On the inequivalence of statistical ensembles

We investigate the relation between various statistical ensembles of finite systems. If ensembles differ at the level of fluctuations of the order parameter, we show that the equations of states can present major differences. A sufficient condition for this inequivalence to survive at the thermodynamical limit is worked out. If energy consists in a kinetic and a potential part, the microcanonical ensemble does not converge towards the canonical ensemble when the partial heat capacities per particle fulfill the relation $c_{k}^{-1}+c_{p}^{-1}<0$.Comment: 4 pages, 4 figure

Microcanonical mean-field thermodynamics of self-gravitating and rotating systems

We derive the global phase diagram of a self-gravitating $N$-body system enclosed in a finite three-dimensional spherical volume $V$ as a function of total energy and angular momentum, employing a microcanonical mean-field approach. At low angular momenta (i.e. for slowly rotating systems) the known collapse from a gas cloud to a single dense cluster is recovered. At high angular momenta, instead, rotational symmetry can be spontaneously broken and rotationally asymmetric structures (double clusters) appear.Comment: 4 pages, 4 figures; to appear in Phys. Rev. Let

Gravitational instability of slowly rotating isothermal spheres

We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plane is deformed by rotation. We find that gravitational instability occurs sooner in the microcanonical ensemble and later in the canonical ensemble. According to standard turning point arguments, the onset of the collapse coincides with the minimum energy or minimum temperature state in the series of equilibria. Interestingly, it happens to be close to the point of maximum flattening. We determine analytically the generalization of the singular isothermal solution to the case of a slowly rotating configuration. We also consider slowly rotating configurations of the self-gravitating Fermi gas at non zero temperature.Comment: Submitted to A&

Statistical mechanics and phase diagrams of rotating self-gravitating fermions

We compute statistical equilibrium states of rotating self-gravitating systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter and angular velocity. We argue that systems described by the Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric structures, in continuity with the case of polytropes with index n>0.808 (the Fermi gas at T=0 corresponds to n=3/2). This contrasts with the study of Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration space and find ``double star'' structures (their model at T=0 corresponds to n=0). We also discuss the influence of rotation on the onset of the gravothermal catastrophe for globular clusters. On general grounds, we complete previous investigations concerning the nature of phase transitions in self-gravitating systems. We emphasize the inequivalence of statistical ensembles regarding the formation of binaries (or low-mass condensates) in the microcanonical ensemble and Dirac peaks (or massive condensates) in the canonical ensemble. We also describe an hysteretic cycle between the gaseous phase and the condensed phase that are connected by a ``collapse'' or an ``explosion''. This notion of hysteresis in self-gravitating systems is new.Comment: submitted to A&

Effect of angular momentum on equilibrium properties of a self-gravitating system

The microcanonical properties of a two dimensional system of N classical particles interacting via a smoothed Newtonian potential as a function of the total energy E and the total angular momentum L are discussed. In order to estimate suitable observables a numerical method based on an importance sampling algorithm is presented. The entropy surface shows a negative specific heat region at fixed L for all L. Observables probing the average mass distribution are used to understand the link between thermostatistical properties and the spatial distribution of particles. In order to define a phase in non-extensive system we introduce a more general observable than the one proposed by Gross and Votyakov [Eur. Phys. J. B:15, 115 (2000)]: the sign of the largest eigenvalue of the entropy surface curvature. At large E the gravitational system is in a homogeneous gas phase. At low E there are several collapse phases; at L=0 there is a single cluster phase and for L>0 there are several phases with 2 clusters. All these pure phases are separated by first order phase transition regions. The signal of critical behaviour emerges at different points of the parameter space (E,L). We also discuss the ensemble introduced in a recent pre-print by Klinko & Miller; this ensemble is the canonical analogue of the one at constant energy and constant angular momentum. We show that a huge loss of informations appears if we treat the system as a function of intensive parameters: besides the known non-equivalence at first order phase transitions, there exit in the microcanonical ensemble some values of the temperature and the angular velocity for which the corresponding canonical ensemble does not exist, i.e. the partition sum diverges.Comment: 17 pages, 11 figures, submitted to Phys. Rev.

Phasenuebergaenge in Kleineren Systemen

Complete version Title & Content General introduction * * * Part I Thermostatistics of "small" systems * * * Chapter 1 Introduction and definitions 1.1 Microcanonical ensemble 1.2 Canonical ensemble 1.3 Microcanonical or canonical ensemble? 1.4 Toy models Chapter 2 Thermostatistics of small systems 2.1 Pure phases 2.2 First order phase transition 2.3 Second order phase transition 2.4 Single event as a signal of phase transition? 2.5 Alternative theories 2.6 Conclusions * * * Part II Liquid-gas transition of metallic clusters * * * Chapter 3 Low pressure and scaling properties 3.1 Introduction 3.2 Model 3.3 Simulation method 3.4 Results 3.5 Summary Chapter 4 Towards the critical point 4.1 Introduction 4.2 MMMC Results 4.3 Lattice model (CL) 3.4 Conclusions * * * Part III Self-gravitating systems * * * Chapter 5 Introduction Chapter 6 Microcanonical properties 6.1 Microcanonical definitions 6.2 Momentum average and dispersion 6.3 Numerical method 6.4 Results 6.5 Discussion and conclusions * * * General conclusion Appendices A Avoided volume B Technical "details" C Momentum distribution D Temperature at constant pressure BibliographyIn conventional thermostatistics there is no phase transition in "small" systems ("small" systems are those where the range of the forces is of the order of the system size). In fact, these systems do not exhibit the usual signals of phase transitions, i.e. Yang-Lee singularities. These singularities (divergences) can only occur at the thermodynamical limit. Nevertheless it is possible to define phases and phase transitions for "small" systems by means of local properties of their microcanonical entropy surface without invoking the thermodynamical limit. In the first part of this thesis, the present status of the theory is summarized. The definitions of phase and phase transitions are recalled. Their relation to the conventional ones is discussed. All these points are illustrated by analytical entropy models. The two other parts are dedicated to original studies of the microcanonical equilibrium properties of two "small" systems. First, the liquid-gas phase transition of sodium clusters composed by a few hundreds of atoms is discussed. At low pressure, their caloric curves as functions of the enthalpy show a region characterized by a negative specific heat capacity. This is the signal of a first order phase transition in ``small'' systems. For certain enthalpy-range, their mass distributions have some peculiarities (multifragmentation) which vanish at the thermodynamical limit. High pressures calculations show for the first time the critical point of this first order phase transition. This critical point is located at higher pressure and smaller temperature compared to the critical point of corresponding thermodynamical limit. The last part deals with self--gravitating systems. Although they are spatially very large they are "small" in the sense given above. These systems are studied in the microcanonical ensemble at constant energy E and total angular momentum L. They are studied without any a priori assumption about their spatial mass distributions (symmetry) and with a "realistic" potential. This is relevant for many astrophysical systems: from galaxies to (multiple-)stars formation. The entropy surface, its derivatives (temperature, angular velocity) and observables probing the mass distribution are worked out for the whole parameter space (E,L). These systems have a rich phase diagram with first order and several second order phase transitions. It is shown that all the properties of (astro-)physical importance are smeared out and lost if the intensive variables are fixed, i.e.\ in the canonical ensemble. Worst, for a given choice of intensive parameters, the partition function diverges for some microcanonical values of these intensive parameters.In der konventionellen Thermostatistik gibt es keine Phasenübergänge in "Kleinen" Systemen. (Systeme mit einer Wechselwirkung von einer Reichweite vergleichbar mit der Systemgrösse.) Diese Systeme zeigen nicht die Yang-Lee Singularitäten in den kanonischen Potentialen. Singularitäten können nur im thermodynamischen Limes auftreten. Dennoch kann man in der mikrokanonischen Statistik Phasen und Phasenuebergänge eindeutig auch für "Kleine" Systeme als lokale Besonderheiten der Entropie definieren. Im ersten Teil der Doktorarbeit wird der augenblickliche Stand der Theorie zusammengefasst. Die Definition der Phasen und der Phasenübergänge wird genannt und ihre Beziehung zur konventionellen Theorie diskutiert. Dies wird an Hand analytischer Modelle illustriert. Zwei weitere Teile der Arbeit behandeln die Eigenschaften des mikrokanonischen Gleichgewichts in zwei Beispielen "Kleiner" Systeme: Zuerst wird der flüssig-gas übergang in Natriumclustern mit einigen hundert Atomen diskutiert. Bei kleinem Druck zeigt die kalorische Kurve als Funktion der Enthalpie einen Bereich mit negativer spez. Wärme. Das ist das Signal für einen Phasenübergang erster Ordnung in einem "Kleinen" System. In bestimmten Bereichen der Enthalpie gibt es Multifragmentation. Diese verschwindet im thermodynamischen Limes. Simulationen von Systemen unter hohen Drucken zeigen das erstemal den kritischen Endpunkt des Phasenueberganges erster Ordnung. Er liegt bei höherem Druck und niedrigerer Temperatur als im Bulk. Der letzte Teil behandelt selbstgravitierende Systeme. Obwohl kosmologische Systeme sehr groß sind, gehören auch sie zu den "Kleinen" Systemen wie wir sie oben definiert haben. Diese Systeme müssen im mikrokanonischen Ensemble bei konstanter Energie und konstantem totalen Drehimpuls studiert werden, ohne irgendwelche a priori Annahmen über ihre räumliche Massenverteilung (Symmetrie) zu machen. Dieses Beispiel ist relevant für viele astrophysikalische Systeme von (Vielfach-)Sternbildung bis hin zu Galaxien. Die Entropiefläche, ihre Ableitungen, die intensiven Größen (Temperatur, Winkelgeschwindigkeit), sowie Observable, die die Massenverteilung kontrollieren, werden in ihrem ganzen Parameterbereich studiert. Diese Systeme haben ein reiches Phasendiagramm: Es gibt alle Arten von Phasenübergängen, erster Ordnung und mehereren von zweiter Ordnung. Es wird gezeigt, daß all diese Eigenschaften von (astro-)physikalischer Bedeutung in dem kanonischen Ensemble als Funktion der intensiven Parameter verwaschen werden oder sogar völlig verloren gehen. Schlimmer noch, für bestimmte Wahl der intensiven Parameter divergiert die kanonische Zustandssumme sogar

Coercitivité des aimants permanents Nd-Fe-B frittés : approches expérimentales et numériques

Nd-Fe-B permanent magnets are the most powerful among all commercially available magnets. They play a significant role in energy applications, such as motors of electric vehicles and generators of windmills. Their outstanding properties come from the excellent intrinsic magnetic properties of the Nd2Fe14B phase and from their microstructure. However, electrical machines operate at about 120-180°C and extrinsic magnetic properties such as coercivity and remanence decrease rapidly with temperature. One way of improving coercivity of Nd-Fe-B sintered magnets is to substitute Nd with a heavy rare earth such as Dy, so as to increase the magnetocrystalline anisotropy. However, Dy is a strategic element and a major objective of the research community is therefore to develop Nd-Fe-B magnets that possess excellent extrinsic magnetic properties with a reduced content of Dy. This requires a better understanding of the link between microstructure and coercivity. The key point is the control of the grain size and the distribution of secondary phases at grain boundaries to prevent magnetization reversal and magnetic coupling. The first part of this thesis concerns a comparison of open-circuit and closed-circuit magnetization measurements carried out on Nd-Fe-B sintered magnets. The observed differences in coercivity values are discussed in terms of magnetic viscosity and demagnetizing field effects. The second part deals with the grain boundary diffusion process performed on Nd-Fe-B sintered magnets using Dy-Co alloys. Microstructural observations and magnetic measurements have been carried out to characterize the diffusion and coercivity profiles and to establish the link between local variations in composition and coercivity. Moreover, micromagnetic simulations have been performed to describe magnetization reversal at the nanoscale in a simple core-shell model. The last part constitutes a discussion about coercivity in graded magnets via a diffusion model and further simulations on a polycrystalline model.Les aimants permanents Nd-Fe-B sont actuellement les plus puissants du marché. Ils sont indispensables pour des applications telles que les moteurs des véhicules électriques ou les générateurs des éoliennes. Leurs propriétés exceptionnelles viennent des propriétés magnétiques intrinsèques de la phase Nd2Fe14B et de leur microstructure. Cependant, les machines électriques fonctionnent entre 120 et 180°C et les propriétés magnétiques extrinsèques telles que la coercitivité et la rémanence diminuent avec la température. Un moyen d’améliorer la coercitivité des aimants frittés Nd-Fe-B est la substitution du Nd par des terres rares lourdes comme le Dy, afin d’augmenter l’anisotropie magnétocristalline. Néanmoins, le Dy est un matériau critique et un objectif majeur de la recherche est actuellement de développer des aimants possédant d’excellentes propriétés magnétiques extrinsèques et contenant peu de Dy. Cela nécessite une meilleure compréhension du lien entre microstructure et coercitivité. Dans les aimants frittés Nd-Fe-B, un des points-clés est le contrôle de la taille de grain et de la répartition des phases secondaires aux joints de grains de façon à limiter la nucléation du retournement de l’aimantation et à garantir un bon découplage magnétique des grains. La première partie de la thèse est une étude comparative des caractérisations magnétiques en circuit ouvert et fermé réalisées sur des aimants frittés Nd-Fe-B. Les différences de coercitivité observées sont expliquées par les phénomènes de viscosité magnétique et d’effets de champ démagnétisant. La deuxième partie traite du procédé de diffusion aux joints de grains appliqué aux aimants frittés Nd-Fe-B et utilisant des alliages Dy-Co. Des caractérisations microstructurales ont été réalisées en complément de mesures magnétiques afin de déterminer les profils de diffusion et de coercitivité, et ainsi d’établir le lien entre les variations locales de composition chimique et le champ coercitif. De plus, des simulations micromagnétiques ont permis de décrire le retournement de l’aimantation à l’échelle nanométrique dans un modèle simplifié cœur-coquille. Enfin, la dernière partie de la thèse constitue une discussion sur la coercitivité des aimants diffusés au Dy-Co (à gradient de champ coercitif) à l’aide d’un modèle de diffusion et de simulations sur un modèle polycristallin

Modeling of demagnetization processes in permanent magnets measured in closed-circuit geometry

International audienceThe hysteresis loops of nucleation-type magnets made of exchange-decoupled grains (i.e. sintered Nd-Fe-B magnets) reflect the discrete character of magnetization switching in such materials. Due to this discrete character, the experimental determination of coercivity depends on the measurement protocol. Finite element modelling allows to investigate how the pattern of reversed grains develops during sample demagnetization performed under closed-circuit conditions, provided that the basic features of the hysteresigraph are known. Numerical modelling provides a quantitative understanding of the collective effects which are very pronounced in the closed-circuit configuration and shows how they affect both the slope of the demagnetizing curve and the sample coercivity. With a grain coercive field standard deviation adjusted to 0.1 T, it is numerically found that the difference in coercivity between closed-and open-circuit configurations is 40 kA/m, in good agreement with previous experimental data

Design, Fabrication and Characterisation of Multi-Parameter Optical Sensors Dedicated to E-Skin Applications.

For many years there has been a strong research interest in soft electronics for artificial skin applications. However, one challenge with stretchable devices is the limited availability of high performance, stretchable, electrical conductors and semiconductors that remain stable under strain. Examples of such electronic skin require excessive amounts of wires to address each sensing element-compression force and strain-in a conventional matrix structure. Here, we present a new process for fabricating artificial skin consisting of an optical waveguide architecture, enabling wide ranging sensitivity to external mechanical compression and strain. The manufacturing process allows design of a fully stretchable polydimethylsiloxane elastomer waveguide with embedded gratings, replicated from low cost DVD-Rs. This optical artificial skin allows the detection of compression forces from 0 to 3.8 N with controllable sensitivity. It also permits monitoring of elongation deformations up to 135%. This type of stretchable optical sensor is highly robust, transparent, and presents a large sensing area while limiting the amount of wires connecting to the sensor. Thus, this optical artificial skin presents far superior mechanical properties compared to current electronic skin