LPTMS Highlights

Geometry and interactions in self-assembled biological systems

In honour of Françoise Livolant, the LPS (Laboratoire de Physique des Solides) and LPTMS (Laboratoire de Physique Théorique et Modèles Statistiques) are organising an international conference on geometry and interactions in self-assembled biological systems.

We will host a two days conference in Orsay, on March 23-24, 2017. Colleagues and friends are welcome.

Participants may submit their contributions to poster sessions.

Registration is free of charge but mandatory.

The conference will be held at the Institut de Physique Nucléaire (IPN), Orsay (Building 100A), close to the train station Orsay Ville on the RER B line.

The link of the conference: https://flivolant2017.sciencesconf.org/


Elaborer une relaxation rapide pour un micro-système

Des physiciens viennent de proposer un nouveau protocole pour changer l’état d’un système mécanique sans qu’il ne s’échauffe. Ce protocole s’avère bien plus rapide que la limite imposée par la relaxation thermique. En l’appliquant au contrôle d’une microparticule piégée par des pinces optiques, ils ont démontré une accélération d’un facteur 100.

Agir vite et attendre ou agir lentement : c’est bien souvent le seul choix de qui veut changer l’état thermodynamique d’un système sans en changer la température. L’archétype de cette situation est la compression d’un gaz : s’il est comprimé rapidement, il s’échauffe et il faut attendre pour que sa température redescende à sa valeur initiale, et si l’on souhaite qu’il ne s’échauffe pas, il faut procéder lentement (le système est en contact avec un thermostat). Des physiciens du Laboratoire de Physique (Univ. Lyon 1/ENS Lyon/CNRS), du Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS, CNRS/Univ. Paris-Sud et Univ. Paris-Saclay) et du Laboratoire Collisions, Agrégats, Réactivité (LCAR, CNRS/Univ. Toulouse 3) viennent de démonter qu’une alternative est possible. Le protocole qu’ils proposent permet d’effectuer une transformation rapide qui amène le système à la fois dans l’état thermodynamique souhaité tout en assurant une température finale égale à la température initiale. Ce résultat a été possible grâce à une collaboration étroite entre les théoriciens et les expérimentateurs des trois laboratoires, et à la qualité du système de piège optique 3D construit au Laboratoire de Physique de l’ENS Lyon. Le principe est d’effectuer une première transformation qui « dépasse » l’état souhaité, pour y revenir dans un second temps en suivant une dynamique très précise (prootocole nommé ESE). In fine, les limites ultimes n’ont plus trait aux échanges thermiques, mais relèvent uniquement de la précision de la modélisation du système physique, et des capacités expérimentales de modulation rapide des paramètres de contrôle. Ce travail est publié dans la revue Nature Physics.

Dans ce travail, les physiciens ont considéré une bille en silice de 2 microns piégée par une pince optique, c’est-à-dire un faisceau laser focalisé qui attire la particule dans la région d’intensité maximale. D’un point de vue thermodynamique, ce système peut être vu comme un gaz à une particule placée dans la boite de taille variable que constitue le piège. En changeant l’intensité du laser, les chercheurs modifient la raideur du piège et en quelque sorte la taille de la boite qui contient la particule. Leur objectif est alors de réduire la taille du piège sans augmenter l’énergie d’agitation thermique de la particule. En modélisant fidèlement ce système, les chercheurs ont déterminé un protocole de transformation 100 fois plus rapide que la solution consistant à réduire rapidement la taille et à attendre la thermalisation de la particule (protocole appelé STEP dans la figure ci-dessus). En pratique leur protocole consiste à augmenter transitoirement la raideur très au-delà de la valeur finale demandée, pour revenir avec la ‘’bonne’’ évolution temporelle à la valeur souhaitée à la fin de la transformation. Cette idée peut s’adapter à de nombreux systèmes, par exemple le contrôle du mouvement de la pointe d’un microscope à force atomique pour de l’imagerie rapide. Elle présente aussi un fort potentiel d’application sur les micro et nano systèmes, où la réduction du temps de réponse est un objectif aussi important que la miniaturisation.

 

image1_ciliberto

Schéma de principe du processus pour lequel le temps d’équilibration a été réduit grâce au protocole ESE. Au temps initial, la particule Brownienne est à l’équilibre. Elle est confinée par un potentiel harmonique de raideur Ki (ligne noire) et sa position a une distribution de probabilité ρ(x) (histogramme bleu) avec une déviation standard σx(ti). A la fin du processus au temps tf, la particule est à l’équilibre dans un potentiel plus confinant. La déviation standard σx(tf) de ρ(x) est par conséquent plus petite que sa valeur initiale. Par la méthode ESE, on peut définir une évolution temporelle appropriée de K(t), qui permet au système d’être à l’équilibre en un temps tf arbitrairement court.

image2_cilibertoRelaxation de la déviation standard σx(t) (normalisée à sa valeur finale) lors d’un protocole STEP (courbe rouge) et ESE (courbe noire). La ligne bleue précise la valeur d’équilibre recherchée. La ligne verticale pointillée indique la durée du protocole ESE. Le temps d’équilibration est 100 fois plus petit pour le protocole ESE que pour le protocole STEP (noter l’échelle log de l’axe des abscisses).

Référence: Engineered swift equilibration of a Brownian particle
I. A. Martínez, A. Petrosyan, D. Guéry-Odelin, E. Trizac et S. Ciliberto
Nature Physics 12, 843–846 (2016).

Lien vers site CNRS

 


Anisotropic collisions of dipolar Bose–Einstein condensates in the universal regime

When confronted with the enormous diversity of material properties, it is absolutely fantastic to believe that such diversity arises from the relatively small group of fundamental elements which constitute the periodic table. Indeed, here on earth we have appreciable quantities of roughly only one hundred different elements. So is it really true that this small group of fundamental building blocks accounts for every single thing we have ever seen, heard, tasted, smelt, and felt?

Science is slowly revealing that in fact it is less the individual atom, and more the interactions among atoms which unlock the door to this diversity. By understanding "what form are these interactions" and "what are their consequences" at the level of just two individual, isolated atoms we make our first step toward understanding material properties. And this can be done with our feet remaining firmly planted in the most fundamental theory at our disposal: Quantum physics.

Each element on the periodic table has its own unique shape. This is determined by the way its electrons are configured. Only elements with either half filled or completely filled electronic sub-shells turn out to be spherical, the rest are aspherical. The rare-earth element dysprosium lies in the lanthanide chemistry group and presents a very interesting extremum in that it's electronic configuration makes it extremely highly ellipsoidal. This ellipticity equates to dysprosium having the strongest interatomic dipole-dipole forces among all common stable elements.

New research published in two recent articles lifts the lid on the consequences of these strong dipolar interactions in ultra-cold dysprosium fluids. When two dysprosium atoms collide, they exchange momenta under the constraints of conserved energy and momentum. The post-collision momenta of the two atoms is statistically determined by the quantum wave-function of the two particles. Indeed, this aspect is similar to all atoms. However, in strongly dipolar atoms such as dysprosium, things get a little more interesting. Here, the distribution of outgoing momenta depends strongly on the orientation of the dipoles relative to the axis along which the atoms collide. The details of such collisions need to be accurately understood in order to understand the dynamics of the entire fluid. Theorists at LPTMS were able to understand these collision events based on an analytic solution to the two-body Schroedinger equation. Furthermore, they went on to incorporate this understanding into a Monte-Carlo simulation algorithm. This technique promotes the information of two-body collisions into an ab-initio theory of the entire gas. Remarkable experimental progress by Professor Benjamin Lev and his team at Stanford University are now able to control isolated mesoscopic clouds of ultra cold dysprosium atoms. Comparisons between the theory and the experiment provided, not only remarkable agreement, but an entirely new method of thermometry in dipolar gases, and a new tool to measure scattering length (particularly in the presence of Fano-Feshbach resonances).

 

v_images

The experiments involve a collision between two oppositely propagating clouds of dysprosium atoms. In the images above (along the bottom row) one can see a snapshot after these clouds have passed through one another. The clearly visible ``large blobs'' are atoms which never experienced a collision. The other atoms, which appear on the ``halo-like'' sphere did experience a collision. While conservation of energy and momentum requires all atoms that scatter end up some where on the sphere, the distribution around the sphere is determined by the nature of the interatomic interaction. Moreover, in the case of dipolar interactions, it is furthermore dependent on the dipole alignment direction relative to the collision axis. So, moving along the images from left to right, the dipoles are aligned parallel, forty five degrees, and perpendicular to the collision axis (as shown schematically in the top row). The experiments analyse the atomic clouds by looking at absorption images of their three-dimensional spatial distribution.

References:

N. Q. Burdick, A. G. Sykes, Y. Tang and B. L. Lev, New J. Phys. 18 113004 (2016)

Y. Tang, A. G. Sykes, N. Q. Burdick, J. M. DiSciacca, D. S. Petrov, and B. L. Lev, Phys. Rev. Lett. 117, 155301 (2016)


Interview et diaporama sur le futur bâtiment

Interview concernant le FLI (bâtiment FAST/LPTMS/IPa):


Comprendre et résoudre l’évolution de modèles en sciences sociales grâce à une équivalence avec des systèmes physiques

Des physiciens viennent de trouver un lien formel profond entre une classe d’équations utilisées en sciences sociales, les jeux à champ moyen quadratiques et une équation omniprésente en physique, l’équation de Schrödinger non linéaire. En transférant les méthodes et solutions de cette équation étudiée depuis près de cent ans en physique, ils apportent de nouveaux outils conceptuels permettant de dépasser les approches actuelles reposant sur la simulation numérique.

 

Il y a une dizaine d’années, des mathématiciens ont proposé un nouveau cadre conceptuel pour modéliser des phénomènes collectifs relevant de la sociologie ou de l’éthologie, tels que le comportement de foules, bancs de poissons et hordes animales, ou des sciences de l’ingénieur, avec par exemple l’optimisation de la consommation électrique pour le chauffage. Dans cette « théorie des jeux à champ moyen », le comportement d’un individu est déterminé par l’anticipation qu’il peut faire, en moyenne sur le bruit, des actions futures de l'ensemble des autres membres du groupe. Cette modélisation qui fait intervenir à la fois le passé et le futur conduit à des équations mathématiques relativement complexes. Excepté quelques cas simples, ces modèles sont par ailleurs le plus souvent abordés par le biais de simulations numériques, ce qui rend difficile une compréhension profonde de leur comportement. Des physiciens du Laboratoire de physique théorique et modèles statistiques (LPTMS, CNRS/ Univ. Paris Sud) et du Laboratoire de physique théorique et modélisation (LPTM, CNRS/Univ. Cergy-Pontoise) viennent d’établir un lien formel entre une classe particulière de ces jeux à champ moyen, les jeux dits « quadratiques », et une équation intervenant dans de nombreux phénomènes physiques et étudiée depuis près d’un siècle : l’équation de Schrödinger non linéaire. Ils ont alors transférer la bonne compréhension de cette équation et la connaissance de solutions à des situations complexes impliquant des groupes sociaux. Ils ont notamment montré comment certains schémas d’approximations telles que les approches variationnelles, bien connus dans le contexte de l’équation de Schrödinger non linéaire, permettent de décrire la formation de ces groupes et d’en comprendre les temps caractéristiques. Ce travail est publié dans la revue Physical Review Letters.

Les chercheurs ont analysé la modélisation d’une population d’individus qui évoluent dans un espace à une dimension pouvant représenter l’espace physique dans lequel ils se déplacent, ou bien la quantité d’une ressource naturelle dont ils disposent ou encore la valeur de leur investissement dans un portefeuille d’actions. La dynamique de ces individus contient une partie déterministe, contrôlée par l’individu lui même et visant à optimiser une certaine fonction de coût ainsi qu’une partie aléatoire. Les physiciens ont montré que sous certaines conditions un changement de variable non linéaire fait apparaître deux nouvelles quantités jouant le rôle de variables duales pour une équation de Schrödinger non linéaire, équation intervenant dans de très nombreux domaines de la physique, allant de la propagation d’une impulsion lumineuse dans une fibre optique aux vagues, en passant par des particules quantiques en interaction. Cette connexion fournit une nouvelle façon de penser ces modèles de jeux à champ moyen et a permis aux chercheurs d’importer dans ce contexte un ensemble de techniques et de méthodes d’approximations, développées au cours du temps par différentes communautés de physiciens. C’est le cas par exemple d’une classe de solutions de l’équation de Schrödinger non linéaire connue sous le nom de solitons. Ces solutions localisées se propagent indéfiniment sans déformation. Les chercheurs ont ainsi montré que des solutions de ce type existent dans certains modèles de jeux à champ moyen et peuvent servir à interpréter le comportement de différents groupes d’animaux tels que des bancs de poissons ou des hordes d’herbivores qui ont la propriété de se déplacer en « paquets localisés », c’est-à-dire comme des solitons.

Reference: Igor Swiecicki, Thierry Gobron, and Denis Ullmo, Phys. Rev. Lett. 116, 128701 (2016).


Fast electronic resistance switching involving hidden charge density wave states

The journal Nature Communications has just published [1] an article “Fast electronic resistance switching involving hidden charge density wave states”. The research is a synthesis of the experimental work performed at the Laboratory of the Complex Matter led by Dragan Mihailovic at the Jozef Stefan Institute in Ljubljana, Slovenia, and the theoretical part performed by Serguei Brazovskii at the LPTMS. The experiments have been performed upon a very popular nowadays material TaS2 which basic units are the planes of strongly correlated electrons showing an intriguing competition of cooperative states such the charge density wave, Wigner crystal, Mott insulator, and polaronic lattice. The article describes an ultrafast non-volatile resistance switching under application of short pulsed current injection, where injected charges create domain walls, converting the material from an insulator to a metal at low temperatures. From a technological viewpoint, this opens up the possibility of low-temperature ultrafast memory devices. The theoretical challenge was to identify and to model the mechanism for formation of a network of charged domain walls originated by the intrincic Coulomb instability of the lattice of self-trapped electrons. This work continues an already well established and very successful collaboration between the two laboratories as it has been certified by earlier prestigious publications [2-4].

[1] Nature Communications, 7, 11442 (16 May 2016), Fast electronic resistance switching involving hidden charge density wave states, I. Vaskivskyi,    I. A. Mihailovic,    S. Brazovskii,    J. Gospodaric,    T. Mertelj,    D. Svetin,    P. Sutar    and D. Mihailovic .

[2] I. Vaskivskyi, J. Gospodaric, S. Brazovskii, D. Svetin, P. Sutar, E. Goreshnik, I.A. Mihailovic, T. Mertelj, and D. Mihailovic, “Controlling the metal-to-insulator relaxation of the metastable hidden quantum state in 1T-TaS2”, Science  Advances, 1, 1500168 (2015).

[3] L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, and D. Mihailovic,
Ultrafast switching to a stable hidden quantum state in an electronic crystal”, Science, 344, 177 (2014).

[4] R. Yusupov, T. Mertelj, V.V. Kabanov, S. Brazovskii, J.-H. Chu, I. R. Fisher, and D. Mihailovic,
“Coherent dynamics of macroscopic electronic order through a symmetry breaking transition”,
Nature Physics,
6, 681 (2010).

 

 


Physics and Biological Systems 2016 October 24-26

Contact: PhysBio2016.lptms@u-psud.fr
The link of the conference: http://lptms.u-psud.fr/physbio2016/

The 3rd International Conference on Physics and Biological Systems will be held on October 24-26th 2016 at École polytechnique in Palaiseau – greater Paris area. It aims to bring together a broad range of physical and life scientists working at the interface between the two disciplines around in-depth talks by first-rate international speakers. Attendance will be limited to 150 participants. We look forward to welcoming you in Palaiseau!


10 million dollars to understand the glass transition

Une collaboration internationale, à laquelle participe Silvio Franz du LPTMS, compte mettre en équation l’état de la matière dans un matériau vitreux. Ses travaux sont financés, au terme d’un appel d’offres très sélectif, par la fondation Simons.

Lien vers l’article sur le site du CEA (français)

Link to the Simons foundation website (english)

Lien vers les actus de l'université Paris Sud (français)


The cell membrane winds up like a watch

Cell membranes are very elastic. They can become distorted when they are asked to do so, when the cell divides, or when a virus detaches itself from the cell. In both cases, the membrane is deformed by a protein complex called ESCRT-III. Up until now, we did not understand how this complex works. Swiss and French researchers say that this protein complex forms a molecular spring at the surface of the cell, and operates like a watch spring. This article was published in Cell.

Just 15 years ago, scientists discovered the ESCRT-III protein complex (pronounced like «escort»). This protein complex actually plays an essential role in the key moments a cell’s life. This complex is behind the final phase of cell division, when the membrane is cut, which allows the daughter cells to divide. ESCRT-III also helps some viruses (such as HIV) to separate themselves from the host cell by cutting the virus bud attached to the cell membrane.

Like a watch spring
Researchers from the University of Geneva (UNIGE) and the NCCR Chemical Biology, INSERM (the French National Institute for Health and Medical Research)/Aix-Marseille University, and the French National Center for Scientific Research (CNRS) have just understood how ESCRT-III operates. Like a lego brick, the proteins fit into each other until they form a spiral. As they pack tightly together, they end up deforming the cell membrane. Similar to a watch spring, the over-compression accumulates the
energy required to start the system.

The latest technology
A combination of skills of biochemists, physicists, and theorists was required to understand the molecular mechanics of this complex. The theoretical estimated stored energy and the spring strength that Martin Lenz from the CNRS estimated were validated by the biophysical experiments conducted in Geneva. With the latest technology, the researchers were able to observe the movements of the complex in real time, and at the nanometer level. This feat was achieved with a high-speed atomic force microscope (AFM), the only one of its kind that can provide nanometric resolution in real time. This microscope was developed by Simon Scheuring, INSERM research director. Aurélien Roux, a  biochemistry professor at the UNIGE Faculty of Science, is pleased to say that this is the first time that this technique has been used for this kind of work, and that this proves, yet again, that interdisciplinary cooperation embarks us on original paths.

 

101772_web_850Electron micrograph showing ESCRT-III spirals bound to a membrane mimicking that of the cell

 

See also CNRS news item (in French)

Ref.: Relaxation of Loaded ESCRT-III Spiral Springs Drives Membrane Deformation,
N. Chiaruttini, L. Redondo-Morata, A. Colom, F. Humbert, M. Lenz, S. Scheuring et A. Roux, Cell 163, 866 (2015)


Caractériser le rayonnement d’un trou noir acoustique dans un condensat de Bose Einstein

Des physiciens viennent de montrer que les techniques expérimentales actuelles devraient permettre de détecter et de caractériser la signature quantique du rayonnement émis par l’horizon d’un trou noir acoustique réalisé dans l’écoulement unidimensionnel d’un condensat de Bose Einstein. L’une des prédictions les plus marquantes du physicien théoricien Stephen Hawking est que les trous noirs gravitationnels émettent un faible rayonnement d’origine quantique. Face aux difficultés théoriques et pratiques posées par l’expérimentation sur un tel objet, les physiciens se sont tournés vers l’étude des systèmes analogues et notamment dans le domaine hydrodynamique. La zone de transition entre une région subsonique et une région supersonique d’un écoulement constitue pour les ondes acoustiques l’analogue de l’horizon d’un trou noir car les ondes acoustiques ne peuvent pas remonter l’écoulement supersonique. Des physiciens du Laboratoire Charles Fabry - LCF (CNRS/IOGS/Univ. Paris-Sud), du Laboratoire de physique théorique d’Orsay - LPT (CNRS/Univ. Paris-Sud), du Laboratoire de physique théorique et de modèles statistiques - LPTMS (CNRS/Univ. Paris-Sud) et du centro Fermi, viennent de montrer dans une étude théorique qu’en choisissant comme fluide un condensat de Bose Einstein il devrait être possible de mettre en évidence un rayonnement sonique émis par l’horizon acoustique qui est l’analogue du rayonnement de Hawking émis par l’horizon d’un trou noir gravitationnel. La méthode proposée, qui consiste à mesurer les corrélations entre les vitesses des particules émises de part et d’autre de l’horizon sonique devrait permettre d’en caractériser les propriétés quantiques en s’affranchissant du bruit thermique. Ce travail est publié dans la revue Physical Review Letters.

Pour ce travail, les physiciens ont analysé théoriquement l’écoulement d’un condensat de Bose-Einstein piégé dans une configuration assurant un écoulement unidimensionnel. Un potentiel extérieur jouant un rôle d’obstacle produit dans le condensat un écoulement subsonique en amont de l’obstacle et supersonique en aval. Les chercheurs ont modélisé l’onde de matière que forme le condensat de Bose-Einstein par un champ quantique dont la dynamique est gouvernée par une équation permettant d’analyser les fluctuations quantiques au voisinage de la solution classique qui décrit l’écoulement du condensat. Dans ce cadre ils ont déterminé la matrice de diffusion permettant de relier les fluctuations repartant du système étudié aux fluctuations qui y parviennent. Ceci leur a permis de montrer que la formation d’un horizon sonique se traduit par l’apparition de deux pics distincts dans la distribution des vitesses des particules. Les physiciens ont également analysé les corrélations entre les vitesses des particules émises de part et d’autre de l’horizon, et ont identifié des signaux de corrélation spécifiques, qui n’apparaissent que lorsqu’un horizon sonique est présent. Certains de ces signaux correspondent alors aux corrélations entre les canaux d’émission des paires issues des fluctuations du vide. Ces corrélations entre les vitesses des particules émises de part et d’autre de l’horizon permettent de remonter à l’origine quantique du phénomène et de prouver que les paires émises sont intriquées. Cette intrication est démontrée en comparant l’intensité des corrélations croisées (sur les canaux d’émission “Hawking” et “partenaire” - voir figure) au simple produit des corrélations intra-canal. Ce critère s’avère particulièrement robuste vis à vis des fluctuations thermiques, et devrait pouvoir être accessible expérimentalement avec des condensats dont les caractéristiques sont semblables à ce qui est usuellement réalisé dans les laboratoires.

See also CNRS news item (in French)

Ref.: Quantum signature of analog Hawking radiation in momentum space, D. Boiron, A. Fabbri, P.-E. Larré, N. Pavloff, C. I. Westbrook, et P. Zin, Phys. Rev. Lett. 115, 025301 (2015)


La planète terre : des mythes à la physique

A réentendre sur France Culture:
X. Campi et H. Krivine : La planète terre : des mythes à la physique  (émission du 9 septembre 2015)

Egalement disponible à la réécoute: "Continent sciences", avec Stéphane Deligeorges
H. Krivine : Les atomes existent-ils vraiment ? (émission du 1er juin 2015)


How does the cell aspirate its own membrane inwards?

A collaboration between LPTMS and the biochemistry department of the University of Geneva sheds new light on endocytosis, a biological process localized at the cell membrane that results in the formation of transport compartments required for exchanges with the outside world. This study is published in the journal Nature Communications.

Clathrin-1

A cage constituted of the protein clathrin engulfs the membrane during endocytosis

 

 

See also CNRS news item (in French)

Ref.: M. Saleem, S. Morlot, A. Hohendahl, J. Manzi, M. Lenz, and A. Roux, Nat. Commun. 6, (2015).


The universality of the Tracy-Widom distribution could originate from a phase transition

Physicists from our lab have recently proposed a mechanism explaining the emergence of the Tracy-Widom distribution in a wide variety of problems in physics and mathematics. They showed that this distribution is associated to the critical behavior of a system of interacting particles, in the vicinity of a third order phase transition. The universality of this distribution would thus be inherited from the expected universality of critical physical systems.

It is well known that, in large systems, probability distributions describing the fluctuations of macroscopic observables often converge to limiting laws. The most famous one is certainly the Gaussian distribution which describes the fluctuations of the sum of independent random variables (of finite variance), but this is not the only one. During the last twenty years, there has been an important activity, both in mathematics and in physics, concerning another universal law: the Tracy-Widom distribution. It was indeed found in a wide variety of systems ranging from the longest increasing subsequence of random permutations, random growth processes and gauge theories (e.g., two-dimensional Yang-Mills theory) to finance. Despite this avalanche of results, the origin of the universality of this law remained a puzzle. By studying a system of particles interacting via a logarithmic potential, two physicists from our lab have recently shown that in such a system, the emergence of the Tracy-Widom distribution is associated to a critical behavior, close to a phase transition. This result reinforces strongly the conjecture that, in general, the Tracy-Widom distribution is associated to the critical fluctuations of certain types of phase transitions. This work was published in Journal of Statistical Mechanics : Theory and Experiment [1].

To understand this phenomenon, the researchers studied the largest eigenvalues of random matrices, whose typical fluctuations are precisely described by the Tracy-Widom distribution. They considered real symmetric or complex Hermitian NxN random matrices whose entries are independent Gaussian random variables, of zero mean and variance 1/N. Thus the average value of the largest eigenvalue has a finite value (=sqrt(2)) when N goes to infinity. Before their work, most of the studies had focused on the typical fluctuations of the largest eigenvalue, of order N^(-2/3) and thus very small, in the vicinity of its mean value sqrt(2). The key idea was to focus, not on the typical fluctuations, which are described by the Tracy-Widom distribution, but on large and rare fluctuations, far away from the mean value. The study of these large deviations can be mapped onto the study of a one-dimensional systems of particles interacting via logarithmic interactions, confined by a harmonic well centered at the origin, and in presence of an impenetrable wall. This wall enforces the particles to stay to its left. When the position of this wall crosses the mean value of the largest eigenvalue at sqrt(2), the system exhibits a transition between a weak coupling phase (when the wall is at the right of sqrt(2)) and a strong coupling phase (when the wall stays at the left of sqrt(2)). They demonstrated that this transition is a third order phase transition, where the third derivative of the free energy (with respect to the position of the wall) exhibits a discontinuity. The authors showed that the critical behavior of this system, close to the transition point, is precisely described by the Tracy-Widom distribution, implying the universality of this distribution.

These results have been recently highlighted in an article by M.  Buchanan in Nature Physics [2] as well as in an article in the online magazine Quanta, from the Simons-fondation [3], and more recently by the CNRS-Institut National de Physique [4].

Reference:

[1] S. N. Majumdar and G. Schehr, Top eigenvalue of a random matrix: large deviations and
third order phase transition, J. Stat. Mech. P01012 (2014).

[2] M. Buchanan, Equivalence Principle, Nature Phys.  10, 543 (2014).

[3] N. Wolchover, At the far ends of a new universal law, Quanta Magazine (October, 2014).

[4] L'universalitŽé de la distribution de Tracy-Widom proviendrait d'une transition de phase.


Mécanique quantique, 2ème édition

par Christophe Texier

www.dunod.com/sciences-techniques/sciences-fondamentales/physique-et-astrophysique/master-et-doctorat-capes-agreg/mecanique-quantiqu-4

LivreMQ-Ch_Texier-Couverture-2ndeEdition


The fate of quantum wavefunction multifractality under perturbation

Ref. : R. Dubertrand, I. García-Mata, B. Georgeot, O. Giraud, G. Lemarié, and J. Martin, Two scenarios for quantum multifractality breakdown, Phys. Rev. Lett. 112, 234101 (2014).

The concept of fractal geometry was introduced by Mandelbrot in the seventies, to describe a range of phenomena characterized by the fact that a certain quantity has a similar distribution at all scales. This notion has proven very useful in the study of many areas of science, including fluid mechanics, biology, economy or geophysics. Multifractals are characterized by the existence of a whole range of fractal dimensions. The application of these ideas to quantum mechanics is much more recent, and the experimental observation of multifractal wavefunctions remains very difficult; in particular, how multifractality can survive under experimental conditions is a challenging question. In a paper recently published in Physical Review Letters, together with collegues from Toulouse, Liège and Mar del Plata, we have investigated theoretically how imperfections unavoidably present in experimental realizations will affect multifractality of quantum wave functions. Multifractality is destroyed under a sufficiently large perturbation in essentially two different ways: either multifractality survives below a certain scale of the quantum fluctuations (in which case one can compensate a small enough perturbation by using high resolution to resolve very small scales),  or multifractality is destroyed at all scales at a similar rate. These results should help interpret or predict experimental results in a real setting.

Lien vers le communiqué INP- CNRS (en français) Seulement deux scénarios pour la disparition de l’invariance d’échelle quantique ?

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Evolution of a complex electronic system to an ordered hidden state: optical quench in TaS2 – theory versus experiment

Contact: S. Brazovskii, brazov@lptms.u-psud.fr

http://lptms.u-psud.fr/membres/brazov/

Femto-second optical techniques addressing pump-induced phase transitions (PIPT) put an ambitious goal to reach “hidden” states of matter – those which are not accessible and not known under equilibrium conditions or more conventional treatments. While there was a case of success in magnetic materials, there has been no success yet in transforming cooperative electronic systems. For the first time, a group from Jozef Stefan Institute in Ljubljana, Slovenia has achieved [1] a bistable switching to a hidden spontaneously ordered conducting state in 1T-TaS2. The hidden state is stable until a laser pulse, electrical current or thermal erase procedure reverts it to the thermodynamic ground state. The theoretical part of this work was developed at the LPTMS, Orsay, France. The theoretical concept and the modeling provided understanding and detailed description of the time evolution under protocols of optical treatments.
The developed theory [1] focuses upon dynamic evolution of electrons and holes as mobile charge carriers, crystallized electrons modifiable by intrinsic defects (interstitials and voids), and the crystal bath. Mutual transformations among the three reservoirs of electrons, together with the heat production, are dictated by imbalances of three partial chemical potentials. The exchange rate among any two reservoirs vanishes when the corresponding chemical potentials become equal. The theory fits and explains all major observations for switching to and from the hidden state. The phenomenological approach sheds a light on very complicated and not yet resolved physics of this material including interplaying effects like CDW, Wigner crystal, polarons, Mott state.
This is already the second fruitful example of cooperation among the LPTMS and the JSI in the field of the PIPT by means of the femtosecond optics [2]. Another branch of theoretical work on this subject is performed also at Orsay, between LPTMS and LPS [3]; it is motivated by experiments performed at the University of Tokyo.

[1] L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, and D. Mihailovic. “Ultrafast switching to a stable hidden quantum state in an electronic crystal”, Science, 11 April 2014.
[2] R. Yusupov, T. Mertelj, V.V. Kabanov, S. Brazovskii, J.-H. Chu, I. R. Fisher, and D. Mihailovic, “Coherent dynamics of macroscopic electronic order through a symmetry breaking transition”, Nature Physics, 6, 681-684 (2010).
[3] S. Brazovskii and N. Kirova, “Excitonic Mechanism of Local Phase Transformations by Optical Pumping”, J. Supercond. Nov. Magn., 27, 1009–1013 (2014).


New solutions to the Boltzmann equation

D. Guéry-Odelin, J. G. Muga, M.J. Ruiz-
Montero and E. Trizac, Exact non-equilibrium solutions of the Boltzmann equation under a time-dependent external force, Physical Review Letters 112, 180602 (2014)

In 1872, Boltzmann established a key bridge between microscopic dynamics and macroscopic irreversibility, through the H theorem: a dilute gas thereby evolves towards equilibrium, where it is ruled by Maxwell-Boltzmann statistics. The Austrian  physicist soon after realized that when the gas is confined by a harmonic trap, more general such statistics could exist, where the mean size and temperature of the gas oscillate with time, without dissipation. A Franco-Spanish collaboration has shown that this class of many-body solutions could be extended, in particular to time-dependent trapping. These new exact solutions for the Boltzmann equation hold for arbitrary short-range interaction potentials, and can be used to propose an original molecular  manipulation technique. The idea, in a reverse engineering perspective, is to work out what time-dependent harmonic confining potential is required to achieve a fast prescribed time evolution of the system's state. It then becomes possible to transform a gas from an equilibrium state to another one in an arbitrarily small time span. This shortcuts the traditional 'adiabatic' technique, which realizes the same goal, but in an unacceptably large duration. Achieving fast cooling of an assembly of particles is desirable in a variety of contexts, in particular to devise accurate atomic clocks.

STA_trizac_english_crop

 

 

Lien en français vers l'actualité scientifique "Changer rapidement l’état d’un gaz piégé" sur le site de l'INP


Dynamics of charged platelet suspensions

S. Jabbari-Farouji, J.-J. Weis, P. Davidson, P. Levitz, and E. Trizac, On phase behavior and dynamical signatures of charged colloidal platelets, Scientific Reports 3, 3559 (2013)

Charged platelet suspensions, such as swelling clays, disc-like mineral crystallites or exfoliated nanosheets are ubiquitous in Nature. Their phase behaviours are nevertheless still poorly understood: while some clay suspensions form arrested states at low densities, others exhibit an equilibrium isotropic-nematic transition at moderate densities. Without electrostatic interactions, hard platelets undergo an isotropic-nematic transition as a result of the competition between orientational and positional entropy, a phenomenon predicted by Onsager. The main question that arises is to understand how electrostatic interactions influence the isotropic-nematic transition and the organisation of charged platelets.

In order to address the influence of the interplay between electrostatic interactions and shape, we used an orientation-dependent effective pair potential and investigated the phase behavior by means of Monte-Carlo simulations. The angular dependence of the effective pair potential has a peculiar form that creates an asymmetry between two states of parallel disks in co-planar and stacked configurations (see figure, left part).

 

trizac-jabbari

This work shows that the original intrinsic anisotropy of the electrostatic potential leads to a rich phase behaviour (see figure, right part), that not only rationalizes generic features of the complex phase diagram of charged colloidal platelets, but also predicts the existence of novel structures. Furthermore, studying the dynamics as a function of density provides strong evidence of slowing-down in the orientationally disordered states. This points to the potential for formation of arrested states in such highly charged systems.

 This is a joint work with Laboratoire de Physique Théorique (LPT Orsay), Laboratoire de Physique des Solides (LPS Orsay) and Laboratoire PECSA (Paris VI).

Contact: Sara Jabbari-Farouji (sara.jabbari@ujf-grenoble.fr)


A new algorithm for discrete tomography

E. Gouillart, F. Krzakala, M. Mezard, L. Zdeborova, Belief propagation reconstruction for discrete tomography, Inverse Problems 29, 3, 035003 (2013)

X-ray tomography is a widely used imaging technique that allows one to produce internal images in materials science or medical applications. An X-ray beam is transmitted through the sample and the projection is recorded on a detector at different angles. The most widely used reconstruction algorithm of the original image from these measurements approximates the inversion of the tomographic inverse linear problem. However, it provides satisfying results only for a well-determined system, with a large number of projections. In order to reduce the acquisition time or the total dose absorbed by the patient, tomography users need other reconstruction algorithms that incorporate known properties of the original image. For example, images with a discrete set of absorption values and few interfaces are frequently encountered in materials science. Using such prior information can make up for the lack of measurements and result in a high-precision reconstruction, an idea popularized by the field of compressed sensing.

 

Inspired by results from statistical physics and graphical models, we have proposed a novel message-passing algorithm for discrete tomography that allows for an efficient computation of pixel expectations under the Bayesian posterior distribution. The algorithm works on two levels: it is iteratively sending probabilistic messages between the different light rays and different set of messages within each light ray in order to satisfy the observed values of measurements. The resulting algorithm is fast, and entirely distributed. We showed that for binary images an accurate reconstruction is obtained for highly undersampled measures, and, equally importantly, the performance is robust to Gaussian measurement noise.

mezard_krzakala


G. Shlyapnikov awarded a 2013 ERC « Advanced Grant »

Georgy Shlyapnikov, Directeur de recherche CNRS at LPTMS, has been awarded an ERC Advanced Grant 2013 for his project on "New collective phases in quantum gases: from few-body to N-body physics".

The ERC Advanced Research Grants are awarded to "exceptional leaders in terms of originality and significance of their research contributions". Their aim is to support "highly ambitious, pioneering and unconventional" projects "that open new directions in their respective research fields or other domains."

More detail (in French) here


C. Nadal awarded 2013 J. Phys. A: Math. Theor. Best Paper Prize

Céline Nadal was awarded one of the three 2013 Best Paper Prizes by the IOP Journal of Physics A: Mathematical and Theoretical, for her paper "Purity distribution for generalized random Bures mixed states", in collaboration with Gaëtan Borot (University of Geneva).

The aim of the prize is "to celebrate and applaud well written papers that make a significant contribution to their field". More detail at http://iopscience.iop.org/1751-8121/page/Best%20Paper%20Prize

Gaëtan Borot and Céline Nadal, Purity distribution for generalized random Bures mixed states, J. Phys. A: Math. Theor. 45, 075209 (2012).


After 20 years the concept of ‘fractal globule’ is experimentally verified in genomics

In the paper “Comprehensive Mapping of Long-Range Interactions Reveals Folding Principles of the Human Genome” (Science 326, 289 (2009)), a team of US authors has reported the existence of a new structural organization of a DNA chain in the human genome in the form of a "fractal globule". This first direct experimental observation of the fractal globule by means of the so-called Hi-C method (genome-wide chromosome conformation capture method) has answered many important questions concerning the functioning and packing of DNA in a chromosome. The concept of fractal globule is currently becoming the new paradigm in genomics. For example this concept has explained why the parts of DNA in the dense globular state can be easily folded out and then retracted back.

The observed fractal ("crumpled") globule has been theoretically predicted more than 20 years ago in a theoretical work by A. Grosberg, S. Nechaev and E. Schakhnovich "The role of topological constraints in the kinetics of collapse of macromolecules" (J. Physique, 49, 2095 (1988)). So, after 20 years this theoretical prediction has found an experimental verification in genomics. The theoretical investigation of the fractal globule was one of the main topics of Sergei Nechaev (LPTMS) during the last decade. In particular, Sergei Nechaev and Oleg Vasilyev have considered the "topological correlations" in unknotted polymer chains in collapsed (globular) phase using methods of algebraic topology, and have proved that due to the topological constraints the trajectory of the chain looks like a statistical Peano curve densely filling the volume on different scales in the 3-dimensional space (S. Nechaev, O. Vasilyev, "On topological correlations in trivial knots: From Brownian bridges to crumpled globules" Journal of Knot Theory and Its Ramifications, 14, 243 (2005)). The works on this subject have been reviewed in: S. Nechaev, O. Vasilyev, "Thermodynamics and topology of disordered knots: Correlations in trivial lattice knot diagrams", in ”Physical and Numerical Models in Knot Theory”, chapter 22, pp. 421-472, Series on Knots and Everything, (WSPC: Singapore, 2005).

The recently developed Hi-C method provides a comprehensive map of spatial contacts between every accessible genomic region of the genome. A map of genomic contacts gives a wealth of information about genome organization in 3D. However inference of the actual 3D conformation (or ensemble of such conformations) and physical principles that govern its folding are challenging problems that is under current investigation by Leonid Mirny Lab (MIT, USA) and Sergei Nechaev group (LPTMS, France). The progress in elucidating principles of 3D genome folding requires deeper understanding of the interplay between topological and physical properties of polymers and connection of these properties to quantities measured by the Hi-C method.

Maps of the genomic contact frequencies obtained by the Hi-C method show several specific features that suggest a possible hierarchical folding of the genome organization. Such hierarchical organization can be revealed by analysis developed in the field of random hierarchical matrices proposed by Sergei Nechaev, Vladik Avetisov et al in the work "On scale free and poly-scale behaviors of random hierarchical networks" (J. Stat. Mech.: Theory and Experiment, 2009: P07008). The mathematical connection between topology and fractal properties of polymers implies new principles of genome organization in 3D.

The digests of Harvard and MIT devoted to the experimental observation of fractal globule can be found here:
http://news.harvard.edu/gazette/tag/fractal-globule
http://web.mit.edu/newsoffice/2009/3d-genome.html


The game of go as a complex network

The study of complex networks has attracted more and more interest in the recent past, fueled in particular by the development of communication and information networks. It turned out that many important aspects of the physical world or of social interactions can also be modelized by such networks. However, these powerful tools have never been applied to the study of games.

Games have been played for millenia, and besides their intrinsic interest, they represent a privileged approach to the working of human decision-making. They can be very difficult to modelize or simulate: only recently were computers able to beat chess champions. The old Asian game of go is even less tractable, as no computer program has been able to beat a very good player.

We have performed the first study of the game of go from a complex network perspective. We built a directed network which reflects the statistics of tactical moves. Study of this network for data sets of professional and amateur games shows that the move distribution follows Zipf's law and the network is scale free, with statistical peculiarities different from e. g. the World Wide Web. These specificities reflect in the outcome of ranking algorithms, such as the PageRank at the basis of the Google search engine. The fine study of eigenvalues and eigenvectors of the matrices used by the ranking algorithms singles out certain strategic situations (see figure), and vary between amateur and different professional tournaments. These results should pave the way to a better modelization of board games and other types of human strategic scheming.

Figure: Moduli squared of right eigenvectors of the 7 largest eigenvalues of the Google matrix for the first 100 most frequent moves, showing that each eigenvector is localised on specific moves: most frequent patterns (first line), ko (second line) and chain connections (third line)

 

See B. Georgeot and O. Giraud , EPL 97 68002 (2012), and CNRS highlight (in French) here


Why does Nature use exactly four letters to encode genetic information?

Genetic information in all life cells is kept within the primary sequences of DNA and RNA molecules. Both of them are heteropolymers consisting of four different nucleotide types. The question ``why nature uses exactly four letters to encode genetic information'' was discussed since the role of DNA and RNA in storing and transmission of genetic information has been understood. Typically, the attempts to answer it are based on the chemistry of interacting nucleotides, or deal with the conjectures lying in the general information theory. We recently presented a new observation concerning the statistics of RNA secondary structures. This observation may be a new contribution to the problem of ``why only four?''.

The RNA molecules are known to form secondary structures (i.e. intra-molecular structures stabilized by theromoreversible hydrogen bonds between non-neighboring nucleotides) which mostly take a ``cactus-like'' (or ``cloverleaf-like'') hierarchically folded form, topologically isomorphic to a tree:

(a) Secondary structure of an RNA gene HAR1F, see http://en.wikipedia.org/wiki/HAR1F; (b) and (c) Schematic cloverleaf structures of RNA-like chains with and without gaps respectively.

(a) Secondary structure of an RNA gene HAR1F, see http://en.wikipedia.org/wiki/HAR1F;
(b) and (c) Schematic cloverleaf structures of RNA-like chains with and without gaps respectively.

Using the methods of statistical physics we demonstrate the existence of a specific morphological transition which occurs in a toy model of random RNA-like chains depending on the number of nucleotide types (letters) c used to construct the chain. This morphological transition occurs at ccr=4 (i.e., at four nucleotide types), making this particular number of nucleotide types special. The transition has a very transparent physical meaning: for ccr in the thermodynamic limit (i.e. for very long chains) there exists a gapless ``perfect'' secondary structure, that is a structure in which the fraction of nucleotides connected to the complimentary ones via hydrogen bonds equals one, while for c>ccr even in the best possible secondary structure there always exists a finite fraction of gaps (i.e., nucleotides which have nobody to connect with).

Such a criticality is specific only for RNA chains and is due to the additional freedom existing in the formation of the complex cactus-like secondary structures. For DNA chains the fraction of matched nucleotides in the optimal alignment of two random sequences is less than 1 for any alphabet with c>1. Thus, the exclusivity of RNA 4--letter alphabets is consistent with the modern opinion that the origin of life could be connected with the template--directed replication of random RNA molecules (the so-called ``RNA world'' hypothesis).

See O.V. Valba, M.V. Tamm and S.K. Nechaev, On exclusivity of alphabets with four nucleotide type, arxiv.org/abs/1109.5410


Superconductor-Insulator transition and energy localization

The problem of decoherence in closed quantum systems is one of the hottest unsolved problems in condensed matter physics. It is important conceptually, as it addresses the basic issue of the nature and decay of many-body states in quantum physics. It also has important implications, as it governs the low temperature noise that makes quantum computing with solid-state devices so difficult.

Usually, in atoms or molecules, one thinks of decoherence as being imposed by the classical environment, evil external agent. However, in some condensed matter systems like strongly disordered superconductors, the interaction with the environment is negligible at very low temperatures: the main source of decoherence comes from the system itself, through its many degrees of freedom. In this paper we predict that the disorder-driven quantum phase transition from superconductor to insulator occurs in two steps. One first finds an insulating phase where the system is able to self-generate its decoherence, leading to a significant dynamics, heat transport and noise. Upon further increase of disorder a second transition occurs, to a completely coherent insulating phase that does not support any transport or noise. A key ingredient driving this physics is the very strong heterogeneity of the electronic states of the system, already present in the superconducting phase, despite the homogeneity of the samples. This prediction has been confirmed by recent experiments by B. Sacepe et al (to be published in Nature Physics).

SeeM.V. Feigel'man, L.B. Ioffe, and M. Mézard,Phys. Rev. B 82, 184534 (2010) or arXiv:1006.5767


Superfluid Motion of Light

Un supraconducteur est un matériau sans résistance électrique ; un superfluide exprime l’absence de viscosité d’un fluide neutre. Ces deux propriétés, fortement reliées, figurent parmi les manifestations les plus spectaculaires de la nature quantique de la matière. Elles ont déjà été observées expérimentalement dans plusieurs systèmes, comme par exemple le mercure (pour la première fois en 1911), l’hélium liquide et, plus récemment, les gaz dilués d'atomes ultrafroids.

Ces propriétés sont à la base de nombreuses applications, comme les électro-aimants supraconducteurs, qu’on retrouve par exemple dans l’imagerie médicale, ou bien dans les trains à sustentation électromagnétique. Un facteur limitant leur développement est celui des faibles températures nécessaires à leur manifestation.

Un autre système, extrêmement prometteur en terme d’applications, est celui de la lumière, qui peut être considérée, d’un point de vue quantique, comme un fluide de photons. C'est en étudiant la propagation, à température ambiante, d'un faisceau laser à travers un cristal non-linéaire, que deux physiciens d'Orsay ont mis en évidence, d’un point de vue théorique, l’existence d’une dynamique superfluide dans la propagation de la lumière. Ils ont démontré la présence d’un régime dans lequel la lumière est capable de se propager à travers un milieu sans être diffusée par des éventuels obstacles présents.

Une réalisation expérimentale possible, basée sur la propagation de la lumière à travers un cristal photonique, permettrait d’observer cet effet. Aussi, le mécanisme responsable de la disparition du régime superfluide a été identifié. Il est relié à des fluctuations de faible amplitude de la lumière, ainsi qu’à des variations fortes très localisées dans l’espace (solitons) qui sont produites lors du passage de la lumière à travers un obstacle. A plus long terme, ce travail ouvre de nouvelles perspectives concernant la réalisation d'un superfluide à température ambiante, avec d'éventuelles applications dans l'optimisation du transport d'information, ou plus généralement le contrôle de la propagation signaux lumineux.

superfluid

Intensité lumineuse oscillant autour d'un obstacle :
A gauche: l'obstacle (ligne horizontale rouge) ne perturbe en rien la propagation lumineuse. C'est le régime Superfluide.
A droite: La présence de l'obstacle entraîne l'émission de solitons et rapidement, la destruction du faisceau. C'est le régime Turbulent

See P. Leboeuf and S. Moulieras, Phys. Rev. Lett. 105, 163904 (2010) or arXiv:1009.2904


Coherent dynamics of macroscopic electronic order through a symmetry breaking transition

See this page

and R. Yusupov, T. Mertelj, V.V. Kabanov, S. Brazovskii, P. Kusar, J.-H. Chu, I. R. Fisher et D. Mihailovic, Nature Physics, Volume 6, 681–684 (2010).


Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

Entanglement is a very fundamental and amazing feature of quantum theory. It measures non-classical non-local correlations between different parts of a quantum system. If the quantum system is in a highly entangled state, measuring an observable on a part of the system can strongly and instantaneously affect the state of another part of the system, even very far away from the point of measurement. Many applications in the field of quantum information and computation, such as quantum teleportation or quantum cryptography, exploit these powerful correlations. They allow to do tasks that are impossible classically. As another example of application, Hawking's radiation is closely related to entanglement between the region inside a black hole and the region outside (the only accessible one for an observer).

When a bipartite quantum system (product Hilbert space HA*HB) is in a pure state, a well-known measure of entanglement is the Von Neumann entropy (or more generally the Renyi entropy) of either subsystem, which is the quantum version of the classical Shannon entropy. The entropy is zero for unentangled ("separable") states and positive for entangled states.

An interesting problem is to study entanglement of random states. In a large system where the Hamiltonian is not known precisely, the wavefunction can be modelled as a random superposition of the basis states, in the same spirit that Wigner introduced random matrices to study large nuclei.

For quantum computation, it is desirable to construct highly entangled states to exploit quantum correlations as best as possible. Random states are suitable candidates as their average entanglement entropy is known to be almost maximal.

We have recently computed the full distribution of entanglement entropy for a random pure state in a bipartite quantum system. For a random pure state in a bipartite system, the eigenvalues of the reduced density matrix of either subsystem are distributed exactly as the eigenvalues of a specific random matrix ensemble (``Wishart'') in presence of an additional constraint (sum of the eigenvalues is one). This allows us to use techniques from random matrix theory such as the Coulomb gas method. The constraint is crucial and leads to unexpected phase transitions in the entropy distribution. We indeed find two critical points at which the entropy distribution changes shape and has a nonanalyticity. These changes are the direct consequence of two phase transitions in the associated Coulomb gas. In particular, at the second transition point, the maximal eigenvalue becomes suddenly much larger than the other eigenvalues. This transition is reminiscent of the Bose Einstein condensation in cold atoms.

With the full distribution of the entropy, we can also get the extreme tails of the distribution. In particular, we show that the common idea that a random pure state is maximally entangled is not quite correct. We indeed find that, although random pure states are highly entangled on average, the probability of an almost maximally entangled state is actually very small.

See C. Nadal, S. N. Majumdar, and M. Vergassola, PRL 104, 110501 (2010), and arXiv:1006.4091, to appear in Journal of Statistical Physics