Gautier Meyer
email: gautier.meyer@francetelecom.com

Vincent Louis
email: vincent.louis@francetelecom.com

Jean-Paul Sansonnet
email: jean-paul.sansonnet@limsi.fr

Yannick Larvor
email: yannick.larvor@francetelecom.com

Two logical operators for representing and handling preferences

Two logical operators for representing and handling preferences

Introduction

The agent paradigm proposes an attractive point of view to model complex information processing systems, all entities being represented in an abstract but nevertheless intuitive way. In particular, rational agents are generally formalized as cognitive entities defined upon primitive concepts (so-called mental attitudes) such as belief, desire or intention [START_REF] Rao | [END_REF], and can be implemented by automatic inference mechanisms [START_REF] Bretier | A rational agent as the Kernel of a cooperative spoken dialogue system: Implementing a logical theory of interaction[END_REF]. Unfortunately, these models do not completely specify the agent's reactions. This is unsuitable in complex situations where s/he has several motivations (or intentions) and/or several means for satisfying them (action plans). For example, as part of a multimodal interaction, an assistant agent has to provide users with itineraries either through a vocal interface (by speaking instructions) or through a graphic interface (by showing a map) according to the context (case of several action plans satisfying the same intention). This has led the authors of [Sadek, 1991;[START_REF] Haddawy | [END_REF] to affirm the need of establishing explicitly the link between the mental state and the reaction of an agent. Thereafter, the authors of [START_REF] Haddawy | [END_REF]Louis, 2002] have shown that this link could be done in two distinct steps (see figure 1): a planning step [Allen et al., 1991;Louis, 2002] and a decision step [START_REF] Doyle | [END_REF][START_REF] Chiclana | Preference relations as the information representation base in multi-person decision making[END_REF]Ha, 2001]. The first one aims at generating the potential reactions of the agent (alternatives); the second one sorts them to choose the actual action to carry out. This last one requires to model preference (see [Öztürk et al., 2005] for an overview).

In a variety of fields, Utility functions are often used for decision steps. The utility of each alternative is computed and the highest rated alternative is selected as the reaction. This method yields a detailed representation of knowledge on the classification of alternatives. Moreover, it makes it possible to take into account the stochastic nature of the environment (concept of expected utility) and has been well studied in the context of decision theory (see [Fishburn, 1999] for a selective survey of numerical representations of preference). Unfortunately, those numerical representations suffer from the known following drawbacks and thus do not seem to be adapted to cognitive agent modeling.

• The determination of utility is difficult in practice. It requires a significant specification effort since describing each alternative requires a global view of the considered problem [START_REF] Doyle | [END_REF]Ha and Haddawy, 1999;[START_REF] Faltings | [END_REF]]. • If the result "optimality" is not crucial, this method is not justified [Brafman and Tennenholtz, 1997;[START_REF] Boutilier | [END_REF]Ha and Haddawy, 1999]. • In order to have a decision method more homogeneous with the agent's internal description via mental attitudes, a qualitative technique would be preferable to a quantitative one [START_REF] Dubois | [END_REF]]. • In the perspective of basing agents' models upon humans' cognitive ones, calling upon qualitative decision techniques seems to be a more natural way [Brafman and Tennenholtz, 1997;[START_REF] Rossi | [END_REF]. To bypass these limitations, we choose to rank alternatives according to a logical preference representation (see [Lang, 2004] and [START_REF] Coste-Marquis | [END_REF] for a non exhaustive comparison of some ordinal representations). More precisely, we define a formal notion of preference, called "partial preference" and associated to a given "point of view", that makes it possible to compare the alternatives with each other without any numerical representation.This notion facilitates the specification of the elementary informations necessary to the ranking and thus to the decision step.

Section 2 globally describes our proposal. Sections 3 and 4 formalize the progressive construction of our model by introducing two logical operators: an operator related to a notion of primitive partial preference and another one related to a notion of extended partial preference.

Overall scheme

Our goal is to provide a formal way both to specify preferences naturally and concisely, and to compare any alternative with any other. This is achieved by modeling the necessary information to make a choice thanks to the concept of partial preference.

Many researchers in AI and philosophy think that the intuitive specification of a preference implicitly makes the Ceteris Paribus assumption (see [START_REF] Boutilier | [END_REF]): all characteristics other than those explicitly specified are supposed to be equal [START_REF] Hansson | [END_REF][START_REF] Doyle | [END_REF]. For example, the statement "I prefer fish dishes to meat ones" implicitly means that characteristics such as price or abundance are the same. Moreover, such an assumption allows for a succinct representation of numerous partial pre-orders [START_REF] Coste-Marquis | [END_REF]. However this assumption is not sufficient for comparing any two objects, in particular those whose unspecified characteristics are not equal: one cannot say whether steamed fish is preferred to grilled meat.

To avoid this problem, our main idea is to split the specification of preferences into two parts (see figure 2). Firstly, the initial data defining each partial preference is interpreted according to the ceteris paribus assumption, which generates a primitive partial preference (generation stage). Secondly, each primitive partial preference is extended (beyond the ceteris paribus assumption) in an extended partial preference (extension stage). This extension stage makes it possible to compare any alternatives. Actually, several partial preferences can be specified in parallel (which result in several primitive and extended partial preferences), each of them representing a "point of view" for classifying alternatives. In our approach, each partial preference is modeled by a binary relation between the objects to compare. These objects, which are called "alternatives", correspond in fact to states of the world. As it is difficult, if not impossible, to represent exhaustively the characteristics of a state of the world (see for example the frame problem [START_REF] Mccarthy | [END_REF]), our model additionally associates with each partial preference an operator over couples of logical formulas that partially describe the two states to be compared. In short, each partial preference is associated with both a binary relation over alternatives and a logical operator over formulas representing alternatives. Our vision is therefore close to Von Wright's according to which the preferences are specified by descriptions but actually apply to "concrete" alternatives corresponding to these descriptions [START_REF] Von | Georg Henrik von Wright. The logic of preference reconsidered[END_REF].

Preliminaries

Language used

In this article, L is a propositional language built upon a set of propositional symbols SYM, the constants (tautology) and ⊥ (contradiction) and the usual logical operators (∧, ∨, ¬, ⇔,⇒). The symbols φ, φ 1 , . . . , ψ, ψ 1 , . . . denote formulas of L . The notation φ means that the formula φ is valid. For any formula φ ∈ L , S(φ) denotes the set of the propositional symbols appearing in this formula. In particular, S( ) = S(⊥) = ∅. S m (φ) denotes the subset of the propositional symbols in S(φ) which are relevant for φ's truth value, thus it is exactly the unique1 set of propositional symbols appearing in the "smallest" (wrt. the number of symbols) formulas logically equivalent to φ:

S m (φ) = S(ψ) iff • ψ ⇔ φ • ∀ψ ∈ L, if ψ ⇔ ψ then S(ψ) ⊆ S(ψ )
φ+ψ denotes the formula whose relevant symbols are those of φ and ψ together, and such that any valuation that validates it associates (1) each relevant symbol for φ with the same truth value and (2) each relevant symbol for ψ but not for φ with the opposite value. The following properties formally define φ+ψ:

       φ+ψ ∈ L S m (φ+ψ) = S m (φ) ∪ S m (ψ) ∀s ∈ S m (φ), φ+ψ ⇒ s iff φ ⇒ s ∀s ∈ S m (ψ)\S m (φ), φ+ψ ⇒ s iff ψ ⇒ ¬s

Alternative

Alternatives are states of the world, i.e. particular interpretations of all symbols of SYM. We note them a, a 1 , . . . , b, . . .. Formally, if ALT is the set of alternatives, for any alternative a of ALT and any symbol s of SYM, either a s or a ¬s.

A formula φ of L denotes a set of combinations of truth values for the propositional symbols of SYM, which can be seen as a set of alternatives: {a such that a ∈ ALT and a φ}. In particular, the formula represents all the alternatives whereas ⊥ does not represent any alternative. Conversely, an alternative can be associated to a formula that specifies a particular truth value for every symbol of SYM. We note f a such a formula that represents exactly the only alternative a.

Preference

A preference is modeled by a binary relation over the set of alternatives. In particular, a partial preference is a preference relation bound to a given point of view. We note the relationships associated to the partial preferences with the symbols G 1 , G 2 , . . . , G i , G 1 , . . . where i is a number identifying the considered point of view. The notation a 1 G i a 2 means that, from the i th point of view, the alternative a 1 is preferred to the alternative a 2 . A preference can thus be seen as a set of couples of alternatives. This set may contain contradictory couples, for example, both aG i b and bG i a.

Indifference

Two alternatives are known as indifferent with respect to a given partial preference (or point of view) if and only if, according to this preference, each one is preferred to the other one or if none of them is preferred to the other one.

Primitive partial preferences 4.1 Formalization Ceteris Paribus

The alternatives a and b of ALT are Ceteris paribus with respect to the formulas φ and ψ of L (noted CP (a, b, φ, ψ)), if and only if the alternatives a and b assign the same truth value to any symbol appearing neither in φ nor in ψ:

CP (a,b,φ,ψ) iff    ∀s ∈ SYM, if s / ∈ S m (φ) ∪ S m (ψ) then a s iff b s   

Definition

Each primitive partial preference relation is associated to a logical operator upon the formulas of L . It is denoted by the symbol i , where i is the considered point of view. The set of all i operators define the language L : if φ and ψ are formulas of L (but not L ) then φ i ψ is a formula of L . φ i ψ means that, from the i th point of view, the alternatives verifying properties expressed in the formula φ and, if it is coherent, the opposite of the ones expressed in ψ, are preferred to the alternatives verifying properties expressed in the formula ψ and, if it is coherent, the opposite of the ones expressed in φ, if they are Ceteris Paribus with respect to the formulas φ and ψ (i.e. all other things, not explicitly specified by φ and ψ, being equal). If G i is the primitive partial preference binary relation associated to the i th point of view (as defined in 3.3), the semantics of the corresponding logical operator i is formalized as follows:

φ i ψ iff • ∃(a,b) ∈ ALT 2
, such that a φ+ψ b ψ+φ

•    ∀(a,b) ∈ ALT 2 , such that a φ+ψ b ψ+φ if CP (a,b,φ,ψ) then aG i b    (1)
In practice, each primitive partial preference relation G i is specified by an initial set of formulas of the form φ i ψ (φ and ψ being themselves formulas of L ): these formulas are the initial data represented in figure 2. They do not necessarily induce coherent sets of preferences: they may specify both φ i ψ and ψ i φ.

To make it possible to compare alternatives naturally, the binary relation supporting each primitive partial preference must be a pre-order2 (i.e. a reflexive and transitive relation). That is why the following axioms are imposed to the corresponding logical operator (supporting the considered primitive partial preference).

Reflexivity axiom (R)

Any consistent formula of L is required to be preferred to itself:

if φ ⇔⊥ then φ i φ (R)
This entails that the binary relation G i supporting the corresponding primitive partial preference is reflexive. This ensures that two similar alternatives are indifferent.

Transitivity axiom (T)

We impose that if the formula φ 1 is preferred to φ 2 and the formula φ 2 is preferred to φ 3 , then φ 1 is preferred to φ 3 :

if φ 1 i φ 2 and φ 2 i φ 3 then φ 1 i φ 3 (T)
This entails that the binary relation G i supporting the corresponding primitive partial preference is transitive. This ensures that if the alternative a 1 is preferred to a 2 and the alternative a 2 is preferred to a 3 , then a 1 is preferred to a 3 .

Remarkable properties

Link between operator i and relation G i As stated in section 3.2, each alternative can be identified with a formula of L that specifies a truth value for each propositional symbol. Consequently, if a couple of alternatives belongs to a preference relation, then the associated formulas are connected by the corresponding logical operator. Moreover, if two formulas, that assign a value to each symbol of SYM, are connected by the logical operator, then the two associated alternatives belong to the corresponding preference relation :

∀(a, b) ∈ ALT 2 , aG i b iff f a i f b
Immediate proof by identifying the formulas φ and ψ with the formulas f a and f b in the definition (1).

Behavior of operator i wrt. logical equivalence If two formulas are logically equivalent, then any occurrence of the first one in a partial preference expression can be substituted for the second one and conversely. Formally:

if φ 1 ⇔ φ 2 ψ 1 ⇔ ψ 2 then φ 2 i ψ 2 ⇔ φ 1 i ψ 1
As a result, if two formulas are equivalent then the former is preferred to the latter and conversely.

Proof by use of the reflexivity and transitivity properties.

Ceteris paribus (CP)

The formula φ is preferred to the formula ψ if one of the characteristics of φ is preferred to one of the characteristics of ψ and if the other characteristics of φ and ψ are equal. Consequently, if a formula is preferred to another one, then the specialization of the former by any criterion not yet specified is preferred to the specialization of the latter by the same criterion. Formally:

if φ i ψ then   ∀γ ∈ L such as (γ ∧φ)+ψ ⊥ (γ ∧ψ)+φ ⊥ (φ∧γ) i (ψ∧γ)   (CP)
Proof by use of definition 1.

Addition of preferences

If the alternatives satisfying a formula φ are preferred to those satisfying a formula ψ when all other criteria not explicitly specified by φ and ψ are equal, and if, similarly, the alternatives satisfying a formula γ are preferred to those satisfying a formula ω when all other criteria are equal, then the alternatives satisfying both φ and γ are preferred to those satisfying both ψ and ω providing that all other criteria are equal:

if φ i ψ γ i ω and
(γ ∧φ)+ψ ⊥ (γ ∧ψ)+φ ⊥ (ψ∧γ)+ω ⊥ (ψ∧ω)+γ ⊥ then (φ∧γ) i (ψ∧ω)
Proof by use of the CP and transitivity properties.

Example

Let L be a propositional language to describe some properties of the world regarding colours, which is built, as specified in section 3.1, upon the set of five propositional symbols SYM = {C g , C b , C r , C o , C y } (meaning respectively "green", "blue", "red", "orange", "yellow").

Let us consider a partial preference supporting a "colour" point of view (identified with i = 1) and defined by the information gathered in the table of figure 3. The resulting language L (see the previous section) includes the 1 logical operator, which makes it possible to express preferences on colours under the Ceteris Paribus assumption. For example, C g 1 C b means that any green object is preferred to any blue object if all other characteristics are equal. Thanks to the properties formalizing primitive partial preferences, the basic data specified in figure 3 additionally entails the following formulas:

Colour

C g 1 C b C r 1 C g C o 1 C y C y 1 C g C o 1 C g
• C r 1 C r ; C b 1 C b ; . . . (R) • C r 1 C b ; C o 1 C b ; C j 1 C b (T) • C g ∧C r 1 C b ∧C r ; C g ∧C o 1 C b ∧C o ; . . . (CP)
For instance, the fact "from the colour point of view, red objects are preferred to blue ones" (i.e. C r 1 C b ) can be explicitly derived from the first two lines of the table and the transitivity property of the primitive partial preference operator. The initial data for specifying a partial preference (according to a given point of view) could also be represented graphically, which is easier to read for humans (see figure 4).

Extended partial preferences 5.1 Motivation

The properties we stated in the previous sections make it possible to practically build primitive partial preference operators (upon formulas) and thus to conveniently specify a partial preference binary relation (upon alternatives).

Nevertheless, in practice the alternatives to be compared are seldom explicitly in relation via a primitive partial preference. Indeed, the description criteria are generally interdependent, so that a difference between alternatives on a criterion implies differences on the other criteria3 . Consequently, it is necessary to be able to compare two alternatives such that the former satisfies a formula φ∧γ and the latter satisfies a formula ψ∧ω, and such that only the formula φ i ψ of L is initially specified. This means one needs a logical operator that can handle couples of formulas of the form (φ∧γ, ψ∧ω) while knowing only that φ i ψ but without knowing anything about the couple of formulas (γ, ω).

Example (continued)

Let us add the C and M propositional symbols (respectively meaning "car" and "motorcycle") to the language L introduced in the previous example. The logical framework we have defined for handling primitive partial preferences makes it possible to formally derive facts such as "from the colour point of view, red cars are preferred to blue cars" (i.e. C r ∧ C 1 C b ∧ C) from the previously inferred formula C r 1 C b and the CP property. In other words, it can be explicitly deduced that an alternative verifying the property "red" is preferred to any alternative verifying the property "blue", provided they assign the same truth values to other criteria (Ceteris Paribus = other things being equal). Unfortunately, if other things are not equal, such a deduction cannot be derived. In particular, nothing can be said about the preference between alternatives verifying the properties "red" and "car" and alternatives verifying "blue" and "motorcycle".

Extension principle

Nevertheless, it should be possible to compare any alternatives on the basis of the initial data. That is why we make a new assumption to extend each primitive partial preference in an extended partial preference. This assumption expresses that a preference is an "argument" for preferring alternatives verifying a formula φ to alternatives verifying a formula ψ.

Formalization Definition

Each extended partial preference relation is associated to a logical operator upon the formulas of L . It is denoted by the symbol ≥ i , where i is the considered point of view. φ ≥ i ψ means that, from the i th point of view, the alternatives verifying properties expressed in the formula φ and, if it is coherent, the opposite of the ones expressed in ψ, are preferred (without the Ceteris Paribus restriction) to the alternatives verifying properties expressed in the formula ψ and, if it is coherent, the opposite of the ones expressed in φ. Formally, if G i is the extended partial preference binary relation associated to the i th point of view (as defined in section 3.3), the semantics of the corresponding ≥ i operator is the following:

φ≥ i ψ iff • ∃(a, b) ∈ ALT 2
, such that a φ+ψ b ψ+φ

•    ∀(a,b) ∈ ALT 2 , such that a φ+ψ b ψ+φ aG i b   

Ceteris Imparibus

The formula φ is Ceteris Imparibus preferred to ψ, according to the i th point of view (noted CI i (φ, ψ)), if and only if there is a couple of formulas (φ , ψ ) such that φ is preferred to ψ according to the i th primitive partial preference (i.e. φ i ψ ) but not conversely, and the formula φ subsumes the formula φ and the formula ψ subsumes the formula ψ . Formally:

CI i (φ, ψ) iff       ∃(φ , ψ ) ∈ L 2 such that φ i ψ ¬(ψ i φ ) φ ⇒ φ ψ ⇒ ψ      

Construction rule

Each extended partial preference operator is constructed as the ceteris imparibus extension of the corresponding primitive partial preference: φ ≥ i ψ iff φ i ψ or CI i (φ, ψ) Properties CI and T should not be imposed together on extended partial preference operators. If they are imposed together, any alternative verifying a formula φ would be preferred (according to the corresponding preference relation) to any other alternative verifying ψ as soon as there would be two formulas φ 1 and ψ 1 such that φ ≥ i φ 1 and ψ 1 ≥ i ψ (proof by using the CI and T properties).

The ¬(ψ i φ) condition in the ceteris imparibus property blocks the construction rule when the formulas φ and ψ are both preferred to each other (according to the considered primitive partial preference). Without this condition, such a rule would involve the symmetry of operators ≥ i and thus of binary relations G i (proof by contradiction with the R and T properties).

Note also that the binary relations induced by the operators i and ≥ i are different. Whereas the first one, G i , is transitive and reflexive, the second one, G i , is only reflexive. This is justified by the fact that G i makes it possible to interpret the initial data and that G i makes it possible to exploit this information for the decision.

Indifference

The ceteris imparibus property implies that two alternatives are indifferent if they have both at least one strictly preferred component according to the corresponding primitive partial preference:

if

φ i ψ ∧ ¬(φ i ψ) γ i ω ∧ ¬(γ i ω) then (φ∧ω) ≥ i (ψ∧γ) (ψ∧γ) ≥ i (φ∧ω)

Example (end)

The new extended partial preference operator preserves all previous primitive partial preferences: red cars are still preferred to blue ones, i.e. C r ∧C ≥ 1 C b ∧C holds. Moreover, new interesting preferences can be now inferred between alternatives whose differences are not initially specified. For example, from the colour point of view, red cars are preferred to blue motorcycles (i.e. C r ∧C ≥ 1 C b ∧M ), meaning that any alternative verifying the properties "red" and "car" is preferred to any alternative verifying "blue" and "motorcycle". The first operator ( 1) is useful to deduce the "strictly" logical consequences of the initial information given about colours (see table 3). The second one (≥ 1 ) relies on these deductions, which only focus on differences in colour between alternatives to compare, to classify (from the colour point of view) alternatives that may differ from each other in other criteria than colour.

Conclusion

The preference concept is often numerically formalized in the literature. Unfortunately, such formalisms are not adapted to our cognitive agent model. In order to overcome this limitation of classical decision theory, we have chosen to formalize the preference concept logically similarly to [START_REF] Von | Georg Henrik von Wright. The logic of preference reconsidered[END_REF]. Even if information logically represented are less precise than those numerically represented, there are more easily elicited4 . Recently, the preference concept has been dealt with in such a qualitative way. In [START_REF] Boutilier | [END_REF][START_REF] Rossi | [END_REF], the concept of Flip (elementary change) makes it possible to compare two alternatives whose differences are not explicitly specified by a single preference expression. This work is based on the following definition: an alternative is preferred to another one if the latter results from the former by a worsening succession of elementary changes: a is preferred to b if there is a sequence of alternatives {a i } such that a 1 = a, a n = b and for all i, a i is obtained from a i-1 via a single elementary change and a i-1 is preferred to a i . Unfortunately, the existence of a worsening succession of elementary changes does not imply the absence of an improving one. What should be concluded in such cases? Also, as the comparisons are computed through all the conceivable alternatives within a worsening sequence, this approach does not seem to be adapted to deal with real cases, where the characteristics describing alternatives, and therefore the alternatives to consider within a sequence, are potentially numerous.

This article provides a new perspective to the problem of preference management by dividing the specification phase in two. The initial intuitive preference specifications are first interpreted by some primitive partial preferences according to the Peteris Paribus assumption. Each of them is then extended in an extended partial preference according to the Ceteris Imparibus assumption. This extended preference makes it possible to compare alternatives whose differences are not explicitly specified by the initial data. Moreover, no explicit definition of equivalence between characteristics is needed.

Next developments concern defining the preferences of an agent by a set of partial preferences (corresponding to different points of view) within our framework. In this view, we consider dynamically solving possible contradictions between these points of view thanks to an aggregation phase (defined for example as in [START_REF] Rossi | [END_REF]), in order to build a new mental attitude that would represent the agent's global opinion. This method is justified by the idea developed in [START_REF] Dubois | [END_REF] according to which the multicriteria decision and the multi-agents decision are two different approaches of the same problem. Each of our partial preferences can be identified with an agent's preference in the model proposed by Rossi et al. Another research direction is to take into account some hierarchical information to specify the partial preferences. For example, the preference "green 4x4 are preferred to red 4x4" may be more important than the preference "red cars are preferred to green cars" because it is more specific. This is related to the problems of default and incomplete information handling in the field of knowledge representation. Finally, it would be also interesting to study the link between preference and desire, and in particular to check whether desiring something could be seen as preferring it to its opposite.

Figure 2 :

 2 Figure 2: Overall scheme for preference construction

Figure

  Figure 3: Specified data

It can be proved by contradiction that any two logically equivalent formulas have the same Sm set.

This relation is not necessarily an order because two indifferent alternatives are not necessarily equal.

For example, the exact starting point of a trip often depends on the used transportation: a trip from Paris may start from the Montparnasse railway station or the Roissy airport.

It is often natural to specify a concept of "desirability" with qualitative information upon standard alternatives (e.g. "I prefer meals with wine to those with water").


G Massacrier
email: gerard.massacrier@ens-lyon.fr

M Böhme
email: m.boehme@hzdr.de

J Vorberger
email: j.vorberger@hzdr.de§militzer@berkeley.edu

F Soubiran

B Militzer

Reconciling ionization energies and band gaps of warm dense matter derived with ab initio simulations and average atom models

Average atom (AA) models allow one to efficiently compute electronic and optical properties of materials over a wide range of conditions and are often employed to interpret experimental data. However, at high pressure, predictions from AA models have been shown to disagree with results from ab initio computer simulations.

Here we reconcile these deviations by developing an innovative type of AA model, AVION, that computes the electronic eigenstates with novel boundary conditions within the ion sphere. Bound and free states are derived consistently. We drop the common AA image that the free-particle spectrum starts at the potential threshold, which we found to be incompatible with ab initio calculations. We perform ab initio simulations of crystalline and liquid carbon and aluminum over a wide range of densities and show that the computed band structure is in very good agreement with predictions from AVION.

I. INTRODUCTION

The electronic and optical properties of materials change drastically with increasing pressure or density, which affects their equation of state, opacity, and transport properties. These changes have profound implications for the interior structure and evolution of stars and planets [START_REF] Oswalt | A lower limit of 9.5 Gyr on the age of the Galactic disk from the oldest white dwarf stars[END_REF][START_REF] Militzer | Understanding Jupiter's interior[END_REF]. The modification of electronic states with temperature and pressure is a very active research subject [START_REF] Graziani | Frontiers and Challenges in Warm Dense Matter[END_REF][START_REF] Flores-Livas | High temperature superconductivity in sulfur and selenium hydrides at high pressure[END_REF]. The characterization of electronic properties is especially challenging in the regime of warm dense matter (WDM) because pressure and temperature are high, particles are strongly interacting, and the system is partially degenerate. One relevant phase transition is the dissociation and metallization of hydrogen [START_REF] Wigner | On the possibility of a metallic modification of Hydrogen[END_REF][START_REF] Saumon | Fluid hydrogen at high density: Pressure ionization[END_REF][START_REF] Weir | Metallization Of Fluid Molecular Hydrogen At 140 gpa (1.4 mbar)[END_REF][START_REF] Knudson | Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium[END_REF] that changes from an insulating, molecular substance to a monoatomic, electrically conducting fluid. This phase change is a manifestation of a Mott transition [START_REF] Mott | Metal-insulator transition[END_REF][START_REF]Metal-to-Nonmetal Transitions[END_REF].

Due to the development of high-power lasers and other facilities, the capabilities to create and diagnose WDM have increased substantially and a variety of new experimental results have been obtained [START_REF] Vinko | Electronic Structure Of An XUV Photogenerated Solid-Density Aluminum Plasma[END_REF][START_REF] Ciricosta | Direct Measurements Of The Ionization Potential Depression In A Dense Plasma[END_REF][START_REF] Vinko | Creation and diagnosis of a solid-density plasma with an X-ray free-electron laser[END_REF][START_REF] Hoarty | Observations Of The Effect Of Ionization-Potential Depression In Hot Dense Plasma[END_REF][START_REF] Vinko | Investigation of femtosecond collisional ionization rates in a solid-density aluminium plasma[END_REF][START_REF] Ciricosta | Measurements of continuum lowering in solid-density plasmas created from elements and compounds[END_REF][START_REF] Kasim | Validating continuum lowering models via multi-wavelength measurements of integrated X-ray emission[END_REF]. Many characterize modifications of the atomic structure due to the plasma environment. A key quantity is the degree of ionization, which may be enhanced by the continuum lowering phenomenon, or ionization potential depression (IPD), that describe the reduction of the ionization energy compared to the value of an isolated ion. X-ray Thompson scattering measurements (XRTS) provide direct access to the IPD [START_REF] Fletcher | X-ray Thomson scattering measurements of temperature and density from multi-shocked CH capsules[END_REF][START_REF] Fletcher | Observations Of Continuum Depression In Warm Dense Matter With X-Ray Thomson Scattering[END_REF][START_REF] Kraus | Characterizing the ionization potential depression in dense carbon plasmas with high-precision spectrally resolved x-ray scattering[END_REF]. The analysis of the scattering signal is based on the Chihara formalism [START_REF] Chihara | Interaction of photons with plasmas and liquid metals -photoabsorption and scattering[END_REF][START_REF] Wünsch | X-ray scattering as a probe for warm dense mixtures and high-pressure miscibility[END_REF], which relies on the separation of bound and free electronic states as well as well-defined direct transitions between the two. There are experimental results both supporting and contradicting the two classic IPD models of Stewart-Pyatt (SP) [START_REF] Stewart | Lowering of ionization potentials in plasmas[END_REF] and Ecker-Kröll (EK) [START_REF] Ecker | Lowering of the ionization energy for a plasma in thermodynamic equilibrium[END_REF]. SP models are widely used to predict plasma properties like ionization balance, equation of state, opacities, electrical and thermal conductivities.

There are a variety of theoretical approaches to study WDM and the IPD [START_REF] Blancard | Equation of state and transport coefficients for dense plasmas[END_REF][START_REF] Hu | Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF][START_REF] Röpke | Ionization potential depression and Pauli blocking in degenerate plasmas at extreme densities[END_REF]. With ab initio simulations, one follows the evolution of many ions in a periodic simulation cell by coupling density functional theory (DFT) [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Mermin | Thermal properties of the inhomogeneous electron gas[END_REF] to molecular dynamics (MD) [START_REF] Car | Unified Approach For Molecular Dynamics And Density-Functional Theory[END_REF]. In comparison, average atom (AA) models are computationally far less demanding because they place only a single nucleus at the center of a cloud of electrons and determine the resulting states. Many variations of the AA approach exist [START_REF] Blancard | Equation of state and transport coefficients for dense plasmas[END_REF][START_REF] Wilson | PURGATORIO-a new implementation of the INFERNO algorithm[END_REF][START_REF] Piron | Variational-average-atom-in-quantumplasmas (VAAQP) code and virial theorem: Equation-of-state and shock-Hugoniot calculations for warm dense Al, Fe, Cu, and Pb[END_REF][START_REF] Starrett | Wide ranging equation of state with Tartarus: A hybrid Green's function/orbital based average atom code[END_REF]. However, when these methods are applied to guide or interpret experiments at extreme densities or highly degenerate plasma conditions, the results appear to deviate from predictions of ab initio methods [START_REF] Hu | Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF][START_REF] Driver | Path integral Monte Carlo simulations of warm dense aluminum[END_REF][START_REF] Iglesias | Comment On "Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF][36], which provided the initial motivation for this study.

In this paper, we use DFT calculations to study the evolution of electronic states of dense C and Al over three orders of magnitude in density and then reproduce them with a novel, general-purpose type of AA model. For the latter, we adopt an alternate view of the continuum. Usually, AA models embed the ion in some kind of jellium and the continuum of states starts at the threshold of the potential. We determine that it is this property that leads to a disagreement with ab initio results. As a matter of fact, valence and conduction bands as derived from DFT-MD simulations exhibit no clear link to such a potential threshold.

II. AB INITIO SIMULATIONS

We performed ab initio calculations for dense carbon and aluminum with the ABINIT code [START_REF] Kang | First-principles investigation to ionization of argon under conditions close to typical sonoluminescence experiments[END_REF][START_REF] Gonze | Recent developments in the ABINIT software package[END_REF][START_REF] Gonze | ABINIT: First-principles approach to material and nanosystem properties[END_REF][START_REF] Bottin | Large-scale ab initio calculations based on three levels of parallelization[END_REF][START_REF] Bruneval | Effect of self-consistency on quasiparticles in solids[END_REF][START_REF] Torrent | Implementation of the projector augmented-wave method in the ABINIT code: Application to the study of iron under pressure[END_REF]. For both materials, we performed static DFT calculation using perfect lattices (T = 0 eV) and DFT-MD simulations at a finite temperature (T = 12.5 eV). For aluminum, we employed the local density approximation (LDA) to incorporate exchangecorrelation (XC) effects [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF] and the PBE-GGA functional was used for carbon [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF]. For both materials, we generated new pseudopotentials with very small core radii using the OPIUM code [START_REF]Opium pseudopotential generation project[END_REF]. The 1s core electrons were frozen in the norm conserving pseudopotential for aluminium. For carbon, we even performed all-electron calculations.

As the predicted crystal structures of aluminium and carbon change with density, we change the structure at T = 0 according to Ref. [START_REF] Benedict | Multiphase equation of state for carbon addressing high pressures and temperatures[END_REF] for carbon and Ref. [START_REF] Pickard | Aluminium at terapascal pressures[END_REF] for aluminium. For carbon, the order of the phases is diamond (3.52 to 7.66 g cm -3 ), BC8 (7.87 g cm -3 ), simple cubic (sc, 15.9 g cm -3 ), simple hexagonal (sh, 23.29 g cm -3 ), and face-centered cubic (fcc, 46.36 to 640.83 g cm -3 ). A 12 3 k-point grid with a plane wave cutoff energy of 600 Hartrees (Ha) was used for diamond. The BC8, sc, and sh calculations used a k-point grid of 32 3 points and a cutoff energy of 700 Ha. The static T = 0 calculations for aluminium were carried out for both the fcc and body-centered cubic (bcc) [START_REF] Fiquet | Structural phase transitions in aluminium above 320 GPa[END_REF] phases to get a better understanding on how the phase affects the valence band gap. For the fcc unit cell a 64 3 k-point grid with 12 bands and a plane wave cutoff energy of 400 Ha has been used. The bcc calculations used a 12 3 k-point grid with 20 bands and a plane wave cutoff at 400 Ha. The band gap reported for these solid phases is the indirect gap over the sampled Brillouin zone.

The MD simulations at T = 12.5 eV were run using a Nosé-Hoover thermostat and with a time step of δt = 0.09 fs and with eight atoms in the periodic supercell which was sampled only at the -point. The number of bands considered was 450 and the plane wave cutoff was 500 Ha for carbon. The aluminum DFT-MD simulations considered 630 to 640 bands and a cutoff energy of 400 Ha. Electronic density of states (DOS) and band-gap results at finite temperatures were derived by averaging over the DOS of 10 sample configurations obtained via DFT-MD and which where postprocessed with a static run using 8 3 k-points. These DOS were averaged after aligning them first at the chemical potential μ (a few eV below the Fermi energy at these low temperatures).

Figure 1 shows the DOS, weighted by the Fermi-Dirac factor 1/(1 + e (ε-μ)/kT ), at T = 12.5 eV and for densities a few times the solid density. For illustration purposes the DOS were smoothed using a Gaussian with a half width at half maximum (HWHM) of 0.8 eV. For carbon, the peaks below 0 0.5 -250 eV represent the 1s states while the 2s and 2p states have already merged with the continuum at these conditions. With increasing density, we find that the continuum edge shifts towards lower energies when compared with the Fermi energy, consistent with earlier findings [START_REF] Driver | Firstprinciples equation of state and electronic properties of warm dense oxygen[END_REF]. For the following discussion, we define the valence band gap (VBG) of carbon to be the gap between the 1s states and the bottom of the continuum band.

For aluminum at the lowest density of 8.03 g cm -3 , the 2s and 2p states form two well-separated peaks. At 22.0 g cm -3 , these peaks have broadened. At yet higher densities, they will merge (not shown). The VBG of aluminum is defined to be the gap between the uppermost 2s/2p state and the continuum band.

Our simulations show consistently that the continuum edge shifts towards lower energies, well below the Fermi energy, as density increases. Eventually gaps between valence and continuum bands close. In traditional AA models, the bottom of the continuum begins at the maximum of the potential and a state is pressure-ionized as soon as it crosses this threshold. This leads to inconsistent results between the DFT and AA approaches because in DFT one derives the free-particle spectrum without introducing such a threshold. To resolve this discrepancy and to reproduce the fully self-consistent many-body DFT calculations, we developed a novel type of AA approach, AVION, that is much more efficient than DFT-MD and can be readily applied to arbitrary materials and thermodynamic conditions. As we will show below, AVION reproduces the DFT-MD predictions very well. It also explains apparent discrepancies between the usual AA models and DFT-MD simulations.

III. AVION MODEL

In ab initio many-body simulations, the band structure is obtained from eigensolutions of the Schrödinger equation in a periodic box for an effective potential resulting from a collection of electrons and nuclei, while taking advantage of the Kohn-Sham scheme [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF]. In our AVION approach, we intend to solve the same equations to obtain the band structure and wave functions around a single nucleus in a limited volume. The only connection with the plasma environment arises from the boundary conditions of the wave functions at the surface of that volume. In an actual many particle system these boundary conditions are highly complex and vary from ion to ion depending on the environment. Still these variations can be represented well enough with approximate methods as long as the predictions are validated through comparisons with many-body simulations.

AVION is a spherical model with a neutrality radius R. To obtain agreement with many-body simulations, we introduce novel boundary conditions at R that allows us to derive bound and free states within a common scheme. As we will show, this has deep implications for the band structure of the continuum states and for the magnitude of band gaps. Conversely, traditional AA models set the boundary conditions at infinity. This leads to a clear distinction between a bound (discrete) and a free (continuous) spectrum, which is not compatible with predictions of many-body simulations at high density.

For an element with nuclear charge Z and at given temperature T , the AVION approach self-consistently determines the electronic density profile n e (r) and an effective potential V (r) within the Kohn-Sham scheme [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF]. The spherical potential

V (r) = - Ze 2 r + v H (r) + v xc [n e (r)] (1)
is the sum of the Coulombic nuclear attraction, the Hartree potential v H resulting from the electronic density through Poisson equation, and an exchange-correlation part in the LDA approximation. (We use Refs. [START_REF] Tanaka | Thermodynamics and correlational properties of finite-temperature electron liquids in the Singwi-Tosi-Land-Sjölander approximation[END_REF][START_REF] Ichimaru | Statistical physics of dense plasmas: Thermodynamics, transport coefficients and dynamic correlations[END_REF][START_REF] Ichimaru | Nuclear fusion in dense plasmas[END_REF].) The potential v H inside R can be written, up to a constant, as a function of the sole charge inside this same volume:

v H (r) = v H (0) -e 2 r 0 Q(r ) r 2 dr , ( 2
)
where Q(r) = r 0 4π r 2 n e (r ) dr is the number of electrons inside radius r. We do not need to assume any definite value for V , and leave v H (0) unspecified.

The electronic density (including all electrons) is, in turn, determined from the spherical eigenstates φ ε m = 1 r P ε (r)Y m (r) in the potential V . Their radial parts obey

d 2 P ε dr 2 = ( + 1) r 2 + 2m h2 [V (r) -ε] P ε ( 3
)
(with m the electron mass, h the Planck constant), are normalized to unity inside the ion sphere, and populated according to a Fermi-Dirac distribution. Accordingly,

4π r 2 n e (r) = 2(2 + 1) dε g (ε) e (ε-μ)/kT + 1 |P εl (r)| 2 , ( 4
)
where the g (ε) are partial DOS for the angular momentum . The chemical potential μ is adjusted so that Z electrons exactly pile up inside of radius R, i.e., Q(R) = Z. From electro-neutrality one has dV dr (R) = 0 (neglecting v xc ). The value V (R) defines the potential threshold on the energy scale.

The distinctive feature of AVION is in the way the DOS is constructed. The continuum in traditional AA models starts above that threshold [setting V (r R) = 0], and includes all ε > V (R) solutions, matching them to a combination of Bessel functions (free solutions) at the R boundary. We show in this paper that the band structure inherent to condensed matter, which persists for crystals under WDM conditions as shown recently by Bekx et al. [START_REF] Bekx | Electronicstructure calculations for nonisothermal warm dense matter[END_REF] (see, in particular, their Fig. 7), or in DFT simulations, challenges this simple view, yet can be recovered with a modified AA model.

For bound states, the concept of bands instead of isolated states [which give a δ-function contribution to some g (ε)] has already been used in a number of AA models [START_REF] Rozsnyai | Relativistic Hartree-Fock-Slater calculations for arbitrary temperature and matter density[END_REF][START_REF] Massacrier | Self-consistent schemes for the calculation of ionic structures and populations in dense plasmas[END_REF][START_REF] Potekhin | Equation of state for partially ionized carbon at high temperatures[END_REF][START_REF] Massacrier | Equation of state for partially ionized carbon and oxygen mixtures at high temperatures[END_REF]. Regarding states above threshold, some preliminary works have explored the modification of the DOS due to neighbors using multiple scattering theory [START_REF] Wilson | Multi-center electronic structure calculations for plasma equation of state[END_REF][START_REF] Starrett | High-temperature electronic structure with the Korringa-Kohn-Rostoker Green's function method[END_REF][START_REF] Starrett | Multiple scattering theory for dense plasmas[END_REF]. While this is in the spirit of our work, our aim is to avoid the huge complexity it adds.

In AVION for every angular momentum, , the partial DOS g (ε) = k g k (ε) is a sum over successive energy bands k. For given , all wave functions within the kth band have k -1 nodes in the interval ]0, R[. In addition, the lower ε - k and upper ε + k energy limits of this kth band result from the eigensolutions of the radial Schrödinger equation [Eq. ( 3)] with the respective boundary conditions d (P ε /r) dr (R) = 0 and P ε (R) = 0. With these two conditions, we intend to mimic the bonding and antibonding character of typical molecular wave functions. Note that the P ε (R) = 0 condition for the upper band limits does not imply that there is a hard wall at radius R, but that the total (over the whole plasma) wave function, a part of which we solve for in the ion sphere, has a node at R. We emphasize that we no longer make a distinction between bound (below threshold) and free (above threshold) states, as is done in traditional AA models. Still we need to make an assumption for the DOS inside each band.

Between the respective limits ε ± k , we choose in this work the Hubbard functional form

g k (ε) = 2 πδ 2 [(ε + k -ε)(ε - ε - k )] 1/2 , where δ = 1 2 (ε + k -ε - k ) [62]
. This is undoubtedly oversimplified. The DOS of the bands is actually controlled by a complex interplay between the ion and its neighbors, and its computation requires involved techniques. For instance the recent work of Starrett and Shaffer [START_REF] Starrett | Multiple scattering theory for dense plasmas[END_REF] used a multiple scattering approach, but requires around one hour of CPU time. Our goal is precisely to bypass this computationally highly demanding step: AVION necessitates a few seconds per point [START_REF]On a laptop equipped with a quad-core 2.5 GHz Intel Core I7 and 16 Gb of RAM[END_REF]. With this DOS at hand and wave functions computed for a set of energies inside the bands, the electronic density can be obtained from Eq. (4). As a summary, the three parameters (Z, T, R) entirely determine an AVION calculation. The iterative procedure is initiated with a trial electronic density as input. The potential is determined through Eqs. ( 1) and ( 2). Bands and wave functions are obtained from the solutions of Eq. ( 3). The resulting electronic density is constructed from Eq. ( 4) with μ adjusted for electroneutrality. The result is compared with the input. If different, a new loop is initiated with a mix of the two densities, and the procedure is continued until full convergence is achieved.

To compare with results at given ion density n i (or mass density ρ = Am u n i , where A is the atomic mass of the element), we set in this paper R equal to the Wigner-Seitz radius, R WS = [3/(4π n i )] 1/3 . Note that this is only one plausible choice among other possibilities that we do not explore in this work [START_REF]R 3 n i could be given an other value, for instance, if R is rather viewed as half the distance to the nearest neighboring ion. This is clear for a crystal structure, but see also[END_REF].

As an illustration of our approach, Fig. 2 shows the evolution of band energies over a large density range for carbon and aluminum at temperatures T = 12.5 eV and T = 0.1 eV. The reference energy in Figs. 2(a) and 2(b) is the potential threshold V (R). To better visualize the VBG, energies are counted from the top of the 1s band for carbon in Fig. 2(c), and the top of the 2p band for aluminum in Fig. 2(d). At low density, bands below the threshold appear as sharp, localized bound states. With increasing density, they widen as they approach the potential threshold, which they eventually cross in a continuous way. Above a density of a few g cm -3 the bottom of the continuum is associated for carbon first with the 2s and later with the 2p band, while for Al, it is first the 3s and later the 3d band. The void region between the top of the valence band (1s for C, 2p for Al) and the continuum defines the VBG, as shown with arrows in the panels.

Our AVION model does not distinguish bound and free states. States rather evolve from being localized to delocalized. To compare with traditional AA and IPD models, we introduce a pseudo-ionization potential (PIP) in AVION, which we define to be the difference between a state's energy and the threshold potential V (R), even though the PIP is not linked to any physical quantity in our approach. We show the PIP for the 1s state of C in Figs. 2(a) and 2(c), and for the 2p state of Al in Figs. 2(b) and 2(d). Both decrease rapidly with density. The VBG predicted by our AVION approach remains much larger because the region void of states extends well above the potential threshold. This explains the discrepancies reported in Ref. [START_REF] Driver | Path integral Monte Carlo simulations of warm dense aluminum[END_REF] between very small gaps in AA models and large VBGs in DFT-MD results, as is demonstrated by further analysis in the following section.

In AVION, we can predict the K-edge by subtracting the 1s state's energy from the chemical potential (or Fermi energy), as is illustrated for carbon in Fig. 2(c). In the same panel, we included as blue dots the values computed from DFT simulations in Ref. [START_REF] Hu | Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF] for T = 1.3 eV. The agreement with the K-edge inferred from our AVION approach is excellent. The increase of the K-edge energy emerges as a "competition between continuum lowering and Fermi surface rising" [START_REF] Hu | Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF][START_REF] Iglesias | Comment On "Continuum Lowering And Fermi-Surface Rising In Strongly Coupled And Degenerate Plasmas[END_REF]36] when including the potential threshold as an intermediate step in the analysis. Still in AVION and in DFT, the K-edge energy can be determined directly.

IV. COMPARISON BETWEEN METHODS

The DOS as obtained from AVION are compared in Fig. 1 to DFT-MD results at T = 12.5 eV and for the lowest densities. To ease comparison, the DOS have been convolved using a Gaussian distribution for the neutrality radii R with a HWHM equal to R WS /40, which slightly broadens the valence peaks. As AVION is based on a quite simple prescription for the DOS shape inside a band, a perfect match is not expected. Nevertheless the model reproduces remarkably the DFT-MD results and their trends, especially since it requires a few seconds against hundreds of hours for DFT. The continuum edges and the valence band positions agree quite well for carbon, and for aluminum at the lowest density of 8.03 g cm -3 . For Al at 22.0 g cm -3 the agreement on the positions of both the 2s and 2p bands and the continuum edge can be made better by upshifting with ∼10 eV the AVION DOS (which amounts to a shift of the chemical potential).

The position of the potential thresholds V (R) are also drawn as vertical bars in the top of the panels. For carbon, the continuum starts just below the threshold at 3.51 g cm -3 , and just above at 21.2 g cm -3 . For aluminum, it is just below the threshold at 8.03 g cm -3 . These situations would be hard to distinguish experimentally from the usual equality in traditional AA models. In contrast, for aluminum at 22.0 g cm -3 , the continuum edge starts some 40 eV above the threshold. These results are easily visualized through inspection of the upper panels of Fig. 2.

In Fig. 3, we compare results of DFT, AVION, and SP methods over three orders of magnitude in density. For both T = 0 conditions and fluid states at T = 12.5 eV, we show results from DFT(-MD) simulations and AVION calculations. The latter use T = 0.1 eV instead of T = 0 to avoid numerical issues, but as a matter of fact the AVION results are fairly similar in these low temperature range except for a slight difference for Al at low density. For carbon, AVION reproduces the DFT band gaps very well but a small deviation remains for the highest densities. AVION predicts a gap closure at a density of ≈500 g cm -3 while the gap in static DFT calculations of fcc carbon has not yet closed. DFT-MD simulations of liquid carbon predict gap closure at a density of ≈620 g cm -3 . (We define the point of gap closure in DFT-MD when the standard deviation of the distribution of gap values overlaps with zero.)

For aluminum, the agreement is very good up to a density of around 20 g cm -3 . Then the AVION band gap increases to higher values than is predicted with DFT methods. However, the following decrease and eventual band gap closure at ≈190 g cm -3 occurs in a range compatible with DFT predictions. The difference in the latter may be traced back to the different crystal or fluid state in the DFT simulations, while the environment in AVION is always with spherical symmetry. Note that AVION shows a marked change of the band-gap slope around 32 g cm -3 , which is linked to the switch from the 3s to the 3d band used to define the continuum edge [see Fig. 2(d)]. This change in slope is also seen in our DFT simulations. A similar slope change, though less dramatic, occurs for carbon at around 35 g cm -3 due to the switch of the lowest continuum band from 2s to 2p.

In Fig. 3, we also plot the PIP computed with AVION for T = 12.5 eV (it is almost the same for T = 0.1 eV on this density range.) Its value goes to zero at already ≈170 g cm -3 for C and ≈30 g cm -3 for Al. From the point of view of traditional AA models, this would imply that the involved bands have already merged with the continuum at these conditions. However, both our AVION and DFT results predict there is still a sizable VBG present under these conditions.

It is useful to compare when AVION and SP models predict a given state to become pressure ionized. From around 1 g cm -3 up to the density where the valence band crosses the threshold, carbon has two localized electrons and aluminum has ten. We hence plot in Fig. 3 the modified IP, E 0 -SP , for the 1s state of C 4+ and C 5+ , and for the 2p state of Al 3+ and Al 4+ , where E 0 is the isolated ion spectroscopic value and SP the IPD predicted by the SP model. It shows little resemblance to the VBG curves computed with AVION and DFT. SP models predict pressure ionization to occur at lower densities, at values in between the point where AVION's PIP goes to zero and AVION and DFT predict the VBG to close.

It should be noted that AA models and DFT simulations involve approximations which do not allow reproduction of the measured energies of isolated ions, which are input parameters in SP models. By shifting SP curves for C 4+ and Al 3+ , i.e., adjusting E 0 , it is possible to match the evolution of AVION's PIP over most of the density range (see lines labeled "shifted" in Fig. 3.) In these degenerate conditions, AVION and SP models make similar predictions for screening effects and SP reduces to the ion sphere expression, 3ze 2 /(2R), where z = 4 for C and z = 3 for Al. The residual discrepancy for the densities where both curves go to zero may be explained by noticing that the PIP is counted from the top of the valence band; the SP model rather refers to an eigenstate that would be computed with the condition P ε (∞) = 0. Hence it is situated in the band below its top and is ionized at a higher density.

V. CONCLUSION

In this paper, we developed a novel type of AA approach, AVION, to obtain consistency with much more computation-ally demanding many-body DFT simulations. We studied solid and liquid carbon and aluminum over three orders of density. With our AVION method, we find excellent agreement with DFT predictions for the continuum edge shift below the Fermi energy, the carbon K-edge increase and the valence band gaps. Traditional AA models on the other hand predict valence bands to enter the continuum at far too low densities. Still, our AVION results agree with a continuum lowering predicted by SP model up to a few times solid density. But at higher densities, drastic errors in the SP models become apparent as the continuum band shifts away from the potential threshold that is incorrectly equated with the beginning of the continuum band in the SP model.

Such predictions may be tested in experiment at high energy laser facilities like the National Ignition Facility or Laser Mégajoule (LMJ) [START_REF] Miquel | LMJ and PETAL status and first experiments[END_REF] that are capable of producing compressed matter of Gbar pressures [START_REF] Kritcher | A measurement of the equation of state of carbon envelopes of white dwarfs[END_REF] and multiple times solid density [START_REF] Smith | Ramp compression of diamond to five terapascals[END_REF] and may be combined with x-ray diffraction [START_REF] Lazicki | Metastability of diamond ramp-compressed to 2 terapascals[END_REF] and x-ray Thomson scattering [START_REF] Chapman | Simulating x-ray Thomson scattering signals from high-density, millimetre-scale plasmas at the National Ignition Facility[END_REF].

Since our AVION approach is as efficient as traditional AA models and can deliver DFT compatible predictions in only a few CPU seconds, it allows to fit and interpret experimental data and infer the temperature and density without resorting to SP models. The capability of AVION will make joint experimental-theoretical explorations of WDM more efficient.

FIG. 1 .

 1 FIG. 1. Density of states (multiplied by the Fermi-Dirac factor) at various densities and T = 12.5 eV for liquid (a) carbon and (b) aluminum. Energies are counted from the chemical potential. Results are from DFT-MD or AVION simulations as shown in the legends.The position of the threshold potential V (R) for each AVION simulation is also shown as vertical bars at the top of each panel. For better readability, we introduced different y scales for the valence bands (left y axis) and the continuum states (right y axis).

FIG. 2 .

 2 FIG. 2. Evolution of band energies computed with AVION as a function of density for carbon (first column) and aluminum (second column), for the temperatures T = 12.5 eV (upper row) and T = 0.1 eV (lower row). In panels (a) and (b), energies are counted from the potential threshold V (R); in (c) from the top of the 1s band for carbon; in (d) from the top of the 2p band for aluminum. Lowest s, p, and d bands are in red, blue, and green colors, respectively. The black line is the potential threshold V (R); the black dots mark the chemical potential μ at computed points. The valence band gap (VBG) and the pseudo-ionization potential (PIP) of the C 1s and Al 2p bands are shown as vertical arrows. In panel (c), an arrow indicates the carbon K-edge, which is in good agreement with Hu's predictions [26] for T = 1.3 eV (blue dots).

FIG. 3 .

 3 FIG.3. Valence band gaps (VBG) of (a) carbon and (b) aluminum. We compare static DFT calculations of solids (T = 0), DFT-MD simulations of liquids at T = 12.5 eV, and AVION results. The pseudo-IP (PIP) from AVION is compared with SP predictions for the IP of various ionization states, as well as a shifted curve for the less ionized ion (dash-dotted). The SP C 4+ curve in (a) has been shifted by -67 eV, and the SP Al 3+ curve in (b) by -21 eV.

ACKNOWLEDGMENTS G.M. acknowledges support from the Programme National de Physique Stellaire (PNPS) of CNRS/INSU, M.B. from the Center of Advanced Systems Understanding (CA-SUS), F.S. from the European Union through a Marie Skłodowska-Curie action (Grant No. 750901) and B.M. from the National Science Foundation-Department of Energy (DOE) partnership for plasma science and engineering (Grant No. DE-SC0016248) and by the DOE-National Nuclear Security Administration (Grant No. DE-NA0003842). The DFT and DFT-MD simulations were performed on a Bull Cluster at the Center for Information Services and High Performace Computing (ZIH) at Technische Universität Dresden.


M Anderlini

D Guéry-Odelin

Thermalization in mixtures of ultracold gases

Keywords: numbers: 03.75.Fi, 05.30.Jp, 67.40.Db

Starting from a set of coupled Boltzmann equations, we investigate the thermalization of a twospecies cold atomic gas confined either in a box or in an isotropic harmonic trap. We show that the thermalization times, by contrast to the collision rate, depend on the interferences between scattering partial waves. The dynamics of thermalization in a harmonic trap is also strongly dependent upon the ratio between the collision rate and the trap frequencies.

I. INTRODUCTION

Ultracold mixtures of alkali atoms have attracted great interest among atomic physicists. The ability to simultaneously cool and trap multiple species offers the possibility to study a large variety of physical phenomena not previously accessible with a single species such as novel degenerate quantum systems made of Fermi-Bose mixtures [1].

Thermalization plays a crucial role in the cooling of a sample by evaporation, and/or in the implementation of sympathetic cooling. The detailed knowledge of the underlying physics is all the more important for experiments where the time available to perform evaporation is limited [2], or for the optimization of the evaporation ramp in the multi-partial-wave regime [3]. In addition the return to equilibrium of a gas initially prepared in an out-of-equilibrium state is a widely used way to infer the value of the scattering length [4,5,6,7,8,9,10].

The purpose of this paper is to provide a theoretical framework to describe the thermalization between two cold gases confined in a box or a harmonic trap. In contrast to previous theoretical studies on this subject [6,7,11], we explicitly take into account all partial waves involved in the collision process, and we work out the role played by the confinement in all collisional regimes. Since the basic mechanism described by the Boltzmann equation is binary collisions, two-component systems exhibit all the complexities of n-component systems. The generalization of the results we obtain to systems containing more than two species is therefore straightforward.

The paper is organized as follows. Section II describes the theoretical framework used to investigate the dynamics of thermalization. Section III is devoted to the simple case of a gas confined in a box. It gives the explicit expression for the rethermalization time including the interference terms between all partial waves involved in the collision process. Section IV addresses the same problem for a gas trapped in an isotropic harmonic trap.

II. THEORETICAL FRAMEWORK

The gas i (i = 1, 2) consists of N i particles of mass m i initially thermalized at T 0 i . The dynamics of the gas is described by a set of two coupled Boltzmann equations for each phase-space distribution function f i (r, v i , t):

∂f i ∂t + {f i , H i } = I ii [f i ] + I ij [f i , f j ], i = j. (1)
The distribution function f i for the species i is governed by the hamiltonian H i through the Poisson bracket {f i , H i }, and by binary collisions. The appearance of two collision terms on the right-hand side of Eq. ( 1) is a result of the possibility of f i to change through both self collisions (i-i collisions) or cross collisions (1-2). The expression for the collision integrals is

I ij [f i , f j ] = d 2 Ω ′ d 3 v B |v B -v A | dσ ij d 2 Ω ′ × [f i (r, v ′ A , t)f j (r, v ′ B , t) -f i (r, v A , t)f j (r, v B , t)], (2)
where i, j = 1, 2. They account for elastic collisions between particles labelled A and B. We denote the velocities of the two atoms before they interact with each other by v A and v B , and after the collision by v ′ A and v ′ B . The expression for the collision terms can be easily extended to include the effects of both Bose and Fermi statistics leading to the Boltzmann-Nordheim equation [12,[START_REF] Kadanoff | Quantum Statistical Mechanics[END_REF]. In fact, most of the results discussed in this paper also hold in the presence of quantum degeneracy, provided the system is not Bose-Einstein condensed. The quantitative estimates of collision effects presented in this work are, however, based on classical statistics. As shown in Refs [11] for bosons and [START_REF] Vichi | [END_REF] for fermions, our approach, based on the result of the classical physics, is valid as soon as the temperature of the sample is larger than twice the temperature for degeneracy, i.e. the critical temperature for bosons and the Fermi temperature for fermions.

The collision problem is simplified by an appropriate change of variables. We introduce the center of mass ve-

locity v 0 = (m A v A + m B v B )/(m A + m B ), the total mass M = m A + m B , the relative velocity v r = v A -v B ,
and the reduced mass µ = m A m B /M . The relative velocity changes from the value v r before the collision to the value v ′ r after the collision, keeping the same value for the modulus v r = v ′ r because of energy conservation. The collision process is described by specifying the scattering direction with the polar angle θ ′ and the azimuthal angle ϕ ′ of the final relative velocity v ′ r with respect to the initial relative velocity v r before the collision. The corresponding solid angle d 2 Ω ′ = dϕ ′ sin θ ′ dθ ′ enters the expression for the integrand of the collision integrals.

The angular dependence of the elastic scattering directly derives from the partial-waves decomposition of the scattering amplitude in the quantum mechanical description of interactions [START_REF] Joachain | Quantum Collision Theory[END_REF]. For non-identical atoms, the asymptotic form of the scattering amplitude f (k, θ ′ ) reads

f (k, θ ′ ) = 1 k ∞ l=0 (2l + 1)e iδ l sin(δ l )P l [cos(θ ′ )], (3)
where k = µv r /h, θ ′ ∈ [0, π) and P l [cos(θ ′ )] are the Legendre polynomials. All the complexity of the interatomic potential is contained in the k-dependence of the phase shifts δ l . The differential cross section for non-identical atoms is given by

dσ 12 /dΩ ′ = |f (k, θ ′ )| 2 .
For identical atoms, we have to take into account the (anti)symmetrization principle. Accordingly the differential cross section takes the form:

dσ ii dΩ ′ = |f (k, θ ′ ) + ǫf (k, π -θ ′ )| 2 , (4)
where ǫ = 1 for bosons and ǫ = -1 for fermions. The integration must be carried out only in the half sphere θ ′ ∈ [0, π/2). As a consequence of the parity of the spherical harmonic, the only partial waves contributing to the scattering cross-section for polarized bosons (respectively, fermions) correspond to even (respectively, odd) values of l. The interference between partial waves are contained in the differential cross section [START_REF] Legere | [END_REF].

A relevant parameter to describe the thermalization of the gases is the collision rate. The average total number of collisions per unit of time Γ ij for atom species i and j is obtained by integrating over space and velocity the output channel term of the collision integral in Eq. ( 2):

Γ ij = d 3 rd 3 v A d 3 v B σ ij |v A -v B |f i (r, v A , t)f j (r, v B , t) (5)
From this quantity we deduce the expression for the mean collision rate per atom γ ij = Γ ij /N i . The integrated cross section σ ij is just the sum of the contributions from partial-waves:

σ ij (v r ) = (1 + δ ij ) l 4π k 2 (2l + 1) sin 2 δ l , (6)
where the factor (1 + δ i,j ) accounts for the constructive interference of undistinguishable scattering channels for identical atoms, δ i,j being the Kronecker delta symbol.

The sum in Eq. ( 6) is taken over even (resp. odd) values for identical (i = j) bosons (resp. fermions), and over all integer values for non-identical atoms (i = j). The expression for the collision rate is just a simple sum over the partial waves, it does not exhibit interference. This is to be contrasted with the expression of the thermalization rate as will be illustrated in the next section.

The determination of the thermalization rate is based on the method of averages [17] combined with an appropriate ansatz. Starting from Eq. ( 1), one readily derives the equation for the average value of a general dynamical quantity O(r, v i ):

d O i dt + O{f i , H i } i = OI ii i + OI ij i , (7)
where the average is taken in both position and velocity space:

O i = d 3 rd 3 v i f i (r, v i , t)O(r, v i )/N i .
As a consequence of the collisional invariants -number of atoms, momentum, and energy -,

O i I ii i = 0 if O i is of the form O i = a(r) + b(r) • v i + c(r)v 2 i .

III. THERMALIZATION IN A BOX

As a first application of the model, we consider the thermalization of a two-species gas in a box of volume V . We assume that atoms can undergo only specular reflection on the walls of the box, which can be realized experimentally by using the dipolar optical force [18]. Initially, the gases are at different temperatures T 0 1 = T 0 2 . After thermalization, the temperature of the gases will be the same

T f = (N 1 T 0 1 + N 2 T 0 2 )/(N 1 + N 2 )
, within the Boltzmann dynamics. To evaluate the thermalization time, we write the equation for the mean total kinetic energy N i m i v 2 i /2 i of species i:

d dt N i m i v 2 i 2 i = N i m i v 2 i 2 I ij i ≡ Σ i , j = i. (8)
The Poisson bracket of Eq. ( 7) vanishes for a homogeneous system, and the self collision integral term does not contribute because of conservation of kinetic energy for elastic collisions. The calculation of the collision integral term requires an ansatz for the distribution function. We choose a Gaussian ansatz for the phase-space distribution function of each species:

f i (v i , t) = N i exp - m i v 2 i 2k B T i , (9)
where

N i = N i m 3/2 i /(V (2πk B T i ) 3/2
) is the normalization factor. The time dependence is contained only in the effective temperatures T i . We assume that (T 0 1 -T f )/T f ≪ 1 and (T 0 2 -T f )/T f ≪ 1. Consequently, we can evaluate the total number of cross collisions per unit of time by setting T 1 = T 2 = T f in Eq. ( 5) with the Gaussian ansatz for the distribution functions:

Γ 12 = 2 √ π N 1 N 2 V c σ 12 , (10)
where we have introduced the velocity c = (2k B T f /µ) 1/2 and the thermally averaged cross section σ 12 = 2 ∞ 0 dx σ 12 (cx)x 3 e -x 2 . The expansion around the final temperature T f of the collision terms yields a set of linear equations. We can rewrite them in the form (see Appendix A):

d(T i -T j ) dt = - N i + N j N i N j T i -T j τ , i = j, (11)
with

1 τ = 8µ 3M σ 12 σ 12 Γ 12 , (12)
where σ 12 = ∞ 0 dx σ12 (cx)x 5 e -x 2 . The quantity σ12 (cx) is defined as:

σ12 (cx) = 2π π 0 sin θ ′ (1 -cos θ ′ ) dσ 12 dΩ ′ dθ ′ . ( 13
)
The new feature of this expression arises from the interference between partial waves resulting from the angular integration. This is in contrast to the expression for the total cross section, which just contains a simple sum over partial waves. We show in Appendix A that interference occur only between partial waves differing by at most one unit of angular momentum.

We thus find that the relaxation corresponds to an exponential decay but the thermalization times τ is not in general proportional to the collision rate because it contains partial wave interferences terms.

At very low temperature where only s-waves contribute to the collision process, no interference can occur and the relaxation time is proportional to the inverse of the collision rate. If the cross section depends on the relative wave vector k of the collision, the factor of proportionality depends on temperature.

For a cross section of the form σ 12 (k) = 4πa 2 12 /(1 + a 2 12 k 2 ), one finds

σ 12 = 4πa 2 12 ξ[1 -ξe ξ Γ(0, ξ)], σ 12 = 2πa 2 12 ξ[1 -ξ + ξ 2 e ξ Γ(0, ξ)], (14)
where ξ = h2 /(2µk B T f a 2 12 ) and Γ(a, z) = ∞ z t a-1 e -t dt. In the very low temperature limit ξ ≫ 1, we have σ 12 = σ 12 = 4πa 2 12 ; in the unitary limit ξ ≪ 1, we obtain σ 12 = 2 σ 12 = 2πh 2 /(µk B T f ). As a consequence, the number (τ Γ 12 ) of inter-species collisions per atom required to equilibrate the temperature of a two-component system made of atoms of the same mass varies from 1.5 in the very low temperature limit to 3 in the unitary limit according to Eq. ( 12).

In the following, we explicitly derive the curves for the thermalization time τ and the inter-species collision rate γ for the same number of 87 Rb atoms N 1 = N 2 = N in two different internal states |5S 1/2 , F = 2, m = 1 and |5S 1/2 , F = 1, m = -1 . This calculation has been performed by taking into account the first five partial wave contributions for the cross collisions [START_REF]The phase shifts for s[END_REF]. We plot in Fig. (1) the average cross section σ 12 normalized to its value with the cross section taken at zero energy. The resonance shape results from a d-wave resonance, which in turn tends to magnify the interference terms between p-d, and d-f partial waves, leading to a significant difference between the thermalization times calculated with and without the inclusion of interference terms.

IV. THERMALIZATION IN AN ISOTROPIC HARMONIC TRAP

We have shown in the previous section that the relation between the collision rate and the thermalization rate is strongly affected by partial wave interferences. In this section we emphasize the role played by the confinement, and we derive the dynamics of the thermalization in all collisional regime [START_REF] Ch | [END_REF].

For the sake of simplicity, we consider the thermalization of an atomic mixture confined in a harmonic and isotropic trap:

U i (x, y, z) = 1 2 m i ω 2 i (x 2 + y 2 + z 2 ). ( 15
)
By application of the method of averages, we derive the following set of six coupled equations starting from the evolution of the square radius:

d r 2 i dt = 2 r • v i i , d r • v i i dt = v 2 i i -ω 2 i r 2 i + r • v i I ij i , d v 2 i i dt = -2ω 2 i r • v i i + v 2 i I ij i . ( 16
)
The momentum and kinetic energy equations for the individual species include a contribution from the interspecies collision term, expressing the fact that it is the total momentum and total kinetic energy of the system rather than the individual species' momentum and kinetic energy that are summational invariants. These dissipative contributions to the species equations do not appear in the equation for the radius since the collision is local and the number of atoms for each species is conserved.

In the absence of inter-species interactions the set of equations ( 16) describes the monopole mode [17,[START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Boltzmann | Wissenschaftliche Abhandlungen[END_REF] of both species independently. The collision terms involving I ii do not contribute to the above equations because all the dynamic quantities are collision invariant for intraspecies collisions. This is why there is no damping for the breathing mode of a single-species classical gas confined in a harmonic isotropic trap. In this particular geometry and in the presence of two species, the relaxation only occurs through the inter-species collisions whatever the collisional regime of a given species may be.

The total number of inter-species collisions per unit of time Γ 12 can be evaluated after the thermalization, once equilibrium has been reached:

Γ 12 = N 1 N 2 c σ 12 π 2 √ 2 µω 1 ω 2 k B T f . (m 1 + m 2 )ω 1 ω 2 m 1 ω 2 1 + m 2 ω 2 2 3/2 . (17)
As the initial state of the mixture is assumed to be not too far from the final state, this expression gives a reliable estimate of Γ 12 during the thermalization process. The set of Eqs. ( 16) is exact within the Boltzmann formalism. To calculate quantitatively the relaxation driven by inter-species collisions, we make a Gaussian ansatz for the phase-space distribution function f i ≡ f i (r, v i , t) of each species, with the inclusion of a factor taking into account the space-velocity correlations:

f i = N i exp - m i v 2 i + m i ω 2 i r 2 2k B T i (1 + η i m i r.v i ), (18)
where N i is the normalization factor. The effective temperature T i and the parameter η i are the only timedependent variables. Such an ansatz is inspired by the exact solution for the phase-space distribution function for the monopole mode, and provides a natural generalization of the local equilibrium distribution. For one species the Gaussian ansatz was shown to be accurate for investigating the damping of the coupled monopolequadrupole oscillations in an anisotropic harmonic trap 0)) (dashed line) and ∆3 = ∆3(t)/∆3(0). They account respectively for the difference in the mean value of the square of the radii, the spacevelocity correlations and the square of the velocity.

from the collisionless to the hydrodynamic regime [17]. At the lowest order, the space-velocity correlations only enter the momentum equations r•v i I ij i , while they can be neglected for the calculation of the term involving the kinetic energy v 2 i I ij i . The details of this calculation can be found in Appendix B. We find:

r • v 1 I 12 1 = - m 1 ω 2 1 r • v 1 1 -m 2 ω 2 2 r • v 2 2 m 1 N 1 ω 1 ω 2 τ , (19)
v 2 1 I 12 1 = - m 1 v 2 1 1 -m 2 v 2 2 2 m 1 N 1 τ , (20)
where the time constants τ and τ are given by:

1 τ = 4 3 µω 1 ω 2 m 1 ω 2 1 + m 2 ω 2 2 σ 12 σ 12 Γ 12 , (21)
1 τ = 8 3 µ M σ 12 σ 12 Γ 12 . (22)
Combining Eq. ( 16), and Eqs. ( 19)-( 22), we obtain a closed set of six linear coupled equations. As a consequence, the relaxation in a trap does not correspond in general to a simple exponential decay. We demonstrate the physics of rethermalization using a specific example in which m 1 = m 2 = m, ω 1 = ω 2 = ω, N 1 = N 2 = N and keeping a constant cross section. However, we emphasize that the conclusions we draw are generic. In order to follow the thermalization it is convenient to introduce the three quantities:

∆ 1 (t) = r 2 1 (t) -r 2 2 (t), ∆ 2 (t) = r • v 1 1 (t) -r • v 2 2 (t), ∆ 3 (t) = v 2 1 1 (t) -v 2 2 2 (t).
We characterize the collisional regime through the quantity ωτ 0 with τ 0 = N τ . The behavior we obtain is reminiscent of the one of the monopole-quadrupole mode relaxation in a harmonic and anisotropic trap [17]. However, the observables r 2 i , r • v i i and v 2 i and their equation of evolution are not the same. The different regimes depend on the dimensionless parameter ωτ 0 .

For ωτ 0 ≫ 1, the gas is in the collisionless regime (see Fig. 2). We can recover formally the results obtained in the previous section, for a confinement of the gas in a box, by neglecting the space-velocity correlation term in the equation for ∆ 3 , leading to an effective equation for thermalization that is identical to Eq. (11). In presence of the harmonic confinement, ∆ 3 obeys a third order differential equation according to the set of equations 16. In the collisionless regime, it can be recast, under our assumptions, in the form:

d∆ 3 dt ≃ - ∆ 3 τ .
The time needed for thermalization is longer for a confined gas(see for comparison Eq. ( 11)). This is in agreement with the results of references [6,7]. This factor of two accounts for the fact that the space of configuration is larger for a non homogeneous gas, and the thermalization affects both the space and velocity degrees of freedom. The decay exhibits oscillations when the collisional regime is such that ωτ 0 > 0.5, with an amplitude scaling as 1/ωτ 0 . In the intermediate regime (ωτ 0 ∼ 1), the space-velocity correlations play a crucial role in the dynamics, and the decay of the ∆ i quantities is not exponential at all as shown in Fig. 3. In the hydrodynamic regime (ωτ 0 ≪ 1), we find two separate timescales: a rapid one for the relaxation in velocity space, and a slow one for the relaxation of the spatial widths (see Fig. 4). We recover in this limit the classical behavior of hydrodynamics, with a short time needed to reach a local equilibrium and a longer time to reach the global equilibrium. The time needed to reach complete thermalization is proportional to 1/(ω 2 τ 0 ). In contrast to the collisionless regime, this time increases with the inter-species collision rate. We have also performed numerical simulations based on molecular dynamics [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] to analyse the rethermalization in all collisional regimes and for arbitrary values of atom numbers, atomic masses, and harmonic trap frequencies for the two species. We have obtained a good agreement with the predictions of our model based on the ansatz (18).

In an anisotropic trap with different trap frequencies ω (x,y,z) i , one can derive a set of equations similar to Eq. ( 16). In this case the behavior of the relaxation is also influenced by the damping of the quadrupole modes [17].

V. CONCLUSION

In this article, we have investigated the thermalization of a non degenerate atomic mixture. In our analysis we have taken into account both the energy and angular dependence of the inter-species elastic differential cross section. We have derived the explicit form for the thermalization rate, which depends on the interference between the partial waves while the collision rate does not. In addition, we have shown how the dynamics of the thermalization is modified by the collisional regime in presence of a confinement.

FIG. 1 :

 1 FIG.1: Averaged cross sections σ12 and σ12 normalized to their value at zero temperature as a function of the final temperature of the mixture made of rubidium 87 in the two different internal states |5S 1/2 , F = 2, m = 1 and |5S 1/2 , F = 1, m = -1 . The resonance shape results from a d-wave resonance. The quantity σ12 is represented in dotted lines. The calculation of σ12 made with interference terms (solid line) differs notably from the calculation where those terms are neglected (dashed line).

FIG. 2 :

 2 FIG.2: Evolution of the normalized moments ∆i as a function of ωt in the collisionless regime ωτ0 = 20: ∆1 = ∆1(t)/∆1(0) (solid line), ∆2 = 2∆2(t)/(∆1(0) + ∆3(0)) (dashed line) and ∆3 = ∆3(t)/∆3(0). They account respectively for the difference in the mean value of the square of the radii, the spacevelocity correlations and the square of the velocity.

FIG. 3 :FIG. 4 :

 34 FIG. 3: Evolution of the normalized moments ∆i as a function of ωt in the intermediate regime ωτ0 = 1, same notation as in Fig. (2).

Acknowledgements

We acknowledge S. Kokkelmans for providing us with the phase shifts in rubidium collisions. We thank D. Roberts for a careful reading of the manuscript. This work was supported by the Délégation Générale de l'Armement (DGA).

APPENDIX A: COLLISION INTEGRALS CALCULATION IN THE HOMOGENEOUS CASE

The global conservation of kinetic energy in a collision leads to Σ + = Σ 1 + Σ 2 = 0. The symmetries of the collision integral permit one to rewrite the quantity Σ -= Σ 1 -Σ 2 in the useful form:

The Gaussian ansatz (9) for the distribution functions f i allows for an expansion of Ξ 1 ′ ,2 ′ 1,2 around the final temperature T f . We obtain

From the calculation of Σ -, one obtains the set of linear Eqs. (11) with the explicit form Eq. ( 12) for the relaxation times τ . The angular integration of Eq. (A2) leads to the quantity σ12 (cx) defined in Eq. ( 13). From the expansion of the scattering amplitude in terms of partial waves (3), we can recast it in the form

(A3) where we have introduced the dimensionless numerical factors α l,l ′ = (2δ l,l ′ )(2l + 1)(2l ′ + 1) 1 -1 (1x)P l (x)P l ′ (x), δ l,l ′ being the Kronecker delta symbol. The expression for the coefficients α l,l ′ explicitly con-tains terms describing interference between different partial waves. The properties of the coefficient α l,l ′ are the following: α l,l > 0, α l,l±1 < 0, α l,l ′ = 0 otherwise.

Finally, we perform the Gaussian integration of Eq. (A2), and we work out the expression for the relaxation time τ of Eq. ( 12) according to the notation of Eq. (11).

APPENDIX B: COLLISION INTEGRALS CALCULATION FOR A HARMONIC CONFINEMENT

The calculation of the quantity v 2 1 I 12 can be carried out in a way similar to the one used in Appendix A. The terms in η i of the ansatz (18) do not contribute at the lowest order.

To calculate the quantity r • v 1 I 12 , we introduce the quantities

The global conservation of momentum ensures Λ + = 0. The first non-vanishing contribution to the linearized expansion of Λ -is obtained by setting T 1 = T 2 = T f in the ansatz (18). One finds:

The ansatz (18) provides the explicit link between the space-velocity correlation moment and the parameters η i : r • v i i = 3η i (k B T f ) 2 /(m i ω 2 i ). This expression in combination with Eq. (B1) permits one to derive the explicit expression [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF] for the relaxation time τ of the space-velocity correlation equation.