x φ on T
and such that T φ(0, •) = 1, φ(0, •) is non negative, and I(φ(0, •)|φ ∞ ) < ∞, where φ ∞ ≡ 1 is the longtime limit of φ. Then, ∀t ≥ 0,

I(φ(t, •)|φ ∞ ) ≤ I(φ(0, •)|φ ∞ ) exp(-8π 2 t).
Proof : Let us denote u = √ φ. We notice that I(φ|φ ∞ ) = T |∂ x ln φ| 2 φ = 4 T |∂ x u| 2 . Moreover, we have from (32)

∂ t u = ∂ x,x u + (∂ x u) 2 u .
Therefore,

d dt T (∂ x u) 2 = 2 T ∂ x,x,x u ∂ x u + 2 T ∂ x (∂ x u) 2 u ∂ x u, = -2 T (∂ x,x u) 2 -2 T (∂ x u) 2 u ∂ x,x u, = -2 T (∂ x,x u) 2 -2 T ∂ x ((∂ x u) 3 ) 3u , = -2 T (∂ x,x u) 2 - 2 3 T (∂ x u) 4 u 2 , ≤ -8π 2 T (∂ x u) 2 ,
where we have used the Poincaré-Wirtinger inequality on T, applied to ∂ x u: For any

function f ∈ H 1 (T), T f - T f 2 ≤ 1 4π 2 T (∂ x f ) 2 .

♦

Let us now consider the case M = R and W = 0 which is such that W ′′ is bounded from below and exp (-βW ) M exp(-βW ) satisfies a logarithmic Sobolev inequality (as stated in [H4']). We know from (20) that ψ ξ satisfies ∂ t ψ ξ = ∂ z W ′ ψ ξ + ∂ z ψ ξ on R, and we would like to show exponential convergence of the Fisher information I(ψ ξ (t, •)|ψ ξ ∞ ).

Lemma 13 (Convergence of the Fisher information when M = R and W = 0) Let φ be a function defined for t ≥ 0 and x ∈ R which satisfies

∂ t φ = ∂ x (W ′ φ + ∂ x φ) on R,
and such that R φ(0, •) = 1, φ(0, •) is non negative, and I(φ(0, •)|φ ∞ ) < ∞, where φ ∞ ≡ exp(-W ) R exp(-W ) is the longtime limit of φ. Let us assume that W ′′ is bounded from below by a constant α and φ ∞ satisfies LSI(r), with r > 0. We can suppose without loss of generality that r ≥ α.

Then there exists I 0 > 0 and r > 0 such that ∀t ≥ 0,

I(φ(t, •)|φ ∞ ) ≤ I 0 exp(-2rt).
More precisely, when α = r > 0, it is possible to take I 0 = I(φ(0, •)|φ ∞ ) and r = α. When α < r, for any ε ∈ (0, r), it is possible to choose r = rε for a well-chosen constant I 0 > 0.

Proof : The fact that r ≥ α is clear since either α ≤ 0, or α > 0 in which case it is well-known that φ ∞ satisfies LSI(α) (see for example [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]), so that one can choose at least r = α.

Let us recall the expression for the entropy

H(φ(t, •)|φ ∞ ) = R ln(φ/φ ∞ )φ and the Fisher information I(φ(t, •)|φ ∞ ) = R |∂ x ln(φ/φ ∞ )| 2 φ. Since φ ∞ satisfies LSI(r), we have H(φ(t, •)|φ ∞ ) ≤ 1 2r I(φ(t, •)|φ ∞ ).
Moreover, by standard computations (see for example [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]), we have

d dt H(φ(t, •)|φ ∞ ) = -I(φ(t, •)|φ ∞ )
and

d dt I(φ(t, •)|φ ∞ ) = -2 R φ φ ∞ ∂ x,x ln φ φ ∞ 2 φ ∞ -2 R φ φ ∞ ∂ x ln φ φ ∞ 2 W ′′ φ ∞ . (36) If α = r, we thus obtain from (36) that d dt I(φ(t, •)|φ ∞ ) ≤ -2αI(φ(t, •)|φ ∞ ) which concludes the proof in this case.
Let us now suppose that α < r. The technique of proof we propose is taken from [START_REF] Villani | Hypocoercivity[END_REF]. For any λ > 0, we have

d dt (H(φ(t, •)|φ ∞ ) + λI(φ(t, •)|φ ∞ )) = - R φ φ ∞ ∂ x ln φ φ ∞ 2 φ ∞ -2λ R φ φ ∞ ∂ x,x ln φ φ ∞ 2 φ ∞ -2λ R φ φ ∞ ∂ x ln φ φ ∞ 2 W ′′ φ ∞ , ≤ - R (1 + 2λW ′′ ) φ φ ∞ ∂ x ln φ φ ∞ 2 φ ∞ , ≤ -(1 + 2λ inf W ′′ )I(φ(t, •)|φ ∞ ), ≤ - 1 + 2αλ λ + 1/(2r) (H(φ(t, •)|φ ∞ ) + λI(φ(t, •)|φ ∞ )) .
We thus obtain that, for any λ > 0,

H(φ(t, •)|φ ∞ )+λI(φ(t, •)|φ ∞ ) ≤ H(φ(0, •)|φ ∞ )+λI(φ(0, •)|φ ∞ ) exp - 1 + 2αλ λ + 1/(2r) t ,
and therefore

I(φ(t, •)|φ ∞ ) ≤ 1 λ H(φ(0, •)|φ ∞ ) + I(φ(0, •)|φ ∞ ) exp - 1 + 2αλ λ + 1/(2r) t .
Since 1+2αλ λ+1/(2r) goes to 2r when λ goes to 0, for any ε ∈ (0, r), one can find a λ > 0 such that 1+2αλ λ+1/(2r) = 2(rε), which concludes the proof. ♦

Convergence of the biasing force

Let us now prove the convergence result (24) for the biasing force.

In the case M = T (and thus W = 0), we can prove the convergence of A ′ t -A ′ L 2 (T) to zero in the following sense (which implies (24), using (21)): for any ε ∈ (0, 1), ∀t ≥ t ε ,

A ′ t -A ′ 2 L 2 (T) ≤ 2 1 -ε M 2 ρ E m (t), (37)
where t ε = min 0, (4π2 ) -1 ln ε -1

T (∂ z ψ ξ (0, •)) 2
. This is obtained using the fact that T (∂ x ψ ξ (t, •)) 2 ≤ T (∂ x ψ ξ (0, •)) 2 exp(-8π 2 t) (the proof of this estimate is similar to the one of Lemma 12) and the fact that for any function f ∈ H 1 (T),

f - T f 2 L ∞ ≤ T (∂ x f ) 2 , applied to f = ψ ξ . Thus we have ψ ξ -1 2 L ∞ ≤ T (∂ x ψ ξ (0, •)) 2 exp(-8π 2
t) which implies that for t ≥ t ε , ψ ξ (t, •) ≥ 1ε which yields (37) from (23).

Let us now prove (24) in the case M = R, under assumption [H4'] on W . Let us introduce a compact K ⊂ M. Since L ∞ (K) ⊂ H 1 (K) (with continuous injection), there exists c > 0 such that

ψ ξ ψ ξ ∞ -1 L ∞ (K) ≤ c ψ ξ ψ ξ ∞ -1 L 2 (K) + ∂ z ψ ξ ψ ξ ∞ -1 L 2 (K) , ≤ c inf K ψ ξ ∞   R ψ ξ ψ ξ ∞ -1 2 ψ ξ ∞ + R ∂ z ψ ξ ψ ξ ∞ -1 2 ψ ξ ∞   .
Thus, for any ε ∈ (0, r), there exists C > 0 such that

ψ ξ ψ ξ ∞ -1 L ∞ (K) ≤ C exp(-rt),
with r = rε. This inequality is obtained from the fact that since ψ ξ ∞ satisfies LSI(r), then ψ ξ ∞ also satisfies a Poincaré inequality with the same constant r (see for example [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]), and a proof similar to that of Lemma 13 for the convergence of the Fisher information

R ∂ z ψ ξ ψ ξ ∞ -1 2 ψ ξ ∞ associated with the Poincaré inequality. Now, we write K |A ′ t -A ′ |ψ ξ ∞ = K |A ′ t -A ′ |ψ ξ - K |A ′ t -A ′ | ψ ξ ψ ξ ∞ -1 ψ ξ ∞ , ≤ R |A ′ t -A ′ | 2 ψ ξ + C exp(-rt) K |A ′ t -A ′ |ψ ξ ∞ .
Thus, for t sufficiently large, K |A ′ t -A ′ |ψ ξ ∞ is bounded from above by some constant times R |A ′ t -A ′ | 2 ψ ξ , which yields (24) (using ( 23) and ( 21)).

Proof of Theorem 2

Let us now prove Theorem 2. We still assume, up to a change of variable, that β = 1.

We have:

dE dt = - D ∇ ln ψ ψ ∞ 2 ψ + D (A ′ t -A ′ ) • ξ∇ξ • ∇ ln ψ ψ ∞ ψ, ≤ - D ∇ ln ψ ψ ∞ 2 ψ + M |A ′ t -A ′ | 2 ψ ξ D ∇ξ • ∇ ln ψ ψ ∞
Since, by Lemmas 10 and 11,

M |A ′ t -A ′ | 2 ψ ξ ≤ M 2 ρ D ∇ Σz ln ψ ψ ∞ 2 ψ,
we thus obtain

dE dt ≤ - D ∇ ln ψ ψ ∞ 2 ψ + M m √ ρ D ∇ Σz ln ψ ψ ∞ 2 ψ D ∇ξ |∇ξ| • ∇ ln ψ ψ ∞ 2 ψ ≤ -1 + M m 2 √ ρ D ∇ ln ψ ψ ∞ 2 ψ,
where we have used the fact that, for any function f :

D → R, |∇f | 2 = |∇ Σz f | 2 + ∇ξ |∇ξ| • ∇f 2
. The logarithmic Sobolev inequality with respect to ψ ∞ (see [H5]) concludes the proof.

A The co-area formula

The aim of this section is to state the co-area formula for a function ξ : D → R p , (where 1 ≤ p < n) such that rank(∇ξ) = p. Classical proofs for the co-area formula can be found in the books [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF][START_REF] Evans | Measure theory and fine properties of functions[END_REF]. These proofs are however quite involved since they assume only Lipschitz-regularity for ξ. The proof is simpler in the case of a smooth ξ: it can be done by an adequate parameterization and a simple change of variables.

Lemma 14 (co-area formula) For any smooth function φ : R n → R,

R n φ(x) det G(x)dx = R p Σz φ dσ Σz dz, (38)
where G is a p × p matrix with G i,j = ∇ξ i • ∇ξ j . In the case p = 1, Equation (38) reads: The measure (det G) -1/2 dσ Σz is sometimes denoted by δ ξ(x)-z in the literature.

B Another possible set of assumptions for the convergence of the adaptive dynamics ( 10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] It is also possible to state a result similar to Theorem 1 for the dynamics ( 10)- [START_REF] Jarzynski | Equilibrium free energy differences from nonequilibrium measurements: A master equation approach[END_REF] under slightly different assumptions than [H2] and [H3] by introducing another Riemannian structure on Σ z (see [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF]) than that induced by the scalar product of the ambient space D. Let us introduce the following scalar product: ∀x ∈ Σ z , ∀u, v ∈ T x Σ z ,

u, v Σz = u • v|∇ξ| 2 (x), (40)
where • denotes as before the scalar product of the ambient space D, and the associated norm: ∀x ∈ Σ z , ∀u ∈ T x Σ z ,

|u| 2 Σz = u, u Σz = |u| 2 |∇ξ| 2 (x).
Accordingly, the definition of the surface gradient is modified as follows 2 (compare with (19)): For f : D → R,

∇ Σz f = |∇ξ| -2 P ∇f. (41)
In particular, we have

|∇ Σz f | Σz = |∇ξ| -1 |P ∇f |.
In this case, the Fisher information between the conditioned measures µ t,z and µ ∞,z is (see [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF]): 2 With a slight abuse of notation, we still use the same notation ∇Σ z to denote the surface gradient, or I(µt,z|µ∞,z) to denote the Fisher information, or dΣ z to denote the geodesic distance, or ρ to denote the microscopic rate of convergence, while these are not the same as in the rest of the paper, since the Riemannian structure has been changed.

I(µ t,z |µ ∞,z ) = Σz ∇ Σz ln ψ(t, •) ψ ∞ 2
where F is defined by [START_REF] Ciccotti | Sampling Boltzmann-Gibbs distributions restricted on a manifold with diffusions: Application to free energy calculations[END_REF]. Notice that

|∇ξ| -1 |P ∇F | = |∇ Σz F | Σz .
Thus, assumption [H2] is modified as:

[H2']
V and ξ are sufficiently differentiable functions such that

|∇ Σz F | Σz L ∞ ≤ M < ∞.
The rest of the proof remains the same, and exponential convergence is thus obtained, assumptions [H2] and [H3] being respectively replaced by [H2'] and [H3']. With this set of assumptions, the rate of convergence is λ = β -1 min(ρ, r).

Theorem 1 (

 1 Exponential convergence of the entropy to zero) Let us assume [H1], [H2], [H3] and [H4]. Then the microscopic entropy E m satisfies:

Corollary 1 (

 1 Convergence of the biasing force) If [H4'] is satisfied and ψ ξ satisfies (20) then [H4] holds.More precisely, if M = T and W = 0, then [H4] is satisfied with I 0 = I(ψ ξ (0, •)|ψ ξ ∞ ) and r = 4π 2 . If M = R, W ′′ is bounded from below and exp(-βW )

Remark 12

 12 This formula shows that if the random variable X has law ψ(x) dx in R n , then ξ(X) has law Σz ψ (det G) -1/2 dσ Σz dz, and the law of X conditioned to a fixed value z of ξ(X) isdµ z = ψ (det G) -1/2 dσ Σz Σz ψ (det G) -1/2 dσ Σz .Indeed, for any bounded functions f and g,E(f (ξ(X))g(X)) = R n f (ξ(x))g(x)ψ(x) dx, = R p Σz f • ξ g ψ (det G) -1/2 dσ Σz dz, = R p f (z) Σz g ψ (det G) -1/2 dσ Σz Σz ψ (det G) -1/2 dσ Σz Σzψ (det G) -1/2 dσ Σz dz.

2 |∇ξ| - 2 V 2 |∇ξ| - 2 2

 22222 Σz ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) , ψ(t, •)|∇ξ| -1 dσ Σz ψ ξ (t, z) ,and the assumption [H3] is stated in terms of this new Fisher information: and ξ are such that ∃ρ > 0, for all z ∈ M, the conditional measure µ ∞,z satisfies LSI(ρ), Σ z being endowed with the Riemannian structure (40).Using this Fisher information, Lemma 11 writes:E m (t) ≤ 1 2ρ D P ∇ ln ψ(t, •) ψ ∞ ψ.The definition for the Wasserstein distance is now stated using the geodesic distance d Σz : ∀x, y ∈ Σ z ,d Σz (x, y) Σz dt w ∈ C 1 ([0, 1], Σ z ), w(0) = x, w(1)= y estimate of Lemma 10 is changed to:|A ′ t (z) -A ′ (z)| = Σz ×Σz (F (x) -F (x ′ ))π(dx, dx ′ ) , ≤ |∇ξ| -1 |P ∇F | L ∞ Σz ×Σz d Σz (x, x ′ ) 2 π(dx, dx ′ ),

  [H2]-[H3] by introducing another Riemannian structure on the submanifolds Σ z . This is made precise in Appendix B (see assumptions [H2']-[H3']).Remark 4 (Other possible assumptions on W ) From Lemma 12 and 13 below (used to prove Corollary 1), it will become clear that [H4] is actually satisfied with W = 0 as soon as M is a bounded domain. If M is an unbounded domain, then a potential W with properties such as those stated in [H4'] is needed. We discuss in this remark other properties on W to satisfy [H4] than those proposed in [H4'], in the case

Such methods can also be applied for other dynamics, like Langevin dynamics. We only consider Brownian dynamics in this paper.

ψ.

Acknowledgements : This work is supported by the ANR INGEMOL of the French Ministry of Research. TL would like to thank Ch. Chipot who initiated this work by a question about the ABF method. Part of this work was completed during a summer school of the GDR CHANT. We would like to thank F. Castella for the organization of this school. We would like to thank C. Villani for pointing out [17] to prove Lemma 13.


Carine Jauberthie

Nathalie Verdière
email: nathalie.verdiere@univ-lehavre.fr

Louise Travé-Massuyès
email: louise@laas.fr

Set-membership functional diagnosability through linear functional independence

Keywords:

Introduction

Before a system is put into operation and eventually diagnosed, diagnosability analysis is an important stage. Diagnosability indeed guarantees that the sensored values delivered by the available instrumentation can be processed into an appropriate set of symptoms allowing to discriminate a reasonable set of faulty situations.

A fault is considered as an additional parameter that impacts the behavior of some components of the system. Its effects may be linear or non-linear. Functional diagnosability, introduced in [START_REF] Verdière | Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations[END_REF], and extended to set-membership (SM) functional diagnosability in [START_REF] Jauberthie | Set-Membership diagnosability: definitions and analysis[END_REF] was analyzed through parameter identifiability. In the proposed work, SMfunctional diagnosability is assessed from the linear independence of SM-functional fault signatures, which results in a much more direct test.

The system is assumed to be represented by the following model :

Γ ẋ(t) = g(x(t), u(t), f, p), x(t 0 ) = x 0 , y(t) = h(x(t), u(t), f, p), (1)
where x(t) ∈ R n and y(t) ∈ R m denote the state variables and the outputs respectively. u(t) is the input vector. The function g is real and analytic on an open set of R n . p denotes the parameter vector belonging to a connected set P ∈ U P , where U P is an open set of R p . The fault vector f ∈ R e belongs to a connected set F ⊆ F SY S , where F SY S is the exhaustive set defining the fault domain. Y (P, F, u) denotes the set of outputs, solution of Γ with the input u, the parameter vector P and the fault vector F . Considering a connected set of faults F ⊆ F SY S , let us denote by F j the vector derived from F by setting all the components equal to zero except the jth component. F j , called a "bounded fault', is a connected set describing a faulty situation characterized by the occurence of a fault whose magnitude is assumed to belong to the bounded set F j .

Definitions and role of analytical redundancy relations

Using elimination theory, some differential polynomials or analytical redundancy relations (ARR) linking system inputs, outputs and their derivatives can be obtained. The use of ARRs makes possible to detect, isolate, and estimate the characteristics of a fault acting on the system. Specific ARRs indexed by i = 1, ..., m can be obtained [START_REF] Verdière | Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations[END_REF] :

w i (y, u, f, p) = m 0,i (y, u, p) -n i k=1 γ i k (f, p)m k,i (y, u) = w 0,i (y, u, p) -w 1,i (y, u, f, p) (2)
where (γ i k (f, p)) 1≤k≤n i are rational in f and p, γ i v = γ i w (v = w) and (m k,i (y, u)) 1≤k≤n i are differential polynomials with respect to y and u. w 0,i (y, u, p) is equal to m 0,i (y, u, p), hence the first part of the polynomial does not contain components of f . It corresponds to the residual computation form whereas w 1,i (y, u, f, p) is known as the residual internal form. Let us notice that (2) can as well be interpreted for bounded faults (vector F ) and uncertain but bounded parameters (vector P ).

One may be interested in distinguishing types of faults, independently of their magnitude. For instance, it may be important to detect a leakage on a pipe but the amount of derived flow may not be relevant. Hence the notion of SM-functional diagnosability that comes through the notion of SM-functional signature.

Definition 1. The SM-functional signature of a bounded fault F j is a function F Sig SM (F j ) which associates to F j the interval vector:

(w 1,i (Y (P, u), u, F j , P )) i=1,...,m .

F Sig (i)

SM (F j ) denotes the ith component of F Sig SM (F j ) and corresponds to an interval function. F Sig (i) SM (F j ) consists in a set of trajectories generated in the presence of F j and can be viewed as a tube of trajectories on the time interval [t 0 , T ].

To distinguish the two tubes of trajectories generated by two different bounded faults, we propose the following definitions. The first one refers to weak SM-functional diagnosability and permits an intersection of the two tubes on a time subinterval whereas the second one refers to strong SM-functional diagnosability and requires the two tubes to be totally disjoint on [t 0 , T ]. Definition 2. Two bounded faults F j and F k are SM-functionally discriminable if F Sig SM (F j ) and F Sig SM (F k ) are distinct, which includes the two following cases:

• there exists at least one index i * ∈ {1, ..., m} and a time interval [t 1 , t 2 ] ⊆ [t 0 , T ] such that for all t ∈ [t 1 , t 2 ], F Sig

(i * ) SM (F 1 ) ∩ F Sig (i * ) SM (F 2 ) = ∅ and F Sig (i * ) SM (F 1 ) ⊆ F Sig (i * ) SM (F 2 ) or F Sig (i * ) SM (F 2 ) ⊆ F Sig (i * )
SM (F 1 ), in which case F j and F k are said to be weakly SMfunctionally discriminable.

• there exists an index i * ∈ {1, ..., m} and a time interval

[t 1 , t 2 ]
such that for all t ∈ [t 1 , t 2 ], F Sig

(i * ) SM (F 1 ) ∩ F Sig (i * )
SM (F 2 ) = ∅, in which case F j and F k are said to be strongly SM-functionally discriminable.

Definition 3. The model Γ given by (1) is weakly (resp. strongly) SMfunctionally diagnosable for F SY S if any two bounded faults F j , F k ⊆ F SY S are weakly (resp. strongly) SM-functionally discriminable.

Analysis of SM-functional diagnosability

To analyse SM-functional diagnosability, a criterion testing the linear independence of SM-functional fault signatures is proposed. Definition 4. Rewriting F S i,j := F Sig (i) SM (F j ), the SM-signature matrix is defined as M Sig = (F S i,j ) 1≤i≤m, 1≤j≤e . Define also the extended following matrix for the case i = 1:

W M Sig =     F S 1,1 F S
Proposition 1. 1. Consider the case i = 1 -If det(W M Sig ) ≡ 0, then the faults of the set F SY S = {F 1 , . . . , F e } are discriminable, as well as any subset of faults within this set1 .

2. Consider the case i = m > 1 and assume that m ≤ e -If the minor M I,J of order α, 2 ≤ α ≤ m, of M Sig is not zero then the faults in the set {F j , j ∈ J} ⊂ F SY S are discriminable, as well as any subset of faults within this set.

Note that Proposition 1 gives conditions on the rank of the interval matrices. Proofs are omitted due to lack of space.

Conclusion

A new approach to test the concept of SM-functional diagnosability is proposed. It leads to a test based on the rank of an interval matrix formed from the SM-functional signatures of the faults.

Note that the determinant of W M Sig is the classical wronskian.


Sujeevan Aseervatham

Aomar Osmani

Mining short sequential patterns for hepatitis type detection

This paper presents the framework we have developed to classify patients according to the type of hepatitis. To detect the type of virus, once the data have been prepared and encoded in a suitable way, we have extracted the sequential patterns for each virus. Temporal differences between hepatitis B and C can then be seen as patterns that are frequent in one data series and infrequent in the other. The B virus and C virus patterns, extracted under specific constraints, were used to classify patients according to the hepatitis virus. In this work, we especially studied the use of very short patterns for virus type detection. However, the framework allows the mining and the use of longer patterns.The results have shown that in more than half of cases, short patterns can reveal the type of virus with a low level of errors.

Introduction

In this paper, we present the work that we have realized in order to discover temporal differences between the hepatitis B and C. This problem was raised by the Chiba University hospital and constitutes one of the subjects of the PKDD Discovery Challenge 2004 and 2005. A hepatitis dataset was provided by the Chiba hospital. It contains long time-series data on the laboratory examinations of hepatitis B and C infected patients. The examinations were realized between 1982 and 2001 on 771 patients. Our approach to discover temporal differences is based on the extraction of short frequent sequential patterns from the in-laboratory examinations only. To reach our aim, we have developed a framework, which discretizes numerical data into categorical data and uses a constraints incorporated frequent sequence mining program based on the SPADE [START_REF] Zaki | Efficient enumeration of frequent sequences[END_REF] algorithm. The frequent patterns are checked against the class labels and then pruned according to their confidence values. Finally, the obtained rules are evaluated on a test database. The results of the experiments are encouraging and can be improved.

Objectives. In this study, our aim was to discover temporal differences between the hepatitis B and C. Given two datasets, one containing the observations of patients infected with hepatitis B and the other with hepatitis C, we can define a temporal difference as a temporal pattern that is frequent in one dataset and infrequent in the other. According to this definition, we have concentrated our work on the analysis of short sequences. Our framework, developed in C++, is able to extract, under user-defined constraints, frequent sequences from the hepatitis dataset. In this work, we were only interested in a classification system that uses short sequences with the constraint that the period between two examinations should not exceed three months. Organization. In section 2, we describe the data preprocessing used to generate a dataset of sequences. The sequential pattern generating algorithm and the classification system are defined in section 3. An experimental evaluation of our classification system is presented in section 4. Finally, we conclude in section 5.

Data Preprocessing

The hepatitis dataset was provided by the Chiba University Hospital. It contains the results of examinations realized from 1982 to 2001 on 771 patients infected with hepatitis B and C viruses.

Dataset cleaning

For the data cleaning operation and the statistical analysis, we used the R environment [START_REF] Development | Core Team: R: A language and environment for statistical computing[END_REF]. Using R, we did the following operations on the data set :

1. cleaning the data of incorrect values, 2. merging the 7 tables of the initial dataset into one table, 3. acquiring basic statistic knowledge of the dataset, 4. selecting relevant variables for hepatitis type discrimination.

The final table was constituted of 548 patients and 37 variables 1 . The distribution of patients according to their sex and the type of virus infection is shown in the table 1

Data Transformation

To extract patterns and classify patients, we needed to transform the dataset into a collection of suitable sequence formatted data. Definition 1. We define a sequence event as a non-empty unordered2 set of measurements realized at the same time. Each event is composed of the pair (variable, measurement). We will use the term examination as a synonym of event.

An event with n measurements will be denoted as {e 1 , . . . , e n }. 1 Variables were scaled according to the interval of "normality". An interval of "normality" for a variable is defined by a lower and upper bounds values between which measurements are considered as being normal. This interval is defined for each variable in the labn table.

For each measurement x of a variable with an interval [L, H], x was scaled using the following formula :

x scaled = x-L H-L .
Definition 2. We shall say that a sequence is a non-empty time-ordered collection of events. A sequence S composed of n examinations and k measurements will be called a k-sequence and it will be denoted as S = s 1 → s 2 → . . . → s n [START_REF] Zaki | Efficient enumeration of frequent sequences[END_REF]. Indeed, a k-sequence S of n elements verifies :

k = sj ∈S | s j |
The processed inlab database was transformed into a collection of sequences. For this, each patient's examinations identified by the value of the M ID variable were grouped together and sorted according to the date of examination. Nevertheless, most of the sequential mining algorithms processing only symbolic data, we discretized measurements into a finite number of intervals. Each interval being a categorical value for a variable.

For the discretization algorithm, a supervised3 global4 method based on a minimal entropy heuristic described in [START_REF] Fayyad | Multi-interval discretization of continuous-valued attributes for classification learning[END_REF] was used. This method uses the class information entropy to find the best cut points. Given a data set S, the attribute A to discretize, a class attribute C, a cut point T , the set S 1 ⊆ S with A-values(S 1 ) ≤ T and the set S 2 = S -S 1 , the best cut point is the one that minimizes E(A, T ; S) :

E(A, T ; S) = | S 1 | | S | Ent(S 1 ) + | (S 2 ) | | S | Ent(S 2 ) Ent(S) = - i P (C i , S) log 2 (P (C i , S))
Once the best cut point is found and accepted, the method recursively partitions S 1 and S 2 . The Minimum Description Length Principle (MDLP) is used as a stopping criterion. Thus, the recursion is stopped and the cut point is rejected iff :

Gain(A, t; S) ≤ log 2 (| S | -1) | S | + ∆(A, T ; S) | S |
with Gain(A, t; S) = Ent(S) -E(A, T ; S) and

∆(A, T ; S) = log 2 (3 k -2) -(kEnt(S) -k 1 Ent(S 1 ) -k 2 Ent(S 2 ))
where k i is the number of class labels in the set S i .

Once all the transformations were done, we have obtained a collection of patients sequences composed of events in which all measurements were encoded with integers. An examination or event was, then, defined as a set of integers with each integer corresponding to a pair (variable, measurement).

Sequence Mining

Mining Algorithm

For the criteria of speed and class decomposition, we have used cSPADE [START_REF] Zaki | Sequence mining in categorical domains: Incorporating constraints[END_REF] in our framework for mining patterns. cSPADE is based on the SPADE algorithm [START_REF] Zaki | Efficient enumeration of frequent sequences[END_REF]. The only difference is that SPADE does not handle the time constraints.

The main key of cSPADE is the use of a vertical database instead of the common horizontal database. In the horizontal database, sequences are represented in a quite natural way : measurements (items) are grouped together by events and events are grouped by patients in a chronological order. Whereas in the vertical database, measurements (items) are lexicographically ordered 5 and with each measurement is associated a set of pairs of patient id and examination date. Therefore, for each measurement, thanks to the patient id we can identify the sequences in which it appears and for each sequence we can identify, thanks to the date, the examinations to which the measurement belongs. cSPADE, using a bottom-up lattice 6 traversal, starts at the first level7 with the set of frequent 1-sequences. It joins the sequences between them and jumps to the minimal upper level.

The vertical database is well suited for the join operations. The support counting for a sequence resulting of a join operation is straightforward. Another advantage of cSPADE is that it can decompose the lattice into small equivalence classes and then mine the frequent patterns, independently, in each classes.

Rules generation

Given D T , the complete training dataset, for each class C, once the frequent sequences have been extracted from the specific class training data D C ⊆ D T , we have calculated for each extracted patterns m the confidence of the rule m ⇒ C from the formula below [START_REF] Zaki | Sequence mining in categorical domains: Incorporating constraints[END_REF] :

conf (m ⇒ C) = σ DC (m) σ DT (m)
where σ D (m) is the number of sequences in D that contain m.

Then, given a user-specified minimum confidence threshold, all the rules that had a confidence value below the threshold were eliminated.

Moreover, as we are dealing with short sequences, we can expect that many rules will be triggered by an input-sequence8 . Therefore we cannot rely on the rules confidences. Thus, we have proposed an heuristic that relies on the ratio of the number of rules triggered for the class to the total number of rules for this class.

For an input-sequence, we compare, for each class, the ratio of triggered rules. Then, we label the input-sequence with the label of the class that had the highest ratio. Nevertheless if the difference between the ratios are less than 10%

Table 1 .

 1 . Final inlab Patient Distribution Table

	Patient's sex Hepatitis B Hepatitis C Interferon
	Female	49	97	60
	Male	204	198	136

However, according to[START_REF] Agrawal | Mining Sequential Patterns[END_REF][START_REF] Srikant | Mining Sequential Patterns: Generalizations and Performance Improvements[END_REF][START_REF] Zaki | Efficient enumeration of frequent sequences[END_REF] we can assume without loss of generality that the events are lexicographically sorted.

A supervised method uses the class information to partition the data.

Global discretization methods produce, for an attribute, partitions that are applied to the entire dataset independently of the other attributes.

Measurements can also be ordered according to other criteria and even not ordered at all.

It is not exactly a lattice but a hyper-lattice. Indeed, the join of two elements of the same level gives three elements. For example, the join of elements A and B will results in elements {{A, B}, A → B, B → A}.

In fact, an optimization approach is to start at the second level after having determined all the frequent 1-sequences as well as all the frequent 2-sequences.

 8 A rule is triggered by a sequence if the rule's antecedent is a subsequence of the sequence according to the various constraints.

then the input-sequence cannot be labelled as the extracted information are not sufficient to classify such sequences.

Experimental Evaluation and Results

The inlab table has been partitioned into three datasets for 3-fold cross-validation. Each subset was extracted from the inlab table by randomly selecting sequences. However, we have extracted the data according to the proportion of the patients infected with hepatitis B. Thus, in inlab there were 41% of the patients who were infected with B virus and the others with C virus. This ratio was kept in each subset. Moreover, we have separated the patients according to the type of virus they were infected with.

For each class B and C, we have extracted patterns using the parameters defined in table 2. As we wanted short sequences, we have selected a high value for the minimum support since we know that higher is the minimum support and shorter will the frequent patterns be. Moreover, since we're interested in detecting the hepatitis virus type within a short period of examinations, we have set, for the desired patterns, the maximum time distance between two successive examinations to 90 days. Nonetheless, we have set a low value for the minimum confidence value of the generated rules because as the extracted patterns will be short, the probability for sequences to contain the patterns will be important. Thus, we cannot expect high confidence values for the rules that will be generated from short sequences.

The statistics about the results of the patterns extraction and rules generation are summarized in table 4. As we have expected, the frequent patterns are very short. Their average length is about 4.1 examinations per pattern. Due to the sliding window and the maximal gap that increase the combinatorial possibilities, the number of generated patterns are very important. However, the rules confidences are low and near 94% of the generated patterns were not suitable to be used in classification rules.

To check the validity of the extracted rules, we have applied them on the test data. The result of the test is shown in table 4. Our classification program has successfully detected the type of virus for about 59% of the tested patients and for nearly 32.7% the rules weren't enough discriminatory and that can be explained by the fact that the extracted patterns are too general to detect specific cases. However, when the discrimination was possible, the system has, correctly, classed 86% of the patients.

Conclusion

We have presented a method for extracting short sequential patterns in order to detect the type of hepatitis virus infection. We focused our work on the problematic of detecting the type of hepatitis within a short period of time. The test done on the hepatitis dataset provided by the Chiba University Hospital allowed us after a learning step, to extract very short patterns, which were sufficient, in half of cases, to detect hepatitis virus type with a low rate of errors. The results were encouraging and can be improved by relevantly setting the various parameters of the mining and classification programs.

As future work, we plan, in a first time, to improve the discretization part of the data transformation. In a second time, we want to test our system using the same database but with different problematic and especially we want to analyse the performance of the system when attempting to extract more specific patterns i.e. long patterns.