Lionel Montagne

Wilfried Blanc

Daniel R Neuville

Éditorial Matériaux désordonnés, verres et leur fonctionnalisation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Le GDR Verres (GDR CNRS 3338) et l'Union pour la Science et la Technologie Verrières (USTV) s'associent pour publier ce numéro thématique de Matériaux et Techniques consacré aux matériaux vitreux. Les articles sont, pour la plupart, écrits sur la base des présentations dans les conférences présentées au congrès Matériaux-2014 de Montpellier, où le GDR et l'USTV ont co-organisé un symposium intitulé (( Matériaux désordonnés, verres et leur fonctionnalisation )).

Le symposium a rassemblé une centaine de participants dont un quart de doctorants dont l'inscription a été financée par le GDR. La présidence scientifique en a été assurée par Étienne Vernaz du CEA, Lionel Montagne, Directeur du GDR-Verres, et Daniel Neuville, Président de l'USTV.

Une première session était consacrée aux verres de chalcogénures et aux verres d'optique. Présidée par Annie Pradel de l'université de Montpellier, elle a été introduite par une conférence invitée d'Andrea Piarristeguy, faisant le point sur les propriétés spécifiques de ce type de verre. Huit exposés oraux ont porté sur le développement d'optiques adaptées à la vision nocturne, de capteurs infrarouge, de fibres destinées à la détection du CO 2 , de matériaux microstructurés aux propriétés optiques optimisées, ou encore sur des matériaux hybrides formés d'un liquide ionique confiné dans un chalcogénure nanoporeux. La seconde session, intitulée (( Verres et amorphes dans tous les états )), était présidée par Etienne Vernaz du CEA. Elle a été introduite par une conférence invitée d'Anne-Lise Faivre qui portait sur la définition du verre et de la transition vitreuse. Même si ce point est débattu depuis plus d'un siècle, on constate que (( l'état vitreux )) garde encore sa part de mystère, ouvrant un champ toujours renouvelé à la recherche, aussi bien fondamentale qu'appliquée. Dix présentations orales ont permis d'en explorer quelques aspects, comme le volume libre interatomique exploré par compression avec des gaz de taille atomique différentes, ou encore l'autocicatrisation de verres composites soumis à des conditions sévères.

La dernière session, présidée par Lionel Montagne de l'université de Lille, portait sur les verres et matériaux de confinement. Elle a été introduite par un exposé général de Florence Bart, chef de projet au CEA Marcoule, et a été suivi de huit exposés portant sur la solubilité Article publié par EDP Sciences de différents éléments (lanthanides, molybdène. . . ) dans les borosilicates ou aluminosilicates complexes (inclusion, cristallisation, séparation de phase), sur le comportement de ces verres à l'auto irradiation, sur la compréhension de la surface réactive formée dans l'eau, ou encore sur la modélisation des propriétés physico-chimiques à partir de modélisation atomistique.

Les articles sélectionnés pour ce numéro spécial illustrent la diversité des activités de recherche et développement menées sur les verres en France, et reflètent bien l'esprit avec lequel le GDR et l'USTV agissent pour fédérer les actions scientifiques sur les verres pour le GDR et pour et assurer un lien entre les communautés académiques et industrielles.

L'USTV et le GDR-Verres assurent ainsi la représentation et l'animation de la communauté scientifique verrière en partenariat avec la Fédération des Chambres Syndicales des Industries du Verre. Des actions sont régulièrement organisées, telles que des ateliers, écoles thématiques et journées sur des problématiques allant du contact alimentaire aux propriétés du verre à chaud ou aux mécanismes de nucléation et cristallisation.

Parmi les prochains évènements organisés, on peut noter les (( Journées Verres )) qui se tiennent à Nice du 18 et 20 novembre 2015. Elles sont précédées par un atelier thématique sur les (( dégâts d'irradiation )) dans les verres, les 17 et 18 novembre. Les participants ont de plus la chance de visiter les verreries de Biot et d'en admirer les productions artistiques. La photo de couverture de ce numéro de Matériaux et Techniques est une oeuvre produite à Biot par Antoine Pierini, qui servira de récompense pour les prix posters décernés lors des Journées Verres.

En conclusion, si vous vous intéressez au verre, au sens le plus large du terme, que vous soyez industriel, chercheur, étudiant, artiste, ou tout simplement curieux, n'hésitez pas à nous rejoindre, toutes les informations sont sur www.ustverre.fr ou http://gdrverres. univ-lille1.fr.

L

  .Montagne et al. : Matériaux & Techniques 103, 401 (2015)

1

  UCCS, Unité de Catalyse et Chimie du Solide, UMR CNRS 8181, Université de Lille 1, École Nationale Supérieure de Chimie

de Lille, Cité Scientifique, Bâtiment C7, BP 90108, 59652 Villeneuve d'Ascq Cedex, France lionel.montagne@univ-lille1.fr 2 Université Nice Sophia Antipolis, CNRS, LPMC, UMR 7336, 06100 Nice, France 3 Géomatériaux, CNRS-Institut de Physique du Globe de Paris, Sorbonne Paris Cité, 1 rue Jussieu, 75238 Paris Cedex 05, France ; Éditeur (( Matériaux et Techniques ))


H Francois-Saint-Cyr

I Martin

W Blanc

Philippe Le Coustumer

C Hombourger

D Neuville

D J Larson

T J Prosa

C Guillermier

P Lecoustumer

Correlative Compositional Analysis of Fiber-Optic Nanoparticles

published or not. The documents may come

L'archive ouverte

Development of new active optical fiber devices requires materials with augmented intrinsic properties, using luminescent ion-doped silica as a host glass. Nanoparticles (NPs) in optical fibers can provide this augmentation as they can combine the sturdiness and low cost of silica with particular spectroscopic behavior that would not appear in a pure silica local environment. Ideally, NPs would fully encapsulate luminescent ions to produce engineered spectroscopic properties. This technology would be of great interest for a large domain of applications: high power fiber lasers, random lasers, light sources with new wavelengths and telecommunications.

As silicate systems have a large phase immiscibility domain when they contain divalent metal oxides (such as Mg), one can take advantage of thermal treatments inherent to the MCVD process to obtain NPs through phase separation. NPs are grown in situ within the material. Although glass-ceramics were discovered in 1950's, there is a great demand on experimental data to understand the early stages of nucleation [1]. Modern glass ceramics are generally obtained through many trial-and-error processing steps, involving variations in composition as well as thermal treatments. Although classical nucleation theory was the first model proposed to explain those phenomena, growth rate mismatches could encompass several order of magnitude. According to this capillary assumption-based model, nuclei properties are the same as those of the bulk (same structure and composition). Recent articles disprove assumption of structure, pointing toward NPs structural changes [2] and transition from amorphous nuclei to crystalline NPs [3]. Compositional changes for small particle sizes (~1-10 nm) have been measured in alloys with Anomalous Small Angle X-Ray Scattering (ASAXS) [4] and in steels with Atom Probe Tomography (APT) [5]. Recent developments in APT allowed the extension of such studies to glass-ceramics, presenting additional challenges in specimen preparation and yield [START_REF] Larson | Local Electrode Atom Probe Tomography[END_REF]. In the current work, we report experimental data disproving the second capillary assumption at the early stage of nucleation-growth process.

Composition of NPs was investigated using APT, NanoSIMS and Transmission Electron Microscopy. The Mg distribution in silica-based glass doped with Mg, P, Ge and Er is reported in Figure 1. When NPs sizes range in the 1-10 nm domain, we can clearly quantify the NPs concentration of Mg, P and Er (Figure 2). These nucleation and growth mechanisms in optical fibers impact the overall material community as experimental results could refine the theories behind these processes.

Figure 1 .

 1 Figure 1. Mg-based dielectric nano-particles (pink) surrounded by silica matrix (blue).

Figure 2 .

 2 Figure 2. Proximity histogram displaying the evolution of Mg, Er, P, and Ge concentrations from the silica matrix toward the center of the Mg-based dielectric nanoparticles.


H Francois-Saint-Cyr

D J Neuville

T J Larson

C Prosa

Guillermier

Fiber Optic Dielectric Nanoparticles Characterization by Atom Probe Microscopy

HAL is

Fiber Optic Dielectric Nanoparticles Characterization by Atom Probe Microscopy

H. Francois-Saint-Cyr1 , I. Martin 1 , W. Blanc2 , P. LeCoustumer3 , C. Hombourger 1 , D. Neuville 4 , The engineered processing of dielectric nanoparticles (DNPs) in optical fibers via luminescent ion-doping of silica-based glass aims at providing an enhanced spectroscopic behavior compared to pure silica. These DNPs should positively impact applications in high power fiber lasers, light sources with new wavelengths and telecommunications.

The prevalence of large phase immiscibility domains in silicate systems containing divalent metal oxides (Mg for instance) promotes the formation of DNPs through phase separation since heat treatments take place during the MCVD process.

Even after 60 years of glass-ceramics research, lack of experimental data concerning early nucleation stages imposes variations in composition and heat treatments as processing steps [1]. Although classical nucleation theory was the first model proposed to explain those phenomena, growth rate mismatches remain wide. According to this capillary assumption-based model, nuclei and bulk share similar structure-composition relationship. Recent articles disprove assumption of structure, pointing toward DNPs structural changes [2] and transition from amorphous nuclei to crystalline DNPs [3]. Compositional changes for small particle sizes (~1-10 nm) have been measured in alloys with Anomalous Small Angle X-Ray Scattering (ASAXS) [4] and in steels with Atom Probe Tomography (APT) [5]. Recent developments in APT has allowed the extension of such studies to glass-ceramics [START_REF] Larson | Local Electrode Atom Probe Tomography[END_REF], and in the current work, we report experimental data disproving the second capillary assumption at the early stage of nucleationgrowth process.

The atomic distribution map of Mg DNPs in silica-based glass doped with Mg, P, Ge and Er is reported in Figure 1 after APT analysis. In addition, quantitative assessment of Mg, P and Er content levels in DNPs smaller than 10nm in diameter (Figure 2) could refine the theories behind nucleation and growth mechanisms.

Figure 1 .

 1 Figure 1. Mg-based dielectric nano-particles (pink) surrounded by silica matrix (blue).

Figure 2 .

 2 Figure 2. Proximity histogram displaying the evolution of Mg, Er, P, and Ge concentrations from the silica matrix toward the center of the Mg-based dielectric nanoparticles.

CAMECA Instruments Inc., 5500 Nobel Drive, Suite 100, Madison, WI, 53711, USA

Université Nice Sophia Antipolis, CNRS, LPMC, UMR7336, 06100 Nice, France

Université Bordeaux 3, Géo-ressources et Environnement, EA4592, 33607 Pessac, France

Institut de Physique du Globe de Paris, 1 rue Jussieu, 75005 Paris, France

National Resource for Imaging Mass Spectroscopy, Cambridge, MA 02139, USA


H Francois-Saint-Cyr

I Martin

P Lecoustumer

C Hombourger

D Neuville

D J Larson

T J Prosa

E Gonthier

L Geai

C Guillermier

W Blanc

Variation of sub-10nm nanoparticle chemical composition in glass revealed by Atom Probe Tomography

INTRODUCTION

Glasses containing rare-earth (RE)-doped nanoparticles are of particular interest for many applications as they can combine (i) mechanical and financial advantages of the glassy host matrix with (ii) the spectroscopic properties of the nanoparticles provided by the local environment of the RE ions [ 1,[START_REF] Blanc | Formation et applications des nanoparticules dans les fibres optiques à base de silice[END_REF]. The material's performance depends on our ability to grow and control the chemical composition of dielectric nanoparticles (DNPs). However, the understanding of the mechanisms involved in the formation of these DNPs is not clear yet due to a lack of information on the characteristics of DNPs at their early stage of formation. Combined imaging techniques at the nanometric scale of nanodomains with chemical analysis of potential chemical gradients are challenging especially when nanoparticules are included into a glass matrix.

In this report, we are taking advantage of a recent technology, Tri-Dimensional (3D) Atom Probe Tomography (APT), to investigate the chemical composition variations in DNPs grown in silicate glass through heat treatments at their early stages of formation. Specifically, we are providing here a comprehensive experimental dataset describing the variation in concentration of elements such as P, Mg, Ge and Er for DNPs which radii spanned in a [1-10 nm] range. This study shows 1-the partition of Er 3+ ions within amorphous oxide nanoparticles smaller than 10 nm, 2-a chemical variation with the size of the DNPs, 3-the first direct observation of a plateau at the early stage of nucleation.

RESULTS

The sample under investigation is an optical fiber drawn from a preform prepared by conventional Modified Chemical Vapor Deposition (MCVD) technique [START_REF] Macchesney | Proc. IEEE[END_REF]. The prevalence of large phase immiscibility domains in silicate systems containing divalent metal oxides (Mg for instance) is well known to promote the formation of DNPs through phase separation mechanism. Here, we take advantage of thermal treatments inherent to the MCVD process to obtain DNP which formation is triggered by the introduction of an alkaline earth element (Mg). Average composition of the fiber core was determined via EPMA analyses. MgO, P2O5, GeO2 mean concentrations estimated in the SiO2-based glass, are 1.5, 1.5 and 2mol%, respectively. Er content is in the 100 ppms range.

Nanoparticles observed by TEM are shown in Figure 1a. The particles size spans few nm to 200 nm. They are spherical in shape despite the drawing stage involved in the fabrication process. top sliced view of a reconstructed 3D volume obtained by APT reveals a high density of very small DNPs (Figure 1c) not observed in a high resolution TEM image (figure 1b). . As we aim at quantifying best the chemical content of the DNPs, which could not be achieved by any other analytical technique, a mathematical approach called proximity histogram or proxigram [START_REF] Hellman | [END_REF] was used to interrogate shells of equal thicknesses in the directions normal to an isoconcentration surface. The proxigram displayed in Figure 2a highlights the enrichment in Mg and P in the order of 100-fold and 10-fold respectively, between the matrix and the particle. We have observed a clear uptake trend in Er, while the Ge content remained steady from the matrix toward the center of the DNPs. Thanks to the high sensitivity of APT, we report on the direct measurement of Mg, P and Ge contents in nanoparticles for the entire [1-10nm] range of radii. The variation of the concentration for Mg and P with the size of the nanoparticles is shown in Figure 2b. Concentration for both elements follows the same trend: the concentration is constant up to a radius of 3.5 nm, then increases forparticles of radii larger than 3.5nm. The set of data shown in this study specifically evidences the existence of a first plateau followed by an increase of the composition of the clusters. These experimental data disprove the capillary approximation of the Classical Nucleation Theory. However, such a plateau is expected in the framework of the Generalized Gibbs Approach formulated by Schmelzer et al. [5]. In order to gather the short-range order (SRO) of the Er atoms, we used the so-called partial radial distribution function (RDF) which consists inprobing the average local environment of each Er atom. s We observed that the close environment of Er is enriched with Mg and P elements when the size of the nanoparticle increases.

CONCLUSION

Thanks to the high sensitivity and resolution of APT, we probed the composition of amorphous sub-10-nm nanoparticles formed via heat treatments in silicate glass. We demonstrate the partition of Mg, P and Er ions in the nanoparticles and describe their concentration trends. We also provide the first set of data which describe Mg and P concentrations increase and erbium environment evolves with the size of the DNPs. An increase in the DNPs sizes corresponds to an intake of Er with Mg and P. This result indicates that the design of the DNPsdoped material does not only consider the size in terms of light scattering but also on their potential environment for luminescent ions.

Figure 1 .Figure 2 .

 12 Figure 1. (a) TEM picture, (b) High resolution TEM picture, (c) XY orientation map of 3D distribution (Mg is in pink and O is in blue)


Alexis Le Couteulx
email: alexis.le-couteulx@rennes.inra.fr

Cédric Wolf

Vincent Hallaire

Guénola Pérès

Burrowing and casting activities of three endogeic earthworm species affected by organic matter location

Keywords: soil structure, behaviour, Allolobophora chlorotica, Allolobophora icterica, Aporrectodea caliginosa

Earthworms are crucial for production and maintenance of soil structure and their activities can strongly impact soil functioning (e.g. water regulation, nutrient dynamics). This laboratory study investigated the bioturbation activity of three endogeic species, A. chlorotica, A. icterica and A. caliginosa, as affected by different locations of organic matter (OM) in the soil profile: OM scattered on the soil surface (surface-OM) or homogeneously mixed into the soil (mixed-OM). Microcosms, each containing a combination of one species (three individuals) and one OM location, were subjected to controlled environmental conditions (temperature, humidity and day/night cycle) for 60 days. At the end of the experiment, microcosms were cut into multiple horizontal cross-sections every centimetre and bioturbation activities were analysed based on the number of burrows, the burrowed area and the percentage of burrowed area totally refilled with casts.

Results showed that regardless of species, there was significantly fewer burrows and a greater percentage of burrowed area refilled with casts under mixed-OM than under surface-OM. A. chlorotica and A. caliginosa had a significantly greater burrowed area under mixed-OM than under surface-OM. Regardless of OM location, as depth increased, burrow number and area decreased for A. chlorotica and generally increased for A. icterica. In contrast, burrowing activity of A. caliginosa was affected by OM location as depth increased: under A c c e p t e d M a n u s c r i p t 2/2 mixed-OM, burrow number decreased but burrowed area remained constant, whereas under surface-OM, burrow number remained constant and burrowed area increased.

These results improve understanding of effects of endogeic species on soil structure and highlight effects of OM location on earthworm bioturbation. Especially this study gives information about the burrowing activity of A. icterica which has so far been little documented, and also informs about refilled burrows which is a major parameter for soil functioning.

Introduction

Earthworms have been described as soil engineers [START_REF] Jones | Organisms as ecosystem engineers[END_REF][START_REF] Lavelle | Soil function in a changing world: the role of invertebrate ecosystem engineers[END_REF] because of their ability to modify their own environment and notably the soil structure via their bioturbation activity which consists of burrowing and producing casts. This ingestionegestion of soil strongly affects soil structure [START_REF] Dexter | Advances in characterization of soil structure[END_REF][START_REF] Lee | Soil fauna and soil structure[END_REF] depending on the context, earthworms increase porosity [START_REF] Pérès | Relationships between earthworm communities and burrow numbers under different land use systems[END_REF][START_REF] Lamandé | Density of macropores as related to soil and earthworm community parameters in cultivated grasslands[END_REF][START_REF] Van Schaik | Linking spatial earthworm distribution to macropore numbers and hydrological effectiveness[END_REF] or increase bulk density [START_REF] Blanchart | Regulation of soil structure by geophagous earthworm activities in humid savannas of Côte d'Ivoire[END_REF]. Consequently, earthworms affect several soil functional properties and ecosystem services, such as soil moisture, water infiltration and water regulation, soil organic matter (OM) availability, nutrient cycling and primary production [START_REF] Jouquet | Soil invertebrates as ecosystem engineers: Intended and accidental effects on soil and feedback loops[END_REF][START_REF] Capowiez | The effect of tillage type and cropping system on earthworm communities, macroporosity and water infiltration[END_REF][START_REF] Blouin | A review of earthworm impact on soil function and ecosystem services[END_REF][START_REF] Crittenden | Effect of tillage on earthworms over short-and medium-term in conventional and organic farming[END_REF]. To understand the functional links between earthworms and soil structure, scientists have focused on one aspect of bioturbation (i.e. burrows) (Pérès et al., 2010;[START_REF] Lamandé | Density of macropores as related to soil and earthworm community parameters in cultivated grasslands[END_REF][START_REF] Van Schaik | Linking spatial earthworm distribution to macropore numbers and hydrological effectiveness[END_REF] but without integrating other aspects of its complexity, especially casts.

Earthworm bioturbation results from complex interactions and can be affected by various parameters, such as the location of OM, which is a food resource for earthworms and affects earthworm foraging activity [START_REF] Jeanson | Essai de pédozoologie expérimentale: morphologie d'un sol artificiel structuré par les Lombricidés[END_REF][START_REF] Martin | Interaction between organic matter in soil and the burrowing activity of three species of earthworms (Oligochaeta: Lumbricidae)[END_REF][START_REF] Pérès | Relationships between earthworm communities and burrow numbers under different land use systems[END_REF]. However, this aspect needs further study, especially in relation to cast production.

Bioturbation properties of anecic earthworms are well described: these dwelling earthworms build a relatively permanent burrow system, vertically oriented, bring soil from the depth to the soil surface and cover their burrow walls with their casts [START_REF] Kretzschmar | Experimental burrow system: pathway patterns and building behaviour[END_REF][START_REF] Kretzschmar | 3D images of natural and experimental earthworm burrow systems[END_REF][START_REF] Daniel | Computer-assisted tomography of macroporosity and its application to study the activity of the earthworm Aporrectodea nocturna[END_REF][START_REF] Jégou | Characterization of the burrow system of the earthworms Lumbricus terrestris and Aporrectodea giardi using X-ray computed tomography and image analysis[END_REF][START_REF] Jégou | Interactions between earthworm species in artificial soil cores assessed through the 3D reconstruction of the burrow systems[END_REF][START_REF] Shipitalo | Occupancy and geometrical properties of Lumbricus terrestris L. burrows affecting infiltration[END_REF][START_REF] Bastardie | X-ray tomographic and hydraulic characterization of burrowing by three earthworm species in repacked soil cores[END_REF][START_REF] Nuutinen | Interaction of Lumbricus terrestris L. burrows with field subdrains[END_REF].

Bioturbation properties of endogeic earthworms, however, are less well known, despite existing studies [START_REF] Bolton | Burrowing, feeding, egestion and energy budgets of Allolobophora rosea (Savigny) (Lumbricidae)[END_REF][START_REF] Capowiez | Burrow systems made by Aporrectodea nocturna and Allolobophora chlorotica in artificial cores: morphological differences and effects of interspecific interactions[END_REF][START_REF] Jégou | Interactions between earthworm species in artificial soil cores assessed through the 3D reconstruction of the burrow systems[END_REF][START_REF] Felten | Earthworm burrowing behaviour in 2D terraria with singleand multi-species assemblages[END_REF], these earthworms are reported to burrow through the soil, creating horizontal and randomly oriented burrows considered to be temporary structures [START_REF] Bouché | Lombriciens de France Ecologie Animale[END_REF]. Only few data exist about their casting activity in soil and especially the proportion of burrows refilled with casts which is an important parameter for soil functioning [START_REF] Schrader | Semi-automatic image analysis of earthworm activity in 2D soil sections[END_REF][START_REF] Francis | The burrowing characteristics of three common earthworm species[END_REF][START_REF] Perreault | Earthworm burrowing in laboratory microcosms as influenced by soil temperature and moisture[END_REF][START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF]. Additionally, some endogeic species have received attention, e.g. A. chlorotica [START_REF] Capowiez | Burrow systems made by Aporrectodea nocturna and Allolobophora chlorotica in artificial cores: morphological differences and effects of interspecific interactions[END_REF][START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF] and A. caliginosa [START_REF] Schrader | Semi-automatic image analysis of earthworm activity in 2D soil sections[END_REF][START_REF] Francis | The burrowing characteristics of three common earthworm species[END_REF][START_REF] Jégou | Interactions between earthworm species in artificial soil cores assessed through the 3D reconstruction of the burrow systems[END_REF][START_REF] Perreault | Earthworm burrowing in laboratory microcosms as influenced by soil temperature and moisture[END_REF][START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF], but other species are not well documented such as A. icterica which has been only assessed once, through its burrow network (Bastardie et al., 2005a).

A study of earthworm bioturbation encounters several challenges, of which difficulty in accessing burrows and casts due to soil opaqueness is one. Several authors have used A c c e p t e d M a n u s c r i p t 4/4 transparent 2D-terrariums [START_REF] Schrader | Semi-automatic image analysis of earthworm activity in 2D soil sections[END_REF][START_REF] Whalen | Quantifying surface and subsurface cast production by earthworms under controlled laboratory conditions[END_REF][START_REF] Perreault | Earthworm burrowing in laboratory microcosms as influenced by soil temperature and moisture[END_REF][START_REF] Felten | Earthworm burrowing behaviour in 2D terraria with singleand multi-species assemblages[END_REF]. Their relative thinness, however, may influence earthworm behaviour and therefore this 2D approach appears less relevant than 3D microcosms [START_REF] Capowiez | Burrow systems made by Aporrectodea nocturna and Allolobophora chlorotica in artificial cores: morphological differences and effects of interspecific interactions[END_REF][START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF]. Several authors have used 3D X-ray tomography [START_REF] Joschko | Assessment of earthworm burrowing efficiency in compacted soil with a combination of morphological and soil physical measurements[END_REF][START_REF] Jégou | Assessment of the burrow system of Lumbricus terrestris, Aporrectodea giardi, and Aporrectodea caliginosa using X-ray computed tomography[END_REF][START_REF] Capowiez | 3D skeleton reconstructions of natural earthworm burrow systems using CAT scan images of soil cores[END_REF], which is relevant for burrow network assessment but this does not allow direct assessment of cast production [START_REF] Joschko | Earthworm burrow system development assessed by means of X-ray computed tomography[END_REF]. As an alternative, the study by [START_REF] Hirth | Volume density of earthworm burrows in compacted cores of soil as estimated by direct and indirect methods[END_REF] is particularly interesting because it used cross sections of cylindrical microcosms to analyse both burrowing and casting by endogeics. Another challenge to studying earthworm bioturbation is correctly identifying which species produced observed burrows and casts, especially in natural conditions, in which several species bioturbate the soil [START_REF] Capowiez | 3D skeleton reconstructions of natural earthworm burrow systems using CAT scan images of soil cores[END_REF][START_REF] Pérès | Identification et quantification in situ des interactions entre la diversité lombricienne et la macro-bioporosité dans le contexte polyculture breton[END_REF]Bastardie et al., 2005b). Thus, despite their artificiality, microcosms remain necessary to describe bioturbation activity of a species (Bastardie et al., 2005b).

The aim of this study was to assess under controlled conditions, burrowing and casting activities of three endogeic earthworms (i) as a function of OM location in the soil profile, and (ii) as a function of soil depth. The destructive method used is based on soil cross sections of microcosms and was used to observe, classify and quantify bioturbation, i.e. number of burrows, burrowed area and percentage of total burrowed area totally refilled with casts.

A c c e p t e d M a n u s c r i p t 5/5

Materials and Methods

Our experimental system takes benefits from previous studies such as [START_REF] Jégou | Interactions between earthworm species in artificial soil cores assessed through the 3D reconstruction of the burrow systems[END_REF] for the design of microcosms and from [START_REF] Hirth | Volume density of earthworm burrows in compacted cores of soil as estimated by direct and indirect methods[END_REF] for the assessment of burrowing activity.

Experimental system

Microcosm characteristics

Twenty four microcosms were built using PVC cylinders (20 cm in length and 15 cm in internal diameter). They were cut lengthwise into two equal halves to facilitate their final opening. A 500 µm nylon mesh was placed at the upper and lower openings to retain earthworms.

Soil and organic matter features

The soil was collected from an arable field in Le Rheu, Brittany, France (N 48°09, W 1°81) and was a silt loam soil [START_REF] Fao | FAO/Unesco Soil Map of the World. Revised Legend with corrections[END_REF] with 16% sand, 69% silt, 15% clay. Soil organic matter content (2%) and pH H2O (6.1) were in accordance with the values observed in cultivated soils in Brittany ("BDAT," 2002). Soil was air-dried before being passed through a 2 mm sieve to remove biostructures already present.

We used ryegrass (Lolium perenne L.) from an unmown and untreated grassland as the OM resource for earthworms. This OM was oven-dried for 48h at 60°C before being ground to a maximum width of 1 mm. OM was supplied at 20.7 g dry weight (dw) per microcosm, i.e. 100 g of soil with 0.6 g dw of OM, corresponding to a non-limiting food resource for earthworms [START_REF] Curry | The feeding ecology of earthworms -A review[END_REF]. Two OM treatments were defined: OM mixed with all the soil (mixed-OM treatment, 12 microcosms) and OM evenly scattered on the soil surface (surface-OM treatment, 12 microcosms).

A c c e p t e d M a n u s c r i p t 6/6

Microcosm filling

Microcosms were filled with five layers of soil. Each layer had a bulk density of 1.3 g.cm -3 : it was made of 690 g dw of soil that was packed down to obtain a 3-cm-high layer. Thus, soil in the columns was 15 cm deep. The bulk density of each layer has been recorded in some cultivated fields [START_REF] Peigné | Earthworm populations under different tillage systems in organic farming[END_REF][START_REF] Bottinelli | Response of soil structure and hydraulic conductivity to reduced tillage and animal manure in a temperate loamy soil[END_REF] and in other microcosm experiments [START_REF] Jégou | Characterization of the burrow system of the earthworms Lumbricus terrestris and Aporrectodea giardi using X-ray computed tomography and image analysis[END_REF]. Additionally, the use of dry soil and thin layers prevented inter-layer smoothing when soil was packed down. Once constructed, microcosms were remoistened by capillary absorption and freely-drained for 48 h to reach field capacity which was kept by re-wetting the surface every two weeks.

Earthworm introduction

Three endogeic species, according to [START_REF] Bouché | Lombriciens de France Ecologie Animale[END_REF], were collected from an arable field:

Allolobophora chlorotica (Savigny, 1826), Allolobophora icterica (Savigny, 1826) and Aporrectodea caliginosa (Savigny, 1826). These species are commonly found in cultivated soil in France [START_REF] Cluzeau | Integration of biodiversity in soil quality monitoring: Baselines for microbial and soil fauna parameters for different land-use types[END_REF]. All earthworms were sub-adults or adults and were acclimated to the soil for one week before being introduced into microcosms [START_REF] Fründ | Using earthworms as model organisms in the laboratory: Recommendations for experimental implementations[END_REF].

Experiments involved use of earthworm monocultures: one endogeic species employing three individuals per microcosm and corresponding to 170 earthworms.m -2 and a mean fresh biomass (± standard deviation) of 0.8±0.1 g for A. chlorotica, 2.1±0.1 g for A. icterica and 1.3±0.1 g for A. caliginosa per microcosm. Earthworm biomass and density values were consistent with those found in cultivated fields in Brittany [START_REF] Pélosi | Reducing tillage in cultivated fields increases earthworm functional diversity[END_REF]. The experimental design had two OM locations with three earthworms species replicated four times (2x3x4=24 microcosms). Earthworms were placed on the soil surface and allowed to burrow down. Then, microcosms were placed on a raised grid in a climatic chamber at 10°C with a day/night cycle corresponding to the external one (18/6h in June and July). Microcosms were maintained for 60 days after earthworm inoculation, at which time they were microwaved to stop earthworm activity by killing them in situ (5 minutes, 400 Watts).

They were then slowly oven dried at 45°C for one week to help cutting cross-sections and to prevent their smoothing.

Bioturbation assessment

Each microcosm was cut from top into eleven cross sections every centimetre corresponding to sections z 0 to z 10 . The surface of each section was lightly brushed using a paintbrush and blown using a compressor at its lowest pressure to remove dust and studied with the following procedure:

1) Outlines of burrows were identified with the naked eye and traced with pen on a transparent sheet of plastic placed on the surface. Casts that completely obstructed the burrows were similarly recorded.

2) After digitizing the drawings (resolution: 600 ppi), they were analyzed with Fiji software [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] and a homemade script in the Jython programming language (http://www.jython.org/). Each burrow was identified and described by its total area and the area occupied by casts. The percentage of burrow refilled with casts (area occupied by casts divided by the burrowed area) was calculated.

Statistical analysis

Data analysis was performed using R software (R. [START_REF] Team | R: A language and environment for statistical computing[END_REF]. If the normality of residues (Shapiro test) and the heteroscedasticity (Bartlett test) were verified, we used multiway ANOVA and post-hoc LSD Tukey's tests with species, depth and OM location as factors.

Otherwise, the Kruskall-Wallis test checked for factor effect, and pairwise Wilcoxon tests with Bonferroni correction were used as post-hoc tests. Linear regressions were calculated to test the relation between bioturbation and depth. If linear regression was not significant, the nls

Results

At the end of the experiment, some OM remained on the surface of the surface-OM treatment, which suggested that excess OM had been applied. We observed that surface casts were still being produced at the end of the experiment which suggests that earthworms were still active.

Bioturbation of earthworms as affected by organic matter location

Number of burrows:

The number of burrows per section was significantly (p < 0.001) affected by earthworm species, OM location, and by 2-way interactions: species × OM location, species × depth, and OM location × depth.

The number of burrows per section was significantly higher under surface-OM vs. mixed-OM regardless of species (Fig. 1a). It was approximately 2.3, 2.1 and 1.5 times as large under surface-OM vs. mixed-OM for A. icterica, A. caliginosa and A. chlorotica, respectively.

Under surface-OM, the number of burrows per section decreased from A. icterica to A. caliginosa to A. chlorotica, with a significant difference between each pair. Under mixed-OM, A. icterica had a significantly higher number of burrows per section than A. chlorotica.

Burrowed area per section

Burrowed area was significantly (p < 0.001) affected by species, OM location, and the 2-and 3-way interactions (species × OM location × depth).

A c c e p t e d M a n u s c r i p t 9/9 A. chlorotica and A. caliginosa had significantly higher burrowed area under mixed-OM than under surface-OM (Fig. 1b). Their burrowed area was approximately 3.7 and 2.2 times as large, respectively, under mixed-OM vs. surface-OM. Conversely, A. icterica burrowed area was not significantly affected by OM location.

Under surface-OM, the area burrowed per section significantly decreased from A. icterica to A. caliginosa to A. chlorotica.

Percentage of burrowed area refilled with casts

The percentage of refilled area was significantly (p < 0.05 for species × depth and p < 0.001 for other factors) affected by species, OM location, and all 2-and 3-way interactions.

Earthworms refilled their burrows more under mixed-OM than under surface-OM (Fig. 1c). This percentage was approximately 3.1, 1.7 and 2.5 times as high under mixed-OM vs. surface-OM for A. chlorotica, A. icterica and A. caliginosa, respectively. Under mixed-OM, A. caliginosa had a higher percentage of burrowed area refilled with casts than the two other species. This percentage was not significantly different for the three species under surface-OM.

Effect of soil depth on bioturbation

Earthworms bioturbated all sections of all microcosms, except one microcosm with A. chlorotica under surface-OM, in which no burrow was found in the two deepest sections. The top section (i.e. z 0 ) of several microcosms could not be analyzed, placing the first analyzed section at a depth of 1 cm (i.e. z 1 ).

Number of burrows as depth increased

Under both OM-location treatments, the number of burrows produced by A. chlorotica decreased as depth increased (Figs. 2a and2d) whereas it increased as depth increased for A. icterica (Figs. 2b and2e). A. caliginosa had a varied response: the number of burrows A c c e p t e d M a n u s c r i p t 10/10 decreased as depth increased under mixed-OM, but remained constant as depth increased under surface-OM (Fig. 2f). For all combinations of species and OM location, except A. caliginosa under surface-OM, regressions predicting the number of burrows as depth increased were statistically significant.

Burrowed area as depth increased

A linear relation exists between burrowed area and depth in most treatments (Fig. 3). The burrowing activity of A. chlorotica was concentrated within the first 3 cm under mixed-OM (Fig. 3a). Unlike the mixed-OM treatment, the upper sections under surface-OM did not noticeably differ from others, even though food resources were located at the surface. However, A. icterica and A. caliginosa under surface-OM had burrowed areas at the surface (i.e. z 1 ) that were higher than those at 1 cm (Figs. 3e and3f). In both cases, this surface point was excluded from linear regression since its standard error was extremely high, and the surface burrowing could reflect a specific behaviour due to OM location on the soil surface.

For both OM locations, burrowing of A. icterica linearly increased as depth increased (Figs. 3b and3e). There was no significant linear correlation between the area refilled with casts and depth (data not shown).

Discussion

Earthworm sensitivity to OM location

Our results showed major differences in the bioturbation of the three species studied, even though all belong to the endogeic group. According to this study, species is a key factor that must be considered when linking earthworms to bioturbation. Additionally, our results highlighted the need to account for effects of OM location on earthworm bioturbation, since it might influence the number of burrows produced, the burrowed area, and the burrowed area refilled with casts. Total burrowed area of A. icterica was noteworthy unaffected by OM A c c e p t e d M a n u s c r i p t 11/11 location, even though its behaviour was changed, notably by increasing the number of burrows. However, A. chlorotica and A. caliginosa were greatly influenced by OM mixed with soil, which led them to increase the area burrowed. These observations are consistent with their food consumption habits: under spruce forest conditions, Bernier (1998) observed that A. icterica's digestive tract contained a higher mineral content than that of other species, including A. caliginosa. Conversely, the diets of A. chlorotica and A. caliginosa are known to be similar [START_REF] Piearce | Gut contents of some lumbricid earthworms[END_REF]. Since OM is a known food resource for the three endogeic species, this study showed that less of it in the soil could lead earthworms to build more burrows supporting previous study under field conditions [START_REF] Pérès | Relationships between earthworm communities and burrow numbers under different land use systems[END_REF]. It should be noted that interspecific competition can affect the burrowing activity of earthworm species. In particular, the burrow system of A. chlorotica was unaffected by Aporrectodea nocturna [START_REF] Capowiez | Burrow systems made by Aporrectodea nocturna and Allolobophora chlorotica in artificial cores: morphological differences and effects of interspecific interactions[END_REF] and A. caliginosa had a significantly lower burrowing activity in the presence of several other species [START_REF] Felten | Earthworm burrowing behaviour in 2D terraria with singleand multi-species assemblages[END_REF]. Thus both OM location and interspecific competition should be included in further studies of earthworm bioturbation activity.

It is worthwhile to note that the measured percentage of burrowed area refilled with casts is far from equal among documented studies. For example, the percentage for A. caliginosa in our study (10-35%) was much lower than that of [START_REF] Francis | The burrowing characteristics of three common earthworm species[END_REF] and Capowiez et al.

(2014) (40-85%) but higher than that observed by [START_REF] Perreault | Earthworm burrowing in laboratory microcosms as influenced by soil temperature and moisture[END_REF]

(<10%).

Differences among studies may stem from the methods used, but also from the influence of food quality, food quantity, temperature and bulk density, which differed among studies. For example, [START_REF] Perreault | Earthworm burrowing in laboratory microcosms as influenced by soil temperature and moisture[END_REF] observed that surface casting of A. caliginosa was greater with wetter soil. Our results showed that OM location is another factor that affects the percentage of burrowed area refilled. We observed a lower percentage of burrowed area refilled with casts under surface-OM vs. mixed-OM. There could be several explanations for this: (i) casts were less stable under surface-OM and were undetectable at the end of the experiment; (ii) casts under the surface-OM treatment had a greater bulk density; (iii) burrows were built by pushing particles; (iv) casts were egested at the soil surface and (v) more casts were crushed against burrow walls under surface-OM vs. mixed-OM. Further studies are needed to explain this difference between the two OM locations and to study the mechanisms of cast production. The measured percentage of burrowed area refilled with casts is an indicator of burrow continuity, notably because reported casts are those that completely obstructed the burrows. Our study suggests that burrows built under mixed-OM are more discontinuous, which can impact on water movements [START_REF] Allaire-Leung | Water and solute movement in soil as influenced by macropore characteristics: 1. Macropore continuity[END_REF] and burrow lifespan [START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF].

Bioturbation activity as depth increased

We assessed differences in burrow number and burrowed area as depth increased. A. chlorotica burrowing activity was concentrated in the top few centimeters which was not observed by [START_REF] Capowiez | Burrow systems made by Aporrectodea nocturna and Allolobophora chlorotica in artificial cores: morphological differences and effects of interspecific interactions[END_REF]. This result agrees with an intermediate position of A. chlorotica between the ecological categories "endogeic" and "epigeic" [START_REF] Bouché | Strategies lombriciennes[END_REF][START_REF] Pérès | Identification et quantification in situ des interactions entre la diversité lombricienne et la macro-bioporosité dans le contexte polyculture breton[END_REF]. Nevertheless, we observed that this epi-endogeic behaviour was more marked with OM mixed into the soil than with OM on the soil surface. This seems counterintuitive, because OM on the soil surface would induce earthworms to burrow at the surface to feed. But our study does not reveal whether earthworms expressed true epigeic behaviour when OM was on the surface by feeding in the thin organic layer at the top of microcosms.

Nevertheless, the effect of A. chlorotica on the soil volume was smaller when OM was on the surface. The burrowed area of A. caliginosa slightly increased as depth increased when OM was located on the soil surface, which contradicts other studies [START_REF] Mckenzie | Size and orientation of burrows made by the earthworms Aporrectodea rosea and A. caliginosa[END_REF][START_REF] Jégou | Assessment of the burrow system of Lumbricus terrestris, Aporrectodea giardi, and Aporrectodea caliginosa using X-ray computed tomography[END_REF], but no trend was found when OM was mixed into the soil. A. icterica is reported to be a typical endogeic earthworm (Bastardie et al., 2005a). This corresponds with A c c e p t e d M a n u s c r i p t 13/13 the observation that OM location significantly influenced A. icterica, but less than the two other species in our study. Like Francis et al. (2001), we observed no significant decrease in the area refilled with casts as depth increased, which is contrary to results of [START_REF] Capowiez | Quantitative estimates of burrow construction and destruction, by anecic and endogeic earthworms in repacked soil cores[END_REF].

Conclusion

Data on the effect of endogeic earthworms on soil structure are rare in the literature, especially concerning A. icterica. Our study provides insights into the burrowing activity of three endogeic species according to organic matter location. Bioturbation is crucial in agriculture particularly in no-till fields which are not mechanically treated to create favourable soil structure and in which structure and soil functioning are strongly affected by biological activity [START_REF] Capowiez | The effect of tillage type and cropping system on earthworm communities, macroporosity and water infiltration[END_REF][START_REF] Peigné | Earthworm populations under different tillage systems in organic farming[END_REF][START_REF] Crittenden | Effect of tillage on earthworms over short-and medium-term in conventional and organic farming[END_REF].

Our results obtained under controlled conditions have now to be confirmed under field conditions, in which soil heterogeneity can be integrated and interspecific competition between earthworms occur. Among other parameters such as bulk density, OM location may change according to tillage practices: for example, no-till systems keep OM on the soil surface and ploughed systems mix OM into the soil. Our results suggest that the three species do not bioturbate the soil in the same way under these tillage practices because of these differences in OM location. However the combined effect of OM location and other parameters, e.g. bulk density, must be studied to confirm this statement. Moreover, even though these species are endogeic, they do not preferentially burrow the soil at the same depth, but are complementary and thus our results suggest that these species do not occupy the same ecological niche. This is one reason why species diversity within the same ecological category must be maintained or increased. Results of this study will be integrated Earthworm ecological categories are still disputed. This study shows that the bioturbation activity of A. chlorotica and A. icterica agrees with their classification as epi-endogeic and true endogeic, respectively. Recent work has begun to focus on addressing the ecological traits of earthworm species [START_REF] Lowe | Life-cycle traits of the dimorphic earthworm species Allolobophora chlorotica (Savigny, 1826) under controlled laboratory conditions[END_REF][START_REF] Fernández | Life cycle and reproductive traits of the earthworm Aporrectodea trapezoides (Dugès, 1828) in laboratory cultures[END_REF][START_REF] Pey | Current use of and future needs for soil invertebrate functional traits in community ecology[END_REF] and this needs to be continued.

  was used to estimate parameters of a non-linear model. The goodness of fit of the non-linear model was assessed with a Pearson test of correlation between estimated and observed values. Significance threshold was set at

  model that simulates impacts of earthworms on soil structure and accounts for tillage practices.

Figure 1 .

 1 Figure 1. Measured bioturbation parameters of three endogeic earthworm species. Error

Figure 2 .

 2 Figure 2. Mean number of burrows per section by depth of three endogeic earthworm

Figure 3 .

 3 Figure 3. Mean burrowed area per section by depth of three endogeic earthworm species.

Acknowledgements

The authors thank Hoël Hotte (UMR Ecobio, Rennes) for his help in sampling earthworms and Alexandre Wotowiec for his help in studying microcosms.


V Jakšić

Y Ogata

C.-A Pillet

The Green-Kubo formula for locally interacting fermionic open systems

We consider a model describing finitely many free Fermi gas reservoirs coupled by local interactions and prove the Green-Kubo formulas and the Onsager reciprocity relations for heat and charge fluxes generated by temperature and chemical potential differentials.

Introduction

This is the fourth in a series of papers [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF][START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF] dealing with derivation of Green-Kubo formulas (GKF) and Onsager reciprocity relations (ORR) in quantum statistical mechanics. The first two papers [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF] were devoted to the abstract axiomatic derivation of GKF and ORR for open systems driven by thermodynamical forces associated to temperature and chemical potential differentials. This paper and [START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF] are devoted to the study of concrete models.

In [START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF] we have studied the well-known spin-fermion model describing the interaction of an N -level atom with finitely many independent free Fermi gas reservoirs [START_REF] Davies | Markovian master equations[END_REF][START_REF] Lebowitz | Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs[END_REF][START_REF] Jakšić | Non-equilibrium steady states for finite quantum systems coupled to thermal reservoirs[END_REF]. Combining the results of [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF] with spectral theory of non-equilibrium steady states developed in [START_REF] Jakšić | Non-equilibrium steady states for finite quantum systems coupled to thermal reservoirs[END_REF] we have established GKF and ORR for this class of models.

In this paper we study a model describing finitely many free Fermi gas reservoirs coupled by local interactions and show that the abstract derivation of [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF] combined with scattering theory of non-equilibrium steady states (see [BM1, AM, BM2, Ru1, FMU]) yields the GKF and ORR for this class of models.

Throughout the paper we shall assume that the reader is familiar with general aspects of linear response theory discussed in [JOP1, JOP2, JOP3] and with the algebraic formalism of quantum statistical mechanics [START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 1[END_REF][START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 2[END_REF]. A modern introduction to these topics can be found in [START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF][START_REF] Fröhlich | Dissipative transport: thermal contacts and tunneling junctions[END_REF] and in the recent lecture notes [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF].

The paper is organized as follows. In Subsection 1.1 for notational purposes we review the description of a free Fermi gas in the algebraic formalism of quantum statistical mechanics. In Subsection 1.2 we introduce the model and state our results. The strategy of the proof is the same as in [START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF] and is described in Section 3.1. This strategy reduces the proof of all our results to a technical estimate formulated in Theorem 3.1. This estimate, which is our main technical result, is established in Section 3.2.

Acknowledgment. The research of V.J. was partly supported by NSERC. A part of this work has been done during V.J.'s visit to CPT-CNRS and during his stay as a Forchheimer Visiting Professor at The Hebrew University of Jerusalem. He would like to thank H. Farkas and Y. Last for the hospitality of the Einstein Institute of Mathematics at The Hebrew University. The research of Y.O. was supported by the Japan Society for the Promotion of Science. A part of this work has been done during the stay of Y.O. at CPT-CNRS, partly supported by the Canon Foundation in Europe and JSPS.

Preliminaries

Let h and h 0 be given Hilbert space and Hamiltonian. The corresponding free Fermi gas is described by the C * -dynamical system (O, τ 0 ) where: (i) O = CAR(h) is the CAR algebra over h. We denote by a * (f )/a(f ) the creation/annihilation operator associated to f ∈ h. As usual, a # stands for either a or a * ;

(ii) τ t 0 is the group of Bogoliubov * -automorphisms generated by h 0 , τ t 0 (a # (f )) = a # (e ith0 f ). We denote by δ 0 the generator of τ 0 ;

The gauge group of the free Fermi gas is the group of Bogoliubov * -automorphisms ϑ ϕ , ϕ ∈ R, generated by the identity operator on h. The physical observables are gauge invariant and hence elements of

O ϑ = {A ∈ O | ϑ ϕ (A) = A for all ϕ ∈ R}. O ϑ is the τ 0 -invariant C * -subalgebra of O generated by {a * (f )a(g) | f, g ∈ h} and 1l.
Let β > 0 and µ ∈ R be parameters and ω βµ the gauge-invariant quasi-free state on O generated by

T βµ = 1 1 + e β(h0-µ) .
The quantum dynamical system (O, τ 0 , ω βµ ) describes a free Fermi gas in thermal equilibrium at inverse temperature β and chemical potential µ. We remark that ω βµ is the unique β-KMS state for the C * -dynamics τ t 0 • ϑ -µt and that ω βµ ↾ O ϑ is a (τ 0 , β)-KMS state on O ϑ .

Let V ∈ O ϑ be a self-adjoint perturbation and τ λ the perturbed C * -dynamics generated by δ λ = δ 0 + iλ[V, • ] where λ ∈ R is a coupling constant. We recall that for A ∈ O and t ≥ 0,

τ t λ (A) = τ t 0 (A) + ∞ n=1 (iλ) n 0≤sn≤•••≤s1≤t [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ t 0 (A)] • • • ]] ds 1 • • • ds n . (1.1)
In this paper we shall consider self-adjoint perturbations of the form

V = K k=1 n k j=1 a * (u kj )a(v kj ), (1.2)
where K and n k 's are finite. We set n = max k n k . Denote D 0 = {u kj , v kj }. By rescaling λ, without loss of generality we may assume that max

f ∈D0 f = 1. (1.3) If n = 1, then τ t λ (a # (f )) = a # (e ith λ f ) where h λ = h 0 + λ k (v k , • )u k ,
and so the C * -dynamics τ λ is also a group of Bogoliubov * -automorphisms. This special case is exactly solvable and has been studied in detail in [START_REF] Aschbacher | Transport properties of quasi-free Fermions[END_REF] (for additional information and references about quasi-free open quantum systems we refer the reader to recent lecture notes [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF][START_REF] Jakšić | Mathematical theory of the Wigner-Weisskopf atom[END_REF]).

The following technical result will play a key role in our paper.

Theorem 1.1 Let A = a # (f 1 ) • • • a # (f m ) be a monomial of order m and C (n) A (s 0 , . . . , s n ) = [V, [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ s0 0 (A)] • • • ]]].
Then for all n ≥ 0 there exist a finite index set P n (A), monomials F

A,p ∈ O, and scalar functions G (n)

A,p such that

C (n) A (s 0 , . . . , s n ) = p∈Pn(A) G (n) A,p (s 0 , . . . , s n )F (n)
A,p (s 0 , . . . , s n ).

(1.4)

Moreover,

1. The order of the monomial F (n)

A,p does not exceed 2(n + 1)(n -1) + m.

The factors of F (n)

A,p are from

{a # (e ish0 g) | g ∈ D 0 , s ∈ {0, s 1 , . . . , s n }} ∪ {a # (e is0h0 g) | g ∈ A},
where

A = {f 1 , • • • , f m }.
The number of factors from the first set does not exceed (n + 1)(2n -1) while the number of factors from the second set does not exceed m -1.

Suppose that

ℓ = ∞ 0 sup f ∈D0,g∈D0∪A |(f, e ith0 g)| dt < ∞, denote ℓ 0 = ∞ 0 sup f,g∈D0 |(f, e ish0 g)| ds,
and set

Λ 0 =      1 2Kℓ 0 if n = 1, 1 2nKℓ 0 (2n -2) 2n-2 (2n -1) 2n-1 if n > 1. If n = 1 and |λ| < Λ 0 or if n > 1 and |λ| ≤ Λ 0 then the sum W = ∞ n=0 |λ| n+1 p∈Pn(A) 0≤sn≤•••≤s0<∞ |G (n) A,p (s 0 , . . . , s n )| ds 0 • • • ds n ,
is finite and satisfies

W ≤ 1 + 2nKℓ|λ| (1 -|λ|/Λ 0 ) + 2n(2n -2)Kℓ 0 |λ| m -1.
(1.5)

Remark 1. Parts (1) and (2) of this theorem are easy to prove and are stated for reference purpose. The key fact is Part (3) which we shall prove using the fundamental Botvich-Gutȃ -Maassen integral estimate [BGM]. Related but weaker results can be obtained using the integral estimates of [START_REF] Botvich | Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas[END_REF][START_REF] Botvich | Asymptotic completeness and all that for an infinite number of fermions[END_REF][START_REF] Fröhlich | Dissipative transport: thermal contacts and tunneling junctions[END_REF]. Remark 2. In our applications we shall not need the explicit form of the bound (1.5).

Our first regularity assumption is

(A1) There exists a dense vector subspace D ⊂ h such that D 0 ⊂ D and that the functions

R ∋ t → (f, e ith0 g), are in L 1 (R, dt) for all f, g ∈ D.
Note that this assumption implies that h 0 has purely absolutely continuous spectrum. A consequence of Theorem 1.1 is Theorem 1.2 Assume that (A1) holds and that |λ| < Λ 0 . Then the limits

γ + λ (A) = lim t→+∞ τ -t 0 • τ t λ (A), (1.6)
exist for all A ∈ O and define a * -automorphism γ + λ : O → O.

Remark.

Under additional regularity assumptions one can also obtain information about the rate of convergence in (1.6), see [START_REF] Jakšić | On the strict positivity of entropy production[END_REF] for details. Although Theorem 1.2 is a well-known result (see [START_REF] Robinson | Return to equilibrium[END_REF][START_REF] Botvich | Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas[END_REF][START_REF] Botvich | Asymptotic completeness and all that for an infinite number of fermions[END_REF][START_REF] Fröhlich | Dissipative transport: thermal contacts and tunneling junctions[END_REF]), for the reader convenience we will sketch its proof in Subsection 2.2.

The model and the result

Our starting point are finitely many, say M , independent free Fermi gasses R j in equilibrium at inverse temperatures β j > 0 and chemical potentials µ j ∈ R. More precisely, R j is described by the quantum dynamical system (O j , τ j , ω j ) where:

(i) O j = CAR(h j ) is the CAR algebra over the single fermion Hilbert space h j ;

(ii) τ t j is the group of Bogoliubov * -automorphisms generated by the single fermion Hamiltonian h j ;

(iii) ω j is the gauge-invariant quasi-free state generated by

T j = 1 1 + e βj (hj -µj ) .
We denote by ϑ j the gauge group of R j . The generators of τ j and ϑ j are denoted by δ j and ξ j .

Let

h = M j=1 h j , h 0 = M j=1 h j , T = M j=1
T j .

The joint system R = R j in absence of interaction is described by the quantum dynamical system (O, τ 0 , ω), where O = CAR(h), τ t 0 is the group of Bogoliubov * -automorphisms generated by h 0 , and ω is the gauge-invariant quasi-free state generated by T . We denote by δ 0 the generator of τ 0 and by ξ the generator of the gauge group ϑ of the joint system. Obviously, δ 0 = j δ j and ξ = j ξ j .

Let V ∈ O ϑ be a perturbation of the form (1.2). This perturbation describes the coupling of the reservoirs, and, possibly, self-interactions within the reservoirs. Let λ ∈ R be a coupling constant and τ λ the C * -dynamics on O generated by δ λ = δ 0 + iλ[V, • ]. The interacting joint system is described by the quantum dynamical system (O, τ λ , ω).

Let γ + λ be as in Theorem 1.2 and

ω λ+ = ω • γ + λ . A consequence
lim t→+∞ η • τ t λ (A) = ω λ+ (A).
The state ω λ+ is the NESS of the quantum dynamical system (O, τ λ , ω) [START_REF] Ruelle | Natural nonequilibrium states in quantum statistical mechanics[END_REF][START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF]. Clearly, this NESS depends on β j and µ j .

Let β eq > 0 and µ eq ∈ R be given (equilibrium) values of the inverse temperature and chemical potential. We are interested in linear response of R to thermodynamical forces

X j = β eq -β j , Y j = β j µ j -β eq µ eq . Let X = (X 1 , • • • , X M ), Y = (Y 1 , • • • , Y M ).
We indicate the dependence on X, Y by denoting

ω XY = ω, ω λXY + = ω λ+ , T XY = T.
Note that by Araki perturbation theory ω λ00+ is the unique β eq -KMS state for the C * -dynamics τ t λ • ϑ -µeqt . We denote this state by ω λeq .

In what follows we shall assume:

(A2) The operators h j are bounded.

Although our method of proof extends to unbounded h j 's (see Remark 2 after Theorem 1.5), the above assumption covers most cases of physical interest to which our results apply and allows for technically simpler exposition of the proofs.

The observables describing the heat and charge flux out of R j are

Φ j = λδ j (V ), J j = λξ j (V ). (1.7) Clearly, Φ j , J j ∈ O ϑ . The conservation laws M j=1 ω λXY + (Φ j ) = 0, M j=1 ω λXY + (J j ) = 0,
hold. The entropy production of the NESS ω λXY + is defined by

Ep(ω λXY + ) = ω λXY +   - M j=1 β j (Φ j -µ j J j )   = M j=1 X j ω λXY + (Φ j ) + M j=1 Y j ω λXY + (J j ).
By the general results of [START_REF] Ruelle | Entropy production in quantum spin systems[END_REF][START_REF] Jakšić | Non-equilibrium steady states for finite quantum systems coupled to thermal reservoirs[END_REF] (see also [START_REF] Tasaki | Fluctuation theorem, nonequilibrium steady states and MacLennan-Zubarev ensembles of a class of large quantum systems[END_REF][START_REF] Fröhlich | Dissipative transport: thermal contacts and tunneling junctions[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF]), Ep(ω λXY + ) ≥ 0. The strict positivity of the entropy production for locally interacting fermionic reservoirs can be established by using either perturbative arguments (see [FMU]) or stability arguments (see Section 4.3 in [START_REF] Jakšić | Mathematical theory of non-equilibrium quantum statistical mechanics[END_REF] and [START_REF] Jakšić | On the strict positivity of entropy production[END_REF]). This point is discussed in more detail in the forthcoming review [START_REF] Jakšić | [END_REF].

To study linear response of ω λXY + , in addition to (A1)-(A2) we need the following regularity assumption.

(A3) For all j and g ∈ D 0 , h j g ∈ D.

Our final assumption concerns time-reversal invariance.

(A4) There exists a complex conjugation c on h which commutes with all h j and satisfies cg = g for all g ∈ D 0 .

If (A4) holds, then the map Θ(a

# (f )) = a # (cf ) extends to an involutive skew * -automorphism of O such that Θ • τ t j = τ -t j • Θ and Θ(V ) = V . This implies that Θ • τ t λ = τ -t λ • Θ for all λ. Note also that Θ(Φ j ) = -Φ j , Θ(J j ) = -J j .
We set

I ǫ = {(X, Y ) ∈ R 2M | |X j | < ǫ, |Y j | < ǫ}, D ǫ = {(X, Y ) ∈ C 2M | |X j | < ǫ, |Y j | < ǫ}, R Λ,δ = {λ ∈ C | |Re λ| < Λ, |Im λ| < δ}.
In the sequel F j stands for either Φ j or J j . Our first result is:

Theorem 1.4 Suppose that Assumptions (A1)-(A3) hold and let 0 < Λ < Λ 0 . Then there exist ǫ > 0 and δ > 0 such that the maps

(λ, X, Y ) → ω λXY + (F j ),
extend to analytic functions on the set R Λ,δ × D ǫ . In particular, for any |λ| < Λ 0 there exists ǫ(λ) > 0 such that the maps

(X, Y ) → ω λXY + (F j ),
extend to analytic functions on D ǫ(λ) .

The kinetic transport coefficients are defined by

L kj λhh = ∂ Xj ω λXY + (Φ k ) X=Y =0 , L kj λhc = ∂ Yj ω λXY + (Φ k ) X=Y =0 , L kj λch = ∂ Xj ω λXY + (J k ) X=Y =0 , L kj λcc = ∂ Yj ω λXY + (J k ) X=Y =0 , (1.8)
where the indices h/c stand for heat/charge. For A, B ∈ O ϑ we set

L λ (A, B) = lim t→+∞ 1 2 t -t ω λeq (τ s λ (A)B) ds,
and

L λ (A, B) = lim t→+∞ 1 β eq t 0 ds βeq 0 du ω λeq (τ s λ (A)τ iu λ (B)),
whenever the limits exist. Our main result is:

Theorem 1.5 Suppose that Assumptions (A1)-(A3) hold and that |λ| < Λ 0 . Then L λ (A, B) is well-defined for A, B ∈ {Φ 1 , • • • , Φ M , J 1 , • • • , J M } and L kj λhh = L λ (Φ k , Φ j ), L kj λhc = L λ (Φ k , J j ), L kj λch = L λ (J k , Φ j ), L kj λcc = L λ (J k , J j ).
(1.9)

Assume in addition that (A4) holds. Then L λ (A, B) is well-defined for A, B ∈ {Φ 1 • • • Φ M , J 1 , • • • J M }, L kj λhh = L λ (Φ k , Φ j ), L kj λhc = L λ (Φ k , J j ), L kj λch = L λ (J k , Φ j ), L kj λcc = L λ (J k , J j ),
(1.10) and

L kj λhh = L jk λhh , L kj λcc = L jk λcc , L kj λhc = L jk λch .
(1.11)

Remark 1.The formulas (1.9) are the GKF without time reversal assumption. The formulas (1.10) are the GKF in the standard form. The formulas (1.11) are the Onsager reciprocity relations. The ORR are an immediate consequence of (1.10) and the KMS condition, see [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF]. Remark 2. If n = 1, then our proofs give that Theorems 1.1-1.5 hold with Λ 0 = 1/2Kℓ 0 . However, since in this case the coupled system is quasi-free, these theorems can be also proven using trace class scattering theory which yields better constants and wealth of additional information about the model. For more information about this special case we refer the reader to [AJPP1, AJPP2, JKP]. Remark 3. With regard to the Green-Kubo formulas (1.10), a natural question is whether the correlation functions

t → ω λeq (τ t λ (A)B) are absolutely integrable for A, B ∈ {Φ 1 , • • • , Φ M , J 1 , • • • J M }.
This is a delicate dynamical problem which is studied in [JPP]. In this paper we only establish the existence of the improper integrals

lim t→+∞ t -t ω λeq (τ s λ (A)B) ds.
Remark 4. By Theorem 1.4, the functions λ → L kj λuv , u, v ∈ {h, c}, are analytic for |λ| < Λ 0 and can be expanded into power series whose coefficients can be computed. Such computations can be used to verify that in specific examples the transport coefficients are non-vanishing. For reasons of space we shall discuss these perturbative computations in the forthcoming review [START_REF] Jakšić | [END_REF]. Remark 5. Our results are tailored for application to tight-binding type models of electronic transport in which Assumption (A2) is usually satisfied. However, all our proofs extend to unbounded h j 's as long as D 0 ⊂ Dom (e a|hj | ) for all j and some a > β eq . It is an interesting technical problem to prove Theorems 1.4 and 1.5 for unbounded h j 's without this additional technical assumption. Remark 6. Theorems 1.1-1.3 are fairly flexible and are easily adapted to a number of different setups involving free Fermi gas reservoirs. The same applies to Theorems 1.4 and 1.5. For example, one may consider the tensor product structure, where the joint system in absence of interaction is described by

O = O 1 ⊗ • • • ⊗ O M , τ 0 = τ 1 ⊗ • • • ⊗ τ M , ω = ω 1 ⊗ • • • ⊗ ω M .
This type of models was studied in [FMU]. Another class of related models are local perturbations of the exactly solvable Electronic Black-Box Model studied in [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF][START_REF] Aschbacher | Transport properties of quasi-free Fermions[END_REF]. Instead of coupled free fermionic systems one may consider coupled X -Y quantum spin chains. Theorems 1.4 and 1.5 extend to these models with only notational changes, see [START_REF] Jakšić | [END_REF] for details. Remark 7. We call A ∈ O centered if ω λXY (A) = 0 for all |λ| < Λ 0 and (X, Y ) ∈ I ǫ . Our proof easily extends to the general Green-Kubo formulas

∂ Xj ω λXY + (A) X=Y =0 = L λ (A, Φ j ), ∂ Yj ω λXY + (A) X=Y =0 = L λ (A, J j ),
for centered observables A which are polynomials in a # (f ) with f ∈ D.

We finish this subsection with some examples to which Theorems 1.1-1.5 apply. Let G be the set of vertices of a connected graph of bounded degree and ∆ G the associated discrete Laplacian acting on l 2 (G). We recall that

(∆ G ψ)(x) = |y-x|=1 ψ(y),
where |y -x| is the distance on the graph. ∆ G is a bounded self-adjoint operator and ∆ G = sup x∈G d(x), where d(x) is the degree of the vertex x. Let δ x be the Kronecker delta function at x ∈ G. We shall call the graph G admissible if there exists γ > 1 such that for all x, y ∈ G,

|(δ x , e -it∆G δ y )| = O(|t| -γ ),
(1.12) as t → ∞. Clearly, the discrete Laplacian of an admissible graph has purely absolutely continuous spectrum. An example of admissible graph is G = Z d for d ≥ 3. In this case γ = d/2. Another example is the half-space

G = Z + × Z d-1 where Z + = {0, 1, • • • } and d ≥ 1 (if d = 1 then G = Z + ). In this case γ = (d + 2)/2.
Tubular graphs of the type Z + × Γ, where Γ ⊂ Z d-1 is finite, are admissible with γ = 3/2. Another well-known admissible graph is a rooted Bethe lattice where γ = 3/2.

Assumptions (A1)-(A4) and Theorems 1.1-1.5 hold if (i) G 1 , . . . , G M are admissible graphs;

(ii) h j = ℓ 2 (G j ) or more generally ℓ 2 (G j ) ⊗ C L to allow for internal degrees of freedom (e.g., spin);

(iii) D is the subspace of finitely supported elements of h;

(iv) h j = -∆ Gj ;

(v) u kj , v kj belong to D.

Allowed interactions include V = V hop + V int where (i) V hop describes tunneling junctions between the reservoirs:

V hop = x,y t(x, y) (a * (δ x )a(δ y ) + a * (δ y )a(δ x )) ,
where t : G × G → R is a finitely supported function (G = ∪ j G j );

(ii) V int is a local pair interaction

V int = x,y v(x, y)a * (δ x )a * (δ y )a(δ y )a(δ x ),
where v : G × G → R is finitely supported.

This concrete model is studied in detail in [START_REF] Jakšić | [END_REF].

Basic properties of the model

In this section we prove Theorems 1.1, 1.2, and 1.3.

Proof of Theorem 1.1

We start with some preliminaries which are of independent interest.

Let A = a 1 • • • a m and B = b 1 • • • b q
where the a k and b j are creation/annihilation operators. Thus, A and B are monomials of order m and q respectively. If q is even it follows from the CAR that

[B, a j ] = b 1 • • • b q a j -a j b 1 • • • b q = b 1 • • • b q a j -({b 1 , a j } -b 1 a j )b 2 • • • b q = -{b 1 , a j }b 2 • • • b q + b 1 (b 2 • • • b q a j + a j b 2 • • • b q ) . . . = q k=1 (-1) k {b k , a j }b 1 • • • b k-1 b k+1 • • • b q ,
and hence

[B, A] = m j=1 a 1 • • • a j-1 [B, a j ]a j+1 • • • a m = m j=1 q k=1 (-1) k {b k , a j }a 1 • • • a j-1 b 1 • • • b k-1 b k+1 • • • b n a j+1 • • • a m .
The anticommutator {b k , a j } on the right hand side is called contraction of the factor b k of B with the factor a j of A. Note that contractions are numbers.

Iterating the last formula we get, for any monomials B 1 , B 2 , . . . B n+1 of even orders q 1 , q 2 , . . . q n+1 and any monomial A of order m

[B n+1 , [τ sn 0 (B n ), [• • • , [τ s1 0 (B 1 ), τ s0 0 (A)] • • • ]]] = p∈Pn(A,B1,...,Bn+1) G p (s 0 , • • • , s n )F p , (2.13)
where the F p are monomials of order q 1 + q 2 + • • • + q n+1 + m -2(n + 1) and the coefficients G p are products of n + 1 contractions. The sum on the right hand side runs over the set P n (A, B 1 , . . . , B n+1 ) whose elements p are contraction diagrams of the type displayed in Figure 1. Each line of this diagram represents a monomial, as labeled on the left. Each dot on a line represents a factor of the corresponding monomial. The dashed lines represent contractions of such factors. From each line of the diagram there is exactly one contraction going up and any factor can belong only to one contraction. To a contraction diagram p we associate its skeleton: a rooted tree T whose nodes are 0 (the root), 1, . . . , n + 1 and whose bonds correspond to the contractions (see Figure 2). The skeleton T is simply obtained by collapsing each line of the contraction diagram p to a single node. If there is an arrow going from the node j to the node k in T we say that j is a child of k or that k is the parent of j (each node has a unique parent and we shall say that the root node 0 is its own parent). We can describe the rooted tree T by the function T : {0, . . . , n + 1} → {0, . . . , n + 1} which to a node j associates its parent T (j). Reciprocally, any function T such that T (0) = 0 and T (j) < j for j = 1, . . . , n+1 defines a rooted tree T . Such a function is called a climber of order n+1 and there is a one-to-one correspondence between climbers and rooted trees.

s 0 s 1 s 2 s 3 s n A B 1 B 2 B 3 B n s n+1 = 0 B n+1
Suppose that all the factors of the monomials B 1 , . . . , B n+1 are from {a # (g) | g ∈ D 0 } and let A and A be as in Theorem 1.1. Then, the factors of the monomials F p are from {a # (e ish0 g) | g ∈ D 0 , s ∈ {0, s 1 , . . . , s n }} ∪ {a # (e is0h0 g) | g ∈ A}.

The number of factors from the first set does not exceed (n + 1)(q -1), where q = max q k . The number of factors from the second set does not exceed m -1. If we denote

S k (t) ≡      sup f ∈D0,g∈A |(f, e ith0 g)| for k = 0, sup f,g∈D0 |(f, e ith0 g)| for k > 0,
then all coefficients G p associated with a given skeleton tree T are bounded by

|G p | n+1 j=1 S T (j) (s T (j) -s j ) ≡ S(T ),
where we set s n+1 = 0. Thus, if N (T ) denotes the number of contraction diagrams with skeleton tree T we have

p∈Pn |G p | T ∈Tn+1

N (T )S(T ),

where T n+1 denotes the set of all rooted trees with nodes 0 (the root), 1, . . . , n + 1. Let us compute N (T ). To this end denote by r j the number of childs of the node j. For the tree of Figure 2 we have for example r 0 = 2, r 1 = 2, r 2 = r 3 = 0, r n = 1 and r n+1 = 0. Clearly, N (T ) = 0 if r 0 > m or r j > q j -1. Otherwise, to construct a

s 0 s 3 s 1 s 2 s n s n+1
Figure 2: The skeleton tree corresponding to Figure 1.

diagram p whose skeleton is T we first have to choose a factor on each line B 1 , . . . , B n+1 . The number of such choices is clearly q 1 q 2 • • • q n+1 . Now on line A we have to choose one factor for each of the r 0 childs of node 0. There are m(m -1) • • • (m -r 0 + 1) such choices. Similarly, on line B 1 we have to chose r 1 factors out of the q 1 -1 remaining. There are (q 1 -1)(q 1 -2) • • • (q 1 -r 1 ) such choices. The same reasoning applies to lines B 2 , . . . , B n , and we conclude that

N (T ) = m! (m -r 0 )! n j=1 q j ! (q j -r j -1)! ≤ m! (m -r 0 )! n j=1 q! (q -r j -1)! ≡ N (T ).
We now turn to the proof of Theorem 1.1. Since V = K k=1 V k where V k are monomials of order q k = 2n k (so q = 2n), we can write

C (n) A (s 0 , . . . , s n ) = K k1,...,kn+1=1 [V kn+1 , [τ sn 0 (V kn ), [• • • , [τ s1 0 (V k1 ), τ s0 0 (A)] • • • ]]],
and Parts (1) and (2) follow immediately with

P n (A) ≡ K k1,...,kn+1=1
{(k 1 , . . . , k n+1 )} × P n (A, V k1 , . . . , V kn+1 ).

To prove (3), we start with the estimate

p∈Pn(A) |G (n) A,p | ≤ K n+1 T ∈Tn+1

N (T )S(T ).

Hence,

W ≡ ∞ n=0 |λ| n+1 p∈Pn(A) 0≤sn≤•••≤s0 |G (n) A,p (s 0 , . . . , s n )| ds 0 • • • ds n , satisfies W ≤ W 0 ≡ ∞ n=1 T ∈Tn N (T ) 0=sn≤sn-1•••≤s0 n j=1 (|λ|KS T (j) (s T (j) -s j )) ds 0 • • • ds n-1 .
We will need the following general result of [BGM].

Theorem 2.1 Let m k , m k be two sequences of nonnegative numbers and g, g two integrable nonnegative functions on [0, ∞[. Denote by g and g their L 1 -norms, set g 0 = g and g k = g for k > 0 and define

M (x) ≡ ∞ k=0 m k k! x k , M (x) ≡ ∞ k=0 m k k! x k .
To any rooted tree T ∈ T n associate the weight (recall that r j is the number of childs of the node j),

w(T ) = m r0 m r1 • • • m rn 0=sn≤sn-1≤•••≤s0 n j=1 g T (j) (s T (j) -s j ) ds 0 • • • ds n-1 .
Then, the sum W = ∞ n=1

T ∈Tn w(T ) is finite if and only if the equation M ( g x) = x has a positive solution x such that M ( g x) < ∞. If x * denotes the least such solution, then W = M ( g x * ).

To apply this result we set m k = 0 for k = 0 and k > m, otherwise

m k = m! (m -k)! , m k = 0 for k ≥ 2n, otherwise m k = (2n)! (2n -k -1)! , and
g(s) = |λ|KS 1 (s), g(s) = |λ|KS 0 (s).
Hence, M (x) = 2n(1+x) 2n-1 , M (x) = (1+x) m -1, g = |λ|Kℓ 0 , and g = |λ|Kℓ. An elementary analysis shows that, if

Λ 0 =      1 2Kℓ 0 for n = 1, 1 2nKℓ 0 (2n -2) 2n-2 (2n -1) 2n-1 for n > 1,
, then, as long as |λ| < Λ 0 for n = 1 and |λ| ≤ Λ 0 for n > 1, the equation M ( g x) = x has a least positive solution x * satisfying

0 ≤ x * ≤ 2n (1 -|λ|/Λ 0 ) + 2n(2n -2)Kℓ 0 |λ| ,
and that

W ≤ W 0 = (1 + Kℓ|λ|x * ) m -1 ≤ 1 + 2nKℓ|λ| (1 -|λ|/Λ 0 ) + 2n(2n -2)Kℓ 0 |λ| m -1.
This ends the proof of Theorem 1.1.

Proofs of Theorems 1.2 and 1.3

Proof of Theorem 1.2. To establish the existence of the limit (1.6) for all A ∈ O it suffices to consider the case

A = a # (f ) with f ∈ D and f = 1. Since τ -t2 0 • τ t2 λ (A) -τ -t1 0 • τ t1 λ (A) = iλ t2 t1 τ -s 0 ([V, τ s λ (A)]) ds,
we have that

τ -t2 0 • τ t2 λ (A) -τ -t1 0 • τ t1 λ (A) ≤ |λ| t2 t1 [V, τ s λ (A)] ds.
(2.14)

The expansion (1.1) yields

[V, τ s λ (A)] = [V, τ s 0 (A)] + ∞ n=1 (iλ) n 0≤sn≤•••≤s1≤s [V, [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ s 0 (A)] • • • ]]] ds 1 • • • ds n .
Our standing assumption (1.3) and the fact that f = 1 implies that F

A,p ≤ 1 and we can estimate

[V, [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ s 0 (A)] • • • ]]] ≤ p∈Pn(A) |G (n) A,p (s, s 1 , • • • , s n )|. Part (3) of Theorem 1.1 yields that for |λ| < Λ 0 , ∞ 0 [V, τ s λ (A)] ds < ∞.
(2.15)

The estimates (2.14) and (2.15) imply the existence of the limit (1.6) for |λ| < Λ 0 . The map γ + λ is obviously a * -morphism. To prove that it is an isomorphism, it suffices to show that the limits

lim t→+∞ τ -t λ • τ t 0 (A),
exist for all A ∈ O. Repeating the above argument we see that it suffices to show that

∞ 0 [V, τ s 0 (A)] ds < ∞, for A = a # (f ), f ∈ D.
But this is a special case of Equ. (2.15). 2

Proof of Theorem 1.3. Since h has purely absolutely continuous spectrum the quantum dynamical system (O, τ 0 , ω) has the property of return to equilibrium: for all ω-normal states η and A ∈ O,

lim |t|→∞ η • τ t 0 (A) = ω(A),
see, e.g., [START_REF] Aschbacher | Topics in non-equilibrium quantum statistical mechanics[END_REF]. The existence of norm-limits (1.6) ensures that

lim t→+∞ η(τ t λ (A)) = lim t→+∞ η • τ t 0 (τ -t 0 • τ t λ (A)) = lim t→+∞ η • τ t 0 (γ + λ (A)) = ω(γ + λ (A)),
and the statement follows. 2

3 Proofs of Theorems 1.4 and 1.5

Strategy

The strategy of the proofs of Theorems 1.4 and 1.5 is based on the arguments in [START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF]. Consider the C * -dynamics σ XY on O generated by

δ XY = δ 0 -µ eq ξ - j X j β eq δ j - j Y j β eq ξ j .
(3.16)

The reference state ω XY is the unique (σ XY , β eq )-KMS state on O. Let σ λXY be the C * -dynamics on O generated by

δ λXY = δ XY + iλ[V, • ].
The Araki perturbation theory [Ar, BR2, DJP] yields that there exists a unique (σ λXY , β eq )-KMS state on O. We denote this state by ω λXY . The states ω XY and ω λXY are mutually normal.

Recall that F j stands for either Φ j or J j . Our main technical result is:

Theorem 3.1 Suppose that Assumptions (A1)-(A3) hold and let 0 < Λ < Λ 0 . Then there exist ǫ > 0 and δ > 0 such that for all t ≥ 0 the functions

(λ, X, Y ) → ω λXY (τ t λ (F j )) have analytic extensions to R Λ,δ × D ǫ satisfying sup λ∈R Λ,δ ,(X,Y )∈Dǫ,t≥0 ω λXY (τ t λ (F j )) < ∞.
This result and the multi-variable Vitali theorem yield Theorem 1.4 (see Theorem 2.3 in [START_REF] Jakšić | The Green-Kubo formula for the spin-fermion system[END_REF]). Moreover, the relations

∂ Xj ω λXY + (F k ) = lim t→+∞ ∂ Xj ω λXY • τ t λ (F k ), ∂ Yj ω λXY + (F k ) = lim t→+∞ ∂ Yj ω λXY • τ t λ (F k ), (3.17) hold for (λ, X, Y ) ∈ R Λ,δ × D ǫ .
The proof of Relations (1.9) is completed by invoking the following identities proven in [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF]:

∂ Xj ω λXY (τ t λ (F k )) X=Y =0 = 1 β eq t 0 ds βeq 0 du ω λeq (τ s λ (F k )τ iu λ (Φ j )), ∂ Yj ω λXY (τ t λ (F k )) X=Y =0 = 1 β eq t 0 ds βeq 0 du ω λeq (τ s λ (F k )τ iu λ (J j )).
Proposition 4.1 in [START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF] yields that (1.9) and time-reversal invariance (A4) imply (1.10). The KMS condition and (1.10) imply (1.11) [START_REF] Jakšić | The Green-Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics[END_REF][START_REF] Jakšić | Linear response theory for thermally driven quantum open systems[END_REF]. Hence, to complete the proofs of Theorems 1.4 and 1.5 we need to establish Theorem 3.1.

Proof of Theorem 3.1

The GNS representation of the algebra O associated to the gauge-invariant quasi-free state ω XY can be explicitly computed [START_REF] Araki | Representations of canonical anti-commutation relations[END_REF][START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 2[END_REF]. Let F be the anti-symmetric Fock space over h. We denote by Ω f the vacuum vector and by N the number operator. Let

H = F ⊗ F, Ω = Ω f ⊗ Ω f .
In the sequel B(H) denotes the C * -algebra of all bounded operators on a Hilbert space H. Let C j be given complex conjugations on h j and C = ⊕ j C j . Without loss of generality we may assume that C j commutes with h j . As usual, we denote Cf = f . The map In what follows we suppose that Assumptions (A1)-(A3) hold. By adding a constant to µ eq without loss of generality we may assume that h j ≥ 0.

π XY (a(f )) = a((I -T XY ) 1/2 f ) ⊗ I + (-I) N ⊗ a * (T 1/2 XY f ),
θ(s, x, y) = -u(s) 1 -i b β -a -i d + b βµ + c β -a , where u(s) = (β -a) s - βµ + c β -a . If (x, y) ∈ M δ with δ < β, then b β -a < δ β -δ , d + b βµ + c β -a < δ β(1 + |µ|) β -δ ,
and it follows that θ(R + × M δ ) is contained in the dashed region of Figure 3. An elementary calculation shows that for δ < ǫ(β, µ) this region does not intersect the half-lines R + ± iπ/2. Another elementary calculation shows that 1 + e θ(R+×M δ ) is contained in a bounded region of the half-plane z ∈ C Re z > 1 -e -πβ(δ -1 -ǫ(β,µ) -1 ) .

Thus, l -is a bounded continuous function on R + × M δ which is clearly analytic in (x, y) for any fixed s ∈ R + . This yields the result since e θ/2 has obviously the same properties and l + = e θ/2 l -for real s, x, y. 2

The spectral theorem and Lemma 3.2 yield

Lemma 3.3 The maps (X, Y ) → (I -T XY ) 1/2 ∈ B(h), (X, Y ) → T 1/2 XY ∈ B(h),
extend to analytic B(h)-valued functions on D ǫ(βeq,µeq) .

Since for X, Y real, π XY (a # (f )) = f , Lemma 3.3 implies Lemma 3.4 For any δ > 0 there exists ǫ(δ) > 0 such that for all f ∈ h the operator-valued function Recall that δ XY is defined by (3.16). Let

(X, Y ) → π XY (a # (f )) ∈ B(H), has an analytic extension to D ǫ(δ) which satisfies sup (X,Y )∈D ǫ(δ) π XY (a # (f )) ≤ (1 + δ) f . iπ/2 -iπ/2 < δ 2β(|µ| + 1) β -δ iδ β(|µ| + 1) β -δ Slope ± δ β -δ
h XY = h 0 -µ eq - j X j β eq h j - j Y j β eq p j = j
β eq -X j β eq h j -β eq µ eq + Y j β eq p j ,

where p j is the orthogonal projection on h j . Clearly, e tδXY (a # (f )) = a # (e ithXY f ) is, for fixed t, an analytic function of X, Y . Set

V XY (s) = K k=1 n k j=1 a * (e -shXY u kj )a(e shXY v kj ), and
G λXY = 1l + n≥1 (-λβ eq ) n 0≤sn≤•••≤s1≤1 V XY (β eq s n ) • • • V XY (β eq s 1 ) ds 1 • • • ds n .
Araki's perturbation theory [Ar, BR2, DJP] yields that for X, Y real the state ω λXY can be expressed in terms of ω XY as

ω λXY (A) = ω XY (AG λXY ) ω XY (G λXY ) . (3.18) Lemma 3.5 The function (t, λ, X, Y ) → π XY (τ t 0 (G λXY )) ∈ B(H)
, extends to a continuous function on R × C × D ǫ(βeq,µeq) which is analytic in (λ, X, Y ) for fixed t. Moreover, for all Λ > 0 and 0 < ǫ < ǫ(β eq , µ eq ), sup t∈R,λ∈C,|λ|<Λ,(X,Y )∈Dǫ π XY (τ t 0 (G λXY )) < ∞.

Proof. Since for X, Y real,

π XY (τ t 0 (G λXY )) = 1l + n≥1 (-λβ eq ) n 0≤sn≤•••≤s1≤1 π XY (V XY (β eq s n , t)) • • • π XY (V XY (β eq s 1 , t)) ds 1 • • • ds n , where π XY (V XY (β eq s, t)) = K k=1 n k j=1
π XY (a * (e -βeqshXY e ith0 u kj ))π XY (a(e βeqshXY e ith0 v kj )), the statement follows from Lemma 3.4. 2 Lemma 3.6 For all t and A ∈ O,

ω λXY (τ t λ (A)) = ω λXY (τ t 0 (A)) + iλ ω XY (G λXY ) t 0 ω XY ([V, τ s λ (A)]τ s-t 0 (G λXY )) ds. (3.19) Proof. Relation (3.18) yields (ω λXY (τ t λ (A)) -ω λXY (τ t 0 (A)))ω XY (G λXY ) = ω XY ((τ t λ (A) -τ t 0 (A))G λXY ).
Since ω XY is τ 0 -invariant we have Proof. For X, Y real, ω λXY (τ t 0 (F j )) =

ω XY ((τ t λ (A) -τ t 0 (A))G λXY ) = ω XY ((τ -t 0 • τ t λ (A) -A)τ -t 0 (G λXY )) = iλ t 0 ω XY (τ -s 0 ([V, τ s λ (A)])τ -t 0 (G λXY ) ds = iλ t 0 ω XY ([V, τ s λ (A)]τ s-t 0 ( G
(Ω, π XY (F j )π XY (τ -t 0 (G λXY ))Ω) ω XY (G λXY ) .

This identity and Lemmas 3.3, 3.5, and 3.7 yield the statement. 2 Lemma 3.9 Let 0 < Λ < Λ 0 be given. Then there exists ǫ > 0 such that for all A = a # (f 1 ) • • • a # (f m ) with f j ∈ D, the map (t, λ, X, Y ) → π XY ([V, τ t λ (A)]) ∈ B(H), π XY ([V, τ t λ (F j )]) dt < ∞.

2

We are now ready to complete: Proof of Theorem 3.1. We start with formula (3.19). By Lemmas 3.7 and 3.8, it suffices to show that for some ǫ > 0 the functions (λ, X, Y ) → By (A2) and (A3), every F j can be written as a finite sum of monomials a # (f 1 ) • • • a # (f m ) with f k ∈ D, and the result follows from Lemma 3.9. 2

Figure 1 :

 1 Figure 1: An element of the set P n .

  uniquely extends to a representation π XY : O → B(H) and the triple (H, π XY , Ω) is the GNS-representation of the algebra O associated to the state ω XY .

Lemma 3. 2

 2 For β > 0 and µ ∈ R set ǫ(β, µ) = πβ π + 4β(|µ| + 1) < β. The functions l ± (s, x, y) = 1 + e ±[(β-x)s-(βµ+y)] -1/2 , are continuous and, for fixed s, analytic in (x, y) on the set {(s, x, y) ∈ R + × C 2 | |x| < ǫ(β, µ), |y| < ǫ(β, µ)}. Moreover, for any δ < ǫ(β, µ) one has sup s∈R+,(x,y)∈C 2 ,|x|<δ,|y|<δ |l ± (s, x, y)| < ∞. Proof. Set x = a + ib and y = c + id with a, b, c, d ∈ R, M δ = {(x, y) ∈ C 2 | |x| < δ, |y| < δ} and write the exponent in l -as

Figure 3 :

 3 Figure 3: The range of the exponent θ(s, x, y).

Lemma 3 . 7 Lemma 3 . 8

 3738 λXY )) ds, and (3.19) follows. 2 For any Λ > 0 there exist ǫ > 0 and δ > 0 such that the function(λ, X, Y ) → ω XY (G λXY ),extends to an analytic function on C × D ǫ which satisfiesinf λ∈R Λ,δ ,(X,Y )∈Dǫ |ω XY (G λXY )| > 0.(3.20)Proof. Since ω XY (G λXY ) = (Ω, π XY (G λXY )Ω), the first statement is a special case of Lemma 3.5. Since ω XY (G λXY ) > 0 for λ, X, Y real, by continuity (3.20) holds for ǫ and δ small enough. 2 For any Λ > 0 there exist ǫ > 0 and δ > 0 such that for all t ∈ R the functions(λ, X, Y ) → ω λXY (τ t 0 (F j )),(3.21) extend to analytic functions on R Λ,δ × D ǫ such that sup λ∈R Λ,δ ,(X,Y )∈Dǫ,t∈R |ω λXY (τ t 0 (F j ))| < ∞.

,

  (3.22) extends to a continuous function onR + × {λ ∈ C | |λ| < Λ} × D ǫ which is analytic in (λ, X, Y ) for fixed t ∈ R. Moreover, ,(X,Y )∈Dǫ π XY ([V, τ t λ (A)]) dt < ∞. (3.23)Proof. The expansion (1.1) yields thatπ XY ([V, τ t λ (A)]) = π XY ([V, τ t 0 (A)]) + ∞ n=1 (iλ) n 0≤sn≤•••≤s1≤t π XY ([V, [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ t 0 (A)] • • • ]]]) ds 1 • • • ds n . Set C (0) XY = π XY ([V, τ t 0 (A)]), and, for n ≥ 1, C (n) XY (t, s 1 , . . . , s n ) = π XY [V, [τ sn 0 (V ), [• • • , [τ s1 0 (V ), τ t 0 (A)] . . . ]]] .Theorem 1.1 yields that for each n there exist a finite index set P n (A), scalar functions G (n)A,p which do not depend on X, Y , and monomialsF (n) A,p ∈ O such that C (n) XY (t, s 1 , . . . , s n ) = p∈Pn(A) G (n) A,p (t, s 1 , . . . , s n )π XY (F (n) A,p ).Recall our standing assumption (1.3) and Part (2) of Theorem 1.1. Let δ > 0 be such thatC 0 ≡ (1 + δ) 2n-1 < Λ 0 Λ . (3.24)Applying Lemma 3.4 with this δ to the factors of π XY (F(n)A,p ) we conclude that there exists ǫ > 0 (which depends on δ) such that for all n the functions(X, Y ) → π XY (F (n) p ) ∈ B(H),extend to analytic functions on D ǫ satisfying sup t,s1,...sn∈R,(X,Y )∈Dǫπ XY (F (n) p ) ≤ C 1 C n+1 0 where C 1 = (1 + δ) m-1 [max(1, f 1 , • • • , f m )]m-1 . By Part (3) of Theorem 1.1, (t, s 1 , . . . , s n )| dtds 1 • • • ds n < ∞, and we conclude that ∞ 0 sup λ∈C,|λ|<Λ,(X,Y )∈Dǫ

  π XY ([V, τ s λ (F j )])π XY (τ s-t 0 (G λXY ))Ω) ds, extend to analytic functions on {λ ∈ C | |λ| < Λ} × D ǫ such that sup λ∈C,|λ|<Λ,(X,Y )∈Dǫ,t≥0 t 0 (Ω, π XY ([V, τ s λ (F j )])π XY (τ s-t 0 (G λXY ))Ω) ds < ∞.By Lemma 3.5, it suffices to show that the functions(t, λ, X, Y ) → π XY ([V, τ t λ (F j )]) ∈ B(H), extend to continuous functions on R + × {λ ∈ C | |λ| < Λ} × D ǫ which,for fixed t, are analytic in (λ, X, Y ) and satisfy the bound ∞ 0 sup λ∈C,|λ|<Λ,(X,Y )∈Dǫ π XY ([V, τ t λ (F j )]) dt < ∞.

  Assume that (A1) holds and that |λ| < Λ 0 . Then for all ω-normal states η and A ∈ O,

	of Theorem 1.2 (see Subsection 2.2) is:
	Theorem 1.3