J.-F Bercher
email: jf.bercher@esiee.fr

A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians

Keywords: Divergence measures, Generalized Rényi and Tsallis entropies, Escort distributions, q-gaussian distributions PACS: 02.50.-r, 05.90.+m, 89.70.+c 1. Introduction

We give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution. When the parameter of the distribution varies, it denes a parametric curve that we call an escort-path. The Rényi divergence appears as a natural by-product of the setting. We study the dynamics of the Fisher information on this path, and show in particular that the thermodynamic divergence is proportional to Jereys' divergence. Next, we consider the problem of inferring a distribution on the escort-path, subject to generalized moments constraints. We show that our setting naturally induces a rationale for the minimization of the Rényi information divergence. Then, we derive the optimum distribution as a generalized q-Gaussian distribution.

Before going into the details of the results, we shall present the context and introduce the main denitions on our main ingredients, that is the escort distributions, information divergences, and Fisher information.

Throughout the paper, we will work with univariate probability densities dened with respect to a general measure µ(x) on a set X. For instance, the Shannon-Boltzmann entropy will be expressed as

H[f ] = -ˆf (x) log f (x)dµ(x). (1)
As particular cases, we have that if X is the real line and µ the Lebesgue measure, then the expression above corresponds to the dierential entropy. When the set X is N or a subset of N and µ the counting measure, then the expression reduces to the standard discrete entropy. When µ is a probability measure, then the expression (1) can also be seen as the relative entropy from the measure with density f to the measure µ.

Let us now turn to the notion of escort distribution. If f (x) is an univariate probability density with respect to µ(x), then we dene its escort distribution of order q, q ≥ 0, by f q (x) = f (x) q ´f (x) q dµ(x) ,

provided that M q [f ] = ´f (x) q dµ(x) is nite. These escort distributions have been introduced as an operational tool in the context of multifractals [START_REF] Chhabra | Direct determination of the f (α) singularity spectrum[END_REF], [START_REF] Beck | Thermodynamics of Chaotic Systems[END_REF], with interesting connections with the standard thermodynamics. Discussion of their geometric properties can be found in [START_REF] Abe | Geometry of escort distributions[END_REF][START_REF] Ohara | A dually at structure on the space of escort distributions[END_REF]. Escort distributions also prove to be useful in source coding where they enable to derive optimum codewords with a length bounded by the Rényi entropy [START_REF] Bercher | Source coding with escort distributions and Rényi entropy bounds[END_REF].

The results presented in this paper are connected to the nonextensive statistical physics introduced by Tsallis, see e.g. [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF]. Indeed, the nonextensive statistical physics uses a generalized entropy, makes use of escort distributions and exhibit generalized Gaussians. All these elements will pop up in our construction, which, therefore could lead to new viewpoints or interpretations in this context. It is particularly remarkable that the derivation of the maximum Tsallis entropy distributions in nonextensive thermostatistics requires a constraint in the form of an escort mean value, that is computed with respect to an escort distribution like (2) [START_REF] Abe | Necessity of q -expectation value in nonextensive statistical mechanics[END_REF][START_REF] Tsallis | Escort mean values and the characterization of power-law-decaying probability densities[END_REF].

One can immediately extend the notion of escort distribution to deal with two probability densities f (x) and g(x) as follows.

Denition 1. Let f and g be two densities with respect to a common measure µ, with g dominated by f . For q ≥ 0 such that M q [f, g] = ´f (x) q g(x) 1-q dµ(x) < ∞, we call generalized escort distribution the function f q (x) = f (x) q g(x) 1-q ´f (x) q g(x) 1-q dµ(x) .

(3)

We will also denote, when non ambigous, by E q [.] the statistical expectation with respect to the generalized escort distribution with index q. This generalized escort distribution is simply a weighted geometric mean of f (x) and g(x), and reduces to f q (x) = f (x) for q = 1 and to f q (x) = g(x) for q = 0. Obviously, if g(x) is a uniform density whose support includes the support of f (x), then the generalized escort distribution gives back the standard one [START_REF] Beck | Thermodynamics of Chaotic Systems[END_REF]. Actually, the generalized escort (3) appeared in Cherno analysis of the eciency of hypothesis tests [START_REF] Cherno | A measure of asymptotic eciency for tests of a hypothesis based on the sum of observations[END_REF], and enables to dene the best achievable exponent in the bayesian probability of error [START_REF] Cover | Elements of Information Theory[END_REF]Chapter 11]. As q varies, the generalized escort distribution denes a curve that connects f (x) to g(x) and further. In the general framework of information geometry [START_REF] Amari | Methods of Information Geometry[END_REF], the generalized escort distribution (3) coincides with the geodesic joining f and g in the case of an exponential connection. Such interpretation also appeared in a work by Campbell [START_REF] Campbell | The relation between information theory and the dierential geometry approach to statistics[END_REF].

Throughout the paper, we will focus on the generalized escort distribution and the path it denes, that we will call the escort-path.

Distances between probability distributions will be measured by means of information divergences. We will use the Kullback-Leibler directed information divergence which is dened as follows.

Denition 2. Let f and g be two univariate densities with respect to a common measure µ, with f absolutely continuous with respect to g. The Kullback-Leibler directed information divergence is given by

D(f ||g) = ˆf (x) log f (x) g(x) dµ(x). (4)
It is understood, as usual, that 0 log 0 = 0 log 0/a = 0 log 0/0 = 0. Note that if we take g(x) = 1 in the expression above, then we obtain minus the Shannon entropy H[f ]. Let us also recall that the minimization of the Kullback-Leibler divergence is a well established inference method, analog to Jaynes' maximum entropy approach and which is supported in particular by large deviation results [START_REF] Ellis | The theory of large deviations: from Boltzmann's 1877 calculation to equilibrium macrostates in 2D turbulence[END_REF]. We will also make use of the Rényi information divergence introduced in [START_REF] Rényi | On measures of entropy and information[END_REF]. Denition 3. Let f and g be two probability densities with respect to a measure µ.

If f is absolutely continuous with respect to g, then, for q ≥ 0 such that M q [f, g] = ´f (x) q g(x) 1-q dµ(x) < ∞, the Rényi divergence is dened by

D q (f ||g) = 1 q -1 log ˆf (x) q g(x) 1-q dµ(x). (5)
Let us recall that the divergence is always non negative D q (f ||g) ≥ 0 with the equality sign i f = g. By L'Hôpital's rule, the Kullback divergence is recovered in the limit q → 1. Taking g(x) = 1 in the expression of the Rényi divergence yields the negative of the Rényi entropy, noted H q [f ].

We will study Fisher information along the escort-path. Indeed, it is well known that the Fisher information metric is a Riemannian metric that can be dened on a smooth statistical manifold [START_REF] Rao | Information and accuracy attainable in the estimation of statistical parameters[END_REF][START_REF] Burbea | Dierential metrics in probability spaces[END_REF]. Furthermore, the Fisher information serves as a measure of the information about a parameter in a distribution. It has intricate relationships with maximum likelihood and has many implications in estimation theory, as exemplied by the Cramér-Rao bound which provides a fundamental lower bound on the variance of an estimator [START_REF] Barankin | Locally best unbiased estimates[END_REF]. It is also used as a method of inference and understanding in statistical physics and biology, as promoted by Frieden [START_REF] Frieden | Physics from Fisher information: A unication[END_REF][START_REF] Frieden | Science from Fisher Information: A Unication[END_REF]. Denition 4. Let f (x; θ) denote a probability density with respect to a measure µ, where θ is a real parameter, and suppose that f (x; θ) is dierentiable with respect to θ. Then, the Fisher information in the density f about the parameter θ is dened as

I[f, θ] = ˆ ∂ ln f (x; θ) ∂θ 2 f (x; θ)dµ(x). (6)
The remaining of the paper is structured as follows. In section 2 we show that the generalized escort presented above arises naturally in a simple probabilistic description of a transition between two states. Interestingly, the Rényi information divergence, and in a particular case the Rényi entropy, emerges as a characterization of the transition.

In section 3, we study the Fisher information, with respect to q, along the escortpath. We show in particular that the integral of the Fisher information along the path, the thermodynamic divergence, is proportional to Jereys' divergence.

In section 4, we consider the problem of inferring the distribution f (x) in [START_REF] Beck | Thermodynamics of Chaotic Systems[END_REF] or (3) on the escort-path when the only available information is given as a mean value. This mean value is the statistical expectation taken with respect to an escort distribution: this is the escort mean value used in nonextensive statistics. Dierent possible approaches, such as minimizing the directed divergence, or Jereys divergence or the thermodynamic divergence, reduce to the minimization of the Rényi information divergence. In this case, the probability distribution that emerges is a generalized Gaussian distribution, which is particularly important in applications.

The escort-path

It has been observed that Tsallis' extended thermodynamics seems particularly appropriate in the case of deviations from the classical Boltzmann-Gibbs equilibrium.

This suggests that the original MaxEnt formulation nd the closest distribution to a reference under a mean constraint may be amended by introducing a new constraint that displaces the equilibrium. The partial or displaced equilibrium can be imagined as an equilibrium characterized by two distributions, say p 0 (x) and p 1 (x). Instead of selecting the nearest distribution to a reference under a mean constraint, we may look for a distribution p q (x) simultaneously close, in some sense, to two distinct references: such a distribution will be localized somewhere `between' p 0 (x) and p 1 (x). Figure 1: Constrained equilibrium between states p 0 and p 1 : the equilibrium distribution is sought in the set of all distributions such that D(p|[p 0 ) = η, and with minimum Kullback distance to p 1 . The equilibrium distribution p q , the generalized escort distribution, is aligned with p 0 and p 1 and intersects the set D(p|[p 0 ) = η.

Displaced equilibrium

We consider two equilibrium states with respective probability densities p 0 (x) and p 1 (x) with respect to a common measure µ, at some point x in the phase space, and we look at intermediate states dened by the following scenario. The system with initial state p 0 , subject to a generalized force, is moved at a distance η = D(p||p 0 ) from p 0 , where D(p||p 0 ) is the Kullback-Leibler divergence (or relative entropy) from p to p 0 . Then, the system is attracted toward the nal state p 1 . Therefore, the new intermediate equilibrium state, say p q , is chosen as the one which minimizes its divergence to the attractor p 1 while being hold on at the distance η from p 0 . As illustrated in Figure 1, the intermediate probability density is located on the straight line p 0 -p 1 and intersects the circle with radius η centered at p 0 . More precisely, the problem can be written as follows:

   min p D(p||p 1 ) s.t. D(p||p 0 ) = η and ´p(x)dµ(x) = 1 (7)
where s.t.

stands for subject to, and where the Kullback-Leibler divergence D(f ||g) is dened by ( 4). The solution is given by the following Theorem.

Theorem 5. Let p 1 a probability density function with respect to µ, and p 0 a non negative function. Assume that p 1 is absolutely continuous with respect to p 0 . Let p q denote the generalized escort distribution with index q ≥ 0 p q (x) = p 1 (x) q p 0 (x) 1-q ´p1 (x) q p 0 (x) 1-q dµ(x) ,

with M q (p 1 , p 0 ) = ´p1 (x) q p 0 (x) 1-q dµ(x) < ∞. If E q log p 1 p 0 is nite, where E q [.] denote the statistical expectation with respect to p q , and if q is chosen such that D(p q ||p 0 ) = η, then the generalized escort distribution ( 8) is the unique solution of problem [START_REF] Abe | Necessity of q -expectation value in nonextensive statistical mechanics[END_REF].

Proof. Let us evaluate the divergence D(p||p q ). For all densities p satisfying D(p||p 0 ) = η, we have

D(p||p q ) = ˆp(x) log p(x) p q (x) dµ(x) = ˆp(x) log p(x) q p(x) 1-q p 1 (x) q p 0 (x) 1-q dµ(x) + log M q (p 1 , p 0 ) (9) = q ˆp(x) log p(x) p 1 (x) dµ(x) + (1 -q) ˆp(x) log p(x) p 0 (x) dµ(x) + log M q (p 1 , p 0 ) (10) = q D(p||p 1 ) + (1 -q)η + log M q (p 1 , p 0 ) (11)
Observe that D(p q ||p 0 ) = qE q log p 1 p 0 -log M q and that D(p q ||p 1 ) = (1-q)E q log p 1 p 0log M q so that both divergences exist. Therefore, taking p = p q , the last equality gives

D(p q ||p q ) = q D(p q ||p 1 ) + (1 -q)η + log M q (p 1 , p 0 ). (12)
Finally, subtracting [START_REF] Amari | Methods of Information Geometry[END_REF] and ( 12) yields D(p||p q ) -D(p q ||p q ) = q (D(p||p 1 ) -D(p q ||p 1 )) .

Since q ≥ 0 and since D(p||p q ) ≥ 0 with equality i p = p q , we obtain that D(p||p 1 ) ≥ D(p q ||p 1 ) which proves the Theorem.

It is interesting to note that (8) is nothing else but a generalized version of the escort or zooming distribution of nonextensive thermostatistics, and that the corresponding statistical expectations are the so-called escort-means or generalized averages. Obviously, one recovers a standard escort distribution like (1) when p 0 (x) is uniform with respect to µ. This is immediate if µ has a compact support. However, if one wants to use a uniform measure on the whole real axis, with µ the Lebesgue measure, then such a measure is no more a probability density since it integrates to innity. In such case, it still possible to modify the formulation to include this case as well. Indeed, with p 0 (x) = 1, the expression of the Kullback-Leibler divergence D(p||p 0 ) becomes nothing but minus the standard entropy

H[p] = -ˆp(x) log p(x)dµ(x).
Therefore, the problem turns into the research of a distribution with a given entropy, which minimizes the divergence to p 1 :

   min p D(p||p 1 ) s.t. H[p] = -η and ´p(x)dµ(x) = 1. ( 13
)
This setting can be illustrated as was done in Figure 1, excepted that the circle now corresponds to the set of distributions with a given level of entropy. Observe that neither the Theorem 5 nor its proof require that p 0 is a probability density. Therefore we can take p 0 (x) = 1 and obtain the solution of (13) as a simple corollary. Corollary 6. Let p q denote the escort distribution with index q, associated with p 1 , dened by

p q (x) = p 1 (x) q ´p1 (x) q dµ(x) , (14)
provided that

M q (p 1 ) = ´p1 (x) q dµ(x) < ∞. If E q [log p 1 ] is nite, where E q [.]
denote the statistical expectation with respect to p q , and if q is chosen such that H[p q ] = -η, then the escort distribution ( 14) is the unique solution of problem [START_REF] Ellis | The theory of large deviations: from Boltzmann's 1877 calculation to equilibrium macrostates in 2D turbulence[END_REF].

When q varies, the function η(q) = D(p q ||p 0 ) is monotonically increasing, and particular intermediate values satisfy the implicit relationship D(p q ||p 0 ) = η. This property will be proved in section 3, corollary 10, as a simple consequence of a result on Fisher information. For q = 0 we have η = 0 and for q = 1, we have η = D(p 1 ||p 0 ). Accordingly, as q varies, p q traces out a curve, the escort-path, that connects p 0 (q = 0) and p 1 (q = 1). In the case q > 1, we have η > D(p 1 ||p 0 ) as shown in Figure 1b.

Interestingly enough, recent results have shown that the average dissipated work during a transition can be expressed as a relative entropy [START_REF] Kawai | Dissipation: The Phase-Space perspective[END_REF][START_REF] Parrondo | Entropy production and the arrow of time[END_REF]. Along these lines, with an Hamiltonian even in the momenta, the minimization of D(p||p 1 ) may be understood as a minimization of the average dissipated work for a transition from p to p 1 .

Rényi and Jereys' divergences as by-products

It is interesting to outline that the Rényi divergence and entropy arise as a byproduct of our construction. Indeed, the minimum of the Kullback-Leibler divergence can be expressed as follows.

Corollary 7. The minimum divergence is given by

D(p q ||p 1 ) = 1 - 1 q (η -D q (p 1 ||p 0 )) (15)
where D q (p 1 ||p 0 ) is the Rényi information divergence with index q, from p 1 to p 0 .

Proof. By direct calculation from the expression of the solution p q (x), or by a direct consequence of relation [START_REF] Campbell | The relation between information theory and the dierential geometry approach to statistics[END_REF].

If p 0 is a uniform distribution, then -D q (p 1 ||p 0 ) = H q (p 1 ), the Rényi entropy, p q is the standard escort distribution and (15) becomes

D(p q ||p 1 ) = 1 - 1 q (η + H q (p 1 )) .
Although it is convenient to think of the Kullback-Leibler divergence D(f ||g) (4) as a distance between f and g, it is not symmetric and does not satisfy the triangle inequality. Kullback and Leibler themselves introduced a symmetrized version, which was also considered before by Jereys. This Jereys' divergence appears here to be a simple ane function of Rényi information divergence D q (p 1 ||p 0 ).

Corollary 8. The Jereys divergence between p 1 and the generalized escort distribution p q is given by

J(p 1 , p q ) = D(p 1 ||p q ) + D(p q ||p 1 ) = (q -1) 2 q (D q (p 1 ||p 0 ) -η) . (16)
Proof. This is a simple consequence of [START_REF] Amari | Methods of Information Geometry[END_REF], which gives D(p

1 ||p q ) = (1 -q)η + log M q (p 1 , p 0 ) if p = p 1
, and of ( 12) that gives D(p q ||p 1 ) = (1-1 q )η -1 q log M q (p 1 , p 0 ).

As an interesting consequence, we see that if one wants to minimize the symmetric divergence between p 1 and p q , subject to additional constraints, then this simply amounts to the minimization of the Rényi information divergence with the same constraints. When p 0 is uniform, this becomes the maximization of the Rényi entropy, or equivalently of the Tsallis entropy. It is thus interesting that our setting induces both an escort distribution and a Rényi divergence (or entropy), and besides with a common index q. Actually, although these two quantities are essential ingredients in nonextensive statistical mechanics, their relationships are discussed, e.g. [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF].

Fisher information along the escort-path

Suppose now that p 0 (x) and p 1 (x) depend on a parameter θ. The Fisher information metric is based on the Fisher information matrix on a vector parameter θ attached to a density p(x; θ). This Fisher information matrix has entries [I(θ)] i,j = ˆp(x; θ) ∂ ∂θ i log p(x; θ) ∂ ∂θ j log p(x; θ) dµ(x).

The derivative of the logarithm of the density with respect to the parameter is called the score function. The mean of the score function is zero, so that the Fisher information matrix is the covariance of the score function.

The length of a curve parametrized by t, from 0 to T, is given by

L = i j ˆT 0 dθ i dt [I(θ)] i,j dθ j dt dt.
In the context of thermodynamics, this quantity is called the thermodynamic length [START_REF] Weinhold | Metric geometry of equilibrium thermodynamics[END_REF][START_REF] Crooks | Measuring thermodynamic length[END_REF][START_REF] Shenfeld | Minimizing thermodynamic length to select intermediate states for free-energy calculations and replica-exchange simulations[END_REF]. A related quantity is the thermodynamic divergence, or energy of the curve, given by

J = i j ˆT 0 dθ i dt [I(θ)] i,j dθ j dt dt.
By Jensen's inequality, we have immediately that J ≥ L 2 . An interesting point, that outlines the importance of these quantities, is the fact that the thermodynamic divergence asymptotically bounds the dissipation induced by a nite time transformation of a thermodynamic system [START_REF] Nulton | Quasistatic processes as step equilibrations[END_REF][START_REF] Crooks | Measuring thermodynamic length[END_REF]. Hence, it is interesting here to study some characteristics of the Fisher information along the escort-path. The general study of the Fisher information on the escort-path with respect to a general parameter θ is interesting in its own right. However, in order to save space, we will focus here on a special case. Let us still simply mention that when p 0 is uniform, the related Fisher information is the escort-Fisher information which has been considered

in [START_REF] Hammad | Mesure d'ordre α de l'information au sens de Fisher[END_REF][START_REF] Pennini | Rényi entropies and Fisher informations as measures of nonextensivity in a Tsallis setting[END_REF][START_REF] Pennini | Escort Husimi distributions, Fisher information and nonextensivity[END_REF]].

As we have seen, the generalized escort distribution describes a geometric path, the escort-path, connecting distributions p 0 and p 1 for the values q = 0 and q = 1.

Clearly, the densities on the escort-path are characterized by the index q. Hence it is quite natural to evaluate the distance between two densities on the path, as well as the Fisher information with respect to q. Let us begin by a general expression of the Fisher information on the path. Then, we will be able to link this Fisher information to information divergences on the path.

Theorem 9. Let p q be the generalized escort distribution as in [START_REF] Tsallis | Escort mean values and the characterization of power-law-decaying probability densities[END_REF]. Then, the Fisher information with respect to q of the generalized escort distribution is given by

I(q) = ˆ1 p q (x) dp q (x) dq 2 dµ(x) = ˆdp q (x) dq log p 1 (x) p 0 (x) dµ(x) (17)
provided that E r log p 1 p 0 2 is nite for r in a compact neighborhood of q. The Fisher information with respect to q can also be written as the variance of the loglikelihood ratio:

I(q) = E q log p 1 (x) p 0 (x) -E q log p 1 (x) p 0 (x) 2 . ( 18
)
Proof. The second order moment condition on the log-likelihood ratio implies, by

Jensen inequality, that both E q log p 1 p 0 and E q log p 1 p 0 are nite. Let us rst consider M q (p 1 , p 0 ) = ´p1 (x) q p 0 (x) 1-q dµ(x). The integrand is clearly dierentiable with respect to q, and this derivative, which is equal to p q log p 1 p 0 is continuous and is absolutely integrable since E q log p 1 p 0 is nite. Furthermore, by the second order moment hypothesis, the last expression is also locally integrable with respect to q.

This enables to use Leibniz' rule and dierentiate under the integral sign, which gives

d log M q dq = ˆpq (x) log p 1 (x) p 0 (x) dµ(x) = E q log p 1 (x) p 0 (x) . ( 19
)
Then, by direct calculation, we also have

dp q (x) dq = p q (x) log p 1 (x) p 0 (x) -E q log p 1 (x) p 0 (x) , (20)
which, inserted in the denition of the Fisher information in [START_REF] Barankin | Locally best unbiased estimates[END_REF] gives [START_REF] Frieden | Physics from Fisher information: A unication[END_REF].

By [START_REF] Kawai | Dissipation: The Phase-Space perspective[END_REF], we have that

ˆ dp q (x) dq dµ(x) ≤ E q log p 1 p 0 + E q log p 1 p 0 < ∞. ( 21
)
Moreover, by the second order moment hypothesis, ( 21) is also locally integrable with respect to q. Since ´pq (x)dµ(x) = 1, then by Leibniz' rule we get that d dq ´pq (x)dµ(x) = ´dpq(x)

dq dµ(x) = 0. Finally, the right hand side of ( 17) is obtained by using ( 20) and the fact that

ˆdp q (x) dq d log M q dq dµ(x) = d log M q dq ˆdp(x) dq dµ(x) = 0.
As a simple consequence, we can now check that η = D(p q ||p 0 ) is indeed a monotone increasing function of q, as announced in section 2.

Corollary 10. Let p q be a generalized escort distribution, with q > 0, and assume that E r log p 1 p 0 2 < ∞ for r in a compact neighborhood of q. Then η(q) = D(p q ||p 0 ) is a strictly monotone increasing function of q, with ∂ ∂q η(q) = q I(q) > 0

Proof. Note that η(q) = ´pq (x) log pq(x) p 0 (x) dµ(x) = q ´pq (x) log p 1 (x) p 0 (x) dµ(x) -log M q .

Under the second order moment condition, one can dierentiate under the integral sign, take into account [START_REF] Frieden | Science from Fisher Information: A Unication[END_REF] and it remains

∂ ∂q η(q) = q ∂ ∂q ˆpq (x) log p 1 (x) p 0 (x) dµ(x) = q ˆdp q (x) dq log p 1 (x) p 0 (x) dµ(x),
where we recognize the Fisher information in [START_REF] Barankin | Locally best unbiased estimates[END_REF]. Therefore, taking into account the fact that both q and the Fisher information are positive, we get [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF].

Finally, an important result is that the integral of the Fisher information, the energy of the curve, is nothing but the Jereys divergence. This result is mentioned in [START_REF] Dabak | Relations between Kullback-Leibler distance and Fisher information[END_REF]. Alternatively, this can also be obtained as a consequence of the general integral representation of the Kullback-Leibler divergence [11, eq. 3.71]. We propose here a direct proof of the result.

Theorem 11. Let p r and p s be two generalized escort distributions. Assume that

E q log p 1 p 0 2 < ∞ for all q ∈ [r, s].
Then, the integral of the Fisher information along the escort-path, from q = r to q = s is proportional to Jereys' divergence between p r and p s :

(s -r) ˆs r I(q)dq = J(p s , p r ) = D(p s ||p r ) + D(p r ||p s ). (23)
With r = 0 and s = 1, we get the integral along the whole path connecting p 0 and p 1 , that is

ˆ1 0 I(q)dq = J(p 1 , p 0 ) = D(p 1 ||p 0 ) + D(p 0 ||p 1 ). (24)
Proof. The Fisher information is nite on the escort path; therefore its integral over a compact interval is also nite. Let us integrate the right equality in [START_REF] Barankin | Locally best unbiased estimates[END_REF]:

ˆs r I(q)dq = ˆs r ˆdp q (x) dq log p 1 (x) p 0 (x) dµ(x) dq. (25)
Since I(q) is positive and ´s r I(q)dq nite, it is possible to apply Fubini's theorem to the right hand side of ( 25), and exchange the order of integrations. Thus, integrating with respect to q yields ˆs r

I(q)dq = ˆ(p s (x) -p r (x)) log p 1 (x) p 0 (x) dµ(x). (26)
On the other hand, the divergence D(p s ||p r ) writes

D(p s ||p r ) = (s -r) ˆps (x) log p 1 (x) p 0 (x) dµ(x) -log M s + log M r ,
and similarly for D(p r ||p s ). Adding the two divergences and taking into account [START_REF] Nulton | Quasistatic processes as step equilibrations[END_REF] give the result [START_REF] Weinhold | Metric geometry of equilibrium thermodynamics[END_REF].

Finally, let θ i , i = 1..M denote a set of intensive variables, which are some functions of the index q. Then, we have that d log p dq = M i=1 ∂ log p ∂θ i dθ i dq and the Fisher information with respect to q can be expressed as

I(q) = ˆp(x) d log p(x) dq 2 dµ(x) = M i=1 M j=1 dθ i dq [I(θ)] i,j dθ j dq ,
where I(θ) is the Fisher information matrix with respect to θ. Therefore, for the escort-path we introduced, we obtain that the thermodynamic divergence is nothing but the Jereys divergence:

J = ˆ1 0 I(q)dq = M i=1 M j=1 ˆ1 0 dθ i dq [I(θ)] i,j dθ j dq dq = D(p 1 ||p 0 ) + D(p 0 ||p 1 ). (27)
4. Inference of a distribution subject to q-moments constraints

In the last section of this paper, we investigate some relationships between escortdistributions, information divergences, Fisher information and generalized Gaussians.

Let us return to the model of states transition as presented in section 2 that led us to the generalized escort distribution [START_REF] Tsallis | Escort mean values and the characterization of power-law-decaying probability densities[END_REF] as the optimum intermediate between p 0 and p 1 . Assume that the distribution p 1 is not exactly known but that the available information is given as an expectation under the escort p q . This expectation is the so-called generalized expectation, or q-average which is largely used in nonextensive statistics, although it is generalized here with the presence of p 0 . In our context, it has the clear meaning of an expectation with respect to the intermediate distribution p q at a given distance of a reference p 0 , c.f. Theorem 5, or with a given entropy, c.f. Corollary 6. Let the observable be given as the absolute moment of order α:

m α,q [p 1 ] = E q [|x| α ] =
´|x| α p 1 (x) q p 0 (x) 1-q dµ(x) ´p1 (x) q p 0 (x) 1-q dµ(x) .

Typically, the observable could be a mean energy, where the statistical mean is taken with respect to the escort distribution. Then, the question that arises is the determination of a general distribution p 1 compatible with this constraint. One may keep the idea of minimizing the divergence to p 1 , as in the original problem [START_REF] Abe | Necessity of q -expectation value in nonextensive statistical mechanics[END_REF] which led us to the generalized escort distribution. Since the Kullback divergence is a directed divergence, we shall keep the notion of direction by minimizing D(p q ||p 1 ) for q < 1 and D(p 1 ||p q ) for q > 1. In both cases, the divergence is an ane function of the Rényi divergence D q (p 1 ||p 0 ), c.f. [START_REF] Rao | Information and accuracy attainable in the estimation of statistical parameters[END_REF]. Therefore, these minimizations are nally equivalent to the minimization of the Rényi divergence under the generalized mean constraint.

In the same vein, we may consider the minimization of the symmetric Jereys' divergence between p q and p 1 . We have noticed ( 16) that this divergence is also an ane function of the Rényi divergence D q (p 1 ||p 0 ). Therefore, its minimization is also equivalent to the minimization of the Rényi divergence under the generalized mean constraint.

Finally, a natural idea is to select the distribution p, thus its escort p q , so as to minimize the thermodynamic divergence ´1 q I(t)dt or ´q 1 I(t)dt from p q to p, while satisfying the constraint [START_REF] Pennini | Rényi entropies and Fisher informations as measures of nonextensivity in a Tsallis setting[END_REF]. We have seen that Jereys' divergence J(p 1 , p q ) is proportional to the thermodynamic divergence, as indicated in [START_REF] Weinhold | Metric geometry of equilibrium thermodynamics[END_REF]. As a consequence, the minimization of the thermodynamic divergence between p q and p 1 is also equivalent to the minimization of the Rényi information divergence D q (p 1 ||p 0 ).

It is known [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF] that the maximization of Rényi entropy subject to generalized qmoments constraints, or equivalently of Tsallis entropy under the same constraints, leads to generalized Gaussian distributions. As far as the minimization of the Rényi information divergence is concerned, a direct proof based on a simple inequality can be derived along the lines in [START_REF] Bercher | On some entropy functionals derived from Rényi information divergence[END_REF]Appendix 1] or in [START_REF] Bercher | Escort entropies and divergences and related canonical distribution[END_REF]Proposition 4]. Therefore, we have the following result.

Proposition 12. Among all distributions with a given q-moment of order α as in [START_REF] Pennini | Rényi entropies and Fisher informations as measures of nonextensivity in a Tsallis setting[END_REF], the distribution p with minimum thermodynamic divergence, or equivalently which minimizes Jereys' or Rényi divergence to its escort, is a generalized Gaussian distribution given by

p(x) =    1 Zq(γ) (1 -(1 -q)γ|x| α ) 1 1-q + p 0 (x) for q = 1 1 Z 1 (γ) exp (-γ|x| α ) p 0 (x) for q = 1, (29)
where we use the notation (x) + = max {x, 0} , and where Z q (γ) is the normalization factor.

When p 0 is uniform the distribution becomes the standard Gaussian distribution, for α = 2, in the limit case q = 1, by l'Hôpital's rule. This gives the rationale for the denomination of generalized Gaussians. For q < 1, the probability density has a compact support, while for q 1, the probability density has heavy tails with a power-law behavior and is analog to a Student distribution. These generalized Gaussians appear in statistical physics, where they are the maximum entropy distributions of the nonextensive thermostatistics [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF]. In this context, these distributions have been observed to present a signicant agreement with experimental data, and also to be the analytical solution of actual physical problems [START_REF] Lutz | Anomalous diusion and Tsallis statistics in an optical lattice[END_REF][START_REF] Schwämmle | q-Gaussians in the porous-medium equation: stability and time evolution[END_REF], [START_REF] Ohara | Information geometry of q-Gaussian densities and behaviors of solutions to related diusion equations[END_REF]. In an other eld, the generalized Gaussians are the one dimensional instances of explicit extremal functions of Sobolev, log-Sobolev or GagliardoNirenberg inequalities on R n , with n ≥ 2 [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diusions[END_REF].

Finally, let us close this paper with the example of a q-variance constraint, i.e. m 2,q [p] = σ 2 q , with p 0 (x) = 1 and µ the Lebesgue measure. We have seen that among all distributions with a given dierential entropy, the standard escort distribution p q minimizes the Kullback-Leibler divergence to p, for some value of the index q (Proposition 6). If p is free but its escort is known to have a given variance, then the distribution p which minimizes the thermodynamic divergence or Jereys' divergence (Proposition 12), or equivalently that maximizes the Rényi entropy, is the generalized Gaussian (29) with α = 2. In this setting, p q is located at the intersection of the set of distributions with a given variance and of the set of distributions with a given Shannon dierential entropy. When q varies, the optimum distributions follow a path indexed by q which is nothing but the path followed by the generalized Gaussians, with compact support for q < 1 and innite support for q > 1. In the limit case q = 1, we obtain a standard Gaussian distribution, which is its own escort distribution, and that has the maximum entropy among all escort distributions with the same variance.

These situations are illustrated in Figure 2.

Conclusions

In this paper, we have presented a simple probabilistic model of transition between two states, which leads naturally to a generalized escort distribution. This For each value of q, the optimum distribution p whose escort p q has a given variance is a generalized Gaussian. Thus, when q varies, the path followed by p is the manifold of generalized Gaussians with index q.

generalized escort distribution enables to describe a path, the escort-path, that connects the two states. Then, we have connected several information measures, and studied their evolution along the escort-path. In particular, we have obtained that the Rényi information divergence appears naturally as a characterization of the transition, and that the notion of escort mean values, as used in nonextensive thermostatistics, receives a clear interpretation. We have studied the properties and the evolution of Fisher information along the escort-path. In particular, we have shown that the thermodynamic divergence on the escort-path is a simple function of Jeffreys divergence. We have also considered the problem of inferring a distribution on the escort-path, subject to a moment constraint on its escort. Looking for the distribution as the minimizer of the thermodynamic divergence, we have shown that this procedure is equivalent to the minimization of Rényi divergence subject to a q-moment constraint, which gives a rationale for this approach. Finally, we have recalled that generalized Gaussian distributions arise as solutions of the previous problem.

Beyond the intrinsic interest of our geometric construction, which enables to connect several quantities of information theory, we have also pointed out possible connections with nite thermostatistics. Furthermore, we have indicated that our ndings interrelates several ingredients of the nonextensive statistics. Let us also add that the literature usually points out that the standard entropy (or divergence) is a particular case of generalized Rényi or Tsallis entropies. Our setting suggests a possible additional layer where the generalized quantities are derived from a construction involving the classical information measures. Therefore, we believe that the presented construction, and the series of observations we made can be useful to workers in this eld. Future work should consider the extension of this setting in the multivariate case. In future work, we plan to look for possible connections with nite time thermodynamics. We also intend to study the information theoretic relationships between generalized moments, Fisher information and generalized Gaussians.

  Case η > D(p1||p0)

Figure 2 :

 2 Figure 2: Path of distributions with maximum Rényi entropy and xed q-variance.For each value of q, the optimum distribution p whose escort p q has a given variance is a generalized Gaussian. Thus, when q varies, the path followed by p is the manifold of generalized Gaussians with index q.


J.-F Bercher
email: jf.bercher@esiee.fr

On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions a)

Keywords: numbers: 02.50.-r, 05.90.+m, 89.70 Generalized entropies, Fisher information, q-Gaussian distributions, inequalities

In the present paper, we would like to draw attention to a possible generalized Fisher information that ts well in the formalism of nonextensive thermostatistics. This generalized Fisher information is dened for densities on R n . Just as the maximum Rényi or Tsallis entropy subject to an elliptic moment constraint is a generalized q-Gaussian, we show that the minimization of the generalized Fisher information also leads a generalized q-Gaussian. This yields a generalized Cramér-Rao inequality. In addition, we show that the generalized Fisher information naturally pops up in a simple inequality that links the generalized entropies, the generalized Fisher information and an elliptic moment. Finally, we give an extended Stam inequality. In this series of results, the extremal functions are the generalized q-Gaussians. Thus, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure in nonextensive thermostatistics.

I. INTRODUCTION

The Gaussian distribution has a central role with respect to standard information measures. For instance, it is well known that the Gaussian distribution maximizes the entropy over all distributions with the same variance; see Dembo et al. [START_REF] Dembo | Information theoretic inequalities[END_REF] , Lemma 5. Similarly, the Stam [START_REF] Stam | Some inequalities satised by the quantities of information of Fisher and Shannon[END_REF] inequality shows that the minimum of the Fisher information over all distributions with a given entropy also occurs for the Gaussian distribution. Finally, the Cramér-Rao inequality, e.g. Dembo et al. [START_REF] Dembo | Information theoretic inequalities[END_REF] , Theorem 20, shows that the minimum of the Fisher information over all distributions with a given variance is attained for the Gaussian distribution.

It is thus natural to inquire for similar results for extended entropies and an associated generalized Fisher information. In the context of the nonextensive thermostatistics introduced by Tsallis [START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF][START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF] , the role devoted to the standard Gaussian distribution is extended to generalized q-Gaussian distributions, see e.g. Lutz [START_REF] Lutz | Anomalous diusion and Tsallis statistics in an optical lattice[END_REF] , Schwämmle et al. [START_REF] Schwämmle | q-Gaussians in the porous-medium equation: stability and time evolution[END_REF] , Vignat and Plastino [START_REF] Vignat | Why is the detection of q-Gaussian behavior such a common occurrence?[END_REF] , Ohara and Wada [START_REF] Ohara | Information geometry of q-Gaussian densities and behaviors of solutions to related diusion equations[END_REF] . The generalized q-Gaussians are also explicit extremal functions of Sobolev, log-Sobolev or GagliardoNirenberg inequalities on R n , with n ≥ 2, see Del Pino and Dolbeault [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diusions[END_REF][START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] . This family of distributions will precisely achieve the equality case in all the information inequalities presented in the sequel.

Denition 1. Let x be a random vector of R n . For α ∈ (0, ∞), γ a real positive parameter and q > (n -α)/n, the generalized q-Gaussian with scale parameter γ has the radially symmetric probability density

G γ (x) =      1 Z(γ) (1 -(q -1) γ|x| α ) 1 q-1 + for q = 1 1 Z(γ) exp (-γ|x| α ) if q = 1 (1)
where |x| denotes the standard euclidean norm, where we use the notation (x) + = max {x, 0}, and where Z(γ) is the partition function such that G γ (x) integrates to one. Its expression is given in (A2).

For q > 1, the density has a compact support, while for q ≤ 1 it is dened on the whole R n and behaves as a power distribution for |x| → ∞. Notice that the name generalized Gaussian is sometimes restricted to the case q = 1 above. In the following, we will simply note G, rather than G 1 , the generalized q-Gaussian obtained with γ = 1. The general expressions of the main information measures attached to the generalized Gaussians are derived in Appendix A.

Let us recall that if x is a vector of R n with probability density f (x) with respect to the Lebesgue measure, then the information generating function introduced by Golomb [START_REF] Golomb | The information generating function of a probability distribution[END_REF] is dened by

M q [f ] = ˆf (x) q dx, (2)
for q ≥ 0. Associated with this information generating function, the Rényi entropy is given by

H q [f ] = 1 1 -q log M q [f ] (3)
while the Tsallis entropy is given by

S q [f ] = 1 1 -q (1 -M q [f ]) . (4)
Finally, we will also call entropy power of order q, or q-entropy power, the quantity

N q [f ] = M q [f ] 1 1-q = exp (H q [f ]) = ˆf q dx 1 1-q , ( 5
)
for q = 1. For q = 1, we let N q [f ] = exp (H q [f ]) . Note that the entropy power is usually dened as the square of the quantity above, with an additional factor. We will also use the denition of the elliptic moment of order α, given by m α [f ] = ˆ|x| α f (x)dx, (6) where |x| denotes the euclidean norm of x.

In the context of nonextensive thermostatistics, the study of the role of Fisher information, or the denition of an extension has attracted some eorts. Several authors have introduced generalized versions of the Fisher information and derived the corresponding Cramér-Rao bounds. Among these contributions we note the series of papers Casas et al. [START_REF] Casas | Fisher information in a Tsallis non-extensive environment[END_REF] , Chimento et al. [START_REF] Chimento | Naudts-like duality and the extreme sher information principle[END_REF] , Pennini et al. [START_REF] Pennini | Rényi entropies and Fisher informations as measures of nonextensivity in a Tsallis setting[END_REF][START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF] where the proposed extended Fisher information involves a power f q of the original distribution f , or a normalized version which is the escort distribution.

We shall point out here the early contribution of P. Hammad [START_REF] Hammad | Mesure d'ordre α de l'information au sens de Fisher[END_REF] that essentially introduced the previous escort-Fisher information. A modied version, that also involves escort distributions, is considered by Naudts [START_REF] Naudts | Generalised exponential families and associated entropy functions[END_REF][START_REF] Naudts | The q-exponential family in statistical physics[END_REF] . A recent contribution in the nonextensive literature and in this journal is the proposal of Furuichi [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF][START_REF] Furuichi | On generalized Fisher informations and Cramér-Rao type inequalities[END_REF] , where the logarithm is replaced by a deformed one and a Cramér-Rao type inequality is given.

In the present paper, we present an extension of the Fisher information measure, the (β, q)-Fisher information I β,q [f ] that depends on an entropic index q and on a parameter β. This information has been introduced by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] . It reduces to the standard Fisher information in the case q = 1,β = 2. We look for the probability density function with minimum generalized Fisher information, among all probability density functions with a given elliptic moment, let us say m α [f ] = m:

I β,q (m) = inf f {I β,q [f ] : f (x) ≥ 0, ˆf (x)dx=1 and m α [f ] = m}. (7)
It will turn out that the extremal distribution is precisely the generalized q-Gaussian with the prescribed elliptic moment. Actually, there is a more precise statement of the result in the form of a Cramér-Rao inequality that links the Fisher information and the elliptic moment m α [f ]. It is shown in section IV that for α and β Hölder conjugates of each other, with α ≥ 1, then

I β,q [f ] 1 βλ m α [f ] 1 α ≥ I β,q [G] 1 βλ m α [G] 1 α (8)
with λ = n(q -1) + 1, and where the equality is obtained for any generalized q-Gaussian distribution f = G γ . This inequality means in particular that the minimum of the generalized Fisher information subject to an elliptic moment constraint is reached by a generalized Gaussian. This complements the classical characterization of the generalized q-Gaussian that are found as the maximum Rényi or Tsallis entropy distributions with a given moment. Incidentally, we will also nd, from a general sharp Gagliardo-Nirenberg inequality, that the generalized entropy power and Fisher information are linked by an inequality that generalizes the Stam inequality

N q [f ]I β,q [f ] n βλ ≥ N q [G]I β,q [G] n βλ , (9)
again with λ = n(q -1) + 1.

Actually, the inequalities (8) and ( 9) have been presented in the monodimensional case in the remarkable paper by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] whose reading has deeply inuenced this work.

Beyond the highlighting on the result in the nonextensive statistics context, a contribution of the present paper is the extension of the denitions and results in the multidimensional case. Furthermore, we show in section III, that the expression of the generalized Fisher information naturally pops up from a simple application of Hölder's inequality, which leads to an inequality that links the generalized Fisher information, the Rényi entropy power and the elliptic moment. We show that this inequality, which is saturated by the generalized q-Gaussian, immediately suggests the Cramér-Rao inequality (8). This Cramér-Rao inequality is derived in section IV with the help of the generalized Stam inequality (9).

II. ON THE GENERALIZED (β, q)-FISHER INFORMATION

We begin by the denition of the generalized (β, q)-Fisher information and its main consequences.

Denition 2. Let f (x) be a probability density function dened over a subset Ω of R n .

Let |x| denote the euclidean norm of x and ∇f the gradient of function f (x) with respect to x. If f (x) is continuously dierentiable over Ω, then for q ≥ 0, β > 1, the generalized (β, q)-Fisher information is dened by

I β,q [f ] = ˆΩ f (x) β(q-1)+1 |∇f (x)| f (x) β dx. (10)
The standard Fisher information (q = 1, β = 2), see for instance Cohen 7 , Huber and Ronchetti [START_REF] Huber | Robust Statistics[END_REF] , Plastino et al. [START_REF] Plastino | Fisher variational principle and thermody-namics[END_REF] is a convex functional of the probability density f. It was shown by Boekee [START_REF] Boekee | An extension of the Fisher information measure[END_REF] that this is also true for the generalized Fisher information with q = 1 and any β > 1. It is also easy to check that this is still true in the case β(q -1) + 1 = β. where ω n is the volume of the n-dimensional unit ball, the expression of the generalized Fisher information becomes

I β,q [f ] = n ω n ˆΩr r n-1 f r (r) β(q-1)+1 f r (r) f r (r) β dr.
In the following, it will also be convenient to use the simple transformation f (x) = u(x) k , with k = β/ (β(q -1) + 1). Indeed, with this notation and taking into account that |∇f | = |k|u k-1 |∇u|, the generalized Fisher information reduces to

I β,q [f ] = |k| β ˆΩ |∇u(x)| β dx, (11)
which corresponds to the β-Dirichlet energy of u(x). In the case k = 2, the function u(x) is analog to a wave function in quantum mechanics.

The variational problem ( 7) can be restated as follows:

I β,q (m) = inf u { ˆΩ |∇u(x)| β dx : ˆΩ u(x) k dx = 1 and m = ˆΩ |x| α u(x) k dx}. (12)
The Lagrangian associated with the problem ( 12) is

L(u; a, b) = ˆΩ |∇u(x)| β dx + a ˆΩ u(x) k dx + b ˆΩ |x| α u(x) k dx ( 13
)
where a and b are the Lagrange parameters associated with the two constraints. It is remarkable to note that the same kind of formulation has been introduced in the book by Frieden [START_REF] Frieden | Science from Fisher Information: A Unication[END_REF] , and solutions derived in the cases k = 2 and k = 1, with the mention of the physical interest of the latter case. The Euler-Lagrange equation, see Brunt [START_REF] Brunt | The calculus of variations[END_REF] , corresponding to the variational problem ( 12) is

div |∇u(x)| β-2 ∇u(x) - k β (a + b|x| α ) u(x) k-1 = 0, (14)
It has the form of a p-Laplace equation as in Lindqvist [START_REF] Lindqvist | Notes on the p-Laplace equation[END_REF] , here with p = β,

β u(x) - k β (a + b|x| α ) u(x) k-1 = 0,
where β denotes the β-Laplacian operator. Observe that in the special case k = 2, (14) is analogous to the Schrödinger equation for the quantum oscillator in a relativistic setting (Klein-Gordon equation). In such case, it is known that the ground state is a normal distribution. In the radial case, with r = |x| and u(x) = u r (r), the nonlinear second order dierential equation reduces to

r n-1 u r (r) β-1 - k β r n-1 (a + br α ) u r (r) k-1 = 0. (15)
In our context, it is thus natural to suspect that the generalized q-Gaussian could be a solution. Actually, it is easy to check that if G γ is the generalized q-Gaussian (1), then

u(x) = G γ (x) 1
k is indeed a solution of this equation, with the following relations between the parameters:

a = -An, b = A(1 + n(q -1))γ where A = β k β γ β -1 β-1 Z(γ) k-β . ( 16
)
It will be shown in the following that the generalized Gaussian is not only a possible solution, but the actual optimum solution of the problem.

Let us nally note that in the case β = 2, the nonlinear dierential equation ( 15) is an instance of the generalized Emden-Fowler equation u (x) + h(x) u(x) γ = 0, where h(x) is a given function. This kind of equations arise in studies of gaseous dynamics in astrophysics, in certain problems in uid mechanics and pseudoplastic ow, see Nachman and Callegari [START_REF] Nachman | A nonlinear singular boundary value problem in the theory of pseudoplastic uids[END_REF] , as well as in some reaction-diusion processes, as mentioned by Covei [START_REF] Covei | Existence and uniqueness of solutions for the Lane, Emden and Fowler type problem[END_REF] .

In the radial case, the minimum Fisher information can be written in terms of the constraints and of the Lagrange parameters associated with these constraints.

Proposition 1. Let Ω be the n-dimensional ball of radius R, possibly innite, centered on the origin. Among all radial densities f = u k dened on Ω and such that f r (R) = 0, the minimum Fisher information in ( 12) can be expressed in terms of the constraints and the Lagrange multipliers as

1 |k| β I β,q (m) = - k β (a + bm) . (17)
Proof. By integration by parts,

n ω n ˆR 0 r n-1 u r (r) β dr = n ω n ˆR 0 r n-1 u r (r) β-1 u r (r)dx (18)
=n

ω n r n-1 u r (x) β-1 u r (x) R 0 -n ω n ˆR 0 u r (r) r n-1 u r (r) β-1 dx (19)
Using the boundary condition and the dierential equation ( 15), we then obtain

n ω n ˆR 0 r n-1 u r (r) β dr = -n ω n k β ˆR 0 r n-1 (a + br α ) u r (r) k dx.
Taking into account the values of the constraints, this reduces to (17) .

It can be noted that the Lagrange parameters are actually (complicated) functions of the constraints, so that the right hand side of ( 17) is not a simple ane function in m.

III. A FISHER-MOMENT-ENTROPY INEQUALITY AND THE EXTENDED CRAMÉR-RAO INEQUALITY

We show here that the generalized Fisher information naturally pops up in a simple inequality that links this generalized Fisher information, the information generating function and the elliptic moment of order α. Then, we show that this inequality suggests the general Cramér-Rao inequality (8).

Theorem 1. Let f (x) = f r (|x|) a radially symmetric probability density on the n-dimensional ball Ω of radius R, possibly innite, centered on the origin. Assume that the density is absolutely continuous and such that lim r→R r n f r (r) q = 0. Let also α ≥ 1 and β be its Hölder conjugate. Then, for q > n/(n + α) and provided that the involved information measures are nite, we have

I β,q [f ] 1 β m α [f ] 1 α ≥ n q M q [f ] (20)
with equality if and only if f is a generalized Gaussian f = G γ for any γ > 0.

Proof. Let us consider the information generating function M q [f ]. Since the density is assumed radially symmetric, the use of polar coordinates reduces the computation to the evaluation of an univariate integral. By integration by parts, we obtain

M q [f ] = ˆΩf (x) q dx = nω n ˆR 0 r n-1 f r (r) q dr = nω n r n n f r (r) q R 0 -nω n q ˆR 0 r n n f r (r) f r (r) q-1 dr (21) = nω n q n ˆR 0 [r] - f r (r) f r (r) f r (r) (q-1) r n-1 f r (r)dr (22)
where we used the fact that lim r→R r n f r (r) q = 0. We can then use the fact that ´g(x)dx ≤ ´|g(x)| dx with equality i g(x) ≥ 0, and the Hölder inequality which states that ˆ|u(x)| α w(x)dx Let us now apply these inequalities to the remaining integral in (22), with g(r) = u(r)v(r)w(r), u(r) = r, v(r) = -f r (r) fr(r) f r (r) (q-1) , and w(r) = r n-1 f r (r). Immediately, we obtain

1 α ˆ|v(x)| β w(x)dx 1 β ≥ ˆ|u(x)v(x)|w(x)dx with α -1 + β -1 = 1, α >
nω n ˆR 0 f r (r) f r (r) β f r (r) β(q-1) r n-1 f r (r)dr 1 β nω n ˆR 0 r α r n-1 f r (r)dr 1 α ≥ n q M q [f ] ( 23
)
which is inequality (20) in polar coordinates.

The conditions for equality give here f r (r) < 0 on the one hand, and r α = k f r (r) fr(r) β f r (r) β(q-1) , k > 0, on the other hand. Finally, using the fact that α/β = α -1, the rst-order nonlinear dierential equation reduces to

r α-1 f r (r) 2-q = -K f r (r). (24)
Then, it is easy to check that the unique normalized solution of the dierential equation ( 24) is nothing but the generalized Gaussian (1).

We shall mention that in the monodimensional case, the radial symmetry hypothesis is not necessary, as the integration can be achieved on the real line, and the previous result holds for all probability densities. In the case q = 1, the information generating function in the right side of (20) equals to 1, so that the inequality reduces to I β,1 [f ]

1 β m α [f ] 1 α ≥ 1
which is a Cramér-Rao inequality that has been exhibited by Boekee [START_REF] Boekee | An extension of the Fisher information measure[END_REF] .

As a direct consequence, we obtain a possible derivation of the more general Cramér-Rao inequality (8). The idea is to lower bound the right side of (20). Let us rst consider the case q > 1. In this case, M q [f ] is a convex functional, and therefore has a single minimizer among all densities with a given moment. It is well known that this minimizer is a generalized q-Gaussian G θ (x) with the same moment:

M q [f ] ≥ inf p/m α,1 [p]=m α,1 [f ] M q [p] = M q [G θ ].
Therefore, in the case q > 1 the inequality (20) also yields

I β,q [f ] 1 β m α,1 [f ] 1 α ≥ n q M q [G θ ], (25)
with equality if and only if f = G θ . Of course, the right hand side term can also be written

I β,q [G θ ] 1 β m α,1 [G θ ] 1 α , or I β,q [G θ ] 1 β m α,1 [G θ ] λ α m α,1 [G θ ]
1-λ α , with λ = (n (q -1) + 1). Therefore, the inequality (25) becomes

I β,q [f ] 1 β m α,1 [f ] 1 α m α,1 [G θ ] -1-λ α ≥ I β,q [G θ ] 1 β m α,1 [G θ ] λ α . Since the optimum distribution G θ is such that m α,1 [f ] = m α,1 [G θ ], we get I β,q [f ] 1 β m α,1 [f ] λ α ≥ I β,q [G θ ] 1 β m α,1 [G θ ] λ α . ( 26
)
By the scaling properties (A10), the right hand side does not depend on θ, so that the bound is attained for any generalized Gaussian distribution. Finally, for q ≥ 1, we have λ > 0 and (26) gives the Cramér-Rao inequality (8).

A similar approach holds in the case q < 1 where we use the fact that the minimum of Fisher information among distributions with a given generalized Fisher information is attained by the generalized q-Gaussian distribution with the same Fisher information. This statement is a consequence of a generalized Stam inequality which will be explicited in Proposition 3. Hence, we have

M q [f ] ≥ inf p/I β,q [p]=I β,q [f ] M q [p] = M q [G θ ]
and we can apply the same approach as above to obtain

I β,q [f ] 1 β m α,1 [f ] 1 α I β,q [G θ ] 1 βλ (1-λ) ≥ I β,q [G θ ] 1 βλ m α,1 [G θ ] 1 α .
Then, since θ is such that I β,q [f ] = I β,q [G θ ] and since the right hand side does not depend on θ, we see that the generalized Cramér-Rao (8) inequality is also true for q < 1. This is summarized in the following Proposition.

Proposition 2. [Cramér-Rao inequality for radially symmetric densities] If the conditions in Theorem 1 hold, then we also have

I β,q [f ] 1 βλ m α [f ] 1 α ≥ I β,q [G] 1 βλ m α [G] 1 α , (27)
with λ = n(q -1) + 1, and where the equality holds if and only if f is a generalized Gaussian

f = G γ .
Let us recall that this generalized Cramér-Rao inequality has been exhibited by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] in the monodimensional case. The proposition above is restricted to radially symmetric densities since the initial Fisher-Moment-entropy inequality in Theorem 1 rests on this hypothesis. However, symmetrization arguments show that the Cramér-Rao inequality (27) should hold more generally. Let v be a function dened on a subset Ω of R n and note v * the symmetric decreasing rearrangement (Schwarz symmetrization) of v, and v * its symmetric increasing rearrangement. The rearrangement inequality of Hardy et al. [START_REF] Hardy | Inequalities[END_REF] , see also Kesavan 20 or Benguria and Linde 2 , states that ´Ω f g ≥ ´Ω f * g * . This can be applied directly to the elliptic moment m α [f ] with g(x) = |x| α . Since |x| α is symmetric increasing, it is its own increasing rearrangement and we obtain that m α [f ] ≥ m α [f * ], which seems indeed a natural inequality. In addition, the famous PólyaSzeg® inequality states that ´Ω |∇u| β ≥ ´Ω |∇u * | β . Therefore, with f = u k and using the fact that f * = (u * ) k , we obtain that I β,q [f ] ≥ I β,q [f * ]. Hence, we obtain that for any f we always have

I β,q [f ] 1 βλ m α [f ] 1 α ≥ I β,q [f * ] 1 βλ m α [f * ] 1 α ,
which means that the minimizer of the left side is necessarily radially symmetric and decreasing. As a result, the extremal function can be sought in the subset of radially symmetric probability densities. Unfortunately, the denition of the increasing rearrangement requires that Ω is a compact subset of R n , so that the argument fails for densities with non compact support (which corresponds to the case q < 1 of the extremal function). However, this still shows that the Cramér-Rao inequality (27) holds for any probability density in the q > 1 case, and suggests to investigate further the general case.

IV. THE GENERALIZED STAM AND CRAMÉR-RAO INEQUALITIES

Proposition 2 relies on a result on the minimization of the Rényi entropy power and on a result on the minimization of the generalized Fisher information. The fact that the generalized Gaussian maximizes the Rényi entropy among all distributions with a given elliptic moment is well known and can be derived by standard calculus of variations. A precise statement under the form of a general inequality is due to Lutwak et al. [START_REF] Lutwak | Moment-Entropy inequalities for a random vector[END_REF] .

Theorem 2. [Lutwak et al. [START_REF] Lutwak | Moment-Entropy inequalities for a random vector[END_REF] -Theorem 3] If α ∈ (0, ∞), q > n/(n + α), and f is a probability density on random vectors of R n , with m α [f ] < ∞ its moment of order α and N q [f ] < ∞ its Rényi entropy power, then

m α [f ] 1 α N q [f ] 1 n ≥ m α [G] 1 α N q [G] 1 n , (28)
with equality if and only if f is any generalized Gaussian in (1).

As a direct consequence we indeed have the maximum entropy characterization of generalized Gaussians: among all probability densities with a given moment, e.g.

m α [f ] = m, we have N q [f ] ≤ N q [G γ ] with γ such that m α [G γ ] = m.
The second important result is the fact that the minimum of the generalized Fisher information, among all probability densities with a given Rényi entropy, is reached by a generalized Gaussian. This generalizes the fact that the minimum of the standard Fisher information for probability densities with a given (Shannon) entropy is attained by the standard normal distribution, as shown by Stam's inequality. Actually, the statement on generalized Fisher information relies on a generalized version of Stam's inequality. We will use a remarkable general result of Agueh 1 that gives the optimal functions in some special cases of sharp Gagliardo-Nirenberg inequalities. For n ≥ 2, Agueh's result recovers the results of Del Pino and Dolbeault [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] , while the case n = 1 can be obtained from a sharp Gagiardo-Nirenberg inequality on the real line established by Nagy [START_REF] Nagy | Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung[END_REF] . It turns out that these optimal functions are the generalized Gaussians. Let us rst recall Agueh's result.

Theorem 3. [Agueh 1 -Theorem 2.1, corollary 3.4] For n, p, r and s such that

n > p > 1 and np n -p > s > r ≥ 1 if n > 1 (29) p > 1 and ∞ > s ≥ r ≥ 1 if n = 1
and if u(x) is a function dened on R n such that the involved norms are nite, then the following sharp Gagiardo-Nirenberg inequality holds

K |∇u| θ p u 1-θ r ≥ u s ( 30
)
where K is an optimum constant, and

θ = np(s -r) s [np -r(n -p)]
.

Let p * = p/(p -1) denote the Hölder conjugate of p. This result enables to obtain a generalization of Stam's inequality in R n , n ≥ 1, involving the q-entropy power and the generalized (β, q)-Fisher information.

Proposition 3. For n ≥ 1, β and α Hölder conjugates of each other, α > 1, q > max {(n -1)/n, n/(n + α)} then for any probability density on R n , supposed continuously dierentiable,

N q [f ]I β,q [f ] n βλ ≥ N q [G]I β,q [G] n βλ , (31)
with λ = n(q -1) + 1, and with equality if and only if f is any generalized Gaussian G γ in (1).

Proof. The result mainly follows from Agueh's theorem above. Take β = p, α = p * and let u(x) = f (x) t .

Let us rst consider case (a), and choose the exponent t > 0 such that st = 1. Observe that the condition s > r gives s > β. Finally, let us denote q = rt = (1 + s/α)t. In such conditions, the sharp Gagliardo-Nirenberg inequality (30) becomes

K ˆf (x) q dx (1-θ)t q ˆf (x) βt |∇f (x)| f (x) β dx θ β ≥ 1,
since f is a probability density. The relationship q = (1 + s/α)t = t + 1 -1/β implies that βt = β(q -1) + 1. Hence, we see that the simple change of variable u(x) = f (x) t yields a relation that involves both the information generating function and the generalized Fisher information. Furthermore, we have β/(β -s) = βt/(βt -1) = t/(q -1) < 0 so that the optimum function f (x) = u(x) 1/t is nothing but the generalized Gaussian. Finally, the direct computation of the exponents leads to

M q [f ] 1 1-q I β,q [f ] n β[n(q-1)+1] a K ≥ 1 with 1 > q > max {(n -1)/n, n/(n + α)} and a = t [n(β(q-1)+1)-(n-β)]
n[β(q-1)+1]-q(n-β) > 0, and where we note that M q [f ]

1 1-q = N q [f ].
Taking into account that the equality sign only holds for the generalized Gaussians and that the corresponding expression is scale invariant (as a consequence of (A10)), we arrive at the inequality (31).

The approach is similar in case (b) where r = α (s -1) . Observe that the condition s > r gives here s < β. Take st = q, with t > 0, and rt = αt (s -1) = 1. These two equalities give βt = β (q -1) + 1 which, since β > s, gives q > 1. Hence, we obtain that f (x) = u(x) 1/t is the generalized Gaussian with exponent 1/(q -1) and compact support. Finally, the simplication of the Gagliardo-Nirenberg inequality shows that the inequality (31) holds.

It remains to examine the case q = 1 in (31). In this case, the result can be obtained as a direct consequence of an Lp log-Sobolev inequality, which can be viewed as the limit case of the Sharp Gagliardo-Nirenberg inequality when p ↓ s, cf. Del Pino and Dolbeault [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF] . For instance, exponentiating the Lp logarithmic Sobolev inequality in Gentil [START_REF] Gentil | The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations[END_REF] , eq. ( 1)

ˆ|f | p log |f | p dx ≤ n p log K p ˆ|∇f | p dx
where K p is an optimal constant, and using the change of variable f = |f | p directly gives the generalized Stam inequality (31), with equality for the generalized Gaussians with q = 1.

Remark. Let us mention that the case q = 1 has already be established as a direct consequence of the Lp logarithmic Sobolev inequality in the recent paper by Kitsos and Tavoularis [START_REF] Kitsos | Logarithmic Sobolev inequalities for information measures[END_REF] . Let us also note that another kind of inequality with a generalized Gaussian extremal can also be derived from the standard Sobolev inequality (that is the inequality (30) with θ = 1).

The monodimensional case of (31) has be derived in a very elegant way in the work by Lutwak, Yang and Zhang in Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] . It can also be derived from Nagy [START_REF] Nagy | Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung[END_REF] 's sharp Gagliardo-Nirenberg inequality on R.

Finally, we obtain that the generalized Fisher information I β,q [f ] and the elliptic moments of order α m α [f ] are indeed linked by a Cramér-Rao inequality, in all generality. The related inequality generalizes the standard Cramér-Rao inequality for the location parameter.

Theorem 4. [Cramér-Rao inequality] For n ≥ 1, β and α Hölder conjugates of each other, α > 1, q > max {(n -1)/n, n/(n + α)} then for any probability density f on R n , supposed continuously dierentiable and such that the involved information measures are nite, we have

I β,q [f ] 1 βλ m α [f ] 1 α ≥ I β,q [G] 1 βλ m α [G] 1 α , (32)
with λ = n(q -1) + 1, and where the equality holds if and only if f is a generalized Gaussian

f = G γ .
Proof. The result is an immediate consequence of the moment-entropy inequality (28) and of the generalized Stam inequality (31): the simple term by term product of both inequalities directly gives (32), with the same equality condition as in the two initial inequalities.

V. CONCLUSIONS

In this paper, we have presented a generalized Fisher information measure that ts well in the nonextensive thermostatistics context. Indeed, just as the maximization of the generalized Rényi or Tsallis entropies subject to a moment constraint yields a q-Gaussian distribution, the minimization of the generalized Fisher information subject to the same constraint also leads to the very same q-Gaussian distribution. A generalized Cramér-Rao inequality corresponds to this result. Furthermore a generalized Stam inequality links the generalized entropies and the generalized Fisher information, with a lower bound attained, again, by q-Gaussian distributions. All these results hold for probability densities dened on R n . Hence, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure associated with Rényi or Tsallis entropies. While this work was under consideration, a similar extension of the generalized Fisher information to the multidimensional case has been proposed by Lutwak et al. [START_REF] Lutwak | Extensions of Fisher Information and Stam's Inequality[END_REF] in the context of information theory.

Future work will include the study of further properties of the generalized Fisher information, namely its convexity properties. Here, the generalized Fisher information has been introduced as the information attached to the distribution. In estimation theory, the Fisher information is dened with respect to a general parameter and characterize the information about this parameter, as well as the estimation performances, as exemplied by the classical Cramér-Rao bound in estimation theory. Hence, it would be of interest to look at general estimation problems that could involve a similar generalized Fisher information. In nonextensive thermostatistics, the notion of escort distributions is an important ingredient related to the generalized entropies. Thus, it would also be of interest to see how these escort distributions can be introduced in the present setting.

αγ|x| α-1 (1 -γ(q -1)|x| α ) -1 + for q = 1, so that the generalized Fisher information has the expression I β,q [G γ ] = (αγ) β (µ 0,1 ) β(q-1)+1 ˆ|x| α (1 -γ(q -1)|x| α ) β(q-1)+1

(q-1) -β + dx (A6) = (αγ) β µ α,1

(µ 0,1 ) β(q-1)+1 .

(A7)

We easily obtain that (A7) also holds in the case q = 1. The explicit expression of the generalized Fisher information in the case of the generalized Gaussian is therefore

I β,q [G γ ] = (α) β 2 α n ω n β(1-q)
|(q -1)| -n β α (1-q)-1 γ β α (n(q-1)+1)

×        B(1+ n α ,-q q-1 -n α ) [B( n α ,-1 q-1 -1 α )]
β(q-1)+1

for max 1 -α, n n+α < q < 1 B(1+ n α , q q-1 ) [B( n α , q q-1 )] β(q-1)+1 for q > 1, (A8) and, for q = 1,

I β,q [G γ ] = (α) β 2 α n ω n β(1-q) γ β α (n(q-1)+1) n α Γ n α β(1-q) . (A9)
Finally, let us note that we have the following simple scaling identities:

          
M q [G γ ] = γ n α (q-1) M q (G), (A10)

I β,q [G γ ] = γ

Unfortunately, the general

  characterization of the convexity or pseudo-convexity properties of the generalized Fisher information remains an open problem. A radially symmetric function f (x) only depends on the euclidean distance r = |x| of the argument from the origin. Hence, in the case of radially symmetric function dened on a symmetric domain Ω, we have f (x) = f r (|x|) = f r (r), where f r is an univariate probability density dened on a subset Ω r of R + . With these notations, we simply have |∇f (x)| = dfr(r) dr . In polar coordinates, with r = |x|, dx = r n-1 drdu and ´du = n ω n ,

1

 1 and where w(x) ≥ 0 is a weight function. The equality case then occurs i |u(x)| α = k|v(x)| β , with k > 0.

  r = 1 + s/p * , then the extremal functions have the form Cu(σ(x -x)) where C, σ and x are arbitrary, and u(x) = 1 + |x| p * p p-s . (b) If r = p * (s -1) , then the extremal functions have the form Cu(σ(x -x)) where C, σ and x are arbitrary, and u(x) = 1 -|x| p * p-1 p-s + .

βα

  (n(q-1)+1) I β,q [G], m α [G γ ] = γ -1 m α [G].

ACKNOWLEDGMENTS

The author thanks the anonymous referee for his valuable comments and suggestions that helped improve the presentation of this article. Thanks are extended to Lodie Garbell for her friendly proofreading of the manuscript.

Appendix A: Information measures of generalized Gaussians

In order to get the expressions of the information measures associated to the generalized Gaussian (1), that in turn give the explicit expressions of the bounds in ( 27), ( 28) and ( 31), (32), we use the following result: Proposition 4. Let

with α, γ > 0. By direct calculation in polar coordinates, one gets

where ω n is the volume of the n-dimensional ball.

From this expression, we immediately identify that the partition function is Z(γ) = µ 0,1 ,

Similarly, we obtain the information generating function M q [G γ ] = ˆGγ (x) q dx = ´gγ (x) q dx ´gγ (x)dx q = µ 0,q (µ 0,1 ) q , (A3) with s = q -1. The information generating function, and thus the associated Rényi and Tsallis entropies are nite for q > n/(n + α).

Likewise, the moment of order p is given by

If p = α, by the properties of the Beta functions, the expressions for the moment of order p

for q = 1, q > n/(n + α)

Let us now consider the generalized Fisher information I β,q [f ] dened in (10). For a radially symmetric function, that is f (x) = f (|x|) = f (r), we simply have |∇f (x)| = df (r) dr , and in the case of the generalized Gaussian (1), we obtain that G γ /G γ = g γ /g γ =


R Bachelard

H Bender

Ph W Courteille

C Stehle

C Zimmermann

S Slama

The role of Mie scattering in the seeding of matter-wave superradiance

Keywords: numbers: 42.50.Ct, 03.75.-b, 42.50.Gy

Matter-wave superradiance is based on the interplay between ultracold atoms coherently organized in momentum space and a backscattered wave. Here, we show that this mechanism may be triggered by Mie scattering from the atomic cloud. We show how the laser light populates the modes of the cloud, and thus imprints a phase gradient on the excited atomic dipoles. The interference with the atoms in the ground state results in a grating that in turn generates coherent emission, contributing to the backward light wave onset. The atomic recoil 'halos' created by the Mie-scattered light exhibit a strong anisotropy, in contrast to single-atom scattering.

Matter wave superradiance (MWSR) [1] and collective atomic recoil lasing (CARL) [2,3] are light-induced instabilities of the density distribution in atomic clouds. More precisely, they are due to correlations between successive scattering events mediated by long-lived coherences in the motional state of an (ultracold) atomic cloud or in the light field of an optical resonator [4]. Despite considerable theoretical efforts having been devoted to the dynamics of MWSR [5][6][7], open questions still remain. One of them concerns the seeding mechanism which is able to start the MWSR instability even in the presence of losses. Thermal and quantum fluctuations will naturally contribute. However, in this paper we point out the particular role of Mie scattering, which turns out to be important at the onset of MWSR. Indeed, Mie scattering is active before any instabilities have developed, and it induces a phase correlation between the atomic dipoles that favors the build up of an instability.

The below-threshold dynamics and the seeding of matter-wave superradiance are interesting problems. As long as we consider the atomic cloud as a homogeneous entity, e.g., a Bose-Einstein condensate (BEC) in the mean field description, no scattering should occur at all. Theoretical models which describe BEC's as matter waves without fluctuations thus fail to explain how MWSR is initiated in the absence of a seeding wave [8]. In contrast, recent work has shown [9,10] how atomic coarse-graining, density fluctuations and Mie scattering from finite-sized clouds can even influence the scattering of a single photon. Also optical cavities may strongly affect the scattering by shaping the angular distribution of the density of modes that are capable of receiving the scattered photons [11]. These processes have a decisive impact on the mode competition preceding the exponential instability, and thereby on the instability itself.

Here we show that Mie scattering, caused by the finite size of the atomic cloud, favors the formation of a matter wave-dipole grating. This has already been suspected in [12]. Indeed, prior to any significant motion of the atoms, their dipoles collectively order, which in turn leads to coherent emission.

In previous papers [13][14][15][16], we discussed the impact of atomic coarse-graining and finite scattering volumes on the radiation pressure force which acts on the cloud's center of mass. Here, we investigate the momentum distribution following the cooperative scattering of the laser light by a BEC. Our theoretical model describes the atomic cloud as a macroscopic matter wave that is homogeneously distributed within a sphere, i.e., the atoms are considered to be strongly delocalized and density fluctuations are neglected. We find that the momentum distribution of the atoms adopts the shape of a recoil halo, very similar to the ones observed experimentally in timeof-flight images of BEC's. The halo indicates the directions into which the atoms are preferentially scattered before the density distribution is noticeably modified. In particular, it exhibits a pronounced peak at 2 k. This corresponds to an increased backscattering of light that acts as a seed for MWSR.

We consider an homogeneous spherical cloud of twolevel atoms illuminated by a laser of wavevector k 0 = k 0 ẑ, Rabi frequency Ω 0 = dE 0 / (where d is the electric dipole matrix element and E 0 the laser electric field) and detuned from the atomic transition by ∆ 0 . The atomic cloud is described as a bosonic ensemble of N two-level atoms with field operator Ψ(r, t) = Ψg (r, t) + Ψe (r, t) (g for the ground state, e for the excited one). We treat the condensate as an ideal gas and consider the scattering between matter waves and optical waves, but we neglect nonlinearities due to atom-atom interaction. In second quantization, the interaction between the atoms and light is described by the Hamiltonian [13,17]:

Ĥ(t) = Ω 0 2 dr Ψ † e (r, t) Ψg (r, t)e -i∆0t+ik0•r + h.c. (1)
+ k g k dr Ψ † e (r, t) Ψg (r, t)â k e -i∆ k t+ik•r + h.c. ,
where

g k = d[ω k /(2 ǫ 0 V v )] 1/2
and V v is the quantization volume. In Eq.( 1), the first (second) line describes the absorption and emission of a pump mode Ω 0 (a vacuum mode âk ). Doppler effects are neglected. Replacing Ψe (r, t) → Ψe (r, t)e i∆0t induces an energy shift of -∆ 0 | Ψe | 2 in the Hamiltonian, from which the Heisenberg equations may be derived:

∂ Ψg ∂t = -i Ψe Ω 0 2 e -ik0•r + k g k â † k e -i(∆0-∆ k )t-ik•r , (2)
∂ Ψe ∂t = -i Ψg Ω 0 2 e ik0•r + k g k âk e i(∆0-∆ k )t+ik•r +i∆ 0 Ψe (3) dâ k dt = -ig k e -i(∆0-∆ k )t dr Ψ † g (r, t) Ψe (r, t)e -ik•r . ( 4
)
For large atom numbers and far detuning from the atomic transition frequency, one can neglect quantum fluctuations and treat the operators as c-numbers ( Ψ → ψ, âk → a k ). Eq.( 4) is integrated over time as

a k = -ig k t 0 dt ′ e -i(∆0-∆ k )t ′ drψ * g (r, t ′ )ψ e (r, t)e -ik•r , (5)
and inserted into Eq.( 3). Furthermore, we switch to a continuous-mode description k → (V ν /(2π) 3 ) dk and obtain:

∂ψ e ∂t = i∆ 0 ψ e (r, t) -i Ω 0 2 e ik0.r ψ g (r, t) -ψ g (r, t) dr ′ (6) × dkg 2 k e ik•(r-r ′ ) t 0 dt ′ e i(∆0-∆ k )(t-t ′ ) ψ * g (r ′ , t)ψ e (r ′ , t ′ ).
We consider time scales over which the atomic density does not significantly change, i.e., ψ g (r, t) ≈ ψ g0 (r) in Eq.( 6), with ρ 0 (r) = |ψ g0 (r)| 2 the initial density of the cloud. Using the Markov approximation, i.e., the photon time of flight through the cloud is much shorter than the atomic decay time, the last integral in Eq.( 6) is replaced by δ(k -k 0 )ψ g0 (r ′ )ψ e (r ′ , t)/c. With the assumption that all the electromagnetic modes are equally present in the system (g k ≈ g k0 ) and by keeping rotatingwave-approximation terms, one can show that [18] dkg

2 k e ik•d ∞ 0 dt ′ e i(∆0-∆ k )(t-t ′ ) = Γ 2ik 0 |d| e ik0|d| , (7) where
Γ = V ν g 2 k0 k 2 0 /πc = d 2 k 3 0 /(2π ǫ 0 )
is the atomic decay rate. For the normalized excitation field β(r, t) = ψ e (r, t)/ψ g0 (r), with |β(r)| 2 the probability for an atom to be excited, one obtains [START_REF](Γ/2)β in Eq.(8) arises from the non-commutative properties of the bosonic operators Ψg and Ψe, analogously to the spontaneous emission from vacuum arising from the commutation rule for âk[END_REF]:

∂β(r, t) ∂t = i∆ 0 - Γ 2 β(r, t) -i Ω 0 2 e ik0•r - Γ 2 dr ′ ρ 0 (r ′ ) exp (ik 0 |r -r ′ |) ik 0 |r -r ′ | β(r ′ , t), (8)
which recovers the model of cooperative scattering of a plane-wave introduced in [13].

In the steady-state regime, Eq.( 8) can be used to describe the scattering of light from a dielectric medium with an index of refraction [16]

m c = 1 -4πρ 0 Γ/k 3 0 (2∆ 0 + iΓ). (9)
In this regime of linear optics, the field β is directly proportional to the electric field, and the ratio between both is the susceptibility. Hence, the excitation pattern inside the cloud can be calculated analogously to Mie's theory [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF][START_REF] Bohren | Absorption and Scatter-ing of Light from Small Particles[END_REF]. The polarization amplitude β inside the cloud is decomposed into modes, thereafter labeled n, which are elementary solution of the Helmholtz equation:

(∆ + m 2 c k 2 0 )β = 0. ( 10
)
In spherical homogeneous distributions (with coordinates r(r, θ, φ), where θ is the angle with respect to the ẑ axis) these solutions are of the form j n (m c k 0 r)P n (cos θ) inside the cloud, with j n the spherical Bessel function and P n the Legendre polynomials. Then the amplitude β n of each mode are calculated according to the method described in Ref. [16], giving for an uniform spherical cloud:

β(r) = Ω 0 Γ ∞ n=0 (2n + 1)i n β n j n (m c k 0 r)P n (cos θ), (11)
where the coefficients β n are given by

β n = j n (k 0 R) (2δ + i)j n (m c k 0 R) + iλ n h (1)
n (k 0 R) , (12)
with R the radius of the sphere and h

n the spherical Hankel function. In general, the radiation is accurately described by considering a number of modes on the order of the size of the system k 0 R [START_REF] Bohren | Absorption and Scatter-ing of Light from Small Particles[END_REF]. Note though that such a coarse-grained approach to describe the cloud cannot capture features associated with disorder (e.g., Anderson localization).

As can be deduced from Eq.( 10), the laser imprints a phase with wavevector m c k 0 to the excited state ψ e , which interferes with the ground state ψ g that has a constant phase. The resultant grating is at the origin of coherent emission and of the backscattering wave that acts as a seed for MWSR. Indeed, the field radiated by the atoms inside the cloud in a direction u(θ, φ) is proportional to the structure factor:

s c (u) = 1 N ρ(r)β(r)e -imc k0u•r dr, ( 13
)
with the index of refraction m c given above. Hence, the coherent emission by the cloud is a direct consequence of the periodic excitation field and the resulting grating. Note that we do not perform an adiabatic elimination of the excited state [START_REF] Gardiner | Quantum Noise[END_REF]; we rather extract the scattering pattern from this state.

The scattered light is difficult to observe directly. However, the radiation pattern is also present in the momentum distribution of the atoms, which can be easily recorded by time-of-flight imaging. Different from an N -body model, the quantum matter field approach yields the momentum distribution simply through the Fourier transform of the matter field, ψ(p) = (2π ) -3 ψ(r)e -ip•r/ dr. Hence, for an homogeneous cloud, the momentum distribution of the excited state is directly proportional to the structure factor: ψ e (p) ∝ s c (p/ ), with p = m c k and k = k 0 u. Using Eq.( 11), it can be deduced that the momentum wavefunction ψ e for an uniform sphere of radius R reads: small cloud (k 0 R ≪ 1), the scattering of light is isotropic, as expected from Rayleigh theory: only the first Mie mode (n = 0) is populated. However, for large manyparticle spheres, Mie scattering turns out to be fundamentally anisotropic, as many modes are populated. Note that the momentum distribution D e (p) = | ψ e (p)| 2 is concentrated along a circle with radius p = m c k 0 rather than k 0 (i.e., that γ n (p) reaches a maximum for p = m c k 0 ). This is a signature of the Minkowski momentum for atomic recoil, that characterizes the momentum exchanged between light and matter in dielectric media [START_REF] Campbell | [END_REF]. The blurring of the momentum wavefunction along the circle originates in the finite size of the cloud that creates a natural momentum spread σ p ∼ /R. The ripples of the distribution are due to the sharp boundary of the cloud's density, yielding a Fourier transform with many secondary peaks.

ψ e (p) = Ω 0 √ ρ 0 Γ(2π ) 3
In optical dilute clouds, almost all of the light is scattered forward, leaving the incoming light almost untouched. Such a scattering pattern is displayed in Fig. 1(b), where the intricate pattern of Mie scattering by an optically dense cloud has disappeared. In this limit, the excitation field is often approximated by the timed Dicke state β(r) ∼ β T DS e ik0•r [23], yet such an ansatz fails to predict any three-dimensional recoil pattern.

After an atom absorbs a photon from the laser with a momentum kick m c k 0 , the photon is reemited according to the Mie pattern s c (p/ ) ∝ ψ e (p). Thus the atom will gain an extra momentum m c k 0 with a direction opposite to the emitted photon, and the momentum pattern of the ground state atoms after the scattering process is given by | ψ e (m c k 0p)| 2 . Experimentally, the columnintegrated momentum distribution is observed in time-offlight images. This leads to defining the projected distribution D y g (p x , p z ) = | ψg (p)| 2 dp y . Such an integrated distribution is presented in Fig. 2(a). The atoms are observed to inhomogeneously fill a circle of radius m c k 0 . In particular, the part of the sphere around p = (3/2) k 0 is weakly populated, which reflects the anisotropic nature of Mie scattering.

Experimentally, the atomic recoil patterns are investigated by using the set-up of [10]. After the interaction with the light, the 87 Rb atoms ballistically expand for t F = 20ms. Since the initial size of the cloud ∼ 20µm is much smaller than its expanded size at the time of imaging, the density distribution of the expanded cloud is recorded by standard absorption imaging yielding the initial momentum distribution before the expansion according to p = m Rb r/t F .

The integrated momentum distribution observed experimentally reproduces the features predicted by Mie scattering (see Fig. 2(b)). The sphere of radius k 0 is filled with atoms, yet it exhibits a region where the probability of the atomic recoil is very low around p z = 1.5 k 0 . Moreover, it can be observed that a large number of atoms recoil around p = 2 k 0 , which is the signature of the backward emitted wave. The presence of preferred directions of emission is associated with the presence of Mie resonances [16]: the increase of emission in these directions can be interpreted as a self-Purcell enhancement [24] because the cloud acts as a cavity on itself to modify its emission. This is in contrast to single-atom scattering, where light is scattered in random directions and not specifically backward and forward. It is important to note that Mie scattering is a threedimensional process, while MWSR only develops along its most unstable direction. Mie scattering is a seeding process that emits light in many directions; including backwards, which is known to be the most unstable direction for MWSR in a cigar-shaped cloud illuminated along its main axis [25].

These two processes are illustrated in Fig. 3. Initially, Mie scattering populates a sphere of radius p ≈ m c k 0 . Then a MWSR instability develops, i.e., the grating induces light emission that, in turn, amplifies this atomic grating. The atoms observed in p ≈ -2m c k 0 and p ≈ 4m c k 0 are not predicted by Mie scattering and can be explained only by the self-consistent matter-wave dynamics. In this paper we showed that Mie scattering induces a grating in the atomic distribution even below the threshold for the MWSR instability. Its signature is an anisotropic three-dimensional halo in the atomic momentum distribution. The atoms observed at p ≈ 2 k 0 generate a seeding wave for MWSR. Indeed, the matter wave modes at 0 and 2 k 0 together form a density grating, at which subsequent light injected from the pump laser is Bragg-scatterered in a self-amplifying process.

Note that quantum fluctuations and disorder may also give rise to backward emission [26], and thus act as a seed for the matter-wave superradiance. These effects will be included in an extended work.

Finally, it is interesting to remark that the off-axis emission of photons should be associated with higher modes (n ≫ 1) that correspond to photons with long lifetime within the cloud [27]. Thus, a time-resolved observation of the off-axis atomic recoils should bear the signature of subradiance.

pp

  FIG. 1. Momentum distribution of the excited state De = | ψe| 2 in logarithmic scale. The emission pattern |sc(k)| 2 concentrates around a circle with radius p = mc k0 (black line). Simulations for a laser detuning ∆0 = -3GHz and a cloud (a) of size k0R = 20, atom number N = 1.15 × 10 6 and refractive index mc = 1.2, (b) of size k0R = 20, atom number N = 100 and refractive index mc = 1 + 2.10 -7 (∆0 = -488MHz). For rubidium, k Rb 0 = 8.05 × 10 6 m -1 and Γ Rb = 6.1MHz.

Fig. 1 (

 1 a) shows a typical momentum distribution of the excited atoms | ψ e (p)| 2 and the associated scattering pattern |s c (k)| 2 of the light. Although most atoms recoil to k = k 0 ẑ, a significant amount of light is scattered backward (k = -k 0 ẑ) and acts as a seed for the MWSR instability.

FIG. 2 .

 2 FIG. 2. (a) Integrated momentum distribution of the ground state, calculated from Eq.14 for a spherical homogeneous cloud of radius k0R = 29.6, ∆0 = -15GHz and with a refractive index mc = 1.067 (a low-pass Gaussian filter was applied to attenuate the ripples due to the sharp boundaries of the homogeneous clouds, since they are irrelevant for comparison with the experiment) (b) Experimental integrated momentum distribution of the ground state for an ellipsoidal cloud of length k0σz ∼ 29.6, transverse radius k0σ ⊥ ∼ 4.7, with N ∼ 147000 atoms and a laser detuning ∆0 = -15GHz and a 20µs laser pulse of 17mW. Its refractive index is mc ≈ 1.067.

FIG. 3 .

 3 FIG. 3. Momentum distribution | ψg| 2 above the threshold of the MWSR. Experiment realized with an ellipsoidal cloud of transverse radius k0σ ⊥ ∼ 3.5, length k0σz ∼ 22, with N ∼ 156000 atoms and a laser detuning ∆0 = -15GHz.

We acknowledge helpful discussion with R. Kaiser. This work has been supported by the Fundação de Amparo Pesquisa do Estado de São Paulo (FAPESP) and the Research Executive Agency (program COSCALI, No. PIRSES-GA-2010-268717). C.S., C.Z and S.S acknowledge support by the Deutsche Forschungsgemeinschaft.