Louisiane Burkart

Anthony Olivier

Olivier Lourdais

Marion Vittecoq

Gabriel Blouin-Demers

Fabrice Alliot

Clément Le Gac

Nicolas Martin

Aurélie Goutte
email: aurelie.goutte@ephe.psl.eu

Determinants of Legacy Persistent Organic Pollutant Levels in the European Pond Turtle (Emys orbicularis) in the Camargue Wetland, France

Keywords: reptiles, PCB, organochlorine pesticides, delta wetlands

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In Europe, the production and use of persistent organic pollutants (POPs) are prohibited or severely restricted since 2004 through the Stockholm Convention (http://www.pops.int/), but POPs still cause environmental concerns.

Polychlorinated biphenyls (PCBs) and organochlorine pesticides (OCPs) remain in the environment for decades because of their slow biodegradation and improper handling of contaminated wastes. Levels of POPs in animals increase with age [START_REF] Vives | Age dependence of the accumulation of organochlorine pollutants in brown trout (Salmo trutta) from a remote high mountain lake Red , Pyrenees)[END_REF][START_REF] Binnington | Clarifying relationships between persistent organic pollutant concentrations and age in wildlife biomonitoring: individuals, cross-sections, and the roles of lifespan and sex: Clarifying POP concentration-age trends in biomonitoring[END_REF] and trophic level [START_REF] Goutte | Trophic transfer of micropollutants and their metabolites in an urban riverine food web[END_REF].

Exposure to POPs causes a wide range of adverse effects, including neurotoxicity, endocrine disruption, immune dysfunction, reproductive impairment, and developmental abnormalities, which may ultimately compromise survival and reproductive output and lead to population declines of wild vertebrates [START_REF] Goutte | Demographic consequences of heavy metals and persistent organic pollutants in a vulnerable long-lived bird, the wandering albatross[END_REF][START_REF] Goutte | Survival rate and breeding outputs in a high Arctic seabird exposed to legacy persistent organic pollutants and mercury[END_REF][START_REF] Salice | Integrative demographic modeling reveals population level impacts of PCB toxicity to juvenile snapping turtles[END_REF].

Pollution, and especially industrial and agricultural discharge, is one of the major threats to freshwater ecosystems [START_REF] Holt | Sources of chemical contaminants and routes into the freshwater environment[END_REF][START_REF] Dudgeon | Freshwater biodiversity: importance, threats, status and conservation challenges[END_REF]. Wetlands support an extremely rich biodiversity, but are among the most transformed and threatened ecosystems of the world [START_REF] Revenga | Prospects for monitoring freshwater ecosystems towards the 2010 targets[END_REF], experiencing rates of population declines and species extinctions far higher than those in forests, grasslands, and coastal ecosystems [START_REF] Dudgeon | Freshwater biodiversity: importance, threats, status and conservation challenges[END_REF]. Deltas are often exposed to high water pollution from intensive agriculture [START_REF] Kuenzer | Climate and Environmental Change in River Deltas Globally: Expected Impacts, Resilience, and Adaptation[END_REF] and from discharge of contaminants in upstream waters. The Camargue, in the Rhône River delta, is the largest wetland in France and is of international importance under the Ramsar Convention (https://www.ramsar.org/), but it is heavily impacted by human activities [START_REF] Cheiron | Rapport d'activité 2017 de la Réserve Naturelle Nationale de Camargue[END_REF]. The contamination of water bodies by agricultural, industrial, and urban discharges has been well studied over the past decades [START_REF] Comoretto | Pesticides in the Rhône river delta (France): Basic data for a field-based exposure assessment[END_REF]. In particular, levels of PCBs in sediments of the Rhône River increase from upstream to downstream, reaching 417 μg/kg dw for the sum of 7 PCB congeners (28, 52, 101, 118, 138, 153, and 180, Mourier et al. 2014).

Mean flux over the 2011-2016 period averaged 14 kg/year for PCB 180 in suspended particulate matter at the outlet of the Rhône River basin [START_REF] Poulier | Combining flux monitoring and data reconstruction to establish annual budgets of suspended particulate matter, mercury and PCB in the Rhône River from Lake Geneva to the Mediterranean Sea[END_REF].

Previous studies conducted in the Camargue confirmed that birds [START_REF] Berny | Impact of local agricultural and industrial practices on organic contamination of little egret (Egretta garzetta) eggs in the rhône Delta, Southern France[END_REF] and fish [START_REF] Roche | Relationships between persistent organic chemicals residues and biochemical constituents in fish from a protected area: the French National Nature Reserve of Camargue[END_REF][START_REF] Roche | Mise en évidence et validation de biomarqueurs écotoxicologiques dans la population d'anguilles d'un étang de la réserve naturelle nationale de Camargue, le Vaccarès, exposée à des polluants organiques persistants[END_REF](Roche et al., , 2009a) ) are exposed to OCPs and PCBs.

Freshwater turtles remain poorly studied in ecotoxicology although they can provide useful information on local contamination because of their longevity [START_REF] Campbell | A logical starting point for developing priorities for lizard and snake ecotoxicology: A review of available data[END_REF][START_REF] El Hassani | Morphological and physiological assessments reveal that freshwater turtle (Mauremys leprosa) can flourish under extremely degraded-polluted conditions[END_REF][START_REF] Gaus | Evaluating internal exposure of sea turtles as model species for identifying regional chemical threats in nearshore habitats of the Great Barrier Reef[END_REF], high trophiclevel, and low dispersal capacity (Ming-ch'eng [START_REF] Adams | Toxicological effects of polychlorinated biphenyls (PCBs) on freshwater turtles in the United States[END_REF]. Moreover, as ectothermic vertebrates, they have a lower ability to metabolize pollutants than endothermic mammals and birds [START_REF] De Solla | Exposure, Bioaccumulation, Metabolism and Monitoring of Persistent Organic Pollutants in Terrestrial Wildlife[END_REF]. The European pond turtle, Emys orbicularis, a long-lived (> 40-80 years) opportunistic predator (fish, amphibian, crayfish) and scavenger [START_REF] Ottonello | Feeding habits of the European pond terrapin Emys orbicularis in Camargue (Rhône delta, Southern France)[END_REF][START_REF] Ficetola | Is the European « pond » turtle Emys orbicularis strictly aquatic and carnivorous? Amphibia-Reptilia[END_REF], is facing significant population declines due to multiple environmental alterations, including water pollution [START_REF] Cheylan | Evolution of the distribution of the European pond turtle in the French Mediterranean area since the post-glacial[END_REF]). This species is listed as "near threatened" (NT) on the UICN Red List of threatened species in France (REF). Pollution by trace metal elements has been studied in E. orbicularis [START_REF] Namroodi | European pond turtle (Emys orbicularis persica) as a biomarker of environmental pollution in Golestan and Mazandaran provinces, Iran[END_REF][START_REF] Guillot | Trace Element Concentrations in European Pond Turtles (Emys orbicularis) from Brenne Natural Park, France[END_REF][START_REF] Beau | Environmental causes and reproductive correlates of mercury contamination in European pond turtles (Emys orbicularis)[END_REF], but data are lacking on burdens of POPs.

We studied contamination levels in two populations of E. orbicularis at the Natural Reserve of the Tour du Valat, in the Camargue, France. Several individuals were of known-age due to capture-mark-recapture (CMR) program, starting more than 20 years ago [START_REF] Olivier | Assessing spatial and temporal population dynamics of cryptic species: an example with the European pond turtle[END_REF][START_REF] Ficheux | Rapid response of a long-lived species to improved water and grazing management: The case of the European pond turtle (Emys orbicularis) in the Camargue, France[END_REF][START_REF] Arsovski | Covariates streamline age-specific early life survival estimates of two chelonian species[END_REF]. The hydraulic system combines irrigation canals originating from the Rhône river and drainage canals, which are exposed to various environmental contaminants from the Rhône river and agricultural plots, especially rice fields [START_REF] Chauvelon | Hydrologie quantitative d'une zone humide méditerranéenne aménagée : le bassin de Fumemorte en Grande Camargue, delta du Rhône[END_REF]. In the present study, we assessed recent (2018) levels of OCPs and PCBs in plasma of the European pond turtle (n = 174). We tested the effects of individual traits (mass, sex, and age) on POP burdens. We expected that older and larger individuals should have higher levels of POPs because of bioaccumulation and length of exposure.

Moreover, we compared POP burdens of turtles from two populations in different locations in the agricultural hydraulic system (irrigation vs drainage). We expected high levels of OCPs in turtles from the drainage site, as a possible consequence of the remobilization of pesticides from agricultural soils, and high levels of PCBs in turtles from the irrigation channel, as a possible consequence of the historical contamination.

Material and methods

Sampling sites and capture

We conducted the study in the Natural Reserve of the Tour du Valat (43°30' N, 4°40' E, Fig. 1) in France. We captured European pond turtles (n = 174) in canals and marshes from 24 April to 26 July 2018 by hand or with funnel traps [START_REF] Olivier | Assessing spatial and temporal population dynamics of cryptic species: an example with the European pond turtle[END_REF][START_REF] Ficheux | Rapid response of a long-lived species to improved water and grazing management: The case of the European pond turtle (Emys orbicularis) in the Camargue, France[END_REF]. We captured turtles at two sites: (1) irrigation canals and their associated marshland (site of Esquineau, n = 126) and ( 2) drainage canals of the Fumemorte basin and marshes filled with water by these canals (site of Faïsses, n = 48). The sex ratio was the same between the two sites (58.3% females at Faïsses and 60.3% females at Esquineau, Pearson's Chi-squared test, p = 0.95).

Individual turtles were identified as part of a long-term CMR program with shallow notches on the marginal and nuchal scales in unique combinations [START_REF] Olivier | Ecologie, traits d'histoire de vie et conservation d'une population de Cistude d'Europe Emys orbicularis en Camargue[END_REF][START_REF] Olivier | Assessing spatial and temporal population dynamics of cryptic species: an example with the European pond turtle[END_REF]. Individuals were weighed and sexed by visual observation of the secondary sexual characters. The year of birth can be determined if the first capture occurred within the first 5 years of life by counting the number of growth streaks [START_REF] Castanet | Les méthodes d'estimation de l'âge chez les chéloniens[END_REF]. This long-term survey starts in 1997, although a few tens of individuals were also marked as adults from 1976. In our data set, 66 of the 174 individuals were of known-aged (5 to 26 year-old, corresponding to a first capture between 1997 and 2018) and some were first captured as adults in 1976, thus being considered as more than 44 year-old. When turtles were captured after their first 5 years of life (N = 108 in the present study), we determined a stage of growth according to growth streaks located on the plastron [START_REF] Olivier | Ecologie, traits d'histoire de vie et conservation d'une population de Cistude d'Europe Emys orbicularis en Camargue[END_REF]. By combining estimated age and stage of growth, we then attributed an age class to each turtle: 4 to 8 year-old (N = 20), 9 to 13 year-old (N = 46), 14 to 22 year-old (N = 54), and 23 to 44+ year-old (N = 54). Turtles were released on the same day at their capture site.

Blood sampling

Plasma is a good matrix to determine individual burdens of POPs because blood sampling is minimally invasive and because circulating levels of POPs in blood are significantly correlated to concentrations of POPs stored in fat [START_REF] Keller | Organochlorine contaminants in sea turtles: Correlations between whole blood and fat[END_REF][START_REF] Dabrowska | Polychlorinated Biphenyl Concentrations, Congener Profiles, and Ratios in the Fat Tissue, Eggs, and Plasma of Snapping Turtles (Chelydra s. serpentina) from the Ohio Basin of Lake Erie, USA[END_REF], liver, kidney, and muscle tissues (Van de [START_REF] Van De Merwe | Using blood samples to estimate persistent organic pollutants and metals in green sea turtles (Chelonia mydas)[END_REF].

We collected blood samples (2 ml, always less than 1% of the turtle body mass) from the dorsal coccygeal vein [START_REF] Innis | Tortoises and Freshwater Turtles[END_REF][START_REF] Keller | Organochlorine contaminants in sea turtles: Correlations between whole blood and fat[END_REF] with a Terumo Company® syringe (Somerset, USA) pre-impregnated with heparin to prevent blood clotting during collection and equipped with a 25G needle (Franklin Lakes, USA).

We then centrifuged the samples to separate the plasma from the red blood cells. We collected blood samples from 174 individuals (126 from Esquineau and 48 from Faïsses), 66 individuals were of known age ranging from 5 to 31 years old (32 females and 19 males from Esquineau; 9 females and 6 males from Faïsses). Samples were stored at -18 °C until analysis in the UMR 7619 METIS of Sorbonne Université, France.

Chemical analyses

We determined the levels of 17 IUPAC numbers # 77,81,105,114,118,123,126,156,157,167,169 and 189) in the 174 plasma samples.

Samples were processed through Solid Phase Extraction (SPE), using hexane: dichloromethane, 9:1, using a validated protocol [START_REF] Tapie | PBDE and PCB contamination of eels from the Gironde estuary: from glass eelsto silver eels[END_REF]) that was adapted for plasma samples (see Supplementary Material for a detailed description of the procedure and method validations). PCBs and OCPs were analyzed using an Agilent 7890 A gas chromatograph (GC) coupled to a 7000 B triple quadrupole mass spectrometer system (MS/MS) (Agilent Technologies, Les Ulis, France). Recovery rates (% RR) of compounds were assessed on replicate plasma samples of European pond turtles (n = 4) with spiked solutions (100 ng of each compound). The repeatability of the method was assessed in terms of relative standard deviation (% RSD) of the recovery. RR and RSD are provided in Table S2 (Supplementary Material). Recovery rates were not satisfactory (< 75% or > 125%) for isodrin, endrin, heptachlor epoxid, heptachlor-endo-epoxide, p,p'-DDD, PCB-77, PCB-189 (Table S2) and they were not further considered.

Statistical analyses

We performed statistical analyses with R software version 3.3.2 (R Core Team, 2016).

Only POPs detected (i.e. > LOQ) in at least 10% of the samples were included in the statistical analyses. Since age and weight were positively correlated (t = 3.65, df = 64, p = 0.0005) and females were significantly heavier than males (W = 474, p < 0.0001), body mass, sex and age were tested separately..

We used generalized linear models (GLM) with a binomial distribution and a logit link function to test for the effects of sex, site, the interaction sex * site, sampling date, and age (or body mass) on contaminant occurrence. For each model, we used a backward elimination to progressively remove non-significant terms (p > 0.05). For all analyses, model specification and validation were based on residual analysis.

The effects of sex, sampling site, date, and age (or body mass) on pollutant concentrations were tested by considering left-censored data, i.e. pollutant levels below the LOQ. To do so, group comparisons and linear regressions were performed using Peto-Prentice tests and tobit models respectively, with the function cendiff of the NADA package and the function tobit of the AER R-package.

Results

POP levels and occurrences

In 37 of the 174 samples, all contaminant levels were below the limit of quantification, which represented 10.4% of turtles sampled in Faïsses and 24.4% of turtles sampled in Esquineau. Eleven compounds were never detected in the plasma samples: PeCB, Lindane, Quintozene, Heptachlor, and the dioxin-like PCBs: PCB-118, -105, -114, -126, -156, -157 and -167. The following seven compounds were detected at least in one sample from Esquineau, but not in samples from Faïsses: alpha-HCH, 44'-DDT and PCBs 28, 101, 123, 169 (Table 1). Overall, the contaminant concentrations in the plasma samples were very low (Table 1).

Influence of individual traits, sampling site, and date on POP occurrences and levels

Only three POPs (HCB, PCB-52 and -153) had substantial occurrence, being detected in more than 10% of individuals (Table 1). Plasma occurrences of PCB-52 and HCB were significantly higher in turtles captured at Faïsses compared to Esquineau, while the occurrence of PCB-153 was higher in turtles captured at Esquineau (Tables 1 and2). HCB occurrence slightly increased with sampling date (Table 2). Sex, mass, and age class did not explain POP occurrence in turtles, except for PCB-153 whose detection frequency was higher in older individuals (Table 2).

Levels of HCB and PCB-52 were significantly higher in turtles from Faïsses, while PCB-153 levels tended to be higher in turtles from Esquineau (Tables 1 and3).

Moreover, levels of PCB-153 were lower in the youngest turtles (Table 3, Fig. 2). Sex, mass, and date did not explain variation in levels of POPs among turtles (Table 3).

Discussion

The aim of this study was to characterize legacy levels of POPs (OCPs and PCBs) in the plasma of European pond turtles in the Camargue, France, and to determine whether these levels were a function of individual traits and habitat types. Plasma levels of POPs were low and often below limits of quantification (from 0.2 to 3.1 ng/mL). We found significant differences in burdens of POPs between turtles from the two sampling sites and these differences could be attributed to the hydraulic system (drainage/irrigation). The occurrence and levels of PCB-153 were higher in older turtles.

The Rhône River has been historically contaminated, leading to an accumulation of POPs in sediments downstream [START_REF] Mourier | Historical records, sources, and spatial trends of PCBs along the Rhône River (France)[END_REF]. Previous studies conducted on several animal taxa in the Camargue confirmed high exposure during the last decades, with high concentrations of PCBs in muscles of eels (Anguilla anguilla) fished between 1997 and 2000 [START_REF] Roche | Recherches écotoxicologiques sur la contamination des peuplements de poissons de la réserve naturelle nationale de Camargue par les polluants organiques persistants[END_REF], but also in eggs of little egrets (Egretta garzetta) collected in 1996 [START_REF] Berny | Impact of local agricultural and industrial practices on organic contamination of little egret (Egretta garzetta) eggs in the rhône Delta, Southern France[END_REF]. We detected low levels of POPs in the pond turtle in Camargue, which could indicate a decline in legacy POP exposure for wild species in the Camargue. Although a longitudinal study of the same species at the same locations would be required to test this hypothesis, decreasing PCB concentrations have been observed in the Rhône River sediments over the last decades [START_REF] Liber | Past and recent state of sediment contamination by persistent organic pollutants (POPs) in the Rhône River: Overview of ecotoxicological implications[END_REF] and water analyses conducted by the Nature Protection National Society (SNPN) have not detected PCBs and OCPs in the Fumemorte canal since 2011 [START_REF] Cheiron | Rapport d'activité 2018[END_REF]. Moreover, organochlorine contamination across food webs tended to diminish in the Vaccarès lagoon (Roche et al., 2009a), the largest water body of the Camargue Biosphere Reserve. Exotic red swamp crayfish (Procambarus clarkii), the main prey of European pond turtles in the Camargue [START_REF] Ottonello | Feeding habits of the European pond terrapin Emys orbicularis in Camargue (Rhône delta, Southern France)[END_REF], were not contaminated by POPs in 2019 (i.e. < LOQ, with LOQ ranging from 0.9. to 1.98 ng/g dw, unpublished data).

One of the first studies looking at the plasma concentration of POPs in turtles documented high PCBs and OCPs levels in snapping turtles (Chelydra serpentina), in Ontario, Canada and sampled in 2001-2004[START_REF] Letcher | Snapping Turtles (Chelydra serpentina) from Canadian Areas of Concern across the southern Laurentian Great Lakes: Chlorinated and brominated hydrocarbon contaminants and metabolites in relation to circulating concentrations of thyroxine and vitamin A[END_REF]: plasma concentrations of OCPs (sum of 17 contaminants) ranged from 0.2 to 236 ng/g w.w.

and the most abundant pesticide was p,p'-DDE (mean ± SE: 27 ± 6 ng/g w.w.). In our study, plasma concentrations of POPs (sum of 16 contaminants) were much lower, ranging from 0 to 2 ng/g w.w and p,p'-DDE levels didn't exceed 2.3 ng/g ww.

Snapping turtles are freshwater turtles with similar feeding habits than European pond turtle: they also consume plant and animal matter, including aquatic invertebrates, fish, frogs, reptiles [START_REF] Ernst | Turtles of the United States and Canada[END_REF].Other studies of turtles documented plasma concentrations of POPs comparable to those in our study; for instance, for PCB 153 and HCB in another freshwater species, the Western pond turtle (Actinemys marmorata) in Sequoia National Park, USA, in 2012 [START_REF] Meyer | Organic contaminants in western pond turtles in remote habitat in California[END_REF], as well as in marine turtle, such as Loggerhead sea turtles (Caretta caretta) in the Eastern Atlantic Ocean in 2011-2012 [START_REF] Bucchia | Plasma levels of pollutants are much higher in loggerhead turtle populations from the Adriatic Sea than in those from open waters (Eastern Atlantic Ocean)[END_REF], and in Green turtles (Chelonia mydas) and Hawksbill turtles (Eretmochelys imbricata) in Cape Verde between 2009 and 2011 [START_REF] Camacho | Monitoring organic and inorganic pollutants in juvenile live sea turtles: Results from a study of Chelonia mydas and Eretmochelys imbricata in Cape Verde[END_REF].

In terms of occurrence of POPs, our results in European pond turtles were low compared to other studies [START_REF] Bucchia | Plasma levels of pollutants are much higher in loggerhead turtle populations from the Adriatic Sea than in those from open waters (Eastern Atlantic Ocean)[END_REF][START_REF] Meyer | Organic contaminants in western pond turtles in remote habitat in California[END_REF]. The occurrences of dioxin-like PCBs in 8% of European pond turtles, as well as p,p'-DDE in 2% of individuals were much lower than those found in plasma of Loggerhead sea turtles that ranged from 63% in the Atlantic Ocean to 100% in the Adriatic Sea for dioxinlike PCBs and 100 % for p,p'-DDE [START_REF] Bucchia | Plasma levels of pollutants are much higher in loggerhead turtle populations from the Adriatic Sea than in those from open waters (Eastern Atlantic Ocean)[END_REF].

Despite low detection frequencies and levels, we found differences in POP burdens between turtles from the two sampling sites. Turtles in Faïsses, a site receiving water from drainage canals, exhibited significantly higher occurrence and levels of HCB compared to turtles from Esquineau, likely because of the remobilisation of trapped OCPs in soils. On the other hand, the occurrence and levels of PCB 153 were higher in turtles from Esquineau, a site with irrigation channels. A previous study has documented higher concentrations of PCBs in the aquatic fauna in irrigation canals compared to the drainage canals downstream of rice fields (Roche et al., 2009b). We also found that concentrations of PCB-52 were higher in turtles at Faïsses, however, which may be due to a lower sedimentation rate of PCB-52 associated with its lower molecular weight and a greater solubility due to a lower level of chlorination compared to PCB-153 [START_REF] Gong | Desorption rates of two PCB congeners from suspended sediments---I. experimental results[END_REF][START_REF] Alkhatib | Parameters affecting partitioning of 6 PCB congeners in natural sediments[END_REF]. The processes of deposition and release of PCBs from the irrigation part to the drainage part of the canals need to be further studied. In addition, other populations of pond turtles in the two types of hydraulic systems should be studied to confirm this pattern.

The occurrence and levels of PCB-153 were lower in the youngest individuals (4 to 8 year old), which could be attributed to a shorter exposure period. PCB-153 is a recalcitrant chemical with a high hydrophobicity (high octanol-water partition coefficient), high resistance to metabolic transformation, and slow respiratory elimination through air-exhalation (high octanol-air partition coefficient). PCB-153 is also more prone to bioaccumulation than many other PCB congeners [START_REF] Kelly | Food web-specific biomagnification of persistent organic pollutants[END_REF], especially in long-lived air-breathing predatory species [START_REF] Rowe | The Calamity of So Long Life »: Life Histories, Contaminants, and Potential Emerging Threats to Long-lived Vertebrates[END_REF]. It is also possible that there is a dietary shift during growth in pond turtles leading to the consumption of higher trophic-level prey with age, and these higher trophic level prey would be more PCB contaminated due to biomagnification. This hypothesis, however, is not supported by dietary studies of E. orbicularis in Camargue), showing a shift to a more herbivorous diet with age, using prey items identification in faecal samples [START_REF] Ottonello | Feeding habits of the European pond terrapin Emys orbicularis in Camargue (Rhône delta, Southern France)[END_REF] or no difference in the proportion of plants, invertebrates and vertebrates, using metabarcoding analyses (Duccotterd et al.

2020).)

We did not detect differences in POP loads between male and female pond turtles, despite males often being more contaminated than females in other turtle species [START_REF] Guirlet | Maternal transfer of chlorinated contaminants in the leatherback turtles, Dermochelys coriacea, nesting in French Guiana[END_REF][START_REF] Bangma | Perfluoroalkyl substances in diamondback terrapins (Malaclemys terrapin) in coastal South Carolina[END_REF][START_REF] Lambiase | Polychlorinated organic pollutants (PCDD/Fs and DL-PCBs) in loggerhead (Caretta caretta) and green (Chelonia mydas) turtles from Central-Southern Tyrrhenian Sea[END_REF] due to a transfer of pollutants from the mother to the eggs through vitellogenesis [START_REF] Moss | Concentrations of persistent organic pollutants in plasma from two species of turtle from the Tennessee River Gorge[END_REF][START_REF] De Solla | Organic contaminants in reptiles[END_REF].

Figure captions

Figure 1 :

 1 Figure 1: Sampling locations for European pond turtles in the regional nature reserve

Figure 2 :

 2 Figure 2: Increase in plasma PCB 153 (ng/mL) levels in European pond turtles with

Acknowledgements

We are indebted to the many people who participated in the turtle captures since 1997.

We thank the French Ministry of Environment for giving us the permission to capture European pond turtles.

Funding

This work was funded by the Water agency Rhône Méditerrannée Corse and the Fondation de la Tour du Valat.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Author contributions

AO, LB, CLG and NM conducted the field work and collected the blood samples. LB and FA conducted the laboratory work to prepare plasma samples for analysis with gas chromatography and tandem mass spectrometry. LB, OL, AG, AO, GBD, and MV conceived and coordinated the study, participated in data analysis, and in writing the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Ethical consideration

This study was performed in accordance with laws relative to capture, transport, and experiments on E. orbicularis (DREAL permit # CERFA_13616-01) and all procedures were approved by an independent ethical committee (APAFIS#17899-201812022345423 v2).


Noëlie Molbert
email: noelie.molbert@upmc.fr

Frédéric Angelier

Fabrice Alliot

Cécile Ribout

Aurélie Goutte

Fish from urban rivers and with high pollutant levels have shorter telomeres

Keywords: chub, telomere, metabolites, phthalate, pesticides

Environmental pressures, such as urbanization and exposure to pollutants may jeopardize survival of free-living animals. Yet, much remains to be known about such an effect on wild vertebrate ectotherms, since most studies still failed to explore the relationship between levels of currently-released pollutants and physiological responses. This study aims at testing the effect of urbanization and pollution (phthalate, pesticides: organochlorine [OCP] and pyrethroid pesticides, polychlorobiphenyles: PCB, polybromodiphenylethers: PBDE, polycyclic aromatic hydrocarbons: PAH and some of their metabolites) on telomere length, a suggested biomarker of life expectancy, in the European chub Squalius cephalus from urban and agricultural rivers of the Marne hydrographic network, France. Our results show that telomere length was reduced in chub from urban rivers. Moreover, among the wide range of anthropogenic contaminants investigated, high levels of PCBs in muscle and metabolites of phthalates, PAH and pyrethroids in liver were associated with shorter telomeres in chub. This study suggests that urbanization and chemical pollution may compromise survival of wild fish, by accelerating telomere attrition.

Introduction

Chronic exposure to complex mixtures of environmental toxicants may have severe consequences in free-living animals by reducing reproductive outputs and survival [1], thereby leading to population collapse [2]. However, we currently lack robust data to link contaminant burden and survival, probably because studying demographic responses to chemical exposure requires long-term (years to decades) monitoring surveys of numerous marked individuals, which are often difficult to achieve in the wild. Aquatic organisms in urban areas are exposed to a wide array of environmental pollutants, because of sewage and runoff from artificialized surfaces.

In that context, the measurement of telomere length has been recognized as a robust molecular tool to predict life expectancy in endotherms [START_REF] Tricola | The rate of telomere loss is related to maximum lifespan in birds[END_REF][START_REF] Whittemore | Telomere shortening rate predicts species life span[END_REF] and to some extent in ectotherms [START_REF] Olsson | Ectothermic telomeres: it's time they came in from the cold[END_REF] and even population vulnerability in wild lizards [START_REF] Dupoué | Shorter telomeres precede population extinction in wild lizards[END_REF]. Located at the end of eukaryote chromosomes, telomeres shorten through successive cell division. Beyond a critical telomere length, the cell starts to senesce, leading to apoptosis and a decline in tissue function [START_REF] Monaghan | Do telomere dynamics link lifestyle and lifespan?[END_REF].

Importantly, this natural process can be accelerated under stressful environmental conditions reviewed in [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF][START_REF] Chatelain | The association between stressors and telomeres in non-human vertebrates: a meta-analysis[END_REF]. In particular, oxidative stress has been recognized as a mechanistic pathway linking environmental stress and telomere erosion in vertebrates [START_REF] Von Zglinicki | Oxidative stress shortens telomeres[END_REF][START_REF] Riechert | Does oxidative stress shorten telomeres in vivo? A review[END_REF].

Chemical pollution is part of multiple stress factors generating or enhancing oxidative stress [START_REF] Yazdani | Comparative toxicity of selected PAHs in rainbow trout hepatocytes: genotoxicity, oxidative stress and cytotoxicity[END_REF], yet the effect of chemical exposure on telomere length is poorly known for wildlife, especially for vertebrate ectotherms [START_REF] Louzon | Telomere dynamic in humans and animals: Review and perspectives in environmental toxicology[END_REF]. To date, studies mostly focused on birds exposed to trace metals [START_REF] Stauffer | Telomere damage and redox status alterations in free-living passerines exposed to metals[END_REF], chlorinated [START_REF] Blévin | Exposure to oxychlordane is associated with shorter telomeres in arctic breeding kittiwakes[END_REF][START_REF] Sletten | Organohalogenated contaminants in whitetailed eagle (Haliaeetus albicilla) nestlings: an assessment of relationships to immunoglobulin levels, telomeres and oxidative stress[END_REF] and halogenated compounds [START_REF] Sletten | Organohalogenated contaminants in whitetailed eagle (Haliaeetus albicilla) nestlings: an assessment of relationships to immunoglobulin levels, telomeres and oxidative stress[END_REF][START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF], which are classes of organic chemicals either banned or phased-out from use and production.

Currently-release pollutants that are ubiquitous in the environment, such as polycyclic aromatic hydrocarbons (PAHs), phthalate plasticizers and pesticides [START_REF] Malaj | Organic chemicals jeopardize the health of freshwater ecosystems on the continental scale[END_REF], are known to impair telomere length in human cohorts [START_REF] Hoxha | Association between leukocyte telomere shortening and exposure to traffic pollution: a cross-sectional study on traffic officers and indoor office workers[END_REF][START_REF] Hou | Lifetime pesticide use and telomere shortening among male pesticide applicators in the Agricultural Health Study[END_REF]. Although exposure pathways and concentrations might differ, similar effects are expected on aquatic vertebrates. Indeed, vertebrates share the ability to regulate the level of internal organic pollutant exposure via metabolic processes [START_REF] Barron | Biotransformation of di(2ethylhexyl)phthalate by rainbow trout[END_REF] that have the potential to produce reactive oxygen species through redox cycling or yield more toxic intermediates than the parent pollutant [START_REF] Wang | A structure-based investigation on the binding interaction of hydroxylated polycyclic aromatic hydrocarbons with DNA[END_REF].

In addition, many vertebrates are particularly vulnerable to habitat alterations [START_REF] Ripple | Extinction risk is most acute for the world's largest and smallest vertebrates[END_REF], other than chemical pollution. With the expanding urbanization, urban river systems have suffered from profound changes through damming, banking and channelization exerting additional stress on aquatic organisms. For instance, environmental harshness, disease prevalence and thermal stress have been associated with disturbed oxidative balance and shorter telomeres in fishes [START_REF] Rollings | Telomeric attrition with age and temperature in Eastern mosquitofish (Gambusia holbrooki)[END_REF][START_REF] Jasinska | Assessment of biomarkers for contaminants of emerging concern on aquatic organisms downstream of a municipal wastewater discharge[END_REF][START_REF] Debes | Is telomere length a molecular marker of past thermal stress in wild fish?[END_REF][START_REF] Mclennan | Interactions between parental traits, environmental harshness and growth rate in determining telomere length in wild juvenile salmon[END_REF][START_REF] Stauffer | Telomere length and antioxidant defense associate with parasite-induced retarded growth in wild brown trout[END_REF]. Non-chemical alterations of river systems, combined to environmental pollution, could therefore lead to a cumulative effect on telomere shortening.

In this study, we examined the effect of organic pollutant burden and habitat alteration (urban and agricultural) on relative telomere length of wild populations of European chub, Squalius cephalus. We expected higher chemical pollution on urban rivers, as previously reported [START_REF] Teil | Phthalate Fate in the Hydrographic Network of the River Seine Basin (France) Under Contrasted Hydrological Conditions[END_REF][START_REF] Peng | Spatial Analysis of PAHs in Soils along an Urban-Suburban-Rural Gradient: scale effect, distribution patterns, diffusion and influencing factors[END_REF] and predicted that telomeres would be shorter in fish from urban rivers compared to agricultural ones and would decline with pollutant burden.

Materials and Methods

(a) Samples collection and chemical analyses

A total of 118 chub, S. cephalus, were caught by electrofishing within twelve days in September 2016 from the Marne River and its tributaries, France, on urban and agricultural areas, representing differently-contaminated riverine habitats (Supplementary material, Appendix A, table S1 and figures S1and S2).Physico-chemical and hydrological parameters of each sampling sites are presented in the supplemental material (Appendix A, table S2). Left pelvic fins were consistently sampled for DNA extraction and scales were removed for age determination (see [START_REF] Molbert | in press) Potential benefits of acanthocephalan parasites for chub hosts in polluted environments[END_REF], for details). Muscle samples (n = 118) were used for the quantification of parent organic contaminants and metabolites of pollutants were analyzed in the liver (n = 93), as the primary organ of xenobiotic biotransformation. Samples were stored in polycarbonate tubes to limit phthalate contamination and frozen at -20 °C until subsequent analyses. Metabolites were quantified in fewer individuals (n = 66; Appendix B, table S3) since some of them did not yield sufficient biological material to carry out chemical analyses.

Analyses of organic pollutants (16 polycyclic aromatic hydrocarbons: PAH; 7 phthalate esters; 7 pyrethroids; 4 organochlorine pesticides: OCP; 7 polychlorinated biphenyls: PCB; 6 polybrominated diphenyl ethers: PBDE) and their metabolites (11 hydroxylated PAHs; 9 phthalate monoesters; 4 metabolites of pyrethroids) were performed in muscle and liver respectively, following previously published protocols [START_REF] Molbert | Multi-residue methods for the determination of organic micropollutants and their metabolites in fish tissues[END_REF].

(b) Telomere analysis

Telomere length was determined by quantitative PCR (qPCR; BioRad CFX 96, Bio-Rad USA) according to [START_REF] Dupoué | Chronic water restriction triggers sex-specific oxidative stress and telomere shortening in lizards[END_REF], adapted for the European chub. Briefly, fin samples were digested with proteinase K and DNA was extracted using the Nucleospin Tissue Kit (Macherey-Nagel), following the manufacturer's instructions. DNA concentration and purity were assessed with a Nanodrop ND1000 spectrophotometer (Thermo Scientific). The telomere primers were similar to those previously used [START_REF] Petitjean | Direct and indirect effects of multiple environmental stressors on fish health in human-altered rivers[END_REF]. The control single-copy gene Recombination Activating Gene 1 (RAG-1) was selected and amplified using specific primers [START_REF] Mclennan | Distinct telomere differences within a reproductively bimodal common lizard population[END_REF] designed for the European chub. Reverse and forward primer sequences for both The efficiency of the telomere and RAG-1 assays ranged from 98.8% to 100% and 88.9% to 92.6%, respectively. All qPCR runs were performed using 2.5 ng of DNA per reaction and using the BioRad SYBR Green Supermix. The universal telomere primers were used at a concentration of 800 nM, and RAG1-F/RAG1-R at 300 nM. To generate a six-point standard curve (from 10.0 ng to 0.31 ng) for controlling the amplifying efficiency of the reactions, serial dilutions of DNA from a pooled sampled of 10 chub were included on the plate. All samples were randomly distributed across the PCR plates. The reference sample ("golden sample") was run in triplicate in all plates to account for inter-plate variation and each sample was run in duplicate on every plate. Intra-plate variation (coefficient of variation: 100 × SD/mean value) for telomere and RAG-1 were 2.07% and 2.80%, respectively. The relative telomere length (expressed as T/S ratio) was calculated as the telomere copy number (T) relative to single-copy gene (S; RAG1), according to [START_REF] Cawthon | Telomere measurement by quantitative PCR[END_REF]. Overall, 20 samples were excluded due to DNA degradation. As a result, telomere length (T/S ratio) was determined for 98 individuals and was relative to the internal single gene control (RAG-1). Additional information about the method are presented in supplemental material (Appendix C).

(c) Statistical analysis

Data were first checked for normality and homogeneity of variances. Differences in agecorrected relative telomere length (RTLc; residuals of RTL against age) between habitats and sampling sites were tested using Student's t-test and analysis of variances (ANOVA), respectively. Prior to analyses, all contaminant concentrations were log-transformed to attain a normal distribution and relationships between contaminant families were assessed using a Pearson's correlation matrix (Appendix D, table S4). To test whether relative telomere length (RTL) was affected by organic pollutant burdens, linear mixed model (LMMs) analyses were conducted using the lme4 and lmerTest packages [START_REF] Pinheiro | Mixed-Effects Models in S and S-PLUS (1st ed)[END_REF][START_REF] Kuznetsova | lmerTest Package: Tests in Linear Mixed Effects Models[END_REF] Indeed, fish from different sampling sites were considered distinct populations given the relatively short-range movements of chubs [START_REF] Penczak | Movement pattern and growth ratio of tagged fish in two lowland rivers of central Poland[END_REF]. Chemicals being highly correlated with each other and being characterized by different chemical properties, we used the sum of each family of organic contaminants to take into account potential mixture effects. Denominator degrees of freedom for fixed effects were calculated using the Satterthwaite approximation.

The significance of random effect was assessed using likelihood ratio tests (Appendix D, tables S5 and S6). We performed diagnostic plots and Shapiro normality tests on residuals to check model assumptions. A significance level of α < 0.05 was used for all tests.

Results

RTL was ~ 9.82% longer in fish near agricultural areas than those closest to Paris, at proximity to urban habitats. Age-corrected relative telomere length (RTLc) of chub significantly differed between habitats (t-test: t = 2.82, p = 0.006, figure 1A) and among sampling sites (ANOVA: F 5,88 = 4.34, p = 0.001; Appendix D, figure S4B). Organic pollutant mean levels ± standard deviation in chub tissues are listed in table S3 (Supplementary material, Appendix B). Fish from urban habitats had higher levels of OCPs (p < 0.001), phthalates (p = 0.045) and pyrethroid pesticides (p = 0.010) relative to agricultural areas, representing a contamination increase of 48.6%, 20.8% and 15.4%, respectively, but not for PAHs, PBDEs, PCBs and ∑metabolites (all p ≥ 0.126).

Age-corrected telomere length (RTLc) significantly decreased with increasing levels of pollutants, especially for pollutants' by-products (∑ Phthalate metabolites, figure 1B; LMM: F 1,49.9 = 5.57, p = 0.022; ∑metabolites of phthalates, LMM: F 1,71.0 = 9.34, p = 0.003 ).

The other chemical families did not show any significant relationships with RTLc (all F ≤ 2.86, p ≥ 0.101; Appendix E, tables S5 and S6).

Discussion

As previously found in birds [START_REF] Penczak | Movement pattern and growth ratio of tagged fish in two lowland rivers of central Poland[END_REF][START_REF] Meillere | Traffic noise exposure affects telomere length in nestling house sparrows[END_REF][START_REF] Salmón | Urban environment shortens telomere length in nestling great tits, Parus major[END_REF][START_REF] Ibáñez-Álamo | Urban blackbirds have shorter telomeres[END_REF], telomeres were shorter in urban habitats compared to agricultural ones, suggesting higher life-threatening situations for fish in urban rivers. In fact, fish from urban and agricultural rivers did not differ in their pollutant load, except for slightly higher plasticizer and pesticide levels in urban watercourses. Urban river systems have however undergone profound changes, such as damming, banking and channelization that have led to the disruption of longitudinal connectivity, loss of wetlands and spawning grounds, but also increased water temperature, pathogens and boat noise [START_REF] Grunst | Proximity to roads, but not exposure to metal pollution, is associated with accelerated developmental telomere shortening in nestling great tits[END_REF][START_REF] Paul | Streams in the urban landscape[END_REF]. Our study suggest that the diverse and profound degradation of urban streams induce deleterious effects in fish by accelerating telomere attrition and probably jeopardizing their survival. Those results are in line with previous findings, stating that environmental stressors accelerate telomere shortening in avian and fish species [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF][START_REF] Jasinska | Assessment of biomarkers for contaminants of emerging concern on aquatic organisms downstream of a municipal wastewater discharge[END_REF][START_REF] Stauffer | Telomere length and antioxidant defense associate with parasite-induced retarded growth in wild brown trout[END_REF].

To the best of our knowledge this is the first evidence that exposure to organic pollutants negatively impacts telomere length in fish. In different species of birds, exposure to environmental contaminants (OCPs, perfluoroalkyl substances: PFAS and trace metals) was associated with a general reduction in telomere length [13,14-16, but see 17]. The originality of this study is to investigate currently-released pollutants and their metabolites in a common freshwater fish species. Among the wide range of analyzed contaminants, the levels of the sum of metabolites (phthalates, PAHs and pyrethroids) were more prone to explain differences in our data than parent pollutants, except for PCBs. In a previous study using the same data set, metabolites of organic pollutants were negatively correlated to antioxidant capacity and peroxidase activity in chub plasma [START_REF] Molbert | in press) Potential benefits of acanthocephalan parasites for chub hosts in polluted environments[END_REF]. Organic pollutants may therefore produce oxidative stress by disrupting the pro-oxidant/antioxidant balance, which is a potential pathway of telomere shortening [START_REF] Riechert | Does oxidative stress shorten telomeres in vivo? A review[END_REF]. Even though the underlying mechanisms are still poorly understood, we hypothesize that electrophilic intermediates generated through the metabolization of parent compounds (PAHs, phthalates and pyrethroid pesticides) could increase oxidative attacks by depleting or weakening defense mechanisms (i.e., antioxidants), ultimately shortening telomeres. Still, some caution is needed to interpret these findings as other factors may mask the effects of environmental contaminants when using a crosssectional approach. Further work is thus advised to support the correlation between levels of organic contaminants and age-corrected telomere length, by using an experimental approach and testing other tissues for telomere length (e.g., liver).

Our results reveal physiological costs to fish living in polluted urban habitats, which may ultimately jeopardize their survival. Moreover, they highlight the importance of considering metabolites of environmental pollutants to better assess the impacts of currentlyreleased chemicals on wildlife.

Ethics

  AGAGAGAGGGGGCTAGATGA-3' and RAG1-R: 5'-CCATGCTTCTCGCTGACAT-3'.

Fish sampling has been

  conducted according to relevant national and European guidelines (L436-9, EN14011). The authorization for the scientific fish capture was delivered by local administration authorities (Departmental Direction of Territories of Seine-et-Marne).

FIGURE CAPTIONS FIGURE 1 .

 CAPTIONS1 FIGURE CAPTIONS

Acknowledgements

The authors are very grateful to Annie Desportes for her technical assistance.

Funding

PhD funding was provided by the Sorbonne University Environmental Transition Institute (SU-ITE). The work described was financially supported by the Seine-Normandy Water Agency (AESN, research program n° 1059502), and by the CPER ECONAT.

Data accessibility

Data related to this article are available on Dryad Digital Repository. Dataset, https://doi.org/10.5061/dryad.bcc2fqz9d

Authors' contributions

A.G. conceived the idea, designed the methodology and acquired the funding. A.G. and


Thierry Bodineau

Isabelle Gallagher

Laure Saint-Raymond

Sergio Simonella

DYNAMICS OF DILUTE GASES AT EQUILIBRIUM: FROM THE ATOMISTIC DESCRIPTION TO FLUCTUATING HYDRODYNAMICS

.

This paper is dedicated to the memory of K. Gawedzki. It shows how noise in hydrodynamic models of perfect gases can emerge from a deterministic microscopic dynamics. It is reminiscent of the concept of spontaneous stochasticity introduced in [START_REF] Bernard | Slow modes in passive advection[END_REF] and formalized in [START_REF] Chaves | Lagrangian dispersion in Gaussian self-similar velocity ensembles[END_REF].

1. The different levels of modeling 1.1. The atomistic description. The microscopic model consists of identical hard spheres of unit mass and of diameter ε. The motion of N such hard spheres is ruled by a system of ordinary differential equations, which are set in (T d ×R d ) N where T d is the unit d-dimensional periodic box with d ≥ 3: writing x ε i ∈ T d for the position of the center of the particle labeled by i and v ε i ∈ R d for its velocity, one has (1.1)

dx ε i dt = v ε i , dv ε i dt = 0 as long as x ε i (t) -x ε j (t) > ε for 1 ≤ i ≠ j ≤ N ,
with specular reflection at collisions:

(1.2) (v ε i ) ′ ∶= v ε i - 1 ε 2 (v ε i -v ε j ) ⋅ (x ε i -x ε j ) (x ε i -x ε j ) v ε j ′ ∶= v ε j + 1 ε 2 (v ε i -v ε j ) ⋅ (x ε i -x ε j ) (x ε i -x ε j ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ if x ε i (t) -x ε j (t) = ε .
This flow does not cover all possible situations, as multiple collisions are excluded. But one can show (see [START_REF] Alexander | The infinite hard sphere system[END_REF]) that for almost every admissible initial configuration (x ε0 i , v ε0 i ) 1≤i≤N , there are neither multiple collisions, nor accumulations of collision times, so that the dynamics is globally well defined.

We will not be interested here in one specific realization of this deterministic dynamics, but rather in a statistical description. This is achieved by introducing a measure at time 0, on the phase space we now specify. The collections of N positions and velocities are denoted respectively by X N ∶= (x 1 , . . . , x N ) in T dN and V N ∶= (v 1 , . . . , v N ) in R dN , and we set Z N ∶= (X N , V N ), with Z N = (z 1 , . . . , z N ), z i = (x i , v i ). A set of N particles is characterized by a random variable Z ε0 N = (z ε0 1 , . . . , z ε0 N ) specifying the time-zero configuration in the phase space (1.3) D ε N ∶= Z N ∈ (T d × R d ) N ∀i ≠ j , x ix j > ε , and an evolution t → Z ε N (t) = z ε 1 (t), . . . , z ε N (t) , t > 0 according to the deterministic flow (1.1)-(1.2) (well defined with probability 1).

1

To avoid spurious correlations due to a given total number of particles, we actually consider a grand canonical state (as in [START_REF] King | BBGKY Hierarchy for Positive Potentials[END_REF][START_REF] Van Beijeren | Equilibrium time correlation functions in the low-density limit[END_REF]), set on the phase space

D ε ∶= ⋃ N ≥0 D ε N
(notice that D ε N = ∅ for N large). This means that the total number of particles is also a random variable, which we shall denote by N .

More precisely, at equilibrium the probability density of finding N particles at configuration Z N is given by

(1.4) 1 N ! W ε N (Z N ) ∶= 1 Z ε µ N ε N ! 1 D ε N (Z N ) M ⊗N (V N ) ,
for N = 0, 1, 2, . . .

for some (large) µ ε to be fixed below, with

M(v) ∶= 1 (2π) d 2 exp - v 2 2 , M ⊗N (V N ) = N i=1 M(v i ) ,
and the partition function is given by

Z ε ∶= 1 + N ≥1 µ N ε N ! T dN i≠j 1 x i -x j >ε dX N .
Here and below, 1 A will be the characteristic function of the set A. The probability of an event A with respect to the equilibrium measure (1.4) will be denoted P ε (A), and E ε will be the expected value. Definition (1.4) ensures that

µ -1 ε E ε (N ) → 1 as µ ε → ∞ with µ ε ε d ≪ 1.
1.2. The kinetic description. Let us define the empirical measure of the hard-sphere model

(1.5) π ε t ∶= 1 µ ε N i=1 δ z ε i (t) .
Under the invariant measure (1.4), it is not hard to see that if

µ ε ε d → 0 then π ε t concentrates on M: for any test function h ∶ T d × R d → R and any δ > 0, t ∈ R, (1.6) P ε π ε t (h) - T d ×R d M(v)h(z) > δ → µε→∞ 0 ,
which can be interpreted as a law of large numbers.

The fluctuations of the empirical density π ε t around its equilibrium value are described by the fluctuation field ζ ε t defined by

(1.7) ζ ε t (h) ∶= √ µ ε π ε t (h) -E ε π ε t (h) ,
for any test function h. Initially ζ ε 0 converges in law towards a Gaussian white noise ζ 0 with covariance

(1.8) E ζ 0 (h 1 ) ζ 0 (h 2 ) = h 1 (z) h 2 (z) M(v) dz .
As the measure is invariant, this covariance is constant in time. Let us define the mean free path

α ∶= (µ ε ε d-1 ) -1 ,
and assume that α -1 ≥ 1 is bounded or slowly diverging, corresponding to the low density scaling. In this scaling it has been proved in [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF][START_REF] Bodineau | Long-time derivation at equilibrium of the fluctuating Boltzmann equation[END_REF] that (ζ ε t ) [0,T ] converges in law for all times T to a weak solution of the fluctuating Boltzmann equation (1.9)

dζ t = -v ⋅ ∇ x - 1 α L ζ t dt + dη t ,
where the linearized collision operator is given by

(1.10) Lg ∶= R d ×S d-1 M(w) (v -v * ) ⋅ ω + g(v) + g(v * ) -g(v ′ ) -g(v ′ * ) dv * dω with notation (1.11) v ′ = v -((v -v * ) ⋅ ω) ω , v ′ * = v * + ((v -v * ) ⋅ ω
) ω for the precollisional velocities obtained upon scattering, and dη t (x, v) is a stationary Gaussian noise, explicitly characterized (see [START_REF] Spohn | Fluctuation theory for the Boltzmann equation[END_REF]). It has zero mean and covariance (1.12) [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] for the velocities obtained upon scattering. Note that this noise is white in time and space, but correlated in velocities.

E T 0 dt dzh (1) (z)η t (z) T 0 dt * dz * h (2) (z * )η t * (z * ) = 1 2α T 0 dt dµ(z, z * , ω)M(v) M(v * )∆h (1) ∆h (2) denoting dµ(z, z * , ω) ∶= δ x-x * (v -v * ) ⋅ ω + dω dv dv * dx and defining for any h ∆h(z, z * , ω) ∶= h(z ′ ) + h(z ′ * ) -h(z) -h(z * ) , where z ′ i ∶= (x i , v ′ i ) with notation (1.
1.3. The hydrodynamic description. It is by now classical (see [START_REF] Bardos | Fluid dynamic limits of kinetic equations. II: Convergence proofs for the Boltzmann equation[END_REF][START_REF] Golse | Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs[END_REF][START_REF] Bodineau | ¿From hard-sphere dynamics to the Stokes-Fourier equations: an L 2 analysis of the Boltzmann-Grad limit[END_REF] and references therein) that the solutions to the scaled linearized Boltzmann equation

(1.13) ∂ t g α + v ⋅ ∇ x g α + 1 α
Lg α = 0 , g α (0) = g 0 converge in the fast relaxation limit α → 0 towards the local thermodynamic equilibrium

g(t, x, v) = ρ(t, x) + u(t, x) ⋅ v + θ(t, x) v 2 -d 2
where ρ, u, θ satisfy the acoustic equations

(1.14) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t ρ + ∇ x ⋅ u = 0 ∂ t u + ∇ x (ρ + θ) = 0 ∂ t θ + 2 d ∇
x ⋅ u = 0 and the initial data is the projection of g 0 onto hydrodynamic modes

ρ t=0 (x) ∶= g 0 (x, v)M(v) dv , u t=0 (x) ∶= g 0 (x, v)vM(v) dv , θ t=0 (x) ∶= g 0 (x, v) v 2 d -1 M(v) dv .
In the linearized equation (1.13), the frequency of collisions 1 α has been tuned according to the hyperbolic scaling. The diffusive regime can then be found by rescaling time by a factor 1 α. In this way, one can also obtain the weak convergence (which actually filters out the fast oscillating acoustic waves)

(1.15) g α τ α , x, v ⇀ u(τ, x) ⋅ v + θ(τ, x) v 2 -(d + 2) 2
towards diffusive fluid models, namely the incompressible Stokes-Fourier equations

(1.16) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ τ u = ν∆ x u, ∇ x ⋅ u = 0 ∂ τ θ = κ∆ x θ ,
where the diffusion coefficients ν and κ depend only on the linearized collision operator L (they are defined explicitly in (3.20) below). The initial data is the projection of g 0 onto non oscillating hydrodynamic modes (1.17)

u τ =0 (x) ∶= P g 0 vM(v) dv , θ τ =0 (x) ∶= g 0 v 2 d + 2 -1 M(v)
dv
where P is the Leray projection on divergence free vector fields. In the following, we refer to non oscillating modes as those satisfying the incompressibility and Boussinesq constraints (see (3.16)).

1.4. Fluctuating hydrodynamics. In the hyperbolic regime corresponding to (1.14), the fluctuation-dissipation principle predicts that there will be no dynamical fluctuation and the fluctuation field tested against hydrodynamical modes (ρ, u, θ) is simply transported by the acoustic equation. In contrast, in the diffusive regime, when taking into account the noise at kinetic level (i.e. starting with (1.9)), we expect to obtain fluctuating hydrodynamics. In the following, we will focus on this more interesting case. We refer to [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF], Section 7.1 for the general theory of hydrodynamic fluctuations, which was first developed for equilibrium states in [START_REF] Landau | Fluid Mechanics[END_REF]. The link with the predictions from kinetic theory in the case of dilute gases was discussed in [START_REF] Kirkpatrick | Fluctuations in a nonequilibrium steady state: Basic equations[END_REF] (see also [3] for a recent contribution).

Let us define a joint process by time rescaling and projecting on non oscillating hydrodynamic modes the fluctuation field ζ ε t defined in (1.7). According to (1.15) we consider, for any pair of test functions (ϕ, ψ) ∈ C ∞ (T d ; R d × R) with ∇ x ⋅ ϕ = 0, the fluctuation field

ζ ε t ϕ ⋅ v + ζ ε t ψ v 2 d + 2 -1 .
To simplify the notation, we denote from now on the couple of test functions by

(1.18) φ = (ϕ, ψ) ∈ C ∞ (T d ; R d × R) , ∇ x ⋅ ϕ = 0
and to recover a diffusive regime, time is rescaled as follows:

(1.19)

ξ ε τ (φ) ∶= U ε τ (ϕ) + Θ ε τ (ψ) ∶= ζ ε τ α ϕ ⋅ v + ζ ε τ α ψ v 2 d + 2 -1 .
We stress the fact that in contrast with ζ ε , the test functions in ξ ε only depend on the space variable. In the limit µ ε → ∞ with α slowly vanishing, we expect the fluctuation fields U ε , Θ ε to converge in the sense of distributions to (U, Θ) solving the fluctuating Stokes-Fourier equations

(1.20) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂ τ U = ν∆ x U + √ 2ν P ∇ ⋅ Ẇt , ∂ τ Θ = κ∆ x Θ + 4κ d+2 ∇ ⋅ Ẇt ,
where W t is a space/time white noise taking values in R d and W t is a d × d matrix with coefficients given by independent white noises. We recall that P stands for the Leray projection on divergence free vector fields. Note that the noise is tuned so that the field has a covariance compatible with the invariance of (1.8). The equations (1.20) should be understood in a weak sense, namely restricting to any pair of test functions

(ϕ, ψ) ∈ C ∞ (T d ; R d × R) with ∇ x ⋅ ϕ = 0 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ U τ (ϕ) = U 0 (e ντ ∆x ϕ) + √ 2ν τ 0 dσ Ẇσ ∇e ν(τ -σ)∆x ϕ Θ τ (ψ) = Θ 0 e κτ ∆x ψ + 4κ d+2 τ 0
dσ Ẇσ e κ(τ -σ)∆x ∇ψ .

We stress that the fluctuations in (1.20) exactly compensate the dissipation according to the fluctuation-dissipation principle. In particular, both Gaussian processes are characterized by their covariances for σ ≤ t

(1.21) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ E U σ (ϕ 1 )U τ (ϕ 2 ) = T d dx ϕ 1 (x) ⋅ e ν(τ -σ)∆x ϕ 2 (x) E Θ σ (ψ 1 )Θ τ (ψ 2 ) = 2 d + 2 T d dx ψ 1 (x) e κ(τ -σ)∆x ψ 2 (x) .
The main result of this paper is that both limits

µ ε → ∞ with µ ε ε d-1 = α -1
, and α → 0 can be combined in order to derive fluctuating hydrodynamics directly from the dynamics of particles, thus solving Hilbert's sixth problem in the particular case of fluctuations of perfect gases at equilibrium. Theorem 1.1. Consider a system of hard spheres at equilibrium in a d-dimensional periodic box with d ≥ 3, with inverse mean free time α [START_REF] Golse | Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs[END_REF]) converges for any T > 0 in law to the solution of the fluctuating Stokes-Fourier equations (1.20).

-1 ∶= µ ε ε d-1 ≤ (log log log µ ε ). Then, in the diffusive limit µ ε → ∞, α → 0, the rescaled joint process (ξ ε τ ) τ ∈[0,T ] defined in (1.
Recall that the microscopic dynamics is completely deterministic, so that stochasticity comes just as a consequence of the sensitivity of the particle system to the microscopic details of the initial configuration. In the low density regime, since the modulus of continuity of trajectories with respect to the initial configurations depends strongly on ε, there is a strong instability as ε → 0, which generates some "spontaneous stochasticity" encoded by the white noise in (1.9). The instability of the microscopic dynamics thus plays a key role in the structure of the noise in Theorem 1.1. At variance, for one dimensional integrable systems, one expects that the dominant contribution is the transport of the initial fluctuations with some additional random shift in the large scale limit, this was pointed out recently in [START_REF] Ferrari | Diffusive Fluctuations in Hard Rods System, Extended abstract of the workshop 'Large Stochastic Dynamics[END_REF] for the the hard rod system (see also [START_REF] Boldrighini | Fluctuations in a One-Dimensional Mechanical System. I. The Euler Limit[END_REF]). The white noise in (1.9) preserves locally the hydrodynamic modes, however at diffusive time scales, it ultimately induces the local noise on the hydrodynamic projections (1.20). Note that a spontaneous generation of noise also holds for the diffusive limits of a tagged particle to a Brownian motion in an equilibrium hard sphere gas [START_REF] Bodineau | The Brownian motion as the limit of a deterministic system of hard-spheres[END_REF][START_REF] Bodineau | Derivation of an Ornstein-Uhlenbeck process for a massive particle in a rarified gas of particles[END_REF] (see also [START_REF] Erdős | Lecture notes on quantum Brownian motion[END_REF] in the quantum case).

The fluctuation field in the low density limit: state of the art, and strategy of proof

The present paper relies on the "weak convergence" approach devised in [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF][START_REF] Bodineau | Long-time derivation at equilibrium of the fluctuating Boltzmann equation[END_REF] in order to prove the convergence of the fluctuation field to the solution of the fluctuating Boltzmann equation (1.9). The proofs of [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF][START_REF] Bodineau | Long-time derivation at equilibrium of the fluctuating Boltzmann equation[END_REF] are quantitative, and the important parameter is the number of collisions, which is proportional to the observation time and inversely proportional to the mean free time α. Thus, the (diffusive) observation time T α and the parameter α -1 can be chosen slowly diverging with µ ε , for instance as O(log log log µ ε ). This will allow us to reach the diffusive regime described in Section 1.4. In the rest of this section, we gather the results of [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF][START_REF] Bodineau | Long-time derivation at equilibrium of the fluctuating Boltzmann equation[END_REF] we shall be using here. We refer to those papers for proofs -see also [START_REF] Bodineau | Dynamics of dilute gases : a statistical approach[END_REF] for an overview.

For the sake of clarity, we will use the following notations for the different time scales described in the previous section : 2.1. Convergence of the covariance for diffusive times. In the analysis of the fluctuation field for diffusive times, the first step is to study the asymptotic behaviour of the time-rescaled covariance

(2.2) Cov ε τ α , g 0 , h ∶= E ε ζ ε 0 (g 0 ) ζ ε τ (h) as µ ε → ∞, µ ε ε d-1 = α -1 .
ε → ∞, µ ε ε d-1 = α -1 ≤ log log log µ ε , the covariance of the fluctuation field ζ ε τ α τ ≥0
defined by (2.2) satisfies the following estimate:

for any T > 0 such that (T α 2 ) ≪ (log log µ ε ) 1 4 , (2.4) sup τ ∈[0,T ] Cov ε τ α , g 0 , h - g α ( τ α )h Mdxdv ≤ C h W 1,∞ g 0 W 1,∞ CT α 2 3 2 (log log µ ε ) -1 4 .
Remark 2.1. In accordance with the diffusive scaling, this estimate depends on T α 2 , which is the ratio between the observation time T α and the mean free time α.

2.2. Convergence of higher order moments for diffusive times. The next step is to prove that the process ζ ε τ α is asymptotically Gaussian when µ ε → ∞ and µ ε ε d-1 = α -1 → ∞. This boils down to showing that the moments are determined by the covariances according to Wick's rule (2.5) lim

µε→∞ α→0 E ε ζ ε τ 1 α (h 1 ) . . . ζ ε τp α (h p ) - η∈S pairs p {i,j}∈η E ε ζ ε τ i α (h i ) ζ ε τ j α (h j ) = 0 , uniformly in τ 1 , . . . , τ p ∈ [0, T ],
where S pairs p is the set of partitions of {1, . . . , p} made only of pairs. Notice that if p is odd then S pairs p is empty and the product of the moments is asymptotically 0. Theorem 2.2 ([13], Gaussian process). Consider a system of hard spheres at equilibrium in a d-dimensional periodic box with d ≥ 3. Let (h i ) 1≤i≤p be a family of p bounded functions on

T d × R d . Then, in the low density regime µ ε → ∞, µ ε ε d-1 = α -1 ≤ log log log µ ε , the fluctuation field ζ ε τ α τ ≥0
defined by (1.7) is almost Gaussian in the sense that for any T > 0

satisfying (T α 2 ) 2p-1 2 ≪ (log log µ ε ) 1 4
, there holds uniformly in τ 1 , . . . τ p ∈ [0, T ],

(2.6)

E ε ζ ε τ 1 α (h 1 ) . . . ζ ε τp α (h p ) - η∈S pairs p {i,j}∈η E ε ζ ε τ i α (h i ) ζ ε τ j α (h j ) ≤ p i=1 h i L ∞ CT α 2 (2p-1) 2 (log log µ ε ) -1 4 .
2.3. Tightness in the kinetic regime. Finally for processes which depend on a continuous variable (the time variable in our setting), the convergence of time marginals is not enough to characterize the convergence in law: possible oscillations with respect to time need to be under control (see [6, Theorem 13.2 page 139]). For the fluctuation field ζ ε , this tightness property has been obtained for short kinetic times, but actually since the equilibrium measure is invariant under the dynamics, a union bound provides the tightness on any finite kinetic time, i.e. times of order O(α).

For times much longer than kinetic times, we actually do not expect the process ζ ε t to be tight. Since the covariance Cov ε t, g 0 , h is close to the solution of the scaled linearized Boltzmann equation (1.13), we expect to see a fast relaxation process with rate O( 1 α ), meaning that only the hydrodynamic part of g α can be compact for t = O(1). Going to diffusive times t = τ α, we also expect to have acoustic waves producing fast oscillations, meaning that only the non oscillating hydrodynamic part of g α can be compact for τ = O(1). Nevertheless, after projecting on the non oscillating modes, we are going to show in Section 4 that the process (ξ ε τ ) τ ≥0 defined by (1. [START_REF] Golse | Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs[END_REF]) is tight on the diffusive scale. 2.4. Strategy of the proof of Theorem 1.1. In view of deriving fluctuating hydrodynamic equations and proving Theorem 1.1, the strategy is now straightforward: we consider the rescaled fluctuation field ξ ε τ projected on hydrodynamic, non oscillating modes (recall (1.19)), and check that with such test functions and this scaling in time, Gaussianity (Theorem 2.2) and tightness still hold, and that the covariance asymptotically converges to the solution to the Stokes-Fourier equation. Note that the projection (1.19) leads to considering test functions which are unbounded in v and therefore there are some technical issues when applying Theorems 2.1 and 2.2. These are dealt with in Section 3.2, thanks to a cut-off in energies introduced in Section 3.1. The tightness of the process on the diffusive time-scale is derived in Section 4.

Finite time marginals

In this section, we are going to characterize the limiting law of the process by proving the following result. We set from now on α -1 = log log log µ ε . Proposition 3.1. For arbitrary times τ 1 , . . . , τ L and test functions φ (1) = (ϕ (1) , ψ (1) ), . . . and φ (L) = (ϕ (L) , ψ (L) ) chosen as in (1.18), the time marginals ξ ε τ ℓ φ (ℓ) ℓ≤L converge in law to the limiting process U τ ℓ (ϕ (ℓ) ), Θ τ ℓ (ψ (ℓ) ) ℓ≤L as µ ε tends to infinity.

3.1.

Truncated hydrodynamic fields. To prove that the limit is Gaussian, Theorem 2.2 cannot be used directly with the process (ξ ε τ ) t≥0 as the test functions are unbounded in L ∞ due to divergences in the velocities. Thus an intermediate cut-off process needs to be introduced. Let us fix an energy cut-off R ≫ 1 to be determined (see (3.2) below). Recalling (1.19), we define the modified joint process ξε τ as follows. For any test function φ as in (1.18), we set

(3.1) ξε τ (φ) ∶= ζ ε τ α χ v 2 R ϕ ⋅ v + ζ ε τ α χ v 2 R ψ v 2 d + 2 -1 ,
where χ is a smooth cut-off function with compact support

χ [0,1] ≡ 1, χ [2,+∞[ ≡ 0 .
We choose R depending on ε and converging to ∞ as µ ε → ∞ as follows

(3.2) R = α -1 = log log log µ ε .
Note that the test functions

h ∶= ϕ ⋅ v + ψ v 2 d + 2 -1 χ v 2
R are smooth and bounded thanks to the cut-off in v:

(3.3) h W 1,∞ x,v ≤ CR 2 ( ϕ W 1,∞ x + ψ W 1,∞ x ) .
The process ξε τ is a good approximation of ξ ε τ when R → ∞. Lemma 3.2. Setting ξ ε,> τ ∶= ξ ε τξε τ then for all 1 ≤ q < ∞ and for ε small enough

(3.4) E ε (ξ ε,> τ (φ)) q ≤ C q φ q L q (T d ) e -R 4 , with L q M defined as in (2.
3) Furthermore, one has also

(3.5) E ε [(ξ ε τ (φ)) q ] ≤ C q φ q L q (T d ) and E ε ξε τ (φ) q ≤ C q φ q L q (T d ) .
As a consequence, the convergence in law of ξε τ ℓ φ (ℓ) ℓ≤L (derived in Proposition 3.3 below) will imply the convergence in law of ξ ε τ ℓ φ (ℓ) ℓ≤L , i.e. Proposition 3.1. Proof of Lemma 3.2. Recall (see Proposition A.1 in [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF]) that for any ε small enough, the following holds under the equilibrium measure for any function h

(3.6) E ε ξ ε τ (h) q ≤ C q h q L q M , with 1 ≤ q < ∞. Since for R ≥ 1 (3.7) ϕ ⋅ v χ v 2 R -1 q L q M ≤ C ϕ q L q (T d ) e -R 4 ψ v 2 d + 2 -1 χ v 2 R -1 q L q M ≤ C ψ q L q (T d ) e -R 4 ,
we find (3.4). For the same reason (3.5) holds. This completes Lemma 3.2.

3.2.

Covariance of the hydrodynamic fields.

Proposition 3.3. For arbitrary times τ 1 , . . . , τ L and test functions φ (1) = (ϕ (1) , ψ (1) ), . . . and φ (L) = (ϕ (L) , ψ (L) ) chosen as in (1.18), the time marginals ξε τ ℓ φ (ℓ) ℓ≤L converge in law to the limiting process U τ ℓ (ϕ (ℓ) ), Θ τ ℓ (ψ (ℓ) ) ℓ≤L as µ ε tends to infinity.

Combined with the approximation Lemma 3.2, this completes the proof of Proposition 3.1. The proof of Proposition 3.3 is split into two parts, first a control of the limiting covariance and then the derivation of Wick's rule to prove that the limiting process is Gaussian.

Step 1. Control of the covariance. Let us define the hydrodynamic, non oscillating projections

(3.8) g 0 (x, v) ∶= u 0 (x) ⋅ v + θ 0 (x) v 2 -(d + 2) 2 , h(x, v) ∶= ϕ(x) ⋅ v + ψ(x) v 2 d + 2 -1 ,
for some smooth divergence free vector fields u 0 , ϕ, and some smooth functions θ 0 , ψ. The scaling in g 0 , h has been tuned asymmetrically so that the initial covariance is given by

E ε ξε 0 (φ 0 ) ξε 0 (φ) → (u(τ ) ⋅ ϕ + θ(τ )ψ)dx , µ ε → ∞ .
We are going to study the covariance of the joint process ξε τ by applying Theorem 2.1 with

(3.9) ḡ0 (x, v) ∶= g 0 (x, v)χ v 2 R , h(x, v) = h(x, v)χ v 2 R .
Setting

φ 0 ∶= (u 0 , d + 2 2 θ 0 ) , φ ∶= (ϕ, ψ) ,
we plug the bounds (3.3) on the test functions into the estimate (2.4) of Theorem 2.1, and recalling the definition (3.1) of the truncated rescaled fluctuation field, we obtain that for any

T > 0 such that (T α 2 ) ≪ (log log µ ε ) -1 2 , (3.10) sup t∈[0,T ] E ε ξε 0 (φ 0 ) ξε τ (φ) - Mg α (t) hdxdv ≤ CR φ 0 W 1,∞ φ W 1,∞ CT α 2 3 2 (log log µ ε ) -1 4 ,
where gα is the solution to the time-rescaled equation

(3.11) α∂ τ gα + v ⋅ ∇ x gα + 1 α Lg α = 0 , gα t=0 = ḡ0 .
To conclude to the convergence of the covariance as α → 0, we just need to identify the limit of Mg α (τ ) hdxdv.

The starting point for the study of hydrodynamic limits of the linearized Boltzmann equation (3.11) is the scaled energy inequality

(3.12) 1 2 gα (τ ) 2 L 2 (Mdvdx) + 1 α 2 τ 0 gα Lg α (τ ′ )Mdvdxdτ ′ ≤ 1 2 ḡ0 2 L 2 (Mdvdx)
. Recall (see [START_REF] Grad | Asymptotic theory of the Boltzmann equation II Rarefied Gas Dynamics[END_REF][START_REF] Hilbert | Begründung der kinetischen Gastheorie[END_REF]) that the linearized collision operator L with hard sphere cross section defined by (1.10) is a nonnegative unbounded self-adjoint operator on L 2 (Mdv) with domain

D(L) = L 2 R d ; (1 + v )Mdv and nullspace Ker(L) = span 1, v 1 , . . . , v d , v 2 .
In particular we recover from (3.12) the uniform

L 2 bound gα (τ ) L 2 (Mdvdx) ≤ ḡ0 L 2 (Mdvdx) ≤ g 0 L 2 (Mdvdx) . This bound implies that there is g ∈ L ∞ τ (L 2 (Mdvdx)
) such that, up to extraction of a subsequence, (3.13) gα ⇀ g weakly in L 2 loc (dτ, L 2 (Mdvdx)) . Moreover the following coercivity estimate holds : there exists C > 0 such that, for each g in D(L) ∩ (Ker(L))

⊥ (3.14) gLg(v)M(v)dv ≥ C g 2 L 2 ((1+ v )Mdv) . The dissipation thus further provides gα -Πg α L 2 ((1+ v )Mdvdxdt) = O(α) ,
where Π denotes the orthogonal projection onto Ker(L) in L 2 (Mdvdx). We deduce from the previous estimate that

(3.15) g(τ, x, v) = Πg(τ, x, v) ≡ ρ(τ, x) + u(τ, x) ⋅ v + θ(τ, x) v 2 -d 2 ⋅
It remains to compute the equations on ρ, u and θ. Denoting ⟨g⟩ ∶= gMdv and recalling (3.11), the moment equations state

α∂ τ ⟨g α ⟩ + ∇ x ⋅ ⟨g α v⟩ = 0 , α∂ τ ⟨g α v⟩ + ∇ x ⋅ ⟨g α v ⊗ v⟩ = 0 , α∂ τ ⟨g α v 2 ⟩ + ∇ x ⋅ ⟨g α v v 2 ⟩ = 0 .
Using (3.13) and (3.15) we deduce from the first two equations that (3.16)

∇ x ⋅ u = 0 , ∇ x (ρ + θ) = 0 ,
referred to as the incompressibility and Boussinesq constraints. We thus have

(3.17) g(τ, x, v) = u(τ, x) ⋅ v + θ(τ, x) v 2 -(d + 2) 2 , ∇ x ⋅ u = 0 .
Note that, up to the cut-off in v which can be removed with a small error thanks to (3.7), the test function h is in the kernel of acoustic operator. It follows that we only need to characterize the mean motion, namely derive the equations for P ⟨g α v⟩ and ⟨g α ( v 2d -2)⟩:

∂ τ P ⟨g α v⟩ + 1 α P ∇ x ⋅ ⟨g α (v ⊗ v - 1 d v 2 Id)⟩ = 0 , ∂ τ 1 d + 2 ⟨g α ( v 2 -d -2)⟩ + 1 α ∇ x ⋅ ⟨g α 1 d + 2 v( v 2 -d -2)⟩ = 0 ,
where we recall that P is the Leray projection on divergence free vector fields. Define the kinetic momentum flux A(v) ∶= v⊗v-

1 d v 2
Id and the kinetic energy flux

B(v) ∶= 1 2 v( v 2 -d-2).
As A, B belong to (Ker L) ⊥ , and L is a Fredholm operator, there exist pseudo-inverses Ã, B in (Ker L) ⊥ such that A = L Ã and B = L B. Then,

∂ τ P ⟨g α v⟩ + 1 α P ∇ x ⋅ ⟨(Lg α ) Ã⟩ = 0 , 1 d + 2 ∂ τ ⟨g α ( v 2 -d -2)⟩ + 1 α 2 d + 2 ∇ x ⋅ ⟨(Lg α ) B⟩ = 0 .
Using the equation

(3.18) 1 α Lg α = -v ⋅ ∇ x gα -α∂ τ gα we get (3.19) ∂ τ P ⟨g α v⟩ -P ∇ x ⋅ ⟨(v ⋅ ∇ x + α∂ τ )g α Ã⟩ = 0 , 1 d + 2 ∂ τ ⟨g α ( v 2 -d -2)⟩ - 2 d + 2 ∇ x ⋅ ⟨(v ⋅ ∇ x + α∂ τ )g α B⟩ = 0 .
Then, plugging the Ansatz (3.15), and taking limits in the sense of distributions, we get the Stokes-Fourier equations

∂ τ u -ν∆ x u = 0 , ∇ x ⋅ u = 0 , ∂ τ θ -κ∆ x θ = 0 ,
with initial data as in (1.17)

u τ =0 (x) ∶= P g 0 (x, v)vM(v) dv , θ τ =0 (x) ∶= g 0 (x, v) v 2 (d + 2) -1 M(v) dv,
and where the diffusion coefficients are given by

(3.20) ν ∶= 1 (d -1)(d + 2) ⟨A ∶ Ã⟩ and κ ∶= 2 d(d + 2) ⟨B ⋅ B⟩ .
We therefore end up with the following convergence as α → 0

(3.21) Mg α (τ ) hdxdv → (u(τ ) ⋅ ϕ + θ(τ )ψ)dx .
Returning to (3.10), we have proved that

(3.22) sup τ ∈[0,T ] E ε ξε 0 (φ 0 ) ξε τ (φ) → (u(τ ) ⋅ ϕ + θ(τ )ψ)dx , µ ε → ∞ .
Remark 3.4. Since the initial data g 0 is well-prepared, both the purely kinetic component and the fast oscillating acoustic waves are negligible, so the convergence of gα can be shown actually to hold in strong sense. Using energy methods, it is even possible to obtain a rate of convergence for (3.21).

Step 2. Wick's rule Consider p times τ 1 , . . . , τ p , possibly repeated. Thanks to the cut-off (3.3), we can apply Theorem 2.2 to obtain (3.23)

E ε ξε τ 1 φ (1) . . . ξε τp φ (p) - η∈S pairs p {i,j}∈η E ε ξε τ i φ (i) ξε τ j φ (j) ≤ C p R p p i=1 φ (i) L ∞ CT α 2 (2p-1) 2 (log log µ ε ) -1 4 .
With the scaling condition (3.2), we get that the right-hand side converges to 0 as µ ε → ∞ which implies the asymptotic pairing of the moments of ξε τ . Since the limiting covariance is characterized by (3.22), this completes Proposition 3.3.

Tightness of hydrodynamic fields on diffusive time scales

Let us first introduce for any k ∈ Z the Sobolev space H k in T d with the norm (4.1)

F 2 k ∶= j∈Z d 1 + j 2 k Fj 2 ,
where ( Fj ) stand for the Fourier coefficients of F .

Proposition 4.1. There exists k > 0 such that, in the diffusive limit

µ ε → ∞, α → 0, with µ ε ε d-1 = α -1 ≤ log log log µ ε , the fluctuation field (ξ ε τ ) τ ≥0 defined by (1.19) is tight in the Skorokhod space D [0, T ], H -k . More precisely, (4.2)
lim δ→0 + lim µε→∞ P ε sup σ-τ ≤δ s,τ ∈[0,T ] ξ ε τ -ξ ε σ -k ≥ δ ′ = 0 , ∀δ ′ > 0 , lim A→∞ lim µε→∞ P ε sup τ ∈[0,T ] ξ ε τ -k ≥ A = 0 .
The tightness property for kinetic times relies on the Garsia-Rodemich-Rumsey inequality on the modulus of continuity of a function ϕ τ ∶ [0, T ] → R, which we recall ( [START_REF] Varadhan | Stochastic processes[END_REF]): for b ≥ 4

(4.3) sup 0≤σ,τ ≤T τ -σ ≤δ ϕ τ -ϕ σ ≤ C T 0 T 0 dσdτ ϕ τ -ϕ σ b τ -σ γ 1 b δ γ-2 b , γ ∈]2, 3[ .
Because of collisions in the Newtonian dynamics, the fluctuation field ξ ε has jumps and this inequality does not apply directly. We therefore start by stating a modified inequality, whose proof is a slight adaptation of [START_REF] Varadhan | Stochastic processes[END_REF] which can be found in [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] (see Proposition 6.2.4).

Proposition 4.2. Let F ∶ [0, T ] → R be a given function and define for a > 0, b ≥ 4

(4.4) B a (F ) ∶= T 0 T 0 dσdτ F τ -F σ b τ -σ γ 1 τ -σ >a , γ ∈]2, 3[ .
Then the modulus of continuity of F is controlled by

(4.5) sup 0≤σ,τ ≤T τ -σ ≤δ F τ -F σ ≤ 2 sup 0≤σ,τ ≤T τ -σ ≤2a F τ -F σ + CB a (F ) 1 b δ γ-2 b .
Proof of Proposition 4.1. To prove the tightness of the joint process (ξ ε τ ) τ ≥0 in D([0, T ], H -k ) for some k large enough, we shall tune the parameter a, introduced in the statement of Proposition 4.2, as a small fraction of the kinetic time, i.e. a ≪ α 2 in the diffusive scaling. More precisely, we shall use (4.4) with the parameters

(4.6) b = 6 , γ = 7 3 , a = (log log µ ε ) -1 10 , α = (log log log µ ε ) -1 .
We deduce from (4.5) that, for arbitrary δ ′ > 0,

P ε ⎛ ⎜ ⎝ sup 0≤τ,σ≤T τ -σ ≤δ ξ ε τ -ξ ε σ 2 -k ≥ δ ′ ⎞ ⎟ ⎠ ≤P ε ⎛ ⎜ ⎝ j C 2 B a ξ ε (φ j ) 1 3
(1

+ j 2 ) k δ γ-2 3 ≥ δ ′ 4 ⎞ ⎟ ⎠ (4.7) + P ε ⎛ ⎜ ⎝ j 4 (1 + j 2 ) k sup σ-τ ≤2a σ,τ ∈[0,T ] ξ ε τ (φ j ) -ξ ε σ (φ j ) 2 ≥ δ ′ 4 ⎞ ⎟ ⎠ ,
where φ j (x) = exp(2iπj ⋅ x) are the Fourier modes used to define the norm (4.1). Since a ≪ α 2 , the two events in the right-hand side of inequality (4.7) control different time scales and their probabilities have to be estimated by different methods :

• for time increments στ ≥ a, by a control on moments using the comparison with the limit process; • for small time increments στ ≤ 2a, by reducing to the estimates on the kinetic times obtained in [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] (see Proposition 6.2.3). To do this, additional cut-off estimates to control divergences at large velocities are necessary.

Step 1. Control of the short hydrodynamic increments.

We are first going to prove that lim

δ→0 lim µε→∞ P ε ⎛ ⎜ ⎝ j CB a ξ ε (φ j ) 1 3
(1

+ j 2 ) k δ γ-2 3 ≥ δ ′ 4 ⎞ ⎟ ⎠ = 0 . (4.8)
Assume that the following bound holds (4.9)

E ε B a ξ ε (φ) ≤ C φ 6 W 2,∞ .
Since for the Fourier basis φ j W 2,∞ ≤ C j 2 , we deduce from (4.9) that for k > d 2 + 2, (4.8) follows from a Markov inequality as γ > 2

P ε ⎛ ⎜ ⎝ j C 2 B a ξ ε (φ j ) 1 3 (1 + j 2 ) k δ γ-2 3 ≥ δ ′ 4 ⎞ ⎟ ⎠ ≤ C δ γ-2 3 δ ′ j 1 (1 + j 2 ) k E ε B a ξ ε (φ j ) 1 3 .
We turn now to the proof of (4.9). As γ = 7 3, this will be a consequence of the following inequality

(4.10) ∀τ, σ ∈ [0, T ] , E ε ξ ε τ (φ) -ξ ε σ (φ) 6 1 τ -σ ≥a ≤ C φ 6 W 2,∞ τ -σ 3 2 .
Applying Lemma 3.2, it is enough to derive (4.10) for the truncated process ξε with cut-off

R = log log log µ ε because ∀τ ≤ T, E ε ξ ε τ (φ) -ξε τ (φ) 6 ≤ C φ 6 L 6 (T d ) e -R 4 ≤ C φ 6 L 6 (T d ) a 2
, with a defined in (4.6).

Our starting point is the asymptotic factorization (3.23) of the moments leading to the following formula for the time increments (4.11)

E ε ξε τ (φ) -ξε σ (φ) 6 -15 E ε ξε τ (φ) -ξε σ (φ) 2 3 ≤ C 6 R 6 φ 6 L ∞ CT α 2 11 2 (log log µ ε ) -1 4 ≤ C φ 6 L 6 (T d ) a 2 , uniformly in τ, σ ∈ [0, T ],
with our choice of scaling (3.2).

Next we are going to use that, by (3.10), the covariance is well approximated by the solution to the linearized Boltzmann equation (3.11). Denoting by gα the solution of the linearized Boltzmann equation (3.11) with truncated initial data (3.9), we get that (4.12)

sup σ,τ ∈[0,T ] E ε ξε τ (φ) -ξε σ (φ) 2 -2 M(ḡ 0 -gα (τ -σ)) g 0 dxdv ≤ CR 2 φ 2 W 1,∞ CT 3 α 6 1 2 (log log µ ε ) -1 4 + C φ 2 L 2 e -R 4 ≤ C φ 2 W 1,∞ a 2 ,
using the time invariance of the equilibrium measure and the control (3.7) to remove the velocity cutoff on (one of) the initial data ḡ0 in the integral. ¿From (3.19) we have

∂ τ P ⟨g α v⟩ -αP ∇ x ⋅ ⟨g α Ã⟩ -P ∇ x ⋅ ⟨v ⋅ ∇ x (g α Ã)⟩ = 0 , 1 d + 2 ∂ τ ⟨g α ( v 2 -d -2)⟩ -2α∇ x ⋅ ⟨g α B⟩ - 2 d + 2 ∇ x ⋅ ⟨v ⋅ ∇ x (g α B)⟩ = 0, so thanks to the uniform L ∞ τ (L 2 (Mdxdv)
) bound on gα we deduce that (4.13)

P ⟨g α v⟩ -αP ∇ x ⋅ ⟨g α Ã⟩ is uniformly bounded in W 1,∞ τ (H -2 ), ⟨g α v 2 -(d + 2) 2 ⟩ -α∇ x ⋅ ⟨g α B⟩ is uniformly bounded in W 1,∞ τ (H -2 ) .
We then have to control the time regularity of the O(α) terms in (4.13). ¿From the uniform L ∞ τ (L 2 (Mdxdv)) bound on gα , we get that for any polynomial p(v) depending only on v (4. [START_REF] Bodineau | Dynamics of dilute gases : a statistical approach[END_REF] ∀τ ∈ [0, T ],

∇ x ⟨g α p(v)⟩ H -1 ≤ C.
Applying the kinetic equation (3.11), we know that

∂ τ ⟨g α p(v)⟩+ 1 α ∇ x ⋅⟨g α p(v)v⟩+ 1 α 2 ⟨Lg α p(v)⟩ = 0 ⇒ ⟨g α (τ )p(v)⟩ -⟨g α (σ)p(v)⟩ H -1 ≤ C τ -σ α 2 .
Replacing p by Ã, B in the previous estimates, we conclude that

αP ∇ x ⋅ ⟨g α (τ ) Ã⟩ -αP ∇ x ⋅ ⟨g α (σ) Ã⟩ H -2 ≤ C min α, τ -σ α ≤ C τ -σ 1 2 , α∇ x ⋅ ⟨g α (τ ) B⟩ -α∇ x ⋅ ⟨g α (σ) B⟩ H -2 ≤ C min α, τ -σ α ≤ C τ -σ 1 2 .
Therefore, applying (4.13), we deduce that the bulk velocity P ⟨g α v⟩ and temperature ⟨g α

v 2 -(d+2) d+2 ⟩ are uniformly bounded in C 1 2 τ (H - 2
x ). Since the initial data g 0 is well prepared (see (3.8)), we deduce that the term involving the linearized equation in (4.12) is controlled by

M(ḡ 0 -gα (τ -σ)) g 0 dxdv ≤ P ⟨g α (τ -σ)v⟩ -P ⟨ḡ 0 v⟩ ⋅ u 0 dx + d + 2 2 ⟨g α (τ -σ) v 2 -(d + 2) 2 ⟩ -⟨ḡ 0 v 2 -(d + 2) 2 ⟩ θ 0 dx ≤ C φ 2 W 2,∞ τ -σ 1 2 .
Combining (4.11)-(4.12) and the time regularity of the covariance, we get that for τ -σ ≥ a

E ε ξ ε τ (φ) -ξ ε σ (φ) 6 1 τ -σ ≥a ≤ C φ 6 W 2,∞ τ -σ 3 2 .
This completes the proof of Inequality (4.10).

Step 2. Control of the very short kinetic times. Finally, it remains to control the second term in (4.7). By splitting the time interval [0, T ] into intervals with kinetic time length scale α 2 , the estimate can be reduced, by using the invariant measure and an union bound, to

P ε ⎛ ⎜ ⎝ j 4 (1 + j 2 ) k sup σ-τ ≤2a σ,τ ∈[0,T ] ξ ε τ (φ j ) -ξ ε σ (φ j ) 2 ≥ δ ′ 4 ⎞ ⎟ ⎠ ≤ T α 2 P ε (A) , (4.15)
with the notation

A ∶= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ j 4 (1 + j 2 ) k sup σ-τ ≤2a σ,τ ∈[0,α 2 ] ξ ε τ (φ j ) -ξ ε σ (φ j ) 2 ≥ δ ′ 4 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ . (4.16)
Recalling that a ≪ α 2 , we are going to show that lim µε→∞ 1 α 2 P ε (A) = 0, (4.17) which is essentially the outcome of Proposition 6.2.3 in [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF], however the proof cannot be applied directly in our context and we explain below the necessary adjustments.

First of all, the test functions are now unbounded in v (contrary to the Fourier-Hermite modes). Thus an energy cut-off is necessary. For technical reasons, we are going to use a larger truncation parameter R = (log µ ε ) 2 instead of R = α -1 introduced in (3.2). The corresponding truncated process is defined as in (3.1) and denoted by ( ξε τ ) τ ≥0 . We are going to check that with high probability both processes coincide because all the velocities remain smaller than R (4.18) lim

µε→∞ 1 α 2 P ε ∃i, sup t≤α v ε i (t) > R = 0.
This can be deduced from a result of [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF] as follows. Fix n = 4d, η = ε 1-1 2d and call microscopic cluster of size n a set G of n particle configurations in

T d × R d such that (z, z ′ ) ∈ G × G if and only if there are z 1 = z, z 2 , . . . , z ℓ = z ′ in G such that x i -x i+1 ≤ 3 R η, ∀1 ≤ i ≤ ℓ -1 .
Let Υ ε N be the set of initial configurations Z ε0 N ∈ D ε N such that for any integer 1 ≤ k ≤ α η , the configuration at time kη satisfies (4. [START_REF] Golse | Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs[END_REF])

∀1 ≤ j ≤ N, v j ≤ R n ,
and any microscopic cluster of particles is of size at most n. Adapting to our framework the proof of Proposition 2.7 of [START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF] implies that (4.20)

P ε c Υ ε N ≤ 1 α n ε d .
We check that for any configuration in Υ ε N , the velocities are bounded from above by R during the kinetic time interval [0, α]. Indeed, at each intermediate time kη, the velocities of configurations in Υ ε N , are smaller than R n by (4.19). Furthermore the clusters are all of size less than n and in the time interval [k η, (k + 1)η] particles within a cluster cannot interact with particles in other clusters. As the total kinetic energy of a finite number of particles is preserved by the hard sphere dynamics, the velocity of each particle will remain less than R. Thus (4.18) is implied by (4.20).

We are now in position to complete the proof of (4.17). Thanks to (4.18), it is enough to replace the event A by the similar event à for the process ( ξε τ ) τ ≥0 . It thus remains to prove The statement of Proposition 6.2.3 from [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] is not precise enough to conclude directly mainly due to the diverging prefactor 1 α 2 . However all the required estimates can be found in [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] and we are going to detail the relevant parts of the argument.

We proceed as in (4.7) and introduce an additional time cut-off µ -7 3 ε instead of a to filter the very small scales 1

α 2 P ε Ã = 1 α 2 P ε ⎛ ⎜ ⎜ ⎝ sup 0≤τ,σ≤α 2 τ -σ ≤2a ξε τ -ξε σ 2 -k ≥ δ ′ 16 ⎞ ⎟ ⎟ ⎠ ≤ 1 α 2 P ε ⎛ ⎜ ⎝ j C 2 Bµ -7 3 ε ξ ε (φ j ) 1 3
(1

+ j 2 ) k a 2γ-4 3 ≥ δ ′ 64 ⎞ ⎟ ⎠ + 1 α 2 P ε ⎛ ⎜ ⎜ ⎜ ⎝ j 4 (1 + j 2 ) k sup σ-τ ≤2µ -7 3 ε σ,τ ∈[0,α 2 ] ξε τ (φ j ) -ξε σ (φ j ) 2 ≥ δ ′ 64 ⎞ ⎟ ⎟ ⎟ ⎠ ,
with the analogous notation of (4.4) on this short time scale Bµ In our procedure, it was necessary to use first a time cut-off a in (4.7) in order to reduce to estimates in the kinetic time scale. Indeed the error term (4.12) occurring in the comparison with the limiting equations on the diffusive time scale [0, T ] was too crude to be efficient up to the smallest time scale µ -7 3 ε . On the kinetic scale better controls can be derived and one can show as in Lemma 6.2.6 of [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] (with the Remark 6.2.8 to take care of the large velocities) that

1 α 2 P ε ⎛ ⎜ ⎝ j C 2 Bµ -7 3 ε ξε (φ j ) 1 3
(1

+ j 2 ) k a 2γ-4 3 ≥ δ ′ 64 ⎞ ⎟ ⎠ ≤ C 2 64 α 2 δ ′ a 2γ-4 3 .
As a ≪ α, this term vanishes in the diffusive limit. By using the proof of Lemma 6.2.5 of [START_REF] Bodineau | Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations[END_REF] (with the Remark 6.2.8 to take care of the logarithmic divergence), we deduce that second term vanishes also in the diffusive limit

1 α 2 P ε ⎛ ⎜ ⎜ ⎜ ⎝ j 4 (1 + j 2 ) k sup σ-τ ≤2µ -7 3 ε σ,τ ∈[0,α 2 ] ξε τ (φ j ) -ξε σ (φ j ) 2 ≥ δ ′ 64 ⎞ ⎟ ⎟ ⎟ ⎠ ≤ C α 2 µ -1 3 ε → 0.
Combining the previous results, (4.21) holds. This completes the proof of Proposition 4.1.

(2. 1 )

 1 kinetic scale : t = αt kin with t kin = O(1), acoustic scale : t = O(1), diffusive scale : t = τ α with τ = O(1).

(4. 21 ) lim µε→∞ 1 α 2 P

 2112 ε Ã = 0.

τ -σ γ 1 τ -σ >µ -7 3 ε

 3 with b = 6 , γ = 7 3.

  The following result states that this covariance is well approximated on R + by

	Mg α ( equation (1.13) starting from g 0 ∈ L 2 τ α )hdxdv where g α is the solution of the scaled linearized Boltzmann M , defined by the norm (2.3) g L 2 M ∶= T d ×R d 1 g 2 Mdxdv 2 .

Theorem 2.1 (

[START_REF] Bodineau | Long-time correlations for a hard-sphere gas at equilibrium[END_REF]

, Linearized Boltzmann equation). Consider a system of hard spheres at equilibrium in a d-dimensional periodic box with d ≥ 3. Let g 0 and h be two Lipschitz functions on T d ×R d and let g α be the unique solution in L ∞ (R + ; L 2 M ) to (1.13) associated with the initial data g 0 . Then, in the low density regime µ