Hugo Malamut

Maxime Sylvestre

Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

We study the convergence of the transport plans γε towards γ0 as well as the cost of the entropy-regularized optimal transport (c, γε) towards (c, γ0) as the regularization parameter ε vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance W2(γε, γ0) is asymptotically greater than C √ ε and the suboptimality (c, γε) -(c, γ0) is of order ε. In the quadratic cost case the compactness assumption is relaxed into a moment of order 2 + δ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance W2(γε, γ0) converges to 0 at rate √ ε. Finally, if in addition the marginals have finite Fisher information, we prove (c, γε) -(c, γ0) ∼ dε/2 and we provide a companion expansion of H(γε). These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.

Contents c

)) E a duality gap function defined by

Σ a set of optimal pairings Σ := {(x, y)/E(x, y) = 0}

OT ε ε-entropic transport cost from µ 0 to µ 1 (see equation (εEOT))

Optimal transport plan for ε-Entropic Optimal Transport (εEOT)

Moment of order k of the probability µ H m Mean entropy H m := 1 2 (H(µ 0 | H d ) + H(µ 1 | H d )) C(a, b, c) A constant that only depends on the terms a,b and c

1 Introduction 2

2 Exact asymptotics from the Schrödinger problem 5

3 Rates for the quadratic cost 8

4 Rates for infinitesimally twisted costs 14

Notations Ω, X

Ω is an open subset of R d and X a compact subset of Ω

P(Ω)

Probability measures on Ω, absolutely continous (P ac (Ω)), with finite variance (P 2 (Ω))

H k Hausdorff measure of dimension k W 2 (µ, ν) 2-Wasserstein distance between µ and ν H(µ | ν) relative entropy of µ with respect to ν (see equation [START_REF] Chizat | Faster Wasserstein distance estimation with the Sinkhorn divergence[END_REF]). Sometimes H(µ) := H(µ | H k ) for some k ∈ N I(µ) Fisher information of µ, see equation [START_REF] Chizat | Faster Wasserstein distance estimation with the Sinkhorn divergence[END_REF] ∥Φ∥ op Operator norm of the affine transformation Φ Π(µ 0 , µ 1 ) Transport plans between µ 0 and µ 1

We would like to thank G. Carlier and P. Pegon for fruitful discussions and helpful advices.

Introduction

We study the regularized optimal transport problem for a cost c

∈ C 2 (R d × R d , R) inf γ∈Π(µ0,µ1) cdγ + εH(γ|µ 0 ⊗ µ 1 ) (1)
where the infimum is taken over all measures γ ∈ P(R d × R d ) with marginals µ 0 , µ 1 . Here H(.|.) is the relative entropy also known as Kullback-Leibler divergence. In this paper we will focus on the case where µ 0 , µ 1 have finite entropy with respect to H d , the Lebesgue measure, H(µ i | H d ) < ∞ and finite order two moments. In that case the minimizer γ ε is the same as if µ 0 ⊗ µ 1 was replaced by the Lebesgue measure H 2d in the entropy term, and hence we consider

OT ε := inf γ∈Π(µ0,µ1) cdγ + εH(γ|H 2d ) (εEOT)
Note that ε = 0 yields the classical optimal transport problem. In this case the minimizer need not to be unique and γ 0 will sometimes denote any of them. We are interested in deriving rates for the cost term cdγ ε = (c, γ ε ) as well as the 2-Wasserstein distance between γ ε and γ 0 , when it is unique.

In the last decade this problem has witnessed a rapid increase in interest. It has proved to be an efficient way to approximate OT problems, especially from a computational viewpoint. The celebrated Sinkhorn's algorithm [START_REF] Sinkhorn | A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices[END_REF] was applied in this framework in the pioneering works [START_REF] Cuturi | Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances[END_REF][START_REF] Benamou | Iterative Bregman Projections for Regularized Transportation Problems[END_REF]. The good convergence guarantees [START_REF] Franklin | On the scaling of multidimensional matrices[END_REF][START_REF] Di | An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm[END_REF] cemented the success of EOT and its applications.

Clearly EOT is a perturbation of classical OT thus it is natural to study the behaviour of this problem as ε vanishes. In this direction several aspects deserved to be studied such as the convergence of optimal values, potentials (optimizers of the dual problem) and optimal plans, possibly with quantitative rates. In the direction of convergence of optimal values recent contributions have thoroughly treated the issue: in the quadratic case, under regularity assumptions, the link between EOT and the Schrödinger problem [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF] has allowed to find a second order [START_REF] Erbar | From large deviations to Wasserstein gradient flows in multiple dimensions[END_REF] and more recently a third order [START_REF] Conforti | A formula for the time derivative of the entropic cost and applications[END_REF][START_REF] Chizat | Faster Wasserstein distance estimation with the Sinkhorn divergence[END_REF] expansion of the value OT ε in ε. The second order expansion has been generalized to other cost functions [START_REF] Pal | On the difference between entropic cost and the optimal transport cost[END_REF], and the first order term has been obtained under very mild assumptions on the cost function and the marginals [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF][START_REF] Eckstein | Convergence Rates for Regularized Optimal Transport via Quantization[END_REF]. Those articles focus on the value problem of (εEOT).

Our main objective is to disentangle the role of the cost cdγ ε and the entropy H(γ ε | L) in order to derive rate of convegence for both. The cost term is of interest itself because it is a faster converging approximation of OT 0 . The entropy term also allows to lower bound W 2 (γ ε , γ 0 ), the Wasserstein distance between the entropic optimal transport plan γ ε and the optimal transport plan γ 0 . The study of the convergence of the cost term in EOT has also been done recently in [START_REF] Altschuler | Asymptotics for Semidiscrete Entropic Optimal Transport[END_REF] for the semidiscrete case which grants an ε 2 rate. They also derive convergence rates for the Kantorovich potentials. In the discrete case the rate of convergence is exponential [START_REF] Cominetti | Asymptotic analysis of the exponential penalty trajectory in linear programming[END_REF][START_REF] Weed | An explicit analysis of the entropic penalty in linear programming[END_REF]. In the continuous case, to our knowledge, no asymptotic rate for the suboptimality cdγ ε -cdγ 0 was known but a rate of order ε was suspected based on simple examples such as Gaussian measures. We tackle the problem of sizing the suboptimality in this article.

Main results

The aim of this paper is to provide tight rates of convergence on the Wasserstein distance W 2 (γ ε , γ 0 ), the cost term (c, γ ε ) and the entropy term

H(γ ε | H 2d ). Let µ 0 , µ 1 ∈ P ac (R d ) such that H(µ i | H d ) < ∞.
It is known [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF][START_REF] Eckstein | Convergence Rates for Regularized Optimal Transport via Quantization[END_REF] that under mild assumptions (see lemma 3.6) on µ 0 , µ 1 ,

OT ε = (c, γ ε ) + εH(γ ε | H 2d ) ≤ OT 0 - d 2 ε ln(ε) + O(ε) (2)
For the quadratic cost and under the stronger assumption of finite Fisher information for the marginals, we have (see [17, claim 4.1], see also [START_REF] Pal | On the difference between entropic cost and the optimal transport cost[END_REF] for other costs and different hypothesis)

OT ε = OT 0 - d 2 εln(2πε) + εH m + o(ε) (3)
where

H m := 1 2 [H(µ 0 | H d ) + H(µ 1 | H d )].
Our goal are to disentangle the role of the cost term (c, γ ε ) and of the entropy term H(γ ε | H 2d ) in those rates rates of convergence. For clarity we present here the three main results present in the article. The first gives a second order expansion of the entropy and the cost of the entropic optimal transport. It relies on known expansions of the Benamou-Brenier (εBB) formulation of the entropic optimal transport problem. The proof of this result can be found in section 2

Theorem (Theorem 2.3). Suppose that the cost is quadratic, that is c(x, y) = 1 2 ∥x -y∥ 2 . Further assume that I(µ i ) < ∞ and Supp(µ i ) compact. Then

H(γ ε | H 2d ) = - d 2 ln(2πε) + H m - d 2 + o(1) (4)
and

(c, γ ε ) = OT 0 + d 2 ε + o(ε) (5)
If we relax the finite Fisher information hypothesis the orders of magnitude still hold. The method, presented in section 3, used to prove that result relies on the Minty reparametrization trick (see definition 3.3) which implies a quadratic detachment for the duality gap function

E = c -(φ ⊕ ψ),
where φ, ψ are Kantorovich potentials.

Theorem (Theorem 3.7). Suppose that the cost is quadratic, that is c(x, y) = 1 2 ∥x -y∥ 2 . Further assume that µ i have finite moment of order 2 + δ then

(c, γ ε ) = OT 0 + Θ(ε), H(γ ε | H 2d ) = - d 2 ln(ε) + O(1), √ ε = O(W 2 (γ ε , γ 0 )) (6)
In the special case where the Monge map ∇f associated to the optimal transport plan γ 0 is Lipschitz then

W 2 (γ ε , γ 0 ) = Θ( √ ε) (7)
Finally for infinitesimally twisted costs we have a similar result but under the stronger assumption of compactly supported marginals. The concept of local quadratic detachment 4.1 is introduced and is key to the proof of the result which can be found in section 4.

Theorem (Theorem 4.8). Suppose that the cost is C 2 and infinitesimally twisted (see definition 4.3). Further assume that µ i is compactly supported then

(c, γ ε ) = OT 0 + Θ(ε), H(γ ε | H 2d ) = - d 2 ln(ε) + O(1), √ ε = O(W 2 (γ ε , γ 0 )) (8)

Definitions and assumptions

First, let recall the definition of the relative entropy H(µ|ν) and the Fisher information I(µ) that provide quantitative estimate of the smoothness of a measure : if µ, ν ∈ P ac (R d ), we note

H(µ|ν) := ln dµ dν dµ if µ ≪ ν +∞ otherwise and I(µ) = ∥∇µ(x)∥ 2 µ(x) 2 dµ(x) if µ ≪ H d +∞ otherwise (9)
Where dµ dν denotes the Radon-Nikodym derivative of µ with respect to ν and µ(x) the density of µ with respect to the Lebesgue measure H d by a slight abuse of notation. Note that a finite Fisher information I(µ) < +∞ implies a finite differential entropy H(µ|H d ) [START_REF] Villani | Topics in Optimal Transportation[END_REF]Chapter 9], that we will often denote simply by H(µ) . We now introduce the Kantorovich potentials which are the solutions to the dual of the classical optimal transport problem. sup φ,ψ φdµ 0 + ψdµ 1 [START_REF] Cominetti | Asymptotic analysis of the exponential penalty trajectory in linear programming[END_REF] Where the sup is taken over all functions φ ∈ L

1 (µ 0 ), ψ ∈ L 1 (µ 1 ) such that for all x, y ∈ R d × R d , we have φ(x) + ψ(y) ≤ c(x, y).
It is well known that in our current setting this problem has maximizers which are continuous [START_REF] Villani | Topics in Optimal Transportation[END_REF]Chapter 1], that is the Kantorovich potentials are continuous. Moreover it is also possible to assume that the constraint φ(x) + ψ(y) ≤ c(x, y) is binding, that is that φ and ψ are c-transform1 one of each other. In the following, we will use the name Kantorovich potentials to denote a pair of continuous and mutual c-transform solutions to the dual problem [START_REF] Cominetti | Asymptotic analysis of the exponential penalty trajectory in linear programming[END_REF]. For a pair (φ, ψ) of Kantorovich potentials, we will call duality gap the quantity E(x, y) := c(x, y) -(φ(x)+ψ(y)) and we will note E = c-(φ⊗ψ). Remark that in the quadratic case, when µ i ∈ P ac (R d ), the Kantorovich potentials are unique up to a constant and so the duality gap function E is unique.

In the general case, we will be interested in the gap between cdγ ε and cdγ 0 which can be restated in terms of any duality gap function E = c -(φ ⊕ ψ) with φ, ψ Kantorovich potentials. Indeed we have by duality theorem that cdγ 0 = φdµ 0 + ψdµ 1 and so

cdγ ε -cdγ 0 = cdγ ε -(φ ⊕ ψ)dγ ε = Edγ ε (11)
As explained in the main results, we will explore three different sets of assumptions, that we present, from the most specific to the most generic. Each set of assumption correspond to one of the theorems presented in the main results.

• (H1)

The cost is quadratic : c(x, y) := 1 2 ∥x -y∥ 2 . The marginal have finite fisher information I(µ i ) < +∞ and compact support • (H2) The cost is quadratic : c(x, y) := 1 2 ∥x -y∥ 2 . The marginals have finite differential entropy H(µ i |H d ) < +∞ and finite moments of order 2 + δ for some δ > 0 • (H3) The cost c belongs to C 2 and is infinitesimally twisted (see definition 4.3).

The marginals (µ i ) have compact support and finite differential entropy.

Since a finite Fisher information implies a finite differential entropy, hypothesis (H2) is an important relaxation of (H1) on the regularity and concentration of (µ i ). Similarly, hypothesis (H3) is an important relaxation of (H1) since the class of cost is broader and the marginals are assumed less regular. A last remark : while the finite entropy or finite information hypothesis are deeply linked with the nature of the problem and essential to our results, the concentration assumptions such as compact support or finite moment must be seen as technical and the results are likely to hold in any case.

Exact asymptotics from the Schrödinger problem

In this section, we assume validity of hypothesis (H1), that is that the cost is quadratic and that the marginals µ 0 and µ 1 have finite Fisher information and compact support. Under finite Fisher information of the marginals, it is known that the value of (εEOT) has a second order expansion in ε (see [17, claim 4.1], see also [START_REF] Pal | On the difference between entropic cost and the optimal transport cost[END_REF] for other costs and different hypothesis)

OT ε = 1 2 W 2 2 - d 2 εln(2πε) + εH m + o(ε) (ε → 0) (12)
In this section, we will disentangle the roles of (c, γ ε ) and H(γ ε |H 2d ) in this asymptotic formula, and hence give a Taylor expansion of both. We will use the dynamic formulation of the entropic optimal transport.

Schrödinger problem and Benamou-Brenier formulation

The quadratic EOT problem can be reformulated as a Schrödinger problem (see [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF])

C ε = min γ∈Π(µ0,µ1) H(γ|m ε ) ( 13
)
where m ε be the measure of density 1 (2πε) d/2 e -c/ε with respect to the Lebesgue measure H 2d . Since for γ ∈ Π(µ 0 , µ 1 ), we can compute explicitely εH(γ|m ε ) = (c, γ) + εH(γ | H 2d ) + d 2 ε ln(2πε), we have that the solutions of ( 13) and (εEOT) are the same and

OT ε = εC ε - d 2 ε ln(2πε) (14)
Moreover there exists an analogous to Benamou-Brenier formula for C ε (see [START_REF] Gigli | Benamou-Brenier and duality formulas for the entropic cost on RCD*(K, N) spaces[END_REF]) :

C ε = H m + min ρ,v 1 ε 1 0 1 2 |v t | 2 dρ t (x)dt + ε 8 1 0 ∥∇ρ t ∥ 2 ρ t dxdt (εBB)
Where the min is taken as in the classical Benamou-Brenier formula on paths from µ 0 to µ 1 that in the weak sense solve the continuity equation ∂ t ρ+div(ρv) = 0. We will denote by v ε , ρ ε the solution. This dynamic formulation and other variations are in fact the core of the Schrödinger problem, that intend to solve for the most probable trajectory of a free process at positive temperature with fixed initial and final marginals µ 0 and µ 1 (see [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF] for a survey). The first term in (εBB) corresponds to the kinetic energy while the second term corresponds to the diffusion (ε is the temperature parameter).

2.2 A precise expansion of H(γ ε ) and (c, γ ε )

Minimization problems (εEOT) and (εBB) give the natural idea of using the envelope theorem to express the ε-derivative of OT ε and C ε . For a minimization problem with a real parameter ε, this theorem ensures that at any differentiability point ε 0 of the value function and for any minimizer, the derivative of the value function coincide with the ε-partial derivative of the objective function with fixed minimizer (see [START_REF] Milgrom | Envelope theorems for arbitrary choice sets[END_REF]). This idea has been followed in [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF] for (εEOT) and in [START_REF] Conforti | A formula for the time derivative of the entropic cost and applications[END_REF] for (εBB). We will use the more general setting of [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF] that only requires compact support for the marginals µ i and obtain that ε → OT ε is C ∞ for ε > 0. From equation [START_REF] Cuturi | Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances[END_REF], it is direct that in that case the function εC ε is also smooth, and so we can apply envelope theorem both to

OT ε = min γ 1 2 ∥x-y∥ 2 dγ +εH(γ | H 2d ) and to εC ε = εH m + min ρ,v 1 0 1 2 |v t | 2 dρ t (x)dt + ε 2 8 1 0 I(ρ t )dt to get d dε OT ε = H(γ ε ) and d dε [εC ε ] = H m + ε 4 1 0 I(ρ ε t )dt ( 15
)
Where H(γ ε ) stands for H(γ ε | H 2d ). But from equation ( 14)

d dε OT ε = d dε εC ε -d 2 ε ln(2πε
) , so we have

H(γ ε ) = - d 2 ln(2πε) + H m - d 2 + ε 4
1 0 I(ρ ε t )dt (16)
This gives a quantitative relation between the entropy H(γ ε ) of the static problem (εEOT) and the average Fisher Information 1 0 I(ρ ε t )dt of the dynamic problem (εBB). It allows to state a complete coupling system between the different terms of (εEOT) and (εBB) as explained in the following proposition : Proposition 2.1. For the quadratic cost, suppose that the support of µ i are compact, and that

H(µ i |H d ) < +∞. Then        (c, γ ε ) = 1 0 1 2 |v ε t | 2 dρ t (x)dt - ε 2 8 1 0 I(ρ ε )dt + d 2 ε H(γ ε ) = ε 4 1 0 I(ρ ε )dt - d 2 ln(2πε) + H m - d 2 (17)
Proof. The second line of the system ( 17) is nothing but equation [START_REF] Erbar | On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces[END_REF]. For the first line, equation [START_REF] Cuturi | Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances[END_REF] implies that

(c, γ ε ) = εC ε - d 2 ε ln(2πε) -εH(γ ε ) (18)
We can inject the expression of H(γ ε ) given by ( 16) and the expression of C ε with the minimal continuous path (ρ ε , v ε ) of equation (εBB) to obtain the first line of the system [START_REF] Erbar | From large deviations to Wasserstein gradient flows in multiple dimensions[END_REF].

Thanks to the system [START_REF] Erbar | From large deviations to Wasserstein gradient flows in multiple dimensions[END_REF], it is enough to study the asymptotics of ε andH(γ ε ). When the Fisher information of the marginal is finite, it turns out that the behaviour of

1 0 I(ρ ε )dt and 1 0 |v ε t | 2 dρ t (x)dt to get results on (c, γ ε )
1 0 I(ρ ε )dt and 1 0 |v ε t | 2 dρ t (x)
dt can be described quite precisely :

Proposition 2.2. Suppose that the cost is quadratic. If µ 0 and µ 1 have finite Fisher information, when ε tends to 0 :

ε 1 0 I(ρ ε )dt → 0 and 1 0 |v ε t | 2 dρ t (x)dt = W 2 2 + o(ε) (19)
Proof. Since the Fisher information of the marginal is finite, from [START_REF] Erbar | From large deviations to Wasserstein gradient flows in multiple dimensions[END_REF]Claim 4.1] the Taylor expansion [START_REF] Conforti | A formula for the time derivative of the entropic cost and applications[END_REF] of OT ε holds. But since OT ε = εC ε -d 2 ε ln(2πε), expansion ( 12) is equivalent to the following :

C ε - 1 2ε W 2 2 -H m → 0
However from the expression of C ε with the minimal continuous path (ρ ε , v ε ) of equation (εBB),

C ε - 1 2ε W 2 2 -H m = 1 ε 1 0 1 2 |v ε t | 2 dρ t (x)dt - 1 2 W 2 2 + ε 8 1 0 ∥∇ρ ε t ∥ 2 ρ ε t dxdt Both terms 1 ε 1 0 1 2 |v ε t | 2 dρ t (x)dt -1 2 W 2 2 (µ 0 , µ 1 ) and ε 8 1 0 ∥∇ρ ε t ∥ 2 ρ ε t
dxdt are positive, the first one because (ρ ε , v ε ) is a path from µ 0 to µ 1 solving the continuity equation, and so the (ε = 0) Benamou-Brenier formula holds. Since the sum tends to zero, we know that both terms tend to zero which is what we needed to prove. Proposition 2.2 can be seen as a refinement of [START_REF] Gentil | An entropic interpolation proof of the HWI inequality[END_REF]Lemma 3.3] where, among other asymptotics, it is shown that ε2 1 0 I(ρ ε )dt → 0 and

1 0 |v ε t | 2 dρ t (x)dt → W 2 2
We can now combine propositions 2.1 and 2.2 to obtain a Taylor expansion of (c, γ ε ) and H(γ ε ) :

Theorem 2.3. Suppose that the cost is quadratic, let µ 0 , µ 1 ∈ P ac (R d ) and suppose that for i = 1, 2, we have I(µ i ) < +∞ and the support of µ i is compact. Then

H(γ ε ) = - d 2 ln(2πε) + H m - d 2 + o(1) (20)
and

(c, γ ε ) = 1 2 W 2 2 + d 2 ε + o(ε) (21)
Proof. Since a finite Fisher information implies a finite differential entropy, the hypothesis of propositions 2.1 and 2.2 hold and we can inject the asymptotics on 19) obtained in the latter in the system (17) of the former.

1 0 I(ρ ε )dt and 1 0 |v ε t | 2 dρ t (x)dt (

Remark 2.4.

The rate in d 2 ε for the subobtimality (c, γ ε ) -(c, γ 0 ) had been already observed in the case of gaussian measures in 1D (see introductions of [START_REF] Altschuler | Asymptotics for Semidiscrete Entropic Optimal Transport[END_REF], [START_REF] Bernton | Entropic optimal transport: Geometry and large deviations[END_REF]). It is astonishing to remark that this first order term does not depend on µ 0 and µ 1 . Hence, the estimator (c, γ ε ) -d 2 ε could provide an interesting estimate of W 2 2 (µ 0 , µ 1 ) with a lack of precision comparable to the one of the Sinkhorn divergences

OT ε (µ 0 , µ 1 ) -1 2 [OT ε (µ 0 , µ 0 ) + OT ε (µ 1 , µ 1
)] (see [START_REF] Feydy | Interpolating between Optimal Transport and MMD using Sinkhorn Divergences[END_REF])

Rephrasing the results with different functionals

The asymptotic formula for the entropy [START_REF] Gentil | An entropic interpolation proof of the HWI inequality[END_REF] and the formula for (c, γ ε ) in the system [START_REF] Erbar | From large deviations to Wasserstein gradient flows in multiple dimensions[END_REF] suggest to introduce two quantities that are linked to our problem. First, the Entropy power function

N d (µ) := e -2 d H(µ|H d ) 2πe
. This functional has various links with our problem : Costa's theorem states that it is concave along the heat flow (see [START_REF] Costa | A new entropy power inequality[END_REF]), √ N 2d is concave along Wasserstein geodesics (see [START_REF] Erbar | On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces[END_REF]) and along Schrödinger bridges (see [START_REF] Ripani | Convexity and regularity properties for entropic interpolations[END_REF]). And it turns out that the asymptotic (20) can be rewritten in the simple way 2

N 2d (γ ε ) ∼ N d (µ 0 ) 1/4 N d (µ 1 ) 1/4 ε 1/2
The other quantity that we need to present is the energy3

E ε := 1 ε 2 1 2 |v ε t | 2 dρ t (x) - 1 8 I(ρ ε t )
As the notation let think and as it is proved in [START_REF] Conforti | A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost[END_REF]Corollary 1.1] and [20, lemma 3.3], this energy does not depend on the time t. With this quantity, the expression of (c, γ ε ) in the system (17

) becomes (c, γ ε ) = ε 2 E ε + d 2 ε
, and so the Taylor expansion (21) expresses

ε 2 E ε = 1 2 W 2 2 + o(ε)
Note that it was already known that

ε 2 E ε → 1 2 W 2 2
(see the introduction of [START_REF] Conforti | A formula for the time derivative of the entropic cost and applications[END_REF] or the proof of [20, lemma 3.3]). The convergence in o(ε) seems new to us. Given the expression for d dε [εC ε ] of equation ( 15), the other Taylor expansion [START_REF] Gentil | An entropic interpolation proof of the HWI inequality[END_REF] corresponding to ε 1 0 I(ρ ε )dt → 0 is also equivalent to the fact that εC ε is C 1 in 0 with derivative H m . It can be seen as an extension in 0 of [START_REF] Conforti | A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost[END_REF]Theorem 1.1].

Rates for the quadratic cost

In the last section, under the strong assumptions (H1) of smoothness (I(µ i ) < +∞) and concentration (Supp(µ i ) compact), the Taylor expansions of (εEOT) and (εBB) allowed us to give precise rates for the entropy as well as the cost term in (εEOT). The main idea was to disentangle the role of the entropy and the cost. In this section we choose the weaker set of assumptions (H2). The cost is still the quadratic one : c(x, y) = 1 2 ∥x -y∥ 2 , but the Fisher information need not to be finite and so the second order Taylor expansion ( 12) of (εEOT) need not to hold. However, the entropy H(µ i ) is still supposed finite. Thanks to the technical assumption that the moments of order 2 + δ of the marginals are finite for some δ > 0, we still have (see lemma 3.6) some rates of convergence for the value of (εEOT) of the form

0 ≤ OT ε -OT 0 ≤ - d 2 ε ln(ε) + O(ε) (22)
From that, it is still possible to disentangle the cost and the entropy (theorem 3.7) through a careful study of the behaviour of H(γ ε ) as ε → 0. It will grant rates for (c, γ ε ) -(c, γ 0 ) of the same order ε as in last section. We start by providing a sketch of the proof : Let E be a duality gap function E = c -(φ ⊕ ψ). Recall that since γ ε has µ i as marginals, we have that Edγ ε = (c, γ ε ) -(c, γ 0 ), and so we will denote by (E, γ ε ) this suboptimality. The goal is to prove (E, γ ε ) ≃ ε and H(γ ε ) ≃ -d 2 ln(ε), so it is natural to try to prove first H(γ ε ) ≃ -d 2 ln(E, γ ε ), or in terms of entropy power, N 2d (γ ε ) ≃ (E, γ ε ). But thanks to the information encoded in inequality [START_REF] Gigli | Benamou-Brenier and duality formulas for the entropic cost on RCD*(K, N) spaces[END_REF], it is enough to show an inequality of the type N 2d (γ ε ) ≤ C (E, γ ε ). In fact, as explained in detail in section 3.1, this kind of inequality is not specific to γ ε and it comes from the following fact : the contact set Σ := {x, y/E(x, y) = 0} has d dimensions less than the ambiant space R 2d and (E, γ) quantifies the average distance of γ to Σ because E has a quadratic detachment in Minty's coordinates (see definitions 3.1 and 3.3). As a matter of fact, W 2 2 (γ, γ 0 ) also quantifies this distance, because γ 0 is supported on Σ. In proposition 3.5 we obtain results of the form

N 2d (γ) ≤ C (E, γ) and N 2d (γ) ≤ CW 2 (γ, γ 0 ) (23)
or equivalently in term of entropy,

H(γ) ≥ - d 2 ln(E, γ) + C and H(γ) ≥ - d 2 ln(W 2 2 (γ, γ 0 )) + C (24)
These inequalities are the key for all our rates of convergence of (E, γ ε ), W 2 2 (γ ε , γ 0 ) and H(γ ε ) detailed in section 3.2.

Lower bounds on the entropy

The results of this section are completely independent of the optimality of γ ε , and even independent of the fact that γ ε transports µ 0 on µ 1 . Hence in the following, we will replace γ ε by a generic γ ∈ P ac (R 2d ). The results presented in this section are lower bounds on its entropy H(γ). These inequalities rely on the comparison of γ with a Gaussian along directions transverse to the support of a generic optimal plan γ 0 for ε = 0. Indeed we know that any optimal plan γ 0 is supported on Σ = {E = 0} which is approximatively a submanifold of (co)dimension d. We will then show that Edγ as well as W 2 2 (γ, γ 0 ) control the variance of γ along transverse directions to Σ. Finally since under variance constraint the Gaussian has the smallest entropy we will conclude with the wanted lower bounds of equations ( 23) and [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF]. The same strategy will work in the case of twisted costs. However in the quadratic case the proof requires only a global change of variable (Minty's trick) which makes the arguments clearer. We start with the fundamental property of a function that allows for estimates like [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF] or [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF], the quadratic detachment

Definition 3.1 (Quadratic detachment). Let G : R d × R d → R + . We say that G has a quadratic detachment if for any u ∈ R d it exists v u ∈ R d such that ∀(u, v) ∈ R 2d G(u, v) ≥ 1 2 ∥v -v u ∥ 2 (25)
This quadratic detachment property allows for a bound on the differential entropy H(γ) := H(γ|R 2d ) of any plan γ ∈ P ac (R 2d ) :

Proposition 3.2. Let G be a function on R d × R d and γ ∈ P ac (R 2d ) and denote C d := -d 2 ln( 4πe d ). If G has a quadratic detachment H(γ) ≥ - d 2 ln Gdγ + H(γ 1 ) + C d ( 26
)
Where γ 1 is the projection of γ on the first coordinate, ie the first marginal of γ, and where H is the differential entropy (H(γ) = H(γ|R 2d ) and H(γ

1 ) = H(γ 1 |R d ))
Proof. If H(γ) = +∞ or Gdγ = +∞ there is nothing to prove. We assume that H(γ) < +∞ and Gdγ < +∞. Let γ = γ 1 ⊗ γ u be the disintegration of γ with respect to the first coordinate. Gdγ < +∞ implies that γ u has a finite moment of order 2, γ 1 almost everywhere, because G has a quadratic detachment. By additivity of entropy we have

H(γ) = H(γ 1 ) + H(γ u )dγ 1 (u) (27)
However under variance constraint the gaussian with independent coordinates has the smallest entropy thus H(γ u ) ≥ H(N (0, Var(γ u )

d I d )) = -d 2 ln (Var(γ u )) -d 2 ln( 2πe d ).
Moreover, the average of a random variable minimizes the average square distance, so Var(γ u ) ≤ ∥v -v u ∥ 2 dγ u (v) ≤ 2 G(u, v)dγ u (v) where the last inequality holds because G has a quadratic detachment. So we have

H(γ) ≥ H(γ 1 ) - d 2 ln 2 G(u, v)dγ u (v) dγ 1 (u) - d 2 ln( 2πe d ) (28)
Now by concavity of the logarithm, exchanging it with the integral against γ 1 gives an even lower quantity, hence we obtain the desired inequality.

This result shows a clear path to get lower bounds on the entropy of γ ε : typically, it is enough to show that the duality gap has a quadratic detachment to show inequality [START_REF] Léonard | A survey of the Schrödinger problem and some of its connections with optimal transport[END_REF]. However, E has no quadratic detachment in the classic (x, y) coordinates, but in transverse coordinates, that we call Minty's coordinates ( [START_REF] Minty | Monotone (nonlinear) operators in Hilbert space[END_REF]) Definition 3.3 (Minty's coordinates). We wil call Minty's coordinates the coordinates (u, v) = 1 √ 2 (x + y, x -y). The change of variable (x, y) → (u, v) is an isometry and for any convex function f , the graph of the subdifferential of f in the (x, y)-coordinates corresponds to the graph of a 1-Lipschitz function defined on the whole space in the (u, v)-coordinates (see [33, theorem 12.12, theorem 12.25]). Moreover, for γ ∈ P(R 2d ), we will denote by γ the pushforward of γ through this change of variable and call Minty's decomposition of γ the disintegration of γ with respect to the orthogonal projection onto the first coordinate γ = μ ⊗ γu .

The quadratic detachment of E in those coordinates follows from the slightly more general inequality [START_REF] Minty | Monotone (nonlinear) operators in Hilbert space[END_REF], known as Minty's trick. It can be generalized for twisted costs (see lemma 4.4). In the quadratic case, it is somehow folklore and we recall the proof, but for more general cost, it was proved in [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF], following arguments of [START_REF] Robert J Mccann | Rectifiability of optimal transportation plans[END_REF]. We also prove the quadratic detachment of the function d(., Σ) 2 that we will use to bound the entropy with the Wasserstein distance to an optimizer. Lemma 3.4. Let µ 0 , µ 1 ∈ P 2 (R d ) with E := c -(φ ⊕ ψ) the duality gap of the quadratic optimal transport problem from µ 0 to µ 1 . Denote by d(z, Σ) the distance of a point z ∈ R 2d to the set Σ = {E = 0}. In Minty's coordinates, the functions (u, v) → E(u, v) and (u, v) → d((u, v), Σ) 2 have quadratic detachment. 4Proof. First note that in the quadratic case E rewrites as E(x, y) = f (x) + f * (y) -⟨x, y⟩ where f is a convex function and f * is its Legendre-Fenchel conjugate. Thus for (x, y),

(x ′ , y ′ ) ∈ R d × R d we have E(x, y) + E(x ′ , y ′ ) = f (x) + f * (y) -⟨x, y⟩ + f (x ′ ) + f * (y ′ ) -⟨x ′ , y ′ ⟩ ≥ ⟨x, y ′ ⟩ -f * (y ′ ) + f * (y) -⟨x, y⟩ + ⟨x ′ , y⟩ -f (y) + f * (y ′ ) -⟨x ′ , y ′ ⟩ ≥ ⟨x -x ′ , y ′ -y⟩ (29)
In Minty's coordinates (u, v) = 1 √ 2 (x + y, x -y) we can see E as a function of (u, v). In these coordinates, inequality [START_REF] Milgrom | Envelope theorems for arbitrary choice sets[END_REF] becomes

E(u, v) + E(u ′ , v ′ ) ≥ 1 2 (∥v ′ -v∥ 2 -∥u ′ -u∥ 2 ) ( 30
)
The set Σ = {x, y/E(x, y) = 0} corresponds to the subdifferential of the convex function f . Hence in the Minty's coordinates (u, v), it corresponds to the graph in u of a 1-lipschitz function that is defined on the whole space R d (see [33, theorem 12.12, theorem 12.25]). So for any u ∈ R d , it exists a unique v u ∈ R d such that E(u, v u ) = 0. And thanks to Minty's trick [START_REF] Minty | Monotone (nonlinear) operators in Hilbert space[END_REF] we have that

E(u, v) ≥ 1 2 |v -v u | 2 .
It remains to show that the distance to the graph of the 1-lipschitz function u → v u has a quadratic detachment. Let u, u ′ , v ∈ R d . From the inequality

∥a∥ 2 + ∥b∥ 2 ≥ 1 2 ∥a + b∥ 2 applied to a = v u -v u ′ and b = v u ′ -v, we get ∥v u -v u ′ ∥ 2 + ∥v u ′ -v∥ 2 ≥ 1 2 ∥v u -v∥ 2
And we can bound

∥v u -v u ′ ∥ 2 by ∥u -u ′ ∥ 2 because u → v u is 1-lipschitz. Hence we obtain ∥u -u ′ ∥ 2 + ∥v u ′ -v∥ 2 ≥ 1 2 ∥v u -v∥ 2
But the left hand side is the distance from (u, v) to (u ′ , v u ′ ) and the inequality holds for any u ′ , so

d((u, v), Σ) ≥ 1 2 ∥v u -v∥ 2
Now that we know that E and d(., Σ) 2 have quadratic detachment, we can use proposition 3.2 to bound the entropy of γ from bellow with Edγ and W 2 2 (γ, γ 0 ). Proposition 3.5. Suppose that the cost is quadratic. Let (µ 0 , µ 1 ) ∈ P 2 (R d ), (φ, ψ) be associated Kantorovich potentials and E := c-(φ⊕ψ) be a duality gap function. Let γ 0 be an optimal transport plan from µ 0 to µ 1 . Then, for any γ ∈ P ac (R 2d )

H(γ) ≥ - d 2 ln Edγ + H(μ) + C d (31)
H(γ) ≥ - d 2 ln W 2 2 (γ, γ 0 ) + H(μ) + C d (32)
where γ = μ ⊗ γu is the Minty's decomposition of γ and C d := -d 2 ln( 4πe d ) (see definition 3.3). If we denote by σ γ (X + Y ) the standard deviation of X + Y when the law of (X, Y ) is γ, we have

N 2d (γ) ≤ σ γ (X + Y ) d Edγ N 2d (γ) ≤ σ γ (X + Y ) d W 2 (γ, γ 0 ) (33)
Proof. By lemma 3.4, the duality gap E and the square distance d(., Σ) 2 have quadratic detachment. So we can apply proposition 3.2 on both to obtain

H(γ) ≥ - d 2 ln Edγ + H(μ) + C d (34)
H(γ) ≥ - d 2 ln d(x, Σ) 2 dγ(x) + H(μ) + C d (35)
The first inequality is exactly [START_REF] Pal | On the difference between entropic cost and the optimal transport cost[END_REF] and the second implies (32) because an optimal plan γ 0 is concentrated on a contact set Σ = {x, y/E(x, y) = 0} by optimality and so W 2 2 (γ, γ 0 ) ≥ d(x, Σ) 2 dγ(x). The second part is a consequence and we provide only the proof for Edγ because the proof for W 2 2 (γ ε , γ 0 ) is stricly similar. Taking the exponential of (31), we get

N 2d (γ) 2 ≤ 2 d N d (μ) Edγ
Since the gaussian minimizes the entropy at fixed variance, the entropy power of μ is bigger than the one of a gaussian of same variance, that is

N d (μ) ≤ Var(μ) d . Moreover Var(μ) = 1 2 Var γ (X + Y ) because μ is the law of the first marginal of γ in the coordinates (u, v) = 1 √ 2 (x + y, x -y)
. This is enough to finish the proof of [START_REF] Rockafellar | Variational Analysis[END_REF].

We have established the bounds (33) on a general plan γ. It is now time to apply these results to the entropic plan γ ε under hypothesis (H2) in order to obtain rates of convergence of the suboptimality (c, γ ε ) -(c, γ 0 ). We will also quantify the weak convergence of γ ε towards γ 0 in W 2 distance.

Rates of convergence

In the sequel of the section, we use the set of assumptions (H2) that imply in particular that the optimizer γ 0 is unique. The main result of this section states that the suboptimality (c, γ ε ) -(c, γ 0 ) converges to 0 at a speed of order ε. We also derive the same speed for W 2 2 (γ ε , γ 0 ) and this matches in both cases the rate found for gaussians. The proofs of the theorems require an upperbound on the rate of convergence of OT ε towards OT 0 . The key point being that the convergence rate is of the form

εH(γ ε ) ≤ OT ε -OT 0 ≤ - d 2 ε ln(ε) + O(ε) (36)
This result has been obtained recently for general costs under some regularity conditions [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF] and in the quadratic case under moments conditions [START_REF] Eckstein | Convergence Rates for Regularized Optimal Transport via Quantization[END_REF]. The following lemma explicits the dependence of the bounded term O(ε) found in the latter.

Lemma 3.6. Assume as in hypothesis (H2) that the cost is quadratic, that µ 0 , µ 1 ∈ P 2+δ (R d ) for some δ > 0 and that H(µ i ) < +∞. Then, as ε → 0

OT ε -OT 0 ≤ - d 2 ε ln(ε) + Cε (37)
for a constant C depending on d, m 2+δ (µ i ) and H(µ i ).

Proof. The constant C introduced in [15, corollary 3.12] is (up to a numerical constant) the quantization constant of the optimal transport γ 0 . It allows for the following upper bound

OT ε -OT 0 ≤ - d 2 ε ln(ε) + (H(µ 0 ) + H(µ 1 ))ε + Cε (38)
However by Minty's trick [15, lemma 3.11] this quantization constant is tied to the quantization constant of a measure admitting a moment of order 2 + δ. In that case [23, corollary 6.7] gives an explicit bound

C ≤ C ′ (m 2+δ (γ 0 ) + d) (39)
where C ′ is independent of dimension. We conclude by using triangular inequality to upper bound m 2+δ (γ 0 ) by 2 2+δ (m 2+δ (µ 0 ) + m 2+δ (µ 1 )).

With the inequality (37) that gives an upper bound on H(γ ε ) and the lower bound [START_REF] Pal | On the difference between entropic cost and the optimal transport cost[END_REF] found in the previous section, we are now able to show that (c, γ ε ) -(c, γ 0 ) = Θ(ε) :

Theorem 3.7. Suppose that the cost is quadratic. Let µ 0 , µ 1 ∈ P 2+δ,ac (R d ) for some δ > 0. And further assume that their entropy is finite. Then in the limit ε → 0,

(c, γ ε ) -(c, γ 0 ) = Θ(ε), H(γ ε ) = - d 2 ln(ε) + O(1) and OT ε = OT 0 - dε 2 ln(ε) + O(ε) (40)
More precisely, there exist c, C > 0, depending on m 2+δ (µ i ), H(µ i ) and the dimension d, such that

cε ≤ Edγ ε = (c, γ ε ) -(c, γ 0 ) ≤ Cε (41)
Proof. Set (E, γ ε ) := Edγ ε , and consider d, H(µ i ) and m 2+δ (µ i ) as fixed. Lemma 3.6 grants the following upper bound on the regularized problem for ε small enough

(E, γ ε ) + εH(γ ε ) ≤ - d 2 ε ln(ε) + Cε (42)
On the other hand, from proposition 3.5, we know that N 2d (γ ε ) ≤ σγ ε (X+Y ) d (E, γ ε ). But we have also σ γε (X + Y ) ≤ σ γε (X) + σ γε (Y ) = m 2 (µ 0 ) + m 2 (µ 1 ) because the marginals of γ ε are µ 0 and µ 1 . And by Hölder inequality, m 2 (µ) ≤ m 2+δ (µ) 2 2+δ . So N 2d (γ ε ) ≤ C ′ (E, γ ε ) or equivalently, by taking the logarithm,

H(γ ε ) ≥ - d 2 ln (E, γ ε ) + C ′′ (43)
Now by combining (43) with (42) we have

(E, γ ε ) ε - d 2 ln (E, γ ε ) ε ≤ C ′′′ (44)
The function x → x -d/2 ln(x) goes to +∞ near 0 and +∞. Thus (E,γε) ε = Θ(1), or more precisely, there exist constants c, C > 0, depending on d, m 2+δ (µ i ) and

H(µ i ), such that cε ≤ (E, γ ε ) ≤ Cε. Injecting (E, γ ε ) = Θ(ε) in inequalities (42) and (43) gives respectively H(γ ε ) ≤ -d 2 ln(ε) + O(1) and H(γ ε ) ≥ -d 2 ln(ε) + O(1)
which concludes the proof. Now that we have proven that the suboptimality (c, γ ε ) -(c, γ 0 ) has speed ε, we want to do the same for the the Wasserstein distance W 2 2 (γ ε , γ 0 ). Inequality [START_REF] Ripani | Convexity and regularity properties for entropic interpolations[END_REF] will give a lower bound on W 2 2 (γ ε , γ 0 ), but we need also an upper bound. If the Brenier map is Lipschitz, the following lemma provides it. 5Lemma 3.8. For the quadratic cost, suppose that the Brenier map T = ∇f is L Lipschitz. Then

W 2 2 (γ ε , γ 0 ) ≤ ∥y -T (x)∥ 2 dγ ε ≤ 2L ((c, γ ε ) -(c, γ 0 )) (45)
Proof. Let S(x, y) = (x, T (x)) with T = ∇f and f the Brenier convex function associated with γ 0 . Then S is a transport map from γ ε to γ 0 thus W 2 2 (γ ε , γ 0 ) ≤ ∥y -T (x)∥ 2 dγ ε . We now use an inequality proved by Berman in [START_REF] Berman | Convergence Rates for Discretized Monge-Ampère Equations and Quantitative Stability of Optimal Transport[END_REF] and Li and Nochetto in [START_REF] Li | Quantitative stability and error estimates for optimal transport plans[END_REF], who have built upon an earlier work of Gigli [START_REF] Gigli | On Hölder continuity-in-time of the optimal transport map towards measures along a curve[END_REF]. The inequality is that whenever the Brenier map is Lipschitz we have ∥y -T (x)∥ 2 ≤ 2LE with E = c -(φ ⊕ ψ) the duality gap. Combining these two inequalities grants the result, since Edγ ε = (c, γ ε ) -(c, γ 0 ).

The hypothesis that the Brenier map is Lipschitz might seem abstract. However, note that Caffarelli's regularity theory ( [START_REF] Caffarelli | The Regularity of Mappings with a Convex Potential[END_REF], [START_REF] Caffarelli | Monotonicity Properties of Optimal Transportation and the FKG and Related Inequalities[END_REF]) ensures it holds under some simple regularity conditions on the marginals. For example as soon as the marginals have Hölder densities bounded away from zero on their supports and the latter are smooth and uniformly convex. In a different direction, recent works [START_REF] Chewi | An entropic generalization of Caffarelli's contraction theorem via covariance inequalities[END_REF] ensure regularity of the transport map between log-concave measures under a variance constraint.

Theorem 3.9. Suppose that the cost is quadratic. Let µ 0 , µ 1 ∈ P 2+δ,ac (R d ) for some δ > 0. And further assume that their entropy is finite and that the Brenier map T = ∇f is Lipschitz. Then W 2 2 (γ ε , γ 0 ) = Θ(ε). More precisely :

1. if µ 0 , µ 1 ∈ P 2+δ,ac (R d ) with finite entropy, then there exist c > 0, depending on the moments m 2+δ (µ i ), the entropies H(µ i ) and the dimension d, such that, as ε → 0

cε ≤ W 2 2 (γ ε , γ 0 ) (46)
2. if moreover the Brenier map is L-lipschitz, then there exist C > 0 depending on d, m 2+δ (µ i ), H(µ i ) and L, such that, as ε → 0

cε ≤ W 2 2 (γ ε , γ 0 ) ≤ Cε (47)
Proof. For the first part, let's recall that from proposition 3.5, we have

N 2d (γ ε ) ≤ σγ ε (X+Y ) d (E, γ ε ).
As in the proof of theorem 3.7, we bound can bound σ γε (X + Y ) with the moments m 2+δ (µ i ) and so we obtain

N 2d (γ ε ) ≤ C(m 2+δ (µ i ), d)W 2 (γ ε , γ 0 ) (48)
But on the other hand, from lemma 3.6, we now that

H(γ ε ) ≤ -d 2 ln(ε) + C(m 2+δ (µ i ), H(µ i ), d). Or equivalently, N 2d (γ ε ) ≥ C(m 2+δ (µ i ), H(µ i ), d) √ ε (49)
So, combining inequality ( 48) and (49), we get W 2 2 (γ ε , γ 0 ) ≥ cε for some constant c depending on d, H(µ i ) and m 2+δ (µ i ). The second part is a direct combination of lemma 3.8 from which

W 2 2 (γ ε , γ 0 ) ≤ 2L ((c, γ ε ) -(c, γ 0 )) and theorem 3.7 which states that (c, γ ε ) -(c, γ 0 ) ≤ C(m 2+δ (µ i ), H(µ i ), d)ε.
Remark 3.10. Note that, as explained in [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF]Proposition 4.5], the method of lemma 3.8 allows to control also the L 2 -gap between T ε , the barycentric projection of γ ε , and the Brenier map T = ∇f .

||T ε -T || 2 L 2 (µ) ≤ |y -T (x)| 2 dγ ε (x, y) ≤ 2L Edγ ε ≤ Cε
In a similar fashion we get a control over the distance between the support of γ ε and the graph of the optimal transport

d(y, supp(γ 0 )) 2 dγ ε ≤ |y -T (x)| 2 dγ ε (x, y) = O(ε) (50)
Hence, for the quadratic case, under technical assumptions, we have obtain generic rates of convergence when the marginals µ i have finite differential entropy :

(c, γ ε ) = (c, γ 0 ) + Θ(ε) H(γ ε ) = - d 2 ln(ε) + O(1) W 2 (γ ε , γ 0 ) = Θ( √ ε) (51)
Can we generalize it to a broader class of costs ? This is the objective of next section.

Rates for infinitesimally twisted costs

In this section, we use the set of assumption (H3) that is adapted to twisted cost functions. As in the quadratic case, we begin by estimates on the entropy based on quadratic detachment.

Lower bound on the entropy

The key ingredient used to lower bound the entropy was the lower bound involving the variance H(γ) ≥ H(N (0, Var(γ)/d)). In fact we used a finer result using a disintegration of γ in two components, one of which had bounded variance. In order to extend the preceding results to more general costs our goal is to have a local version of the proposition 3.5 that states a precise lower bound for H(γ). Indeed, imagine that E ≥ 0 is a function such that there is a direction along which E grows quadratically locally around Σ = {E = 0}. For any measure γ, it is then possible to bound the variance of γ along that same direction which in turns give a bound on the entropy of γ in a similar fashion to lemma 3.4. In order to state this idea formally we introduce the concept of local quadratic detachment for a positive function.

Definition 4.1 (Local quadratic detachment). Let X ⊂ R d a compact set and E : X × X → R + continuous. Assume that Σ = {E = 0} ̸ = ∅. We says that E has a local quadratic detachment of parameters ((U i ) i , (Φ i ) i , κ) where κ > 0, (U i ) i is a finite open covering of Σ, and (Φ i ) i a family of volume preserving affine functions Φ i : x → (u, v) ∈ R d × R d such that for all (u, v), (u, v ′ ) ∈ Φ i (U i ) E(u, v) + E(u, v ′ ) ≥ κ∥v -v ′ ∥ 2 (52)
Where E denotes E • Φ -1 i by a slight abuse of notation.

This local quadratic detachment property is a direct generalisation of the notion of quadratic detachment (definition 3.1). And it happens that the same result holds: the entropy of γ is lower bounded by the log of the integral of a function having a local quadratic detachment. The idea of proof is the same as in the quadratic case with some added technical issues associated to the locality of the quadratic detachment.

Proposition 4.2. Let X ⊂ R d a compact set and E a continuous function on X × X that has a local quadratic detachment with parameters

((U i ) N i=1 , (Φ i ) N i=1 , κ). Set R = X × X \ i U i .
There exists a constant C depending on X, ∥Φ i ∥ op and N such that for any γ ∈ P(X × X) we have

H(γ) ≥ - d 2 ln E(x)dγ - d 2 ln 4πe κd + d 2 ln 1 ∧ inf x∈R E(x) + C (53)
Proof. First if H(γ) = +∞ or Edγ = +∞, there is nothing to prove. We assume that H(γ) < +∞ thus γ has a density that we denote ρ and Edγ < +∞. Let κ > 0, (U i ) N i=1 and Φ i as in the definition 4.1 of the local quadratic detachment. Let ζ i a partition of unity subordinate to the family (U i ) i . Denote by

U = i U i . If p i = (ζ i ρ)(U ) > 0 let ρ i = 1 pi ζ i ρ, else ρ i = 0.
Then we have the following decomposition ρ = 1 U i p i ρ i + 1 U c ρ which we can inject in the definition of the entropy

H(γ) = ρ ln(ρ) = U i p i ρ i ln( j p j ρ j ) + U c ρ ln(ρ) = i p i ρ i ln( j p j ρ j ) + U c ρ ln(ρ) ≥ i p i ρ i ln(p i ρ i ) + U c ρ ln(ρ) ≥ i p i ln(p i ) + i p i ρ i ln(ρ i ) + U c ρ ln(ρ) ≥ i p i H(ρ i ) - 1 e N + H 2d (X × X) (54)
Where the first inequality holds because p i ρ i ≤ j p j ρ j and the last inequality holds because under support constraint the uniform distribution has the smallest entropy. Now for i such that ρ i ̸ = 0 set

τ i = (Φ i ) # ρ i , since Φ i is volume preserving we have H(τ i ) = H(ρ i ).
We now disintegrate τ i with respect to the projection onto the first coordinate u and denote the disintegration τ i = µ i ⊗ τ u i . By additivity of the entropy we have

H(τ i ) = H(µ i ) + H(τ u i )dµ i (55)
However let u in the support of µ i and let v within the support of

τ u i . Take v ′ ∈ π 2 Φ i (X × X) such that E(u, v ′ ) = min π1Φi(X×X) E(u, .
), which is possible by compactness of π 2 Φ i (X × X) and continuity of E. Then by the local quadratic detachment of E we have

E(u, v) ≥ 1 2 (E(u, v ′ ) + E(u, v)) ≥ κ 2 ∥v ′ -v∥ 2 (56)
Moreover since the variance is the distance to the constants in L 2 space we have

E(u, v)dτ u i ≥ κ 2 ∥v ′ -v∥ 2 dτ u i ≥ κ 2 Var(τ u i ) (57)
Using this control of the variance in conjonction with the fact that the gaussian has lowest entropy under variance constraint we have

H(τ u i ) ≥ - d 2 ln (Var(τ u i )) - d 2 ln 2πe d ≥ - d 2 ln E(u, v)dτ u i - d 2 ln 4πe κd (58)
On the other hand we know that µ i is supported in π 1 Φ i (U i ) the diameter of which is smaller than a constant time that of X, thanks to the continuity of Φ i . Thus since on a bounded support the uniform distribution has the smallest entropy we have

H(µ i ) ≥ -ln(H d (π 1 Φ i (U i ))) = C(∥Φ i ∥ op , X).
Now coming back to equation (55) we have

H(ρ i ) = H(τ i ) = H(µ i ) + H(τ u i )dµ i ≥ - d 2 ln E(u, v)dτ u i dµ i - d 2 ln 4πe κd + C(∥Φ i ∥ op , X) ≥ - d 2 ln E(u, v)dτ i - d 2 ln 4πe κd + C(∥Φ i ∥ op , X) ≥ - d 2 ln E(x)dρ i - d 2 ln 4πe κd + C(∥Φ i ∥ op , X) (59)
where the second to last inequality holds by concavity of the logarithm and the last inequality holds because Φ i is volume preserving. We now multiply equation ( 59) by p i and sum over i in order to combine it with the lower bound term in equation ( 54)

H(γ) ≥ i p i H(ρ i ) - 1 e N + H 2d (X × X) ≥ i p i - d 2 ln E(x)dρ i - d 2 ln 4πe κd + C(∥Φ i ∥ op , X) - 1 e N + H 2d (X × X) ≥ - d 2 ρ(U ) ln U E(x)dρ - d 2 ln 4πe κd + C(N, ∥Φ i ∥ op , X) (60)
where we used the concavity of the logarithm one last time. It remains to prove that there is a constant such that -ρ(U ) ln U Edρ ≥ -ln Edρ + C.

Let E 0 = 1 ∧ inf x∈U c E(x) > 0 because Σ ∩ U c = ∅ is a compact set and E is continuous. Then U c Edρ ≥ E 0 (1 -ρ(U )) which implies -ln Edρ ≤ -ln U Edρ + E 0 (1 -ρ(U ))
. For α ∈ [0, 1] and x > 0 set f : x → -α ln(x) + ln(x + E 0 (1 -α)). We have f (x) ≥ f (αE 0 ) = -α ln(α) + (1α) ln(E 0 ) ≥ ln(E 0 ). This allows us to conclude that

H(γ) ≥ - d 2 ln E(x)dρ - d 2 ln 4πe κd + d 2 ln(E 0 ) + C(N, ∥Φ i ∥ op , X) (61)

Rates of convergence of OT ε

The quadratic case showed that the duality gap function E = c -(φ ⊕ ψ) has a global quadratic detachment, where φ, ψ are any Kantorovich potentials. In fact the duality gap still has a quadratic detachment, though local, in the more general case of infinitesimally twisted costs, which includes situations where the optimal plans are not necessarily given by a map. This is the point of [7, lemma 4.2], which relies on a generalized version of Minty's trick also stating that in charts the support of γ 0 is the graph of a Lipschitz function. They also provide a quadratic lower bound on the duality gap in these local charts. Our goal is to then apply proposition 4.2 in order to control the entropy of the optimal regularized transport plans as ε → 0. We will then be able to conclude by finding a rate of order ε for (c, γ ε ) -(c, γ 0 ) in a similar fashion to the quadratic case. Throughout this section we will work on X a compact subset of R d . We denote by Ω an open set containing X. We will assume that the marginals µ 0 , µ 1 are supported in X and have finite entropy. We start by recalling the results found in [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF].

Definition 4.3 (Infinitesimal twist). Given c ∈ C 2 (Ω 2 ) we say that c is infinitesimally twisted if ∇ 2 xy c(x, y) = (∂ 2 xiyj c(x, y)) i,j ∈ M d (R) is invertible for every (x, y) ∈ Ω 2 .
We now recall the quadratic detachment lemma [7, lemma 4.2]. Their proof closely follow an earlier proof found in [START_REF] Robert J Mccann | Rectifiability of optimal transportation plans[END_REF], however they are interested in points that do not belong to the support of the optimal transport plan.

Lemma 4.4. Let c ∈ C 2 (Ω 2 ) be an infinitesimally twisted cost, and (φ, ψ) ∈ C(X) 2 be a pair of c-conjugate functions. We denote by E = c -φ ⊕ ψ ≥ 0 the duality gap function defined on X × X, by Σ = {E = 0} the contact set and for every r > 0,

τ (r) = sup x,x ′ ∈X×X ∥x ′ -x∥≤r ∥∇ 2 xy c(x ′ ) -1 ∇ 2 xy c(x) -Id∥ ∈ [0, ∞) (62) If x ∈ X × X and x, x ′ ∈ B r (x) ∩ (X × X), then E(x) + E(x ′ ) ≥ ∥∆v∥ 2 -∥∆u∥ 2 -τ (r)(∥∆v∥ 2 + ∥∆u∥ 2 ) ( 63
)
where ∆u = u(x ′ ) -u(x), ∆v = v(x ′ ) -v(x), and

u(x) = 1 2 (x -∇ 2 xy c(x)y), v(x) = 1 2 (x + ∇ 2 xy c(x)y) (64)
for every x = (x, y).

Remark 4.5. The following remark was made in [START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF]. What directly follows from the lemma is that locally, up to a change of variable, the support of γ 0 lies within the graph of a Lipschitz function, with Lipschitz constant arbitrarily close to 1. Indeed for (u, v), (u

′ , v ′ ) ∈ Φ(B(x, r)) ∩ Φ(suppγ 0 ) we have 0 = E(u, v) + E(u ′ , v ′ ) ≥ (1 -τ (r))∥∆v∥ 2 -(1 + τ (r))∥∆u∥ 2 (65)
Thus 1 + τ (r) 1 -τ (r) ∥∆u∥ ≥ ∥∆v∥ (66)
This lemma essentially says that locally around the set Σ = {E = 0} the function E grows at least quadratically along the direction Im(Id + ∇ 2 xy c(x)). In particular when the functions (φ, ψ) are Kantorovich potentials of the optimal transport problem, this gives a quadratic lower bound of the duality gap close to Σ. The following lemma restates lemma 4.4 in the language of the quadratic detachment of the duality gap function.

Lemma 4.6. Using the notations of lemma 4.4. Let (φ, ψ) ∈ C(X) 2 be a pair of c-conjugate functions. Then E = c-(φ⊕ψ) has a local quadratic detachment of parameters ((B(x i , r)) i , (Φ i ) i , κ) where r is such that τ (r) ≤ 1 2 , and

κ = 1 4 inf x∈X 2 |det(∇ 2 xy c(x))| 1/d ).
The familly (B(x i , r)) i is a finite covering of Σ, and

Φ i (x) = α i (u i (x)-u i (x i ), v i (x)-v i (x i )) with α i chosen such that J Φ i = 1 and u i (x) = 1 2 (x -∇ 2 xy c(x i )y), v i (x) = 1 2 (x + ∇ 2 xy c(x i )y) (67)
Proof. We use the notations of lemma 4.4. Let r > 0 such that τ (r) ≤ 1/2 which is possible since by continuity of ∇ 2 xy c over the compact set X 2 . We have τ (r) → 0 as r → 0. Note that E is continuous because φ, ψ are c-conjugate functions and c is continuous. Thus Σ is closed, hence compact. We can choose (x i ) N i=1 ∈ Σ N such that (B(x i , r)) i is a finite covering of Σ. For i ∈ [1, N ] and for every x, x ′ ∈ B(x i , r) such that u i (x) = u i (x ′ ), we have by lemma 4.4

E(x) + E(x ′ ) ≥ (1 -τ (r))∥∆v i ∥ 2 ≥ 1 2α 2 i ∥∆(α i v i )∥ 2 (68)
where v i (x) = 1/2(x + ∇ 2 xy c(x i )y). Note that Φ i is an affine volume preserving map thus we have 1 =

α 2d i 2 d |det(∇ 2 xy c(x i ))|.
The determinant and the cross derivative are continuous which implies

I = inf x∈X 2 | det(∇ 2
xy c(x))| > 0 by the infinitesimal twist condition on c. We conclude that

E(x) + E(x ′ ) ≥ 1 2α 2 i ∥∆(α i v i )∥ 2 ≥ I 1/d 4 ∥∆(α i v i )∥ 2 (69)
We will use this lemma in the specific case ofa duality gap function where φ, ψ are Kantorovich potentials for the optimal transport problem. Note that there is no dependence in ε and thus the quadratic detachment framework can be used to derive the rates found in the quadratic cost case.

It is known (see [START_REF] Eckstein | Convergence Rates for Regularized Optimal Transport via Quantization[END_REF][START_REF] Carlier | Convergence rate of general entropic optimal transport costs[END_REF]) that for infinitesimally twisted cost and compactly supported marginals of finite entropy, the regularized problem εEOT satisfies the following inequality for some real

M OT ε -OT 0 ≤ - d 2 ε ln(ε) + M ε (70)
We formally recall that convergence result in the following lemma.

Lemma 4.7. [15, Theorem 3.8, Lemma 3.13] Assume that µ i are compactly supported and c ∈ C 2 (Ω 2 ) is infinitesimally twisted. Then there is C > 0 such that, as ε → 0

OT ε -OT 0 ≤ - d 2 ε ln(ε) + Cε (71)
Proof. Let X compact such that supp(µ i ) ⊂ X. Then by continuity, ∇ 2 c is bounded. Moreover since µ i are compactly supported [15, lemma 3.13] ensures that the optimal transport plan γ 0 satisfies the right quantization property for [15, theorem 3.8] to apply and grant a constant C > 0 such that

OT ε -OT 0 ≤ - d 2 ε ln(ε) + Cε (72)
Note that OT ε -OT 0 = Edγ ε + εH(γ ε ). Thus in combination with lemma 4.7 and the lower bound on the entropy 4.2 we have the following result.

(µ i ) < ∞. Then (c, γ ε ) -(c, γ 0 ) = Θ(ε), H(γ ε ) = - d 2 ln(ε) + O(1) and OT ε = OT 0 - dε 2 ln(ε) + O(ε) (73)
Proof. We denote by X a compact subset of Ω such that µ i (X) = 1. It is known that the Kantorvich potentials are c-conjugate functions and thus are continuous on Ω. In particular the duality gap function E(x, y) = c(x, y) -φ(x) -ψ(y) is continuous on Ω × Ω. By lemma 4.6 we thus know that E has a local quadratic detachment. Thus proposition 4.2 implies that there is a constant C such that

H(γ ε ) ≥ - d 2 ln Edγ ε + C (74)
Now combining this inequation with the upper bound on the rate of convergence (70) we have

- d 2 ε ln(ε) + Cε ≥ Edγ ε + εH(γ ε ) ≥ Edγ ε - d 2 ε ln Edγ ε + Cε (75)
Dividing both sides by ε and adding ln(ε) we get

Edγ ε ε - d 2 ln Edγ ε ε ≤ C (76)
And we conclude as in theorem 3.7.

4.3 Lower bound on W 2 (γ ε , γ 0 )

We now focus on the lower bound of the Wasserstein distance between γ ε and γ 0 . Unlike the quadratic case we won't be able to directly use a quadratic detachment for the Wasserstein distance.

For ε > 0, by construction γ ε is absolutely continuous with respect to the Lebesgue measure, thus there is an optimal transport map for the quadratic cost from γ ε to γ 0 . In particular this map is the gradient of a convex function f . We are now able to write the Wasserstein distance as W 2 2 (γ ε , γ 0 ) = ∥x -∇f (x)∥ 2 dγ ε (x). Inspired by the quadratic case and the last section we could say that x → ∥x -∇f (x)∥ 2 has a local quadratic detachment, but this function is not necessarily continuous which prevents us from applying the results on quadratic detachment. However the spirit of the proofs remains true and the results of this section are essentially an adaptation of the results presented before. First the Wasserstein distance between two measures satisfies a property close to a quadratic detachment whenever one of the measure is supported on a Lipschitz graph. Lemma 4.9. Let E a subspace of R n , T : E → E ⊥ and µ = (Id × T ) # µ 0 with µ 0 a probability on E with finite moment of order 2. Suppose that T is L-Lipschitz. Then for any ν ∈ P 2,ac (R d )

W 2 2 (µ, ν) ≥ (1 -L)W 2 2 (µ 0 , ν 0 ) + 1 L + 1 V ar(ν x )dν 0 (x) ( 77
)
where ν = ν 0 ⊗ ν x is the disintegration of ν with regard to the orthogonal projection on E. In particular

(1 + L) 2 W 2 2 (µ, ν) ≥ V ar(ν x )dν 0 (x) (78)
Proof. Let π be the optimal coupling between µ, ν. Let (X ′ , Y ′ ) ∼ µ and (X, Y ) ∼ ν such that (X ′ , Y ′ , X, Y ) ∼ π. Thus

W 2 2 (µ, ν) = E π ∥X -X ′ ∥ 2 + ∥Y -T (X ′ )∥ 2 (79)
. Now using that T is L-Lipschitz:

L∥X -X ′ ∥ + ∥Y -T (X ′ )∥ ≥ ∥Y -T (X)∥.

(80)

and taking the square

L 2 ∥X -X ′ ∥ 2 + ∥Y -T (X ′ )∥ 2 + 2L∥X -X ′ ∥∥Y -T (X ′ )∥ ≥ ∥Y -T (X)∥ 2 . ( 81
)
Thus we have

W 2 2 (ν, µ) ≥ E π ∥X -X ′ ∥ 2 + ∥Y -T (X ′ )∥ 2 ≥ (1 -L 2 )E π ∥X -X ′ ∥ 2 + E π ∥Y -T (X)∥ 2 -2LE π [∥X -X ′ ∥∥Y -T (X ′ )∥] (82)
but the last term satisfies by Young inequality ab ≤ 1 2 (a 2 + b 2 )

E π [∥X -X ′ ∥∥Y -T (X ′ )∥] ≤ 1 2 (E π ∥X -X ′ ∥ 2 + E π ∥Y -T (X ′ )∥ 2 ) ≤ W 2 2 (µ, ν) 2 (83)
Inequality (82) becomes

W 2 2 (µ, ν)(1 + L) ≥ (1 -L 2 )E π ∥X -X ′ ∥ 2 + E π ∥Y -T (X)∥ 2 (84)
and finally we have

W 2 2 (µ, ν) ≥ (1 -L)E π ∥X -X ′ ∥ 2 + 1 1 + L E π ∥Y -T (X)∥ 2 ≥ (1 -L)W 2 2 (µ 0 , ν 0 ) + 1 1 + L E νx ∥Y -E [Y | X = x] ∥ 2 dν 0 ≥ (1 -L)W 2 2 (µ 0 , ν 0 ) + 1 1 + L V ar(ν x )dν 0 (85)
Where the second inequality is true since the mean is the L 2 orthogonal projection of the random variable on the space of constants. Now since W 2 2 (µ, ν) ≥ W 2 2 (µ 0 , ν 0 ) we have (1 + L)W 2 2 (µ, ν) ≥ W 2 2 (µ, ν) + LW 2 2 (µ 0 , ν 0 ) ≥ W 2 2 (µ 0 , ν 0 ) + 1 1 + L V ar(ν x )dν 0 (86)

Which finally grants (1 + L) 2 W 2 2 (µ, ν) ≥ V ar(ν x )dν 0 . The proof can be slightly modified to show that the function x → d(x, Γ) 2 where Γ is the graph of a Lipschitz function has a quadratic detachment. As stated before, to use the local quadratic detachment apparatus one could show that x → ∥x -∇f (x)∥ 2 has a local quadratic detachment where ∇f is the optimal transport from γ ε to γ 0 . And thanks to remark 4.5, γ 0 is supported on a Lipschitz graph on charts. In particular if on an open set U , γ 0 is the graph of a Lipschitz function, then x → ∥x -∇f (x)∥ 2 satisfies a quadratic detachment on (∇f ) -1 (U ). However since we have no information on the regularity of the transport map we cannot conclude that (∇f ) -1 (U ) is open and observe a local quadratic detachment. Thus we have to adapt the proof of proposition 4.2 in order to manage this issue and derive a lower bound for the entropy of γ ε using W 2 2 (γ ε , γ 0 ). Proposition 4.10. Let µ i ∈ P(Ω) two compactly supported measures of finite entropy. Then there exists C > 0 such that as ε → 0

H(γ ε ) ≥ - d 2 ln W 2 2 (γ ε , γ 0 ) + C (87)
Proof. Throughout we use the notations of lemma 4.4. Since γ ε has a density, there is a transport map T for the optimal transport problem with quadratic cost from γ ε to γ 0 . Let r > 0 such that τ (r) ≤ 1 2 . Let (U i = B(x i , r) i an open covering of the support of γ 0 with x i ∈ suppγ 0 . Let (ζ i ) i a partition of unity subordinate to the covering (U i ) i . For i set γ i ε = 1 pi ζ i • T γ ε and γ i 0 = 1 pi ζ i γ 0 , where p i is the normalization constant which is independent of ε. Note that T transports γ i ε to γ i 0 . Thus we have W 2 2 (γ ε , γ 0 ) ≥ i p i W 2 2 (γ i ε , γ i 0 ) We now introduce Φ i : Φ i (x, y) = α i (u i (x, y)u i (x i ), v i (x, y) -v i (x i )) where α i is such that Φ i is volume preserving. Let τ i ε , τ i 0 the pushforwards of γ i ε and γ i 0 with respect to this map. We disintegrate τ i ε with respect to the projection on the variable u and denote it µ i ε ⊗ τ i,u ε . Note that Φ i is not an isometry, however it is an affine map thus W 2 2 (γ i ε , γ i 0 ) ≥ C i W 2 2 (τ i ε , τ i 0 ), where C i = 1/∥Φ i ∥ op . Note that C i only depends on ∇ 2 xy c(x i ), thus it is independent of ε. It was pointed out in remark 4.5 that for every i, τ i 0 is supported on the graph of a √ 3-Lipschitz function. Thus thanks to lemma 4.9

H(γ i ε ) = H(τ i ε ) = H(µ i ε ) + H(τ i,u ε )dµ i ε ≥ H(µ i ε ) - d 2 ln V ar(τ i,u ε )dµ i ε + C ≥ H(µ i ε ) - d 2 ln W 2 2 (γ i ε , γ i 0 ) + d 2 ln C i √ 3 + 1 + C ≥ - d 2 ln W 2 2 (γ i ε , γ i 0 ) + C(X) (88)
Where the last inequality holds because µ i ε is supported on a compact set of diameter comparable to that of X, as seen in the proof of theorem 4.2. We proceed as in the proof of proposition 4.2 . Thus by summing over i we get

H(γ ε ) = i p i ln(p i ) + i p i H(γ i ε ) ≥ - d 2 i p i ln W 2 2 (γ i ε , γ i 0 ) + C ≥ - d 2 ln W 2 2 (γ ε , γ 0 ) + C (89)

Theorem 4 . 8 .

 48 Let c ∈ C 2 (Ω 2 ) infinitesimally twisted. Let µ i ∈ P ac (Ω) two compactly supported measures satsifying H

φ(x) = infy c(x, y) -ψ(y) and ψ(y) = infx c(x, y) -φ(x)

Remark that since the functional N is multiplicative, the right hand side is the 4d entropy power of µ 0 ⊗ µ 1 ⊗ N (0, ε 2d I 2d )

see the introduction of[START_REF] Conforti | A formula for the time derivative of the entropic cost and applications[END_REF] for a discussion on the name energy

here, by a slight abuse of notation, E denotes E • Φ -1 where Φ : (x, y) → (u, v) is Minty's change of variable (definition3.3)

This lemma and its proof was suggested to us by P. Pegon

Where the last inequality holds by concavity of the logarithm.

It remains to combine the last result with the rate of convergence of (εEOT) in order to retrieve the negative result on the Wasserstein distance between γ ε and γ 0 . Theorem 4.11. Let c ∈ C(Ω 2 ) infinitesimally twisted. Let µ i ∈ P(Ω) two compactly supported measures of finite entropy. Then there exists c > 0 such that as ε → 0

Proof. Lemma 4.7 ensures that

And proposition 4.10 grants the following lower bound for the entropy

Combining the two equations together we have

Taking the exponential grants the result.

Remark that in the general case, we obtain only the domination ε = O(W 2 2 (γ ε , γ 0 )) whereas in the quadratic case, even if it is under strong assumptions, we could obtain W 2 2 (γ ε , γ 0 ) = Θ(ε) (theorem 3.9). The reason is not that the solution are not smooth enough. Indeed, there exist some precise assumptions, such as the Ma-Trudinger-Wang conditions (see [START_REF] Ma | Regularity of potential functions of the optimal transportation problem[END_REF]), that guarantee regularity of the solutions of the dual problem. The true difficulty is to replace the formula W 2 2 (γ ε , γ 0 ) ≤ L(E, γ ε ) of lemma 3.8 that transform this regularity (the lipschitz constant L of the transport map) into a bound of W 2 2 (γ ε , γ 0 ).


R Ravaud

G Lemarquand

V Lemarquand

Force and Stiffness of Passive Magnetic Bearings Using Permanent Magnets. Part 2 : Radial Magnetization

Keywords: Magnetic forces, analytical calculation, ring permanent magnet, magnetic bearing

This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is radial. Such a configuration corresponds to a passive magnetic bearing. The magnetic force exerted between ring permanent magnets is determined by using the coulombian model. The expressions obtained are semi-analytical and we show that it is not possible to find an exact analytical expression of the force between two ring permanent magnets. Then, thanks to these semi-analytical calculations, the ring dimensions are optimized in order to have a great force or a great stiffness. Moreover, we show that the relative position of the rings for which the force is the strongest depends on the air gap dimension. This result is new because the curvature effect is taken into account in this paper.

We can say that such semi-analytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite element method. Moreover, semi-analytical expressions have a low computational cost whereas the finite element method has a high one. Thereby, as shown in this paper, such calculations allow an easy optimization of quadripolar lenses or devices using permanent magnets.

I. INTRODUCTION

The first studies concerning passive magnetic bearings with permanent magnets have been done by Yonnet [START_REF] Yonnet | Passive magnetic bearings with permanent magnets[END_REF] [START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF]. Such passive magnetic bearings used only permanent magnets radially or axially magnetized. Permanent magnets are commonly used in many electrical devices and engineering applications. Most engineering applications need several ring permanent magnets and the determination of the magnetic force between them is thus required. It is to be noted that several way of obtaining the magnetic forces between rings are possible. Authors generally use either numerical means or 2D analytical calculations for determining magnetic fields or magnetic forces created between ring permanent magnets [3]- [START_REF] Akoun | 3d analytical calculation of the forces exerted between two cuboidal magnets[END_REF]. One of the major problem of a numerical approach lies in the fact that it has a high computational cost. Moreover, numerical algorithms are not so precise as analytical calculations. Alternative solutions are thus required. As it is difficult to obtain fully analytical expressions of the magnetic field created by ring permanent magnets [START_REF] Kim | Mutual inductance of noncoaxial circular coils with constant current density[END_REF]- [START_REF] Kim | Restoring force between two noncoaxial circular coils[END_REF], the determination of the forces between them is still more difficult. Consequently, authors use semi-analytical expressions which are very important steps in order to evaluate either the magnetic field created by ring permanent magnets or the magnetic forces exerted between them. The determination of the magnetic field created by ring permanent magnets have been studied a lot with such approaches [START_REF] Babic | New expression for calculating magnetic fields due to current-carrying solid conductors[END_REF]- [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF].

Another way of calculating the forces between ring permanent magnets can be done by using the 2D analytical expressions of the magnetic field created by infinite parallelepipedic magnets. Assuming that the magnetic field created by ring permanent magnets can be approximated by the magnetic field created by infinite parallelepipedic magnets either to the faces of the magnets (2D analytical 1. Representation of the two ring permanent magnets which are radially centered. The inner radius of the outer ring is denoted r in , the outer one is rout, its height is h. The inner radius of the inner ring is denoted r in2 , the outer one is r out2 , its height is z b -z a .

approach) [START_REF] Lemarquand | Nonsinusoidal torque of permanent-magnet couplings[END_REF]- [START_REF] Zhu | Comparison of halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet[END_REF], the magnetic forces can be calculated more easily with the 2D analytical approach.

However, these formula are not valid when the ring radius is small [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF]- [START_REF] Ravaud | Discussion about the analytical calculation of the magnetic field created by permanent magnets[END_REF]. In any case, the magnetic field created by ring permanent magnets can be determined in terms of elliptic integrals [START_REF] Masson | Electrostatique[END_REF] for ring permanent magnets axially magnetized. Such an approach is appropriate because the algorithms used to calculate elliptic integrals are both very robust and fast. Consequently, the expressions obtained have a very low computational cost (less than 0.2 s to determine the magnetic components of the field created by ring permanent magnets whose polarization is axial).

First, this paper presents useful semi-analytical expressions of the force and the stiffness exerted between two ring permanent magnets whose polarization is radial. Then, this paper explains why we cannot reduce some numerical integrations of the semianalytical expressions of the force between two ring permanent magnets. Eventually, this paper deals with the optimization of the ring dimensions in order to have either a great force or a great stiffness between the rings. We show that the relative position between the rings is of great importance in the design of devices using ring permanent magnets. All the expressions determined in this paper are available online [27].

II. CALCULATION OF THE AXIAL FORCE BETWEEN TWO RING PERMANENT MAGNETS WHOSE POLARIZATION IS RADIAL

This section presents a semi-analytical calculation of the force exerted between two ring permanent magnets whose polarization is radial and which are radially centered. Such configuration corresponds to an axial passive magnetic bearing using ring magnets radially magnetized for mutual attraction. We can say that the devices realized with ring permanent magnets radially magnetized were the first to be built. Moreover, they are usually made of several sections which are stacked together.

A. Notation and geometry

The geometry considered is shown in Fig 1 . A two dimensional representation of the passive magnetic bearing is shown in Fig 2.

The outer radius of the outer ring is r out and the inner one is r in . The outer ring height is h. The outer radius of the inner ring is r out2 and the inner one is r in2 . The inner ring height is z b -z a . It is to be noted that the coulombian model of a permanent magnet is used. Consequently, each ring permanent magnet is represented by both two curved planes which correspond to the inner and outer Fig. 2. Radial magnetic bearing using magnets radially magnetized for mutual attraction. The inner radius of the outer ring is denoted r in , the outer one is r out , its height is h. The inner radius of the inner ring is denoted r in2 , the outer one is r out2 , its height is z b -z a .

faces of the rings which are charged with a surface magnetic pole density σ * and a magnetic pole volume density σ * v . For each case, the inner face is charged with the surface magnetic pole density +σ * and the outer one is charged with the surface magnetic pole density -σ * . It is noted that all the illustrative calculations are done with σ * = J. n = 1T where J is the magnetic polarization vector and n is the unit normal vector which is directed towards 0. Moreover, it is noted that the magnetic pole volume density exits for ring permanent magnets whose polarization is radial in order to to have a charge equilibrium of the ring magnet.

B. Semi-analytical expression of the magnetic force

The two ring permanent magnets which form an axial passive bearing are supposed to be radially centered. Consequently, there is only the axial component of the magnetic force which is exerted between the two rings. We can call it the axial force F z . This axial force can be determined by integrating the magnetic field created by the outer ring on the contributions of the inner one. We must take into account both the magnetic pole surface density and the magnetic pole volume density of each ring. Consequently, as there are two magnetic pole surface densities and one magnetic pole volume density for each ring permanent magnet, we have nine terms for determining the axial force between the two ring permanent magnets. By denoting H z the axial component of the magnetic field produced by the outer ring permanent magnet, Thus, this axial force F z can be written as follows:

F z = (S in ) H z σ * 2 d S - (S out ) H z σ * 2 d S + (V ) H z σ * 2 r 2 d Ṽ (1)
where σ * 2 is the magnetic pole surface density owing to the inner ring permanent magnet, (S in ) is the inner face of the inner ring permanent magnet and (S out ) is the outer face of the inner ring permanent magnet and (V ) is the volume of the inner ring permanent magnet. Therefore, the axial force F z can be expressed as follows:

F z = - 2π θ 1 =0 h z 1 =0 2π θ 2 =0 z b z 2 =z a a(r in , r out2 )dz 1 dθ 1 dz 2 dθ 2 + 2π θ1=0 h z1=0 2π θ2=0 z b z2=za a(r out , r out2 )dz 1 dθ 1 dz 2 dθ 2 + 2π θ 1 =0 h z 1 =0 2π θ 2 =0 z b z 2 =z a a(r in , r in2 )dz 1 dθ 1 dz 2 dθ 2 - 2π θ1=0 h z1=0 2π θ2=0 z b z2=za a(r out , r in2 )dz 1 dθ 1 dz 2 dθ 2 - r out r 1 =r in 2π θ 1 =0 h z 1 =0 2π θ 2 =0 z b z 2 =z a b(r out2 , r 1 )dr 1 dz 1 dθ 1 dz 2 dθ 2 + rout r1=rin 2π θ1=0 h z1=0 2π θ2=0 z b z2=za b(r in2 , r 1 )dr 1 dz 1 dθ 1 dz 2 dθ 2 + r out2 r 2 =r in2 2π θ 1 =0 h z 1 =0 2π θ 2 =0 z b z 2 =z a b(r in , r 2 )dz 1 dθ 1 dr 2 dz 2 dθ 2 - rout2 r2=rin2 2π θ1=0 h z1=0 2π θ2=0 z b z2=za b(r out , r 2 )dz 1 dθ 1 dr 2 dz 2 dθ 2 + r out r 1 =r in r out2 r 2 =r in2 2π θ 1 =0 h z 1 =0 2π θ 2 =0 z b z 2 =z a c(r 1 , r 2 )dr 1 dz 1 dθ 1 dr 2 dz 2 dθ 2 (2) with a(α, β) = σ * 1 σ * 2 4πµ 0 (z 2 -z 1 )αβ (α 2 + β 2 -2αβ + (z 2 -z 1 ) 2 ) 3 2 (3)
where σ * 1 is the magnetic pole surface density owing to the inner ring permanent magnet. and

b(α, β) = a(α, β) β (4) c(α, β) = a(α, β) αβ (5)
The next step is to evaluate (2). If we determined directly F z for given values with all the integrals presented in (2), the computational would be too high because several numerical integrals should be determined at a time. Consequently, we must reduce the number of integrals by integrating analytically a(α, β), b(α, β) and c(α, β) according to the integral variables. Unfortunately, it is not possible to find a fully analytical expression of the magnetic force between two ring permanent magnets but we can use a useful semi-analytical expression which is expressed as follows:

F z = σ * 1 σ * 2 2µ 0 2π θ 1 =0 Sdθ 1 + σ * 1 σ * 2 2µ 0 2π θ 1 =0 M dθ 1 + σ * 1 σ * 2 2µ 0 2π θ1=0 rout2 r2=rin2 V dθ 1 dr 2 (6)
where S corresponds to the magnetic interaction between the magnetic pole surface densities of each ring permanent magnet.

Then, M corresponds to the magnetic interaction between the magnetic pole surface densities of one ring permanent magnet and the magnetic pole volume density of the other one. At least, V corresponds to the magnetic interaction between the magnetic pole volume densities of each ring permanent magnet.

The first contribution S is given by [START_REF] Selvaggi | Calculating the external magnetic field from permanent magnets in permanent-magnet motors -an alternative method[END_REF].

S = -f (z a , z b , h, θ 1 , r in , r out2 ) +f (z a , z b , h, θ 1 , r out , r out2 ) +f (z a , z b , h, θ 1 , r in , r in2 ) -f (z a , z b , h, θ 1 , r out , r in2 ) (7)
with

f (α 1 , α 2 , α 3 , θ 1 , α 5 , α 6 ) = α 5 α 6 log α 3 -α 1 + α 2 5 + α 2 6 + (α 3 -α 1 ) 2 -2α 5 α 6 cos(θ 1 ) +α 5 α 6 log α 1 + α 2 5 + α 2 6 + α 2 1 -2α 5 α 6 cos(θ 1 ) -α 5 α 6 log α 3 -α 2 + α 2 5 + α 2 6 + (α 3 -α 2 ) 2 -2α 5 α 6 cos(θ 1 ) -α 5 α 6 log α 2 + α 2 5 + α 2 6 + α 2 2 -2α 5 α 6 cos(θ 1 ) (8)
The second contribution M is given by [START_REF] Conway | Noncoaxial inductance calculations without the vector potential for axisymmetric coils and planar coils[END_REF].

M = -t(r in , r out , r out2 , h, z a , z b , θ 1 ) +t(r in , r out , r in2 , h, z a , z b , θ 1 ) +t(r in , r out , r in , h, z a , z b , θ 1 ) -t(r in , r out , r out , h, z a , z b , θ 1 ) (9)
with

t(β 1 , β 2 , β 3 , β 4 , β 5 , β 6 , θ 1 ) = t (2) β 1 , β 2 , β 4 -β 5 , β 2 3 + (β 4 -β 5 ) 2 , 2β 3 cos(θ 1 )
+t (2)

β 1 , β 2 , β 5 , β 2 3 + β 2 5 , 2β 3 cos(θ 1 ) -t (2) β 1 , β 2 , β 4 -β 6 , β 2 3 + (β 4 -β 6 ) 2 , 2β 3 cos(θ 1 ) -t (2) β 1 , β 2 , β 6 , β 2 3 + β 2 6 , 2β 3 cos(θ 1 ) (10)
and

t (2) (β 1 , β 2 , q, d, f ) = t (3) (β 2 , q, d, f ) -t (3) (β 1 , q, d, f ) (11)
and

t (3) (s, q, d, f ) = -s + 4d -f 2 -4q 2 2 arctan -f + 2s 4d -f 2 -4q 2 - f 4 log d -q 2 -f s + s 2 +s log q + d -f s + s 2 + q log -f + 2(s + d -f s + s 2 ) - 4d -f 2 -4q 2 + f η log [u 1 ] 4η - -4d + f 2 + 4q 2 + f η log [u 2 ] 4η (12
)

u 1 = - 2 f 2 + 4f q 2 -f 2 (η + 2s) + 4d(-f + η + 2s) q 2 (-4d + f 2 + 4q 2 -f η)(-f + η + 2s) - 8q(-2qs + η d -f s + s 2 ) q 2 (-4d + f 2 + 4q 2 -f η)(-f + η + 2s) ( 13
)
u 2 = - 2 f 2 + 4f q 2 -f 2 (η -2s) -4d(f + η -2s) q 2 (-4d + f 2 + 4q 2 + f η)(f + η -2s) - -8q(2qs + η d -f s + s 2 ) q 2 (-4d + f 2 + 4q 2 + f η)(f + η -2s) (14)
with

η = -4d + f 2 + 4q 2 (15)
The third contribution V is given by [START_REF] Azzerboni | Magnetic field evaluation for disk conductors[END_REF].

V = th (1) (r out , r 2 , z a , z b , h, θ 1 ) -th (1) (r in , r 2 , z a , z b , h, θ 1 ) (16)
with

th (1) = t (3) (r 1 , h -z a , r 2 2 + (h -z a ) 2 , 2r 2 cos(θ 1 )) +t (3) (r 1 , z a , r 2 2 + z 2 a , 2r 2 cos(θ 1 )) -t (3) (r 1 , h -z b , r 2 2 + (h -z b ) 2 , 2r 2 cos(θ 1 )) -t (3) (r 1 , z b , r 2 2 + z 2 b , 2r 2 cos(θ 1 )) (17)

C. Expression of the axial stiffness between two rings whose polarization is radial

The stiffness K exerted between two ring permanent magnets can be determined by calculating the derivative of the axial force with respect to z a . We set z b = z a + b where b is the height of the inner ring permanent magnet. Thus, the axial stiffness K can be calculated with [START_REF] Rakotoarison | Using coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization[END_REF].

K = - ∂ ∂z a F z ( 18
)
where F z is given by (2). We obtain :

K = K S + K M + K V ( 19
)
where K S corresponds to the stiffness determined with only the magnetic pole surface densities of each ring permanent magnet. Then, K M corresponds to the stiffness determined with the magnetic interaction between the magnetic pole surface densities of one ring permanent magnet and the magnetic pole volume density of the other one. At least, K V corresponds to the stiffness determined with the magnetic interaction between the magnetic pole volume densities of each ring permanent magnet. Thus, the first contribution K S is expressed as follows:

K S = η 31 1 √ α 31 K * - 4r 3 r 1 α 31 - 1 √ β 31 K * - 4r 3 r 1 β 31 + 1 √ δ 31 K * - 4r 3 r 1 δ 31 - 1 √ γ 31 K * - 4r 3 r 1 γ 31 +η 41 1 √ α 41 K * - 4r 4 r 1 α 41 - 1 √ β 41 K * - 4r 4 r 1 β 41 + 1 √ δ 41 K * - 4r 4 r 1 δ 41 - 1 √ γ 41 K * - 4r 4 r 1 γ 41 +η 32 1 √ α 32 K * - 4r 3 r 2 α 32 - 1 √ β 32 K * - 4r 3 r 2 β 32 + 1 √ δ 32 K * - 4r 3 r 2 δ 32 - 1 √ γ 32 K * - 4r 3 r 2 γ 32 +η 42 1 √ α 42 K * - 4r 4 r 2 α 42 - 1 √ β 42 K * - 4r 4 r 2 β 42 + 1 √ δ 42 K * - 4r 4 r 2 δ 42 - 1 √ γ 42 K * - 4r 4 r 2 γ 42 (20)
with

η ij = 2r i r j σ * µ 0 ( 21
)
α ij = (r i -r j ) 2 + z 2 a ( 22
)
β ij = (r i -r j ) 2 + (z a + h) 2 (23)
γ ij = (r i -r j ) 2 + (z a -h) ( 24
)
δ ij = (r i -r j ) 2 + (b -h) 2 + z a (2b -2h + z a ) (25)
K * [m] = π 2 0 1 1 -m sin(θ) 2 dθ ( 26
)
The second contribution K M is expressed as follows:

K M = σ * 1 σ * 2 2µ 0 2π θ=0 udθ ( 27
)
with

u = f (r in , r out , r in2 , h, z a , b, θ) -f (r in , r out , r out2 , h, z a , b, θ) +f (r in2 , r out2 , r in , h, z a , b, θ) -f (r in2 , r out2 , r out , h, z a , b, θ) (28)
and

f (α, β, γ, h, z a , b, θ) = -γ log α -γ cos(θ) + α 2 + γ 2 + z 2 a -2αγ cos(θ) +γ log α -γ cos(θ) + α 2 + γ 2 + (z a + b) 2 -2αγ cos(θ) +γ log α -γ cos(θ) + α 2 + γ 2 + (z a -h) 2 -2αγ cos(θ) -γ log α -γ cos(θ) + α 2 + γ 2 + (b -h) 2 + 2z a (b -h) + z 2 a -2αγ cos(θ) +γ log β -γ cos(θ) + β 2 + γ 2 + z 2 a -2αγ cos(θ) -γ log β -γ cos(θ) + β 2 + γ 2 + (z a + b) 2 -2αγ cos(θ) +γ log β -γ cos(θ) + β 2 + γ 2 + (z a -h) 2 -2αγ cos(θ) -γ log β -γ cos(θ) + β 2 + γ 2 + (b -h) 2 + 2z a (b -h) + z 2 a -2αγ cos(θ) (29)
The third contribution K V is expressed as follows:

K V = σ * 1 σ * 2 2µ 0 2π θ=0 r out r 1 =r in δdθ (30) with δ = -log r in2 -r 1 cos(θ) + r 2 1 + r 2 in2 + z 2 a -2r 1 r in2 cos(θ) + log r in2 -r 1 cos(θ) + r 2 1 + r 2 in2 + (z a + b) 2 -2r 1 r in2 cos(θ) -log r in2 -r 1 cos(θ) + r 2 1 + r 2 in2 + (b -h) 2 + 2bz a -2hz a + z 2 a -2r 1 r in2 cos(θ) + log r in2 -r 1 cos(θ) + r 2 1 + r 2 in2 + (z a -h) 2 -2r 1 r in2 cos(θ) + log r out2 -r 1 cos(θ) + r 2 1 + r 2 out2 + z 2 a -2r 1 r out2 cos(θ) -log r out2 -r 1 cos(θ) + r 2 1 + r 2 out2 + (z a + b) 2 -2r 1 r out2 cos(θ) -log r out2 -r 1 cos(θ) + r 2 1 + r 2 out2 + (z a -h) 2 -2r 1 r out2 cos(θ) + log r out2 -r 1 cos(θ) + r 2 1 + r 2 out2 + (b -h) 2 + 2bz a -2hz a + z 2 a -2r 1 r out2 cos(θ) (31)
We can say that the expression of the axial stiffness can be determined analytically if we only take into account the magnetic pole surface densities of each ring.

III. DISCUSSION ABOUT THE POSSIBILITY OF REDUCING THE NUMBER OF NUMERICAL INTEGRATIONS FOR THE AXIAL FORCE

EXPRESSION

The aim of this section is to explain why we cannot reduce the number of numerical integrations of the semi-analytical expression of the force exerted between two ring permanent magnets radially magnetized. They correspond physically to three kinds of interactions between two ring permanent magnets whose polarization is radial.

A. Interaction between the contributions of the surface densities

The first kind of physical interaction is due to the surface contributions of each ring permanent magnet. This physical interaction corresponds to the case when the surface densities of the outer ring are integrated with the surface densities of the inner one. In Eq.

(2), these surface contributions correspond to the integrand denoted a(α, β) where α and β can be r in , r out , r in2 and r out2 . Let us consider the integrand a(α, β). As we can see in Eq. ( 2), the integration variables depend on θ 1 , θ 2 , z 1 and z 2 . The integration according θ 2 does not change the form of the integrand a(α, β) because a(α, β) does not depend on θ 2 . Consequently, we can say that the form of the force F s between the surface density contributions of each ring permanent magnet is given as follows :

F s = θ1 z1 z2 a 1 (z 2 -z 1 ) (a 2 -a 3 cos(θ 1 ) + (z 2 -z 1 ) 2 ) 3 2 dθ 1 dz 1 dz 2 (32)
where a 1 , a 2 and a 3 are constant. For example, we can have

a 1 = 2π(h-z a ), a 2 = 2π(r 2 in +r 2 out +(h-z a ) 2
) and a 3 = 2π(-2r in r out2 ) for our illustration here. In short, these parameters are given by (2) and correspond to the case when four integrals must be determined.

After having integrated according z 1 and z 2 , we obtain a semi-analytical expression with only one numerical integration whose variable depends on θ 1 :

F s = θ 1 log a 4 + a 5 + a 6 cos(θ 1 ) dθ 1 (33)
These parameters are given by ( 8) and they are not a function of the angle θ. It is to be noted that (33) cannot be integrated analytically.

B. Interaction between the contributions of the surface and volume densities

The second kind of physical interaction is due to both the surface and the volume contributions of each ring permanent magnet.

This physical interaction corresponds to the case when the surface densities of each ring permanent magnet are integrated with the volume densities of the other ones. In Eq. ( 2), these surface contributions correspond to the integrand denoted b(α, β) where α and β can be r in , r out , r in2 , r out2 , r 1 and r 2 . Let us consider the integrand b(α, β). As we can see in Eq. ( 2), the integration variables depend on θ 1 , θ 2 , z 1 , z 2 , r 1 and r 2 . The integration according θ 2 does not change the form of the integrand b(α, β) because b(α, β)

does not depend on θ 2 . Consequently, we can say that the form of the force F vs between the surface density contributions and the volume density contributions of each ring permanent magnet is given as follows :

F vs = θ 1 r i z 1 z 2 b 1 (z 2 -z 1 ) (b 2 -b 3 cos(θ 1 ) + (z 2 -z 1 ) 2 ) 3 2 dθ 1 dz 1 dz 2 dr i ( 34
)
where b 1 , b 2 and b 3 are constant. These parameters are given by ( 2) and correspond to the case when five integrals must be determined.

After having integrated according z 1 , z 2 and r 1 , we obtain a semi-analytical expression with only one numerical integration whose variable depends on θ 1 : this semi-analytical expression is in fact given by ( 33) where s depends on θ 1 . We see that (34) owns a term which has the same form as the one presented in Eq.( 33). Consequently, it cannot be integrated analytically as well.

C. Interaction between the contributions of the volume densities

The third kind of physical interaction is due to the volume contributions of each ring permanent magnet. This physical interaction corresponds to the case when the volume densities of each ring permanent magnet are integrated together. In Eq. ( 2), this volume contribution correspond to the integrand denoted c(α, β) where α and β can be r 1 or r 2 . Let us consider the integrand c(α, β). As we can see in Eq. ( 2), the integration variables depend on θ 1 , θ 2 , z 1 , z 2 , r 1 and r 2 . The integration according θ 2 does not change the form of the integrand c(α, β) because c(α, β) does not depend on θ 2 . Consequenlty, we can say that the form of the force F v between the volume density contributions is given as follows :

F v = θ 1 r 1 r 2 z 1 z 2 (z 2 -z 1 ) (c 1 -c 2 cos(θ 1 ) + (z 2 -z 1 ) 2 ) 3 2 dθ 1 dz 1 dz 2 dr 1 dr 2 (35)
where c 1 and c 2 are constant. These parameters are given by ( 2) and correspond to the case when six integrals must be determined.

After having integrated according z 1 , z 2 and r 1 , we obtain the same integrand as the one obtained previously in the case of the study of the force exerted between the surface and volume charge contributions. Consequently, we deduct that we cannot integrate analytically F v according to θ 1 . Moreover, the analytical integration according to r 2 does not seem possible. As a conclusion, we can say that the obtaining of a fully analytical expression of the force between two ring permanent magnets whose polarization is radial does not seem possible but a semi-analytical expression can be used to determine this axial force.

IV. OPTIMIZATION OF THE INNER RING PERMANENT MAGNET DIMENSIONS

This section discusses the optimal dimensions of the rings in order to have either a great force or a great stiffness.

A. Influence of the air gap dimension on the force and the stiffness

First, we study the influence of the air gap dimension on the force and the stiffness between the two ring permanent magnets. For this purpose, we represent the axial force versus the axial displacement of the inner ring for different air gaps in Fig. 3. It is noted that, in our configuration, the air gap corresponds to the difference between r in and r out2 . Furthermore, the width and the height of each ring permanent magnet are constant. We take r in = 0.025m, r out = 0.028m, r out2 -r int2 = 0.003m, J = 1T, h = 0.003m, zb -z a = 0.003m. Fig. 3 shows three important points.

First, we see that the smaller the air gap between the ring permanent magnets is, the greater the axial force is. Consequently, it is necessary to have the smaller air gap between ring permanent magnets if a great force is searched. This result is well-known. It was shown with the two dimensional approach.

Second, we see that the exact position of the maximal force exerted between two ring permanent magnets depends slightly on the air gap dimension. This result is new because our study uses a three-dimensional approach of the magnetic force whereas the previous ones used a two-dimensional approach. The magnet curvature must be taken into account in order to obtain precisely the position of the maximal force exerted between two ring magnets. The exact position of the maximal force is represented in Sti ness [N/m] Fig. 5. Representation of the axial stiffness of the magnetic force exerted between two ring permanent magnets versus the axial displacement of the inner ring permanent magnet for different air gaps; r in = 0.025m, r out = 0.028m,r out2 -r int2 = 0.003m, J = 1T, h = 0.003m, zb -z a = 0.003m.

B. Determination of the optimal height of the inner ring permanent magnet

Another parameter which can be optimized is the height of the inner ring permanent magnet. To do so, the axial component of the magnetic force is plotted versus the axial displacement of the inner ring permanent magnet for several inner ring heights in Fig. 6.

The values taken for the parameters are still the same as the previous ones. Fig. 6 shows that the axial component of the magnetic force is the greatest if its height equals the outer ring height. Indeed, if the ring inner height is smaller than the ring outer one, the smaller the inner ring width is, the smaller the axial component of the magnetic force is. If the ring inner height is greater than the ring outer one, the greater the inner ring width is, the smaller the axial component of the magnetic force is. Consequently, if a great force is searched, the two ring permanent magnets must have the same height. Furthermore, we see that the gradient of the curves is the more important when the inner ring height is the same as the outer one. This result is consistent with Fig 7 where the axial stiffness is represented versus the axial displacement of the inner ring permanent magnet.

C. Determination of the optimal width of the inner ring permanent magnet

The third parameter which can be optimized in our configuration is the width of the inner ring permanent magnet. To do so, the axial component of the magnetic force is plotted versus the axial displacement of the inner ring permanent magnet for several inner ring widths in Fig. 8. The values taken for the parameters are the same as the previous ones. Fig. 8 shows that the greater the inner ring width is, the greater the axial component of the magnetic force is. However, it is noted that a compromise in the ring dimensions must be found because the cost of the magnet must be taken into account. A good compromise can be found as follows : if the inner ring width equals two times its height, the axial component of the magnetic force is 72N when z = 0.0015m. If the inner ring width equals three times its height, the axial component of the magnetic force is 73N. Consequenlty, we deduct that it is not necessary to have an inner ring width which is greater than two times its height.

The optimal stiffness depends also on the inner ring width. To see that, we have represented in

V. OBTAINING THE BEST CONFIGURATION

The previous section shows that the ring dimensions must be optimized in order to create a very good passive magnetic bearing.

The air gap must be the smallest, the ring heights must be the same and we have shown that the inner ring width must equal two times its height. All these parameters have been determined with a outer ring whose cross-section is a square. However, we can also optimize the outer ring in order to improve the passive bearing. By taking into account the optimal dimensions found in the previous section, we can compare three configurations. The first one, shown in Fig 10-A, consists of two rings whose cross-section is a square.

VI. CONCLUSION

This paper has presented new three-dimensional semi-analytical expressions allowing us to determine both the axial force and the axial stiffness between two ring permanent magnets whose polarization is radial. This paper also discusses the reason why we cannot find a fully analytical expression of the axial component of the magnetic force. Eventually, we have discussed the optimal dimensions of the ring permanent magnets which allow us to have either a great axial force or a great axial stiffness. We have shown that a good compromise can be found when the cross-section of a ring is a rectangle. However, the relative position of the two rings is not constant but depends on the air gap dimension. Such results can be very useful for people involved in the design of magnetic bearings.

  Fig.1. Representation of the two ring permanent magnets which are radially centered. The inner radius of the outer ring is denoted r in , the outer one is rout, its

Fig. 3 .Fig. 4 .

 34 Fig.3. Representation of the axial component of the magnetic force exerted between two ring permanent magnets versus the axial displacement of the inner ring permanent magnet for different air gaps; r in = 0.025m, r out = 0.028m,r out2 -r int2 = 0.003m, J = 1T, h = 0.003m, zb -z a = 0.003m.

Fig 4 .

 4 Such result is very useful because it clearly shows that if a great axial force is searched, the relative height between the two rings inner ring depends on the air gap dimension. Eventually, Fig 3 shows that the stiffness depends greatly on the air gap dimension. Fig 3 shows that the smaller the air gap dimensionis, the greater the stiffness is because the gradient of the curve is the more important for small air gaps. This result is consistent with the representation of the axial stiffness versus the axial displacement of the inner ring permanent magnet (Fig5). Indeed, we see that the smaller the air gap dimension is, the greater the axial stiffness is. Moreover, we see that when the axial force is maximal in Fig 3, the axial stiffness equals zero in Fig 5, which is still consistent.

Fig. 6 .

 6 Fig.6. Representation of the axial component of the magnetic force exerted between two ring permanent magnets versus the axial displacement of the inner ring permanent magnet for different heights; r in = 0.025m, rout = 0.028m,r out2 = 0.0249m, J = 1T, h = 0.003m, za = 0m.

Fig. 7 .Fig. 8 .

 78 Fig.7. Representation of the axial component of the magnetic force exerted between two ring permanent magnets versus the axial displacement of the inner ring permanent magnet for different heights; r in = 0.025m, r out = 0.028m,r out2 = 0.0249m, J = 1T, h = 0.003m, z a = 0m.

  Fig 9 the axial stiffness versus the inner ring width when z a = 0. Fig 9 shows that the larger the inner ring width is, the greater the axial stiffness is.

Fig. 9 .Fig. 10 .

 910 Fig.9. Representation of the axial stiffness exerted between two ring permanent magnets versus inner ring width r in = 0.025m, rout = 0.028m,r out2 = 0.0249m, J = 1T, h = 0.003m, zb -z a = 0.003m.

Figs 11 and 12

 12 Figs 11 and 12 show that the best configration is the configuration B presented in Fig 10. However, the relative height between the two ring permanent magnets depends on the air gap dimension (Fig 4). Consequently, this last parameter must be taken into account in the design of passive bearings using ring permanent magnets.

 12. Representation of the axial stiffness exerted between two ring permanent magnets versus inner ring width r in = 0.025m, rout = 0.028m,r out2 = 0.0249m, J = 1T, h = 0.003m, zb -z a = 0.003m.

[3] B. Azzerboni, E. Cardelli, and A. Tellini, "Computation of the magnetic field in massive conductor systems," IEEE Trans. Magn., vol. 25, pp. 4462-4473, november 1989.