Maqsood Ahmed

Muhammad Yar

Ayoub Nassour

Benoit Guillot

Claude Lecomte

Christian Jelsch
email: christian.jelsch@univ-lorraine.fr

Experimental and theoretical charge density analysis of a bromoethyl sulfonium salt

Keywords: Electron density, Electrostatic potential, Nucleophile, Chemical reaction

Bromoethyl sulfonium trifluoromethanesulfonate is a salt complex in which a sulphur atom makes three covalent bonds. This molecule has been proved to act as an efficient annulation reagent which results in the formation of synthetically challenging and pharmaceutically important 4-, 5-, 6-and 7-membered heterocycles in excellent yields. The charge density of the molecule was determined from both experimentally and theoretically derived diffraction data.

The stereochemistry and electron density topology of the sulfonium group was analysed. To understand the chemical reactivity of the molecule, the electrostatic potential difference between the two carbon atoms of the bromoethyl group was investigated. It has been considered that the hydrogen atoms on the carbon atom bound to sulphur are more acidic in character due to their vicinity with the triply covalently bonded positively charged sulphur atom. The electropositivity of the S-attached and Br-attached methylene groups are compared in the experimental and theoretical charge densities using topological atomic charges and electrostatic potential at the molecular surface.

Introduction.

Bromoethyl sulfonium triflate 1 (or BEST, Scheme 1) has been found to be an effective annulation reagent in the synthesis of various medicinally important compounds [1][2][3][4][5][6][7] .

Diphenylvinylsulfonium triflate 2 (or DVST, Scheme 1) has been extensively used and found to be very reactive towards various nucleophiles. A variety of nitrogen, sulphur, oxygen, and carbon-based nucleophile molecules undergo conjugate addition to vinyl sulfonium salts in the presence of base 3,5,8 the resulting ylide have two possible reaction pathways depending upon the substrates type. In a first reaction type, the resulting ylides can be trapped intra or intermolecularly by aldehydes or imines to produce epoxides or aziridines respectively 1,2,5,9 . In another reaction type, ylides undergo intramolecular proton transfer from an acidic site and the anion generated then displaces the sulfide to effect a ring-closure to generate required heterocycles 7,10 . DVST is good alternative of 1, 2-dihaloethanes for the synthesis of heterocycles. 1, 2-dihalo derivatives tend to be poorer electrophile groups and their reactions are often accompanied by competing elimination processes. For example, the direct synthesis of 1,4-diheterocyclic compounds by alkylation of -amino alcohols/thiols/amines with 1,2-dihaloethanes often gives low yields and side reactions 11 . DVST (soft electrophile, good Michael acceptor) operates under less basic conditions and minimizes competing elimination pathways. Thus, the efficient synthetic procedure for DVST and its synthetic applications has made it an attractive annulation agent.

OTf

It was later discovered that these annulations can be indeed conducted using BEST compound 2 which possibly generates diphenylvinylsulfonium triflate in situ. BEST is a crystalline solid and is easier to handle and store as compared to free flowing oil DVST 4 . BEST works effectively with a modification of the nature of the base; this method gave access to a range of heterocycles in high yields (Scheme 2). The range of nitrogen substituents demonstrated to be suitable was expanded to include sulfinamides, N-aromatics and N-heteroaromatic substituents 4,6 . Concerning the challenging synthesis of seven-membered heterocycles (diazepines and oxazepines), the annulation reaction of 1,3-aminoalcohols and 1,3-diamines with BEST gave 7-membered ring heterocycles in good yield 4 . Employing N-tosyl 1, 3-amino alcohols or 1, 3-diamines and bromoethyl sulfonium salt yielded 1, 4-oxazepines or 1, 4diazepines respectively in moderate-to-excellent yields. A mixed N-tosyl/N-SES 1, 3-diamine gave a heterocycle bearing orthogonally protected amines which can be deprotected and derivatized as required.

Br S Ph

Ph OTf When BEST is employed in the transformations described above, one can draw a plausible mechanism involving successive SN 2 displacements of diphenylsulfide and bromide. It has been suggested that the reaction proceeds through in situ formation of DVST followed by conjugate addition as shown in Scheme 3 [1][2][3][4][5] . As the reactivity of a compound such as BEST is related to its electron distribution, a precise charge density analysis (either experimental or theoretical) is a method of choice to recover molecular properties. In particular, it is of interest to know how the two CH2 carbon units present in compound BEST between the bromine and the sulphur atom are different from a charge distribution point of view. It has been assumed 4 , that the carbon atom attached to the sulphur atom is probably more electron deficient due to the positive charge of S and hence would be the site of attack by a base, owing to its comparatively acidic hydrogen atoms (Scheme 4). To test this hypothesis, an experimental (Exp) and theoretical (Theo) charge density determination bromoethyl sulfonium trifluoromethanesulfonate 1 (BEST) was carried out.

1 NH OH R 1 R 2 R 3 N O R 3

Materials and Methods.

Crystallization.

The synthesis of the BEST compound was described before [3][4][5][START_REF] Yar | In Encyclopedia of Reagents for Organic Synthesis[END_REF] . Being an organic salt, the compound crystallizes very easily in a number of solvents including alcohols, acetone, and chloroform. For the current experiment, the crystals were grown by slow evaporation of a toluene solution of BEST compound in a few days at room temperature. A single, colorless crystal of dimensions 0.18×0.16×0.15 mm was selected for the diffraction experiment.

Data Collection

Single crystal X-rays high resolution and highly redundant data collection of BEST was performed on an Oxford SuperNova 13 diffractometer using Mo Kα radiation ( = 0.71073 Å). Details of data collection and refinement procedure are given in Table 1. The crystal was mounted on a glass needle using silicone greasFige and cooled from room temperature to 100(1) K over a period of one hour under a stream of liquid nitrogen using the Oxford Cryosystems gas flow apparatus. The SuperNova diffractometer works under the software CrysAlisPro 13 which calculates the strategy to optimize the angular positions of detector and the goniometer head during the data collection. A first diffraction data set was collected under ω scans of 1° intervals. To improve the quality of the data, a second data set was later collected using the same crystal at the same temperature, but using different exposure times in the experiment. In the first experiment, the exposure time was 10 s and 25s for low high resolution data. In the second experiment, a 45s exposure time was used for high angle data collection whereas the low angle data were also collected at the 10 s exposure time. The image frames were indexed and integrated using CrysalisPro package.

An analytical absorption correction [START_REF] Clark | [END_REF] was applied on the basis of the face indexes of the crystal.

The Friedel mates were merged during data processing, the crystal being centrosymmetric space group P21/n. The two data sets were then merged using SORTAV 15 . Although intensity peaks were observed up to d=0.44Å resolution, the data resolution was truncated at d>0.50Å as the very high resolution reflections were very weak and the <Iobs>/<Icalc> ratio became significantly larger than unity (reaching 1.16) as verified with XDRKplot 16 software (see supplementary materials).

Structure solution and refinement

The structure was solved in the monoclinic P21/n space group using the SIR92 17 software. An initial Independent Atom Model (IAM) refinement was undertaken using the SHELXL97 software 18 . The model was subsequently imported to MoPro 19 software. The hydrogen atom positions were constrained to the standard neutron distances as available in the International Tables of Crystallography 21 . The bromine atom was modelled using an anharmonic thermal motion description up to fourth order of Gram-Charlier parameters, the resulting (root mean square)/<sigma> ratio is 7.0. An ORTEP 22 The Fourier residual electron density maps obtained after the experimental IAM refinement are shown in Supplementary Materials. Examination of these maps demonstrate the quality of the experimental data, as deformation bonding electron densities can be found on all covalent bonds, as well as around atoms holding electron lone pairs (S, Br). The crystallographic statistics are given in Table 2.

Experimental multipolar refinement

The least squares multipole refinement was carried out with MoPro software 19 on the basis of

Hansen & Coppens 20 model. The parameters of the Slater radial functions used for the multipolar description of atoms are given in Table Sup3. Reflections up to d=0.50Å resolution and with I>3sigma were used in the refinement. The different structural and charge density parameters were refined iteratively. The coordinates and thermal displacement parameters of all non-hydrogen atoms were refined using high-order diffraction data only (d<0.7Å) in the initial stages of the refinement. After a first round of charge density refinement, the anisotropic thermal displacement parameters of H atoms were calculated with the SHADE server 29 and were kept fixed subsequently.

Constraints and restraints.

The C-H covalent bonds were restrained to standard neutron distances 21 (d=0.002Å). The target C-H distances were 1.083(2) for aromatic groups, 1.059(2) for the bromomethyl group and 1.092(2) Å for the methylene group.

The following restraints were applied in the experimental charge density refinement. The  parameters of all the H atoms were restrained to 1.16 (0.01) 30 .

R-free refinements were performed to estimate the best weight to be applied to the chemical equivalence and multipoles local symmetry restraints 31 . The sigma value r=0.02 was found to be optimal as it gave the lowest values of wR 2 (F) free factor (Fig. Sup4). A mirror symmetry restraint was imposed on the S1 atom as it is linked to two sp 2 and one sp 3 carbon atoms. A 3m symmetry restraint was imposed on the triflate S2 atom (C-SO3 -type) (see Fig. 1). Similarly 3m symmetry was imposed on atom C15 bearing the three fluorine atoms. All other carbon atoms had one single mirror symmetry. The charge density parameters of the chemically equivalent atoms were restrained to be similar with the same sigma value r. The two ions forming the asymmetric unit were constraint to have a formal charge of ±1.

It was observed that the multipolar refinement did not allow reaching a proper charge distribution model for the three fluorine and oxygen atoms of the triflate anion. This was probably due to the higher thermal motion within the triflate which has a rotation degree of freedom around the S-C axis, the F2 atom has a Ueq value reaching 0.03 Ǻ 2 . Chemical equivalence constraints were therefore applied to the three fluorine and oxygen atoms, which yielded realistic experimental deformation electron density.

At the end of the multipolar refinement against the experimental data, the R (F) factor was 1.68 % and the goodness of fit was 0.99. All the fractional coordinates, bonds lengths, bond angles and thermal displacement parameters are listed in the CIF file, in Supplementary Materials.

Theoretical calculations

CRYSTAL09 22 program package was used to perform periodic quantum mechanical calculations. The crystal geometry observed experimentally was used as a starting geometry and optimization of the hydrogen atom positions was performed with density functional theory (DFT) method 23 and with the B3LYP hybrid functional 24,25

Bond Critical points.

The covalent bond critical points (BCP) were searched and are shown in Fig. 3 while the topological properties are listed in Table 3 and Sup1. It can be observed in Table Sup1 that the topological values from experiment and theory are in general agreement with each other. It can also be noticed that the BCP is closer to the carbon atom in the sulfonium C-S bonds while the reverse occurs in the triflate C-S bond, denoting the stronger electron depletion of the sulfonate S atom. The three sulfonium S-C bonds have ellipticities of 0.06±2 and the S1-C13 bond involving a sp 3 C atom is slightly longer than the two others which involve aromatic C atoms (Table 3). The values of ellipticity for the aromatic C(sp 2 )-C(sp 2 ) bonds on the phenyl rings are higher which is related to the partial double bond character. The average value is ε=0.20±2 and 0.16±2 for the twelve C-C covalent bonds found in the two phenyl rings, for the Exp and Theo models, respectively.

Intermolecular interactions.

There is a large number of intermolecular interactions of different nature present in the crystal structure. The Hirshfeld surface analysis was performed with program CRYSTALEXPLORER 38 to find out the respective proportions of the various interactions. A Hirshfeld surface 39 of the molecule (Fig. Sup6) shows the areas where intermolecular interactions are shorter than the sum of atomic van der Waals radii.

The 'fingerprint' plots of the interactions are shown in Supplementary Materials and a breakdown of the interaction surfaces is listed in 38 and represents a significant interpenetration of atoms beneath van der Waals contact ( d = 2.61 Å). The oxygen atom O3 on the -SO3 -group is a strong acceptor, while H-bonds are shorter when the H-C donors are acidic; this is the case for H13A as its electrons are withdrawn by the nearby sulfonium atom.

Table 5: Selection of hydrogen bonds and their topological properties at the CP for the experimental model. Definitions are in Table 3. G and V are the kinetic and potential energy density at the CP (kJ/mol/bohr 3 ). The symmetry codes are the:

(i) X ; Y-1 ; Z (ii) -X+1 ; -Y ; -Z+1 (iii) X+½ ; -Y-½ ; Z-½ (iv) -X+3/2 ; Y-½ ; -Z+3/2 (v) X-½ ; -Y-½ ; Z-1/2 (vi) X+½ ; -Y-½ ; Z-½
The sulfonium S1 atom shows no intermolecular CP, although it is in contact with the sulfonate O3 atom at a 3.169(1) Å, which is a distance below the van der Waals contact.

There 5, 6 and 7 critical points related respectively to H Sup2).

•••H, H•••F and H•••C interactions (Table

Atomic charges

On the basis of the multipolar refinement result, an analysis of the valence populations was made, as atomic charges can be derived from them. The kappa refinement formalism 33 was considered to avoid the possible charge transfers between atoms due to the multipoles in the Bond Path The Bader Quantum Theory of Atoms in the Molecules (QTAIM) 34 provides another reliable method to calculate atomic charges on the basis of the total electron density only. It is based on a topological analysis of the total electron density to provide bond topological properties but also atomic properties like charges, volumes and de-localization indexes used to find the bondorder of the reactive surfaces delineating regions of charge concentration from charge depletion 37 . The QAIM charges were integrated for a promolecule using the VMoPro software 19 .

d 12 d 1cp d 2cp ρ  2  ε G V O1ꞏꞏꞏH6 (iv) 2.
The charges on the two sulphur atoms are worth comparing. The S1 atom has three covalent In order to analyse the important role of the bromoethyl hydrogen atoms in the proposed mechanism, the charges of C13 and C14 atoms and those of their bonded hydrogen atoms deserve special attention. The C13 methylene CH2 group was found to bear globally a slightly less positive charge compared to the C14 methylene for three out of the four charge types given in Table 6, presumably due to the presence of a bromine atom bound to C14 vs. a sulfonium on C13 atom. The relative acidity of H14(A,B) vs. H13(A,B) atoms varies with the charges model considered.

The bromine atom is negatively charged in three charge models, except for QAIM_EXP. The possible charge distribution of atoms mainly involved in the reaction mechanism, represented in Scheme 10, is compatible with the same three types of charges. A refinement with the charges of the anion and cation not constrained to a formal unitary charge was also performed. The asymmetric unit having been kept neutral electrically during the refinement, the two ionic moieties necessarily hold complementary charges. The resulting model should reflect the ionic character of both molecules in the asymmetric unit. This is indeed the case, the triflate moiety turns out to be significantly negative as the sum of its atomic valence populations led to a total charge of -0.84 e in the Exp model, while the Theo charge -0.62e is more attenuated.

Electrostatic Potential

As the BEST compound is an ion pair, the triflate molecule has an overall negative The ESP issued from the Exp multipolar refinement is more positive around the two H14A,B hydrogen atoms (Fig. 5), due to more positive charges on the hydrogen atoms and on the bromine atom. The converse is observed for the theoretical model, where the potential is more positive around the H13A,B atoms due to the influence of the vicinal positive sulfonium atom.

The electrostatic potential is much more electronegative in the theoretical model on the bromine atom compared to the experimental one.

The isosurface coloured according to the ESP issued from the refinements without formal 1e charge constraint on the cation/anion pair is shown in Fig. Sup7. The unconstrained Exp electrostatic potential is slightly attenuated compared to the constrained one, as the charge transfer between the ions is lower, but is qualitatively similar to the constrained one, notably around the bromoethyl group. The ESP from the theoretical unconstrained model is also attenuated except for the bromine atom and the region around which is far more electronegative.

Dipole moment.

The molecular dipole moments μ are calculated, based on atomic net charges qi derived from Pval and on the atomic dipoles μi (from the multipole formalism), according to the formula:

    i i i i Total q r μ μ   
Due to the electroneutrality constraint which kept the total charge of the asymmetric unit equal to zero, μ is actually independent of the origin. The atomic dipole moment contributions are dependent on seven variables: the net charge derived from Pval, the atomic coordinates (corresponding to the location in space of atomic charges) and the three dipole populations P10, P11+, P11-.

The BEST salt compound, being a polar entity, possesses relatively high dipole moment, which is found to be 25.2 D and 28.8 D for the Exp and Theo models, respectively. The dipole moment vectors are shown in the Fig. 6 for both models. The system being composed of anionic and cationic molecules, the dipole moment orientation and magnitude is largely ruled by this property. Hence the dipole moments vectors are roughly oriented in the direction joining the negative triflate moiety to the positive region located next to the C13-C14 group. The dipolar charge distribution explaining the dipole moment orientation can also be seen on the electrostatic potential surfaces (Fig. 6), with a negative potential around the triflate anion.

The value of the dipole moment for the Exp model is found to be 15% lower than the theoretical one. If the formal charges of the two ions were not constrained to be 1, the contrary would be observed, in accordance with the fact that molecular dipole moments derived from experimental charge density studies often present pronounced enhancements compared to independent theoretical estimates 41.

Conclusion.

An experimental charge density analysis of the compound bromoethylsulfonium triflate was performed as well as a high level ab initio periodic computation of the system. The analysis of the charge distribution on the basis of the Exp model did reveal very interesting results.

Generally, it is believed that the C13 atom being attached to a triply covalently bonded S1 + atom is comparatively more electron deficient. So in this case, a base should preferentially abstract an acidic hydrogen from atom C13. The presence of the bromine atom adjacent to C14 renders the H14(A,B) hydrogen atoms also more acidic. The electrostatic potential at the molecular surface shows a strongly electropositive region around the sulphur atom, which is favourable to attract a nucleophile base. The electrostatic potential is also positive around the H14(A,B) hydrogen atoms compared to the H13(A,B) hydrogen atoms.

The distribution of charges raises again the question how the reaction actually proceeds.

Previously Yar et al. 4 have indentified the in situ transformation of the bromoethyl sulfonium salt to vinyl sulfonium salt in the presence of a base. In this process, a base removes a proton from a carbon atom of the bromoethyl group, and then an elimination reaction occurs (the HBr group being removed).

The question raised by our study is whether the proton is removed from C14 or C13 atom. Most likely if H + is removed from the C13 which is adjacent to the positively charged S1 atom, the carbanion would be more stable than if the proton is taken from the C14 atom, located next to the bromine atom.

A possibility remains that the proton extraction occurs first on C14, which holds more acidic hydrogen atoms according to the experimental electrostatic potential map. Then, the relative stability of the C14 and C13 carbanions being in favour of the second one, the C14 atom may be re-protonated by exchange with the neighbour atom C13. At this point, the C13 carbanion being stabilized by the S1 + atom, no reverse protonation would take place. In particular, a possible perspective would be to analyze, in the same way, what the DVST vinyl homologue is producing in situ (Scheme 2).

Figure Sup7.

Isosurface of static total electron density at ρ=0.01 e/Å 3 coloured according to the electrostatic potential generated by the BEST promolecule. The bromoethyl sulfonium moiety is in the foreground while the negatively charged triflate group is behind. The P val valence populations were freely refined, with no 1 e charge constraint on the cation and anion. the electron density (e.Å -3 );  2 ρ(rcp) is the Laplacian (e.Å -5 ); λ1, λ2, λ3 are the eigenvalues of Hessian matrix (e.Å -5 ); ε the ellipticity whereas G and V denote the kinetic and potential energies, respectively. The symmetry codes are the following: (i) -X+2; -Y ; -Z+1 (ii) -X+1 ; -Y ; -Z+1 (iii) X+1/2 ; -Y-1/2 ; Z-1/2 (iv) -X+3/2 ; Y-1/2 ; -Z+3/2 (v) X-1/2 ; -Y-1/2 ; Z-1/2 (vi) X-1/2 ; -Y-1/2

Scheme 1 :

 1 Scheme 1: Chemical diagram of bromoethyl sulfonium trifluoromethanesulfonate 1 (BEST) and diphenylvinylsulfonium triflate 2 (DVST)

Scheme 2 :

 2 Scheme 2: Schematic illustration of reactions involving BEST to give various heterocycles.

Scheme 3 :

 3 Scheme 3:The reported proposed mechanism of annulation reaction involving BEST (see references above)

Scheme 4 :

 4 Scheme 4:The proposed 4 distribution of charges on C13 and C14 atoms in BEST compound.

Figure 1 .

 1 Figure 1. ORTEP diagram of the molecule showing the atom numbering scheme and the thermal displacement ellipsoids, drawn at 50% probability level.

using 6 -Fig. 2 .

 62 Fig.2. An electron lone pair is visible on the S1 sulphur atom and forms a tetrahedral geometry with the three C-S bonds. The sulfonate S2 atom of the triflate anion has a different configuration as it is bonded with three oxygen atoms and one carbon atom and is arranged in a nearly tetrahedral geometry (Fig.1).

Figure 2 .

 2 Figure 2. 3D view of the experimental static deformation electron density showing the electron lone pair of the sulphur atom. Blue colour shows deformation electron density accumulation and red colour shows the depletion. Isosurfaces levels are ±0.2 eÅ -3 .

Figure 3 :

 3 Figure 3: The covalent bond critical points shown in dark brown colour for the experimental model.

2 0. 07 The  2

 072 cp values of the Laplacian at the BCPs are found to be negative except on the sulfonate S-O bonds and on the C-Br bond. This is the case for both experimental and theoretical cases and denotes the polar character of these bonds. A negative minimum of the Laplacian is however found on the C-O bond path to the oxygen atoms and on the C-Br bond closer to the electronegative Br atom (Fig.4). The values of the ellipticity on the S-O bonds are small (0.03±3 on average) and reflect the nearly three-fold symmetry character of the O-SO2C bonds; the three S-O distances are very similar 1.446±5Å.The negative Laplacian is stronger on the triflate S2-C15 BCP than on the S1-C BCPs of the sulfonium, despite a longer bond length for S2-C15 (1.8320(5)Å vs 1.79±2 Å). The same tendency is observed for the Theo charge density. Contrarily to non-polar bonds, the Laplacian  2 cp cannot be directly related to the bond strength in polar bonds such as S-C or S-O. This result can be related to the experimental charge density study of K2 sulphate34 , where the longest S-O bonds have also the strongest  2 cp Laplacian values. When the S-O distance is decreased in sulphate, the S-O bond unexpectedly becomes less covalent because of a counterbalancing effect arising from the large enhancement of the bond polarity. In theoretical studies of S-O and N-O polar covalent bonds in several small molecules in vacuo, Love

Figure 4 .

 4 Figure 4. Electron density Laplacian  2 ρ maps of the experimental multipolar model.

  bonds and has a positive topological QAIM_EXP charge of +0.50e. The sulfonate S2 atom in the triflate anion is bonded to three oxygen atoms and has three fluorine atoms as its second neighbours. This position of S2 renders it highly deficient in electrons resulting in a net integrated Exp charge largely positive of +3.14e. The three oxygen atoms display similar S-O bond lengths 1.446±0.004Å. Their three atomic charges, which were not constrained to be equivalent in the Theo model, are similar. The cumulative negative QAIM_EXP charge on the three oxygen atoms riding on S2 atom is -3.80e (1.26 each as they were constraint to be equivalent) while that on the three fluorine atoms is -1.74 e.The C15 atom holding the three fluorine atoms bears a large positive charge (QAIM_EXP =+1.41e). For the two charge definitions and the two models, this carbon atom is positively charged, while the fluorine atoms are negatively charged for all charge definitions.The charge values of atoms in the triflate moiety are larger for the QAIM definition than for the Qval_  one.

Scheme 10 :

 10 Scheme 10: Distribution of charges in the light of charge density analysis.

Figure 5 .

 5 Figure 5. Isosurface of static total electron density at ρ=0.01 e/Å 3 coloured according to the electrostatic potential generated by the BEST promolecule. The bromoethyl sulfonium moiety is in the foreground while the negatively charged triflate group is behind. The view highlights the difference between the H atoms bound to C13 and C14 atoms. Properties are computed using (a) experimental data (b) theoretical multipolar model.

Figure 6 :

 6 Figure 6: Total dipole moment for the BEST molecule computed from experimental (blue) and theoretical (red) models.

  Left: ORTEP diagram of the BEST molecule showing the atom numbering scheme and the thermal ellipsoids (drawn at 50% probability level).Right: 3D electron density surface coloured according to the electrostatic potential of the bromoethyl sulfonium cation.

Figure Sup3 .

 Sup3 Figure Sup3. Residual Fourier electron density on various parts of BEST molecule obtained after IAM experimental refinement. The contours are drawn at ±0.05 e/Å 3 , positive and negative contours in blue and red, respectively. The reciprocal resolution is truncated at sinθ/λ <0.7 Å -1 .

Figure Sup4 :

 Sup4 Figure Sup4:Curves showing the statistical crystallographic indices evolution along the variation of weight applied on the charge density restraints (similarity and local symmetry). The optimal weight corresponding W=0.02 to the lowest free R factors was kept along the refinement.

Figure Sup5 :

 Sup5 Figure Sup5: Experimental static deformation electron density (left) and Fourier residual electron densities (right) for the experimental multipolar model on the aromatic rings. Contour intervals are 0.05eÅ -3 ; Blue solid lines are positive; red dot lines are negative; green dash line is a zero contour. The Fourier series have been truncated at sinθ/λ <0.7 Å -1 for the computation of residual maps.

  Figure Sup6:

  Properties are computed using (a) experimental multipolar model (b) theoretical multipolar model. -0.3 +0.3e/Ǻ 3

Table 1 :

 1 Crystal and data collection statistics.

Table 2 :

 2 Crystallographic agreement factors and residual electron density values obtained after multipolar refinements of the BEST molecule.

	Data	EXP	DFT DZP
	Resolution (Å)	0.50	0.40
	R(F) %	1.68	1.10
	wR2(F) %	1.15	1.07
	G. o. f	0.99	/
	Δρmax (e/Å 3 )	0.36,
	Δρmin	-0.41
	Δρrms	0.055

Table 3 :

 3 Topological

	the total electron density (e.Å -3 );  2 ρ(rcp) is the Laplacian (e.Å -5 ) and ε the ellipticity. The
	values are from the experimental model. Other BCPs and values from theory are described in
	supplementary materials.

properties at the bond critical points in the bromoethyl-sulfonium and triflate parts of the BEST molecule (experimental multipolar model). d12 is the interatomic distance, d1cp and d2cp (Å) is the distance between the first / second atom and the CP; ρ(rcp) is

Table 4

 4

. The H•••O hydrogen bonds constitute the largest contact area, followed by H

•••H, H•••F and H•••C interactions. The H•••Br interactions

are present in a significant ratio of 10.6%. A small but interesting proportion of interactions of Br•••F type is also observed although the bromine and the F2 atoms are distant by more than 3.45Å in the crystal packing.

Table 4 :

 4 Breakdown of the most prevalent (>1%) intermolecular interactions in the BEST There are only C-H•••O type hydrogen bonds in the crystal packing of BEST compound and those with H•••O distance significantly below the van der Waals contact distance are reported in Table5. The topological values reveal that some of the interactions are remarkably strong.

	crystal packing with their respective proportions:
	contacts	%	contacts	%
	HꞏꞏꞏO	23.2	HꞏꞏꞏBr	10.6
	HꞏꞏꞏH	20.8	C...F	6.7
	HꞏꞏꞏF	19.5	Br...F	1.7
	HꞏꞏꞏC	14.3	C...O	1.5
	A topological analysis of the total electron density in the intermolecular interactions regions on
	the basis of Bader theory 35 of "Atoms In Molecules" (AIM) is a very informative way to study
	the nature of interactions. A topological analysis was performed for all the significant and weak

interactions in the crystal packing and the intermolecular CPs were located. The topological values for all the (3,-1) type interactions are listed in Table

Sup2

.

Table 6 .

 6 Coppens modelling. The valence populations Pval_  results of the kappa refinements are mentioned in Table6. Valence populations Pval_  derived from the kappa refinements and PAIM derived from topology, using experimental (Exp) and theoretical (Theo) models (e). PAIM = QAIM -Ncor is the number of electrons QAIM integrated over the Bader atomic basins (multipolar models) minus Ncor, the number of core electrons. Atomic charges can be retrieved by difference Nval-Pval with the number of valence electrons Nval of the neutral atom. Nval = 7, 6, 7, 6, 4, 1e for Br, S, F, O, C and H respectively and Nval + Ncor = Z, the atomic number.

		396	1.465	0.969	0.068	1.08 0.43 23.1 -16.9
	O1ꞏꞏꞏH13B (iv) 2.738	1.625	1.130	0.021	0.44 0.37 8.4	-4.95
	O1ꞏꞏꞏH12 (iv) 2.311	1.403	0.932	0.066	1.17 0.01 24.7 -17.4
	O2ꞏꞏꞏH4 (v)	2.661	1.539	1.164	0.042	0.63 0.13 13.0 -8.87
	O2ꞏꞏꞏH10 (iii) 2.692	1.617	1.085	0.021	0.48 0.16 9.2	-5.53
	O2ꞏꞏꞏH14B (i) 2.483	1.469	1.037	0.044	0.77 0.06 15.7 -10.4
	O3ꞏꞏꞏH13A (ii) 2.151	1.290	0.877	0.107	1.72 0.03 38.8 -30.8
	F2ꞏꞏꞏH9 (vi)	2.675	1.495	1.210	0.027	0.48 0.27 9.5	-5.88
	F1ꞏꞏꞏH10 (vi)	2.667	1.480	1.212	0.037	0.62 0.07 12.5 -8.19

Table Sup1 :

 Sup1 List of the bond critical points and topological properties (see Table5) of the BEST molecule. The values in upper/lower lines are from experiment/theory.

		1.3940	0.7139	0.6802 2.0648	-17.49	-14.66 -12.29	9.46	0.19
	C8-C9	1.3936(7) 0.6945	0.6991 2.1112	-18.34	-15.71 -13.32 10.69	0.15
	Bond BR1-C14 1.9458(6) 1.0948 d 12 d 1cp 1.3936 0.7032 C9-C10 1.3920(7) 0.6830 1.3911 0.7075	d 2cp 0.6904 2.0802 ρ(r cp ) 0.8513 0.9662 0.7090 2.1457 0.6836 2.0909	 2 ρ(r cp ) -17.90 0.03 -18.59 -18.25	λ 1 -15.04 -12.62 λ 2 -4.60 -4.55 -16.47 -13.48 11.35 λ 3 9.76 9.17 -15.28 -12.79 9.82	0.19 ε 0.18 0.01 0.19
	1.9454 C10-C11 1.3925 (7) 0.7024 1.0883	0.8572 0.8885 0.6901 2.1873	0.57 -19.72	-4.01 -17.20 -14.37 11.86 -3.87 8.46	0.04 0.16
	S1-C1	1.7823(5) 0.9493 1.3923 0.7094	0.8330 1.2739 0.6829 2.0837	-5.70 -18.07	-7.08 -15.23 -12.75 -6.71	8.09 9.92	0.05 0.19
	1.7813 C11-C12 1.3917 (7) 0.7049 0.9741	0.8074 1.2719 0.6868 2.1500	-5.77 -19.17	-6.81 -16.60 -13.89 11.32 -6.46 7.5	0.05 0.16
	S1-C7	1.7796(5) 0.9508 1.3941 0.7017	0.8291 1.2880 0.6924 2.0804	-5.52 -17.65	-7.14 -15.09 -12.63 10.07 -6.89 8.51	0.04 0.19
	1.7789 C13-C14 1.5126(7) 0.7855 0.9783	0.8009 1.2631 0.7278 1.6641	-5.40 -10.22	-6.81 -11.16 -10.69 11.62 -6.32 7.73	0.08 0.04
	S1-C13	1.8158(5) 0.9785 1.5144 0.7615	0.8372 1.1891 0.7534 1.6278	-4.49 -9.46	-6.57 -10.6	-6.06 -10.24 11.37 8.14	0.08 0.04
	1.8149 C2-H2 1.0823	1.0117 0.7352	0.8033 1.1773 0.3771 1.8119	-4.40 -16.79	-6.27 -17.36 -16.61 -5.84	7.71 17.19 0.04 0.07
	S2-C15	1.8320(6) 0.8981 1.0069 0.6636	0.9341 1.3348 0.3435 1.8942	-9.24 -17.26	-7.73 -17.96 -17.47 -7.55	6.03 18.15 0.03 0.02
	1.8329 C3-H3 1.0823	0.8018 0.7335	1.0312 1.3112 0.3489 1.7585	-10.15 -15.84	-6.01 -16.77 -16.23 -5.90	1.76 17.15 0.03 0.02
	S2-O1	1.4508(8) 0.5918 1.0278 0.6729	0.8591 2.1894 0.3549 1.8789	3.33 -17.2	-14.89 -14.02 32.24 -17.75 -17 17.55 0.04 0.06
	1.4511 C4-H4 1.0829	0.5781 0.7317	0.8732 1.9489 0.3511 1.8229	19.12 -17.39	-10.82 -10.54 40.48 -18.06 -17.16 17.83 0.05 0.03
	S2-O2	1.4467(9) 0.5921 1.0398 0.6809	0.8552 2.1741 0.3589 1.8809	3.79 -17.67	-14.41 -14.28 32.49 -17.68 -17.04 17.06 0.04 0.01
	1.4455 C5-H5 1.0826	0.5778 0.7307	0.8681 1.9243 0.3519 1.7777	20.22 -16.24	-10.71 -10.24 41.17 -17.05 -16.20 17.01 0.05 0.05
	S2-O3	1.4410(9) 0.5924 1.0287 0.6747	0.8503 2.2165 0.3541 1.8782	3.78 -17.46	-14.39 -14.29 32.46 -17.65 -17.05 17.24 0.04 0.01
	1.4398 C6-H6 1.0822	0.5752 0.7335	0.8648 2.0058 0.3487 1.8199	20.41 -17.68	-11.47 -17.79 -16.66 -11.1	42.98 16.77 0.06 0.03
	F1-C15	1.3415(6) 0.8238 1.0071 0.6662	0.5180 1.9470 0.3409 1.9019	-15.61 -17.55	-15.92 -15.20 15.50 -18.19 -17.42 18.05 0.04 0.05
	1.3442 C8-H8 1.0826	0.8146 0.7312	05302 1.8995 0.3515 1.8272	-15.19 -17.46	-15.13 -13.45 13.39 -17.53 -16.59 16.67 0.05 0.13
	F2-C15	1.3334(7) 0.8263 1.0054 0.6663	0.5074 1.9328 0.3391 1.9081	-13.94 -17.71	-15.15 -14.18 10.37 -18.35 -17.61 18.25 0.04 0.06
	1.3309 C9-H9 1.0822	0.8157 0.7284	0.5152 1.9687 0.3539 1.7731	-18.56 -15.94	-16.36 -14.93 12.73 -16.69 -16.08 16.83 0.04 0.10
	F3-C15	1.3340(6) 0.8236 1.0155 0.6612	0.5109 1.9784 0.3545 1.8811	-16.23 -17.32	-16.35 -15.25 15.36 -17.66 -16.93 17.27 0.04 0.07
	1.3340 C10-H10 1.0822	0.8077 0.7280	0.5269 2.0195 0.3542 1.8427	-13.61 -17.22	-16.35 -14.54 17.28 -17.95 -17.29 18.01 0.04 0.11
	C1-C2	1.3911(4) 0.7146 1.0117 0.6771	0.6766 2.1410 0.3346 1.8968	-18.64 -16.95	-16.19 -13.65 11.20 -18.47 -17.87 19.4	0.16 0.03
	1.3917 C11-H11 1.0823	0.7073 0.7330	0.6846 2.0855 0.3493 1.8147	-17.95 -16.73	-14.98 -17.63 -16.89 -12.4	9.43 17.79 0.04 0.21
	C1-C6	1.3911(7) 0.7085 1.0257 0.6707	0.6828 2.1426 0.355 1.8745	-18.80 -17.14	-16.55 -13.55 11.30 -17.65 -16.89 17.54 0.04 0.18
	1.3922 C12-H12 1.0823	0.7153 0.7404	0.677 2.0816 0.3420 1.8048	-17.92 -17.55	-14.83 -12.39 -17.69 -16.88	9.3 17.02 0.05 0.20
	C2-C3	1.3948(7) 0.6848 1.0067 0.6696	0.7101 2.0993 0.3373 1.9015	-17.89 -17.26	-15.62 -13.20 10.92 -18.39 -17.7 18.93 0.04 0.15
	1.3968 C13-H13A 1.0907	0.6943 0.7393	0.7025 2.0737 0.3515 1.7932	-17.48 -15.92	-15.0 -17.02 -17.01 -12.61 10.12 18.11 0.00 0.19
	C3-C4	1.3925(9) 0.6702 1.0023 0.6775	0.7223 2.1798 0.3249 1.896	-19.37 -16.77	-17.10 -14.09 11.82 -18.46 -18.35 20.03 0.01 0.18
	1.3937 C13-H13B 1.0914	0.6818 0.7309	0.712 2.0912 0.3606 1.7418	-18.19 -14.87	-15.23 -16.25 -16.22 -12.8	9.83 17.60 0.00 0.19
	C4-C5	1.3919(8) 0.7228 1.0224 0.6783	0.6691 2.1769 0.3442 1.8445	-19.83 -15.32	-17.05 -14.19 11.41 -17.22 -17.01 18.91 0.01 0.17
	1.3916 C14-H14A 1.0914	0.6865 0.7338	0.7051 2.0982 0.3577 1.8048	-18.57 -16.52	-15.29 -12.86 -16.68 -16.51	9.57 16.67 0.01 0.19
	C5-C6	1.3917(7) 0.6891 1.0480 0.6902	0.7027 2.1351 0.3579 1.8453	-19.06 -16.07	-16.44 -13.40 10.78 -17.05 -16.65 17.63 0.02 0.19
	1.3930 C14-H14B 1.0919	0.6824 0.7308	0.7106 2.0793 0.3611 1.7789	-17.85 -15.89	-14.99 -12.59 -16.30 -16.08	9.73 16.48 0.01 0.19
	C7-C8	1.3955(6) 0.7136 1.0536 0.6987	0.6821 2.1295 0.3549 1.8482	-18.44 -16.55	-16.18 -13.70 11.44 -17.13 -16.98 17.56 0.01 0.15
		1.3966	0.715	0.6818 2.0722	-17.7	-14.77 -12.29	9.35	0.20
	C7-C12	1.3933(6) 0.7140	0.6794 2.1401	-18.36	-16.60 -13.51 11.75	0.19

Table Sup2 :

 Sup2 List of intermolecular CPs and topological properties. The upper line stands for the experimental and the lower line stands for the theoretical values. d12 is the interatomic distances, d1cp and d2cp (Å) are the distance between the first / second atom and the CP; ρ(rcp) is

Acknowledgements. M.A. thanks the Higher Education Commission of Pakistan for PhD

funding.

Supporting Information Available:

Figures: plot of observed vs. expected residual electron density; plot of average Fobs / average Fcalc, residual and static electron density maps; Rfree plotted as a function of restraints weight; Hirshfeld surface; fingerprint plots of the Hirshfeld surface; molecular surface coloured according to ESP, after refinement without formal charge constraint on anion/cation. Tables: list of the bond and intermolecular critical points with topological properties. CIF file. This material is available free of charge via the Internet at http://pubs.acs.org.

Supplementary Materials.

Experimental and theoretical charge density analysis of a bromoethyl sulfonium salt.

Maqsood Ahmed, Muhammad Yar, Ayoub Nassour, Benoit Guillot, Claude Lecomte and Christian Jelsch.


Gustavo Castillo

Simon Merminod

Eric Falcon

Michael Berhanu

Tuning the distance to equipartition by controlling the collision rate in a driven granular gas experiment

In a granular gas experiment of magnetized particles confined in a thin layer, the rate of dissipative collisions is tuned by adjusting the amplitude of an external magnetic field. The velocity statistics are analyzed using the dynamic and static structure factors of transverse velocity modes. Using the fluctuating hydrodynamics theory we measure the deviation from kinetic energy equipartition in this out-of-equilibrium system as a function of the dissipative collision rate. When the collision rate is decreased, the distance to equipartition becomes smaller meaning that the dynamical properties of this granular gas approach by analogy those of a molecular gas in thermal equilibrium.

I. INTRODUCTION

Statistical mechanics succeeds in predicting the macroscopic states of systems composed of many interacting particles, mainly when these systems are in thermal equilibrium. Such systems have a reversible dynamics and do not dissipate energy. In contrast, for out-of-equilibrium systems like turbulent flows, biological living systems, active fluids or electrical circuits, only few general results statistically describe the nonequilibrium steady states [START_REF] Derrida | Non-equilibrium steady states: fluctuations and large deviations of the density and of the current[END_REF], in which energy must be continuously injected to compensate for energy dissipation [2]. Among them, granular gases refer to an assembly of athermal macroscopic particles mechanically agitated which undergo dissipative collisions. They are relevant model systems to investigate non-equilibrium steady states theoretically [START_REF] Barrat | Granular gases: dynamics and collective effects[END_REF][START_REF] Van Noije | Velocity Distributions in Homogeneously Cooling and Heated Granular Fluids[END_REF][START_REF] Goldhirsch | Clustering Instability in Dissipative Gases[END_REF], numerically [START_REF] Mcnamara | Energy nonequipartition in systems of inelastic, rough spheres[END_REF][START_REF] Moon | Steady-state velocity distributions of an oscillated granular gas[END_REF][START_REF] Aumaitre | Collision frequencies and energy flux in a dilute granular gas[END_REF][START_REF] Opsomer | Phase transitions in vibrated granular systems in microgravity[END_REF] and experimentally [START_REF] Olafsen | Clustering, Order, and Collapse in a Driven Granular Monolayer[END_REF][START_REF] Losert | Velocity statistics in excited granular media[END_REF][START_REF] Reis | Caging dynamics in a granular fluid[END_REF][START_REF] Mujica | Dynamics of Noncohesive Confined Granular Media[END_REF][START_REF] Noirhomme | Threshold of gas-like to clustering transition in driven granular media in lowgravity environment[END_REF]. Taking into account the inelasticity of collisions and assuming that the forcing acts as a stochastic noise [START_REF] Kadanoff | Built upon sand: Theoretical ideas inspired by granular flows[END_REF][START_REF] Goldhirsch | Scales and kinetics of granular flows[END_REF], kinetic theories propose a method for predicting the large scale behavior of many particle systems. For instance, mode coupling theory models velocity structure factors and predicts long range spatial correlations caused by dissipative collisions [START_REF] Van Noije | Randomly driven granular fluids: Large-scale structure[END_REF]. More recently, for a driven granular system fluidized by a stochastic bath with friction, the fluctuating hydrodynamics theory [START_REF] Gradenigo | Fluctuating hydrodynamics and correlation lengths in a driven granular fluid[END_REF] also derives the static velocity structure factors and finds correlation lengths related to energy dissipation, in order to model vibrated granular experiments. Due to the dissipative collisions, the kinetic energy per particle at large scales called the "bath temperature" is higher than the one at the particle scale, the "grain temperature". The energy equipartition is thus violated through the space scales. This approach has been successfully validated in a quasi-twodimensional experiment of homogeneously driven granular particles [START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. In that work, the level of dissipation is varied by changing the number of particles per area unit, the area fraction. However, their structure factors remain dominated by collision effects because the particle area fraction must be kept high enough to maintain the validity of the hydrodynamics approach.

In a previous work [START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF], we introduced a different experimental setup realizing a homogeneously driven quasi two-dimensional granular gas, in which tunable interparticle repulsive forces have been added by means of an external magnetic field. When these forces are strengthened, the rate of dissipative collisions decreases because collisions are progressively replaced by elastic dipolar interactions. In a range of moderated applied magnetic Typeset by REVT E X field, the statistical properties of the granular gas approach those of a molecular gas in thermal equilibrium. To our knowledge, the effect of repulsive interactions on the dissipation rate has been firstly investigated theoretically and numerically by Scheffler and Wolf [START_REF] Scheffler | Collision rates in charged granular gases[END_REF] for a granular gas of electrically charged particles in ballistic motion. Here, we quantify the distance to kinetic energy equipartition in our experiment throughout the transition from a dissipative granular gas to a quasi-elastic system of particles. Specifically, we use the methods from the fluctuating hydrodynamics theory [START_REF] Gradenigo | Fluctuating hydrodynamics and correlation lengths in a driven granular fluid[END_REF] to compute the difference between the bath and the grain temperatures to define a distance to equipartition. We observe that once the collision rate vanishes, these two temperatures are nearly equal. Indeed, we show that the deviation from equipartition is caused by energy depletion at small scales due to dissipative collisions.

II. TRANSITION TOWARDS A COLLISIONLESS GRANULAR GAS

First, we recall the features of the experimental device [START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF][START_REF] Merminod | Transition to a labyrinthine phase in a driven granular medium[END_REF][START_REF] Merminod | Self-organization of vibrated and magnetized particles: Structure, dynamics and transitions[END_REF] whose schematic is given in Fig. 1 (a). An assembly of 2000 soft magnetic spheres of diameter a = 1 mm and of mass m = 4.07 × 10 -3 g is confined in a square cell of surface S 0 = 90 × 90 mm 2 with a vertical gap of 1.42 a. The cell is vertically vibrated at a frequency f v = 300 Hz with a r.m.s. acceleration Γ = (2πf v ) 2 A 0 /g = 1.6 g,where A 0 is the amplitude of vibration and g is the gravitational acceleration. Particles perform Brownian-like motion in the horizontal plane due to the roughness of the bottom surface of the cell, whose r.m.s. rugosity is measured to be 20 µm. By imaging with a high-speed camera (Phantom V10) the area S (50.36 × 50.36 mm 2 ) through a transparent and smooth lid, the trajectory of each particle is reconstructed in two-dimensions, in the horizontal plane. The ratio of the surface of the imaged area S to the surface of the entire cell S 0 reads S/S 0 = 0.315. When immersed in the external vertical magnetic field B, each sphere behaves as an induced dipole. In a first approximation, two particles whose centers are separated by a distance r i,j interact according to the repulsive potential

U i,j = 4π µ0 B 2 (a/2) 6 r 3
i,j [START_REF] Merminod | Self-organization of vibrated and magnetized particles: Structure, dynamics and transitions[END_REF][START_REF] Jackson | Classical Electrodynamics[END_REF], where µ 0 is the permeability constant. The relevant parameters of our experiments are the area fraction φ = N πa 2 4S , with N the average number of spheres imaged in S (here φ ≈ 0.2), the mean kinetic energy per particle

E k = m 2N N i=1 v 2 i
(with v i the velocity of particle i in the horizontal plane and the brackets denote time averaging) and the mean magnetic energy per particle

E m = 1 N N i=1 N j=i+1 U i,j .
The dimensionless number ε = E m /E k quantifies the competition between the interaction strength and kinetic energy. B is varied in [-0.11, 430] G, corresponding to ε ∈ [6.5 × 10 -4 , 1.5 × 10 3 ]. Each measurement is averaged over five independent realizations. After an equilibration time of 100 s, images are acquired using the high-speed camera during 3.85 s at a frame rate of 780 Hz. Using tracking algorithms [START_REF] Shattuck | Particle tracking[END_REF], the position, trajectory and velocity of each particle in the horizontal plane are computed in the window of observation S. For this vertical confinement distance of approximately 1.42 a and area fraction φ ≈ 0.2, it has been shown [START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF][START_REF] Schockmel | Melting of a confined monolayer of magnetized beads[END_REF][START_REF] Schockmel | Frustrated crystallization of a monolayer of magnetized beads under geometrical confinement[END_REF], that the dipolar interaction remains purely repulsive and the system can be described as twodimensional. For larger gap, three-dimensional effects must be taken into account for the interaction potential and the spatial distribution of spheres, leading to a large variety of phases [START_REF] Merminod | Transition to a labyrinthine phase in a driven granular medium[END_REF][START_REF] Merminod | Self-organization of vibrated and magnetized particles: Structure, dynamics and transitions[END_REF][START_REF] Opsomer | Herringbone Patterns in Confined Vibrated Magnetic Granular Matter[END_REF]. When the external magnetic field is increased, the strength of magnetic interactions quickly overcomes kinetic agitation as shown in the plot of ε = E m /E k as a function of B (Fig. 2 (a)). As ε is increased, the competition between repulsive interactions and kinetic agitation results in a transition from a granular gas towards a hexagonal crystal. Snapshots of a window inside S are shown without applied magnetic field in Fig. 1 (b) (ε ≈ 0) and with a moderate value of B = 62 G (ε ≈ 8.80) in Fig. 1 (c). In both cases, the assembly of spheres is in a granular gas state, but in the second snapshot the particles do not come into contact anymore. We note also a smaller number of particles in the second case. In Fig. 2 (b), we show indeed a decrease of φ with ε in the observation window S, which is due to increasing particle repulsion while boundaries are non-repulsive. The crystallization towards a hexagonal crystal is monitored by the sixfold bond-orientational order parameter per particle

Ψ j 6 = 1 n j nj k=1 e 6iθ jk , (1)
where n j is the number of nearest neighbors of particle j, and θ jk is the angle between the neighbor k of particle j and a reference axis. The corresponding global average,

|Ψ 6 | = 1 N N j=1 Ψ j 6 , (2)
where the vertical bars denote a modulus, measures the degree of hexagonal order of the particle assembly. We will refer to |Ψ 6 | as the hexagonal order parameter. In Fig. 2 (c) |Ψ 6 | is plotted as a function of ε. |Ψ 6 | is of order 0.4 in the granular gas phase, to reach 0.9 in the hexagonal crystal phase. The transition towards the hexagonal phase is located at ε c ≈ 62, corresponding to a maximal susceptibility, i.e the maximal variation of the hexagonal order parameter |Ψ 6 | to changes of ε. Here, our study is focused on the granular gas phase, thus for ε < 62, i.e. B 165 G. Although the system remains in a fluid-like phase for this range of B, it undergoes important structural changes. In Fig. 2 (d), the radial pair distribution function g(r) (or radial pair correlation function) [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Barrat | Basic Concepts for Simple and Complex Liquids[END_REF] is plotted as a function of the center-to-center distance between spheres r for selected values of ε. For ε = 0, (f) g(r) displays a strong peak at contact between spheres, i.e. r = a due to hard-sphere repulsion. For larger values of ε, the amplitude of this peak decreases due to strengthened inter-particle repulsion. In particular, g(r) becomes nearly flat for ε = 6.1 denoting the quasi absence of spatial correlations. When repulsion is further increased, the contacts become unlikely and a depleted zone appears for r slightly larger than a. At ε = 62, g(r) displays spatial oscillations characteristic of an emerging hexagonal order [START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF]. For roughly ε > 5, the rate of collisions between particles, f c , defined as the average number of distinct events per second for which r < 1.03 a, is strongly re-duced by the increasingly repulsive interactions (Fig. 2 (e)). It nearly vanishes for ε ≈ 20. For even larger ε, the dissipative collisions disappear. Then, particles interact with each other only through magnetic dipolar interactions, which are elastic (i.e. conservative). The decrease of f c with ε is roughly approximated by a decaying exponential function, f 0 exp(-b ε), with f 0 = 28 s -1 and b = 0.25. To collide two particles must overcome an energy barrier due to magnetic repulsion. Hence, the collision rate should be proportional to an Arrhenius factor exp(-ε) [START_REF] Scheffler | Collision rates in charged granular gases[END_REF]. Factor b being different from unity in our experiments may be explained by the exclusion of vertical motion from our analysis or by collective effects if the granular gas is not dilute enough [START_REF] Scheffler | Collision rates in charged granular gases[END_REF]. Note also that, the variations of φ and E k are too limited in this set of measurements to test their expected influence on the collision rate. However, a possible dimensional scaling is f c ∼ E k /m φ a -1 g eq (r = a) exp(-b ε), by assuming that f c is given by the product of the mean quadratic velocity by the mean free path, whose expression is given for hard-spheres by (a √ π)/(2 g eq (r = a) φ) [START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF] with g eq (r = a) the value of the equilibrium pair correlation function at contact.

B = 0 G , ε = 0 B = 20 G , ε = 3.4 B = 42 G , ε = 6.1 B = 62 G , ε = 8.8 B = 93 G , ε = 17 B = 165 G , ε = 62 (e)
10 0 10 1 10 2 2 3 4 5 6 •10 -10 ε E k (J) Data Model (g) -6 -4 -2 0 2 4 6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 v/σ v PDF ε ≈ 0 ε = 3.4 ε = 8.8 ε = 62 Gaussian (h)
The measured kinetic energy per particle E k has a nonmonotonous behavior, as a function of ε (Fig. 2 (f)). For 0 < ε < 10, E k increases due to the decrease of the collision rate. Then for ε > 10, i.e. for stronger magnetic repulsion, E k significantly decreases. For a granular gas in stationary regime, by balancing energy injection with dissipation [START_REF] Aumaitre | Collision frequencies and energy flux in a dilute granular gas[END_REF], the theoretical mean kinetic energy per particle can be written as:

E th k = P [(1 -r 2 ) f c + δ] ( 3
)
where P is the average injected power per particle, δ E th k is the average dissipation due to the collisions of particles with the bottom and top walls, whereas (1r 2 ) f c E th k is the dissipation caused by the inelastic collisions between particles, and r = 0.9 is a realistic restitution coefficient [START_REF] Opsomer | Herringbone Patterns in Confined Vibrated Magnetic Granular Matter[END_REF]. Using the experimentally measured collision rate f c , the growth of E k as a function of ε is well described by Eq. 3 with fitted parameters P = 1.1 × 10 -9 W and δ = 1.9 (red curve in Fig. 2 (f)). For ε ≥ 10, the magnetic repulsion constrains the horizontal motions perpendicular to the applied magnetic field, to favor the vertical motions and thus decreases the effective injected power. Finally, as reported for other quasi-two-dimensional vibrated granular gas experiments [START_REF] Losert | Velocity statistics in excited granular media[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF][START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF][START_REF] Olafsen | Velocity distributions and density fluctuations in a granular gas[END_REF][START_REF] Rouyer | Velocity fluctuations in a homogeneous 2D granular gas in steady state[END_REF][START_REF] Reis | Forcing independent velocity distributions in an experimental granular fluid[END_REF][START_REF] Scholz | Velocity Distribution of a Homogeneously Driven Two-Dimensional Granular Gas[END_REF], the distribution of particle velocities v for ε = 0 deviates from the Gaussian distribution expected for a molecular gas in equilibrium (Fig. 2 (g)). When ε is increased, the distance to the Gaussian distribution decreases due to the diminution of the particle collision rate. Indeed, when the collision rate decreases, the kurtosis of the velocity distribution, or the "flatness", F = v 4 /σ 4 v , where σ v is the standard deviation of the velocity distribution, approaches the value 3 expected for a Gaussian distribution (Fig. 2 (h)). We note that the velocity distributions are satisfactorily fitted by a stretched exponential function

f (v) ∝ exp(-A|v/σ v | β ) , (4)
with A and β varying respectively from 0.81 to 0.51 and 1.40 to 1.97 as f c decreases (Fig. 2 (i)). For a homogeneously driven granular gas with dissipative collisions kinetic theory predicts an exponent β = 3/2 for the high energy tail of the velocity PDF [START_REF] Van Noije | Randomly driven granular fluids: Large-scale structure[END_REF]. In our experiment, the fitted value of β is 1.48 for f c = 18.7 s -1 the largest collision rate. β is thus close to this theoretical prediction. In contrast, when f c ≈ 0, we find A = 0.520 and β = 1.94. These values approach those expected for a Gaussian distribution A = 1/2 and β = 2. Therefore, as B is increased and f c diminishes, the continuous increase of β from 1.40 to 1.97, shows unambiguously that the shape of the velocity distribution depends on the collision rate in a granular gas. Similarly, the velocity distribution can be fitted using the one-dimensional Sonine poynomial corrections to a Gaussian distribution [START_REF] Reis | Forcing independent velocity distributions in an experimental granular fluid[END_REF][START_REF] Coppex | On the first Sonine correction for granular gases[END_REF][START_REF] Brilliantov | Breakdown of the Sonine expansion for thevelocity distribution of granular gases[END_REF]. The experimental velocity distributions are approximately reproduced using the development at the second order (not shown), with Sonine coefficients a 1 = 0, a 2 in the range [0.01, 0.26] [START_REF]v) = π -1/2 exp(-c 2 ) (1 + a1 S1(c 2[END_REF]. The coefficient a 2 decreases indeed with f c and the kurtosis verifies accurately F = 3 (1 + a 2 ).

III. EFFECTIVE DISSIPATION PARAMETERS

We showed in previous Section that the dynamical behavior of this assembly of magnetized spheres is strongly controlled by the particle collision rate for moderate ues of ε, by studying the statistics of individual velocities. We now investigate the collective dynamics to obtain a different characterization of the dissipative processes at work. For that, velocity correlations can be computed in the spatial Fourier space and analyzed in the framework of linearized hydrodynamics [START_REF] Hansen | Theory of Simple Liquids[END_REF] which provides effective transport coefficients. A fruitful approach is to compute the dynamical transverse velocity structure factor J t (k, t) [START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF][START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Castillo | Order and density fluctuations in the vicinity of a granular solid-liquid-like phase transition[END_REF] (also called the transverse current correlation function):

J t (k, t) = 1 N N i,j=1 ( k × v i )(t) ( k × v j )(0) e i k (ri(t)-rj (0))
where t is the time, k is the wave vector, k is the unitary vector directed along k, v i (resp., v j ) is the velocity vector of particle i (particle j), and r i (resp., r j ) is the position vector of particle i (particle j). denotes a time average. These structure factors are computed for a stationary forcing for each test value of k = [k x , k y ] chosen in a horizontal grid of size 50×50 in the domain 0.0624 < k x,y < 3.12 mm -1 . Each pair (k x , k y ) is discretized according to (n x π/L x , n y π/L y ), where n x , n y ∈ N and L x = L y = 50.36 mm. A loop and a nested loop over the image numbers perform sweeps of initial times t = 0 and time lags t of the correlation function. Then, the quantity ( k × v i )(t) ( k × v j )(0) e i k (ri(t)-rj (0)) is computed by separating the real and imaginary parts for each pair of particles i at time t and j at time 0. J t (k, t) is obtained by first ensemble averaging and then time averaging. Assuming isotropy of particle motions, the quadrants for negative k x or k y are reconstructed. The angular average is computed as J t (k, t) = (2π) -1 2π 0 J t (k, t) dθ, where k = ||k||. J t (k, t) is finally averaged over five independent runs with identical experimental parameters.

The typical decay time of the transverse current J t (k, t) characterizes the dissipative processes at play. In the granular gas phase (ε < 62), the short-time decrease of J t (k, t) is well approximated by a decaying exponential ∼ e -t/τ (k) , where τ (k) is the typical life time of an excitation at the scale k (Fig. 3 (a) and(b)). For vibrated granular layers, energy dissipation is often modeled by the combination of a viscous drag and of a Coulomb friction leading to the equation τ -1 (k) = ν k 2 + γ b [START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF][START_REF] Castillo | Order and density fluctuations in the vicinity of a granular solid-liquid-like phase transition[END_REF], with ν a kinematic viscosity and γ b a friction coefficient. For each value of ε, parameters ν and γ b are obtained by fitting this equation to the measured values of τ -1 (k) (Fig. 3 (c) and(d)). Note that such a modeling of en-ergy dissipation becomes invalid for high ε, when the hexagonal phase is reached. Moreover, the fit quality worsens at small k a, where the statistical convergence is lesser and finite size effects may interfer. Parameters ν/a 2 and γ b are plotted as a function of ε in the granular gas phase (0 < ε < 62) in Fig. 3 (e). For this set of experiments, dissipation is dominated by friction. A characteristic length ξ = ν/γ b of order 0.3 mm can be defined. Surprisingly, this value is significantly smaller than the one found in Puglisi et al. [START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. Important differences between their system and ours include that their particles do not remotely interact and are more strongly agitated, and that their experimental cell has no lid.The fluctuating hydrodynamics theory interprets this length as a spatial correlation length of excitations [START_REF] Gradenigo | Fluctuating hydrodynamics and correlation lengths in a driven granular fluid[END_REF][START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF].

The relation between particle diffusion and dissipation is studied as follows. The self-diffusion coefficient, D, is obtained by fitting the mean-squared displacements of particles using the equation [r i (t)r i (t = 0)] 2 = 4Dt. Standard diffusion remains valid in the granular gas phase until approximately ε < 50. By analogy with the Einstein relation [START_REF]D of a molecular fluid writes D = kB T /Λ[END_REF], we propose and experimentally test the relation

D ≈ τ a T e = E k m(ν/a 2 + γ b ) , (5)
where τ a is the characteristic dissipative time at the scale 1/a, τ a = τ (k = 1/a), and T e is the usual granular temperature T e = v 2 /2 = E k /m. Indeed, by analogy with the physics of molecular systems [START_REF] Barrat | Granular gases: dynamics and collective effects[END_REF][START_REF] Andreotti | Granular Media: Between Fluid and Solid[END_REF], for granular gases the kinetic energy per particle is often expressed in terms of an effective granular temperature T e = E k /m. The relation Eq. 5 works especially well in our experiments for ε < 10, when E k increases with ε , as shown in Fig. 3 (f). The measured value of D is close τ a T e until a turning point at its maximal value corresponding to ε = 7.30. However, for ε > 10, D does not verify as well this scaling law anymore. This result suggests that the Einstein relation (which is a particular case of the fluctuation-dissipation theorem) holds in granular gases with dissipative collisions (for our system ε 10), when the dissipation coefficient is estimated from the velocity correlation functions at the scale k = 1/a of few sphere sizes. When the hexagonal crystal phase is approached (ε = 62), we note that D nearly vanishes, which is consistent to the transformation of a fluid-like phase into a solid-like phase.

IV. DEVIATION FROM ENERGY EQUIPARTITION

The static transverse velocity structure factor, J t (k, t = 0), provides the distribution across the spatial scales k of the kinetic energy associated to the transverse modes. Hence, it can be viewed as a kinetic energy power spectrum. A flat spectrum denotes equipartition of energy over the modes. For out-of-equilibrium, dissipative and driven granular gases, the fluctuating hydrodynamics theory defines a "bath temperature", T b = J t (k → 0, 0), and a "granular temperature" at the particle scale, T g = J t (k ≈ 2π/a, 0) [START_REF] Gradenigo | Fluctuating hydrodynamics and correlation lengths in a driven granular fluid[END_REF][START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. Due to the dissipative collisions acting at the particle scale, T b > T g . In a system in thermal equilibrium with elastic collisions, these three temperatures T e = E k /m, T b and T g must be equal. In addition, this theory predicts the shape of J t (k, 0) as a function of the characteristic correlation length ξ = ν/γ b :

J t (k, 0) = T g + T b -T g 1 + ξ 2 k 2 (6)
Our experimental measurements of J t (k, 0) are plotted for selected values of ε in Fig. 4 (a). Consistently with the non-monotonic evolution of E k as a function of ε (Fig. 1 (c)), the average level of J t (k, 0) increases with ε until ε ≈ 0, then decreases. In the granular gas phase, (ε < 62), J t (k, 0) is well fitted by Eq. ( 6), using the experimentally obtained values of ξ, except for the smallest k (Fig. 4 (a)). The largest peak at k ≈ 0 may be attributed to a global system vibration rather than to particle dynamics. Note also that since our values of ξ are smaller than those of Puglisi et al. [START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF], J t (k, 0) display less variations than the sigmoidal shapes, that they reported. Fitting J t (k, 0) by Eq. 6 provide an estimation of T b and of T g , whose difference quantifies the distance to equipartition [START_REF] Gradenigo | Fluctuating hydrodynamics and correlation lengths in a driven granular fluid[END_REF][START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. In Fig. 4 (b), we compare the temperatures T b , T g and T e from the kinetic energy as a function of ε. For moderate ε, we verify that T b > T e > T g and we observe that the distance to equipartition T b -T g decreases with ε to vanish at ε = 13.5 near the maximum of T g . For larger ε, T g is greater than T b . The intersection of T g with T b corresponds to an inversion of the slope of J t (k, 0) and likely the limit of the validity domain of the fluctuating hydrodynamics theory. For ε > 13.5, the magnetic interactions oppose the large scale fluctuations before inducing crystallization at ε ≈ 62. Fluctuating hydrodynamics attributes the difference between T b and T g to dissipative collisions. We verify this statement for ε ≤ 17 in Fig. 4 (c). We find indeed that when f c decreases, T b -T g essentially monotonically decreases, to nearly vanish when f c = 0. Moreover, the inverse of the dissipative time 1/τ a = 1/τ (k = 1/a), extracted from the time decay of J t (k, t), is plotted in Fig. 4 (d) as a function of ε. We find that for ε ≤ 10, 1/τ a is nearly equal to f c shifted by a positive constant τ -1 0 = 21 s -1 . Note that τ -1 0 = 21 s -1 equals roughly twice the dissipation coefficient δ/(1r 2 ) ≈ 10 extracted from the fit of the kinetic energy E k by E th k (see Eq. 3). Therefore, the phenomenological dissipation coefficients extracted from the fluctuating hydrodynamics, τ -1 a reveals that in addition to the collisions between the particles, a supplemental dissipation mechanism, the particles collisions with the bottom and top walls must be also taken into account. The decrease of the particle collision rate, induced by increasing the magnetic field, is precisely reported in 1/τ a at small ε. For larger values of ε, 1/τ a significantly increases although f c nearly vanishes, in correlation with the decrease of E k and the beginning of crystallization. Our interpretation is that since magnetic repulsion becomes then very strong vertical bead motion is favored, thus reducing the injected power into horizontal motion.

To summarize, by tuning the collision rate, dissipation can be adjusted although not canceled. The distance to equilibrium can be thus varied, but even at the maximum of the kinetic energy, the system remains out-ofequilibrium and a continuous energy input is needed to maintain a stationary state. However, our study highlights the peculiar role of dissipative collisions between particles in tuning the distance to equipartition. Because they generate dissipation at small scales and are uncorrelated with the forcing (the cell mechanical agitation), these dissipative collisions induce small scale correlations which reduce the kinetic energy spectrum at large k. Close to the maximum of the kinetic energy at ε ≈ 10, the spectrum is nearly flat, corresponding to equipartition of the velocity modes. For this quasielastic granular gas [START_REF] Merminod | Transition from a dissipative to a quasi-elastic system of particles with tunable repulsive interactions[END_REF], spatial structural correlations disappear (Fig. 2 (d)). Additionally, the velocity fluctuations become Gaussian (Fig. 2 (g) (h) and (i)), as for a molecular gas with elastic interactions in thermal equilibrium. For the largest values of ε, the assembly of vibrated spheres becomes structured into a crystalline phase by the magnetic repulsion. Consequently, the kinetic energy at large scale is reduced.

V. DISCUSSION AND CONCLUSIONS

Our experimental study confirms the validity of the fluctuating hydrodynamics theory and extends previous works [START_REF] Gradenigo | Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics[END_REF][START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. We show a clear relationship between the dissipation of hydrodynamics modes and the rate of dissipative collisions between particles. By tuning this collision rate, we demonstrate that the deviation from the kinetic energy equipartition is a consequence of the small scale dissipation induced by the collisions. When the magnetic energy becomes large compared to the horizontal kinetic energy, i.e. ε > 10, the limit of the validity domain of the fluctuating hydrodynamics theory is reached. Magnetic repulsive forces modify the system structure, increase its rigidity and induce correlations in the velocity modes. Although a complete description of the system would require to incorporate the magnetic interactions in the fluctuating hydrodynamics, in this work we use a perturbation approach assuming that in first approximation the interactions influence only the collision rate. Our results show the relevance of this hypothesis at least for ε 10. Therefore, the fluctuating hydrodynamics theory satisfactorily describes the dynamical properties of such quasi-two-dimensional vibrated granular gas for a large range of collision rates. The influences of the packing fraction and of the agitation strength have been tested by Puglisi et al. [START_REF] Puglisi | Structure factors in granular experiments with homogeneous fluidization[END_REF]. However, it is a strong hypothesis to equate the energy injection by consecutive particle collisions on the bot-tom rough wall with a thermal-like noise. Comparisons with other theoretical methods like the mode coupling model [START_REF] Van Noije | Randomly driven granular fluids: Large-scale structure[END_REF] or with three-dimensional molecular dynamic simulations would be useful to characterize energy injection in quasi-two-dimensional driven granular gas, given the absence of a measurement of particle vertical motions. Finally, we have shown that the out-of-equilibrium specificity of granular gases, unlike molecular gas, is related to the emergence of spatial correlations, here caused by the dissipative collisions. We would encourage to examine similar questions in other out-of-equilibrium systems with heterogeneous energy dissipation or injection such as turbulent flows or assemblies of active particles.

FIG. 1 .

 1 FIG. 1. (a) Schematic of the experimental setup (not to scale). Magnetized granular spheres of diameter a = 1 mm are imaged from the top with a high-speed camera in the region of interest S. The spheres are immersed in a transverse magnetic field B and vertically shaken with acceleration Γ. (b) Snapshot of the experiment for B ≈ 0 G (ε ≈ 0), and (c) for B = 62 G (ε = 8.80). Shaker acceleration is Γ = 1.6g and snapshot size is 17.5 × 17.5 mm 2 .

FIG. 2 .

 2 FIG. 2. (a) Ratio between magnetic and kinetic energy per particle ε = Em/E k as a function of the applied magnetic field B. (b) Area fraction φ as a function of ε. (c) Hexagonal order parameter |Ψ6| as a function of ε. Shaded area corresponds to the hexagonal phase (described in [21]). (d) Radial distribution function g(r) for various values of ε. (e) Collision rate or collision frequency between particles fc as a function of ε. Red line shows the modeled exponential decay of the collision rate: fc = f0 exp(-b ε), with f0 = 28 s -1 and b = 0.25. (f) Kinetic energy experimentally measured (E k , blue) and modeled (E th k from Eq. 3, red) as a function of ε. The model relates the increase of E k to the decrease of fc. (g) Probability distribution function (PDF) of the particle horizontal velocities v, normalized by the standard deviation σv for selected values of ε. The black line corresponds to the Gaussian distribution. Both spatial coordinates (x and y) are statistically equivalent and are used to compute the velocity PDF. (h) Kurtosis or flatness of the velocity distributions (F = v 4 /σ 4 v ) minus 3 as a function of the collision rate fc for 1 < ε < 100. (i) Fit parameters of modeled velocity PDF, A and β (see Eq. 4) versus the collision rate fc. Blue dashed line, value A = 1/2 = 0.5 expected for a Gaussian distribution. Red dashed lines, β = 2 expected for a Gaussian distribution and β = 3/2 = 1.5 for a homogeneously driven granular gas.

2 FIG. 3 .

 23 FIG. 3. Temporal decay of the transverse current correlation functions Jt(k, t), for selected k increasing from top to bottom for ε = 0 (a) and ε = 8.80 (b). Dashed lines, exponential fits measuring the dissipative time τ (k). The behavior is similar for larger values of ε in the granular gas phase (ε 50). Inverse of the fitted typical dissipative time τ (k) as a function of k a for ε = 0, and (d) for ε = 8.80. (e) Evolution of the friction coefficient γ b and of the rescaled kinematic viscosity ν/a 2 with ε. (f) Self-diffusion coefficient D as a function of τa Te, with τa = τ (k = 1/a) and Te = E k /m. Each data point corresponds to a value of ε (for ε < 62). Red arrowheads point towards greater ε. Dashed line corresponds to D = τaTe.

1 0FIG. 4 .

 14 FIG. 4. (a) Static transverse velocity structure factor Jt(k, t = 0) for selected values of ε. The dashed lines are the fits obtained using Eq. (6). (b) Comparison of the usual granular temperature Te = E k /m, the "bath temperature" T b , and the "granular temperature" at the particle scale Tg as a function of ε. The shaded region indicates the transition to the hexagonal phase. (c) Distance to kinetic energy equipartition T b -Tg as a function of the collision rate fc. (d) Inverse of the dissipative time, 1/τa and fc shifted of an arbitrary constant τ -1 0 = 21 s -1 as a function of ε.

ACKNOWLEDGMENTS

We thank Thierry Hocquet and Martin Devaud for letting us use their computing facilities at MSC, Université Paris Diderot. We acknowledge Nicolás Mujica and Rodrigo Soto from University of Chile for discussions and Sébastien Aumaître from CEA Saclay, SPEC for comments. Frédéric van Wijland from MSC, Université Paris Diderot is also gratefully thanked for comments and proofreading of the manuscript. This research was supported by Université Paris Diderot and Fondecyt Grant No. 3160032 (G.C.).