M2 Systèmes Complexes

This lecture introduces the students to the use of computing tools, with applications ranging from statistical physics to complex systems and modern subjects such as machine learning. The course is accompanied by tutorials, and students are given homework to program by themselves. We also attempt to introduce the students to modern applied mathematics and statistics (frequentist and Bayesian) and neural networks, and seek a fined-tune balance between mathematics, physics, algorithms and programming.

Statistical Mechanics: Algorithms and Computations, W. Krauth, Oxford University Press. The Elements of Statistical Learning, T. Hastie, ?R. Tibshirani & ?J. Friedman, Springer. All of statistics a concise course in statistical inference, L. Wasserman, Springer.

Septembre - Octobre - Novembre - Décembre.

Nonlinear dynamics and chaos, S. Strogatz, Westview Press. Structures dissipatives, chaos et turbulence, P. Manneville, Aléa Saclay. Course Notes for Nonlinear Dynamics, L. Tuckerman, https://www.pmmh.espci.fr/~laurette/explanatory.

Nonlinear systems are ubiquitous in real world. In this case the superposition principle is no more valid meaning that the response of the system is not proportional to the input it receives; the sum of different solutions is not a solution anymore. After a brief introduction, we study systems with increasing complexity starting from one dimensional systems to multiple dimension ones. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations using linearization about a basic state. Multiple solutions can exist that are visited through bifurcations when a parameter of the system is varied and can lead to interesting phenomena such as chaos.

Septembre - Octobre - Novembre - Décembre.

26 lectures.

While statistical mechanics deals with averages and fluctuations of global quantities, statistical field theory copes with their spatial fluctuations and correlations. By the so-called coarse-graining procedure, one can define for any quantity of interest a field, and, based on symmetry arguments, an effective Hamiltonian that describes its fluctuations. Advanced tools, such as functional integrals, Feynman diagrams, Wick theorem, and renormalization group (RG), are used to perform calculations. Statistical field theory is particularly adapted to study critical phase transitions, in which scale-free fluctuations are responsible for universal behaviours. The course introduces percolation as a geometrical example of critical phenomena, and develops detailed statistical field theoretical calculations for magnetic systems described by the O(n) and ?4 coarse-grained models, which are found in many other areas of condensed and soft matter physics.

Statistical physics, Quantum mechanics, mathematical tools, M1 level.

Lecture on Phase Transitions and the Renormalization Group, N. Goldenfeld, Frontiers in Physics. Statistical Physics of Fields, M. Kardar, Cambridge University Press. Principles of Condensed Matter Physics, P. Chaikin and T. C. Lubensky, Cambridge University Press.

Septembre - Octobre - Novembre - Décembre.

This course provides an introduction to stochastic processes, i.e., processes whose dynamical evolution displays random features and, thus, whose description calls for the use of a probabilistic approach. Many phenomena fall under such a situation: tossing coin, stock market, motion of gas molecules, waiting queue, spreading of diseases, weather, chemical reactions, etc. In fact almost all processes have, at some scale, a random component. We provide here the necessary technical background to tackle these stochastic phenomena. After presenting the most important phenomenon, the Brownian motion, we provide a reminder of probabilities. We then derive the different equations that allow quantitative predictions. Many examples, taken from physics, chemistry, ecology, epidemiology, are discussed to illustrate the formal notions and to show their universal character.

Stochastic Methods a handbook for the natural and social sciences, C. W. Gardiner, Springer. The Fokker-Planck Equation: Methods of Solutions and Applications, H. Risken, Springer. Stochastic processes in physics and chemistry, N. G. van Kampen, Elsevier.

Septembre - Octobre - Novembre - Décembre.

N. Pavloff, L. Foret (UPMC).

Lectures.

Nonlinear wave propagation are studied on the basis of the mathematical method of characteristic and illustrated on simple examples. On this basis, the phenomenon of “wave breaking” is presented and it is shown how viscosity and/or dispersion regularizes the phenomenon and underlies the theory of shock waves. The theory of weak shock waves is presented on the basis of the Burgers equation, which allows for a complete analysis. The balance between nonlinearity and dispersion provides the basis for introducing the fundamental concept of soliton. The basic properties of solitons is studied in the framework of the theory of the Korteweg-de Vries equation. The universality of this equation is clarified. The concept of topological soliton is also illustrated by the Sine-Gordon equation. A similar universal role is revealed for the nonlinear Schrödinger equation. The physics of the Bose-Einstein condensation of ultracold vapors is the opportunity to study specific aspects of the mathematical physics of solitons in a modern experimental setting. Other examples are discussed, such as the physics of conducting polymers and of magnetic chains.

Linear and nonlinear waves, G. B. Whitham, Wiley-Blackwell. Physique des solitons, M. Peyrard & T. Dauxois, EDP sciences.

Septembre - Octobre - Novembre - Décembre.

PARIS

Beyond its intrinsic beauty and usefulness, the motion of living cells is a puzzle to the physicist. How does a cell harness its internal mess of proteins under strong thermal fluctuations to effect useful work? Do these processes teach us fundamental things about how matter functions out of equilibrium? We discuss these questions over a spectrum of length and time scales, from individual proteins to living tissues. While the nanometer-scale components of these systems, as well as their large-scale behaviors, are well characterized experimentally, the connection between the two levels is far from understood. This course discusses this relation through the prism of statistical mechanics, and takes the students to some state-of-the-art questions in the field.

C. P. Broedersz and F. C. MacKintosh. Modelling semiflexible polymer networks. Rev. Mod. Phys., 86, 995 (2014). J. Prost, F. Jülicher & J.-F. Joanny. Active gel physics. Nat. Phys., 11(2), 111 (2015).

Septembre - Octobre - Novembre - Décembre.

PARIS

G. Foffi (PSaclay), G. Durand (UPS).

At school, we generally learn that matter exists in three phases, solid, liquid and gas. If you think about it, you will however encounter a great many examples of materials that seem to fall between the categories solid and liquid. Soft matter, or complex fluids, is the subfield of condensed matter that aims to study these systems. Examples include colloids, polymers, foams, gels, granular materials, liquid crystals, and biological materials. In this course, we introduce the essential principles of soft matter physics. Soft matter systems typically consist of a large number of small elements whose interaction energies are comparable with thermal energy. At this weak energy scale, entropy is often an important key player in controlling the materials behaviour, in contrast to traditional hard matter. As a result, soft matter systems display an extraordinary complex and, sometimes, counter-intuitive behaviour, even at room temperature. The same material may behave like a fluid or like a solid, depending on the experimental conditions. Materials may harden with increasing temperature or under mechanical solicitation, and substances may become more soluble upon decreasing the temperature.

Basic Concepts for Simple and Complex Liquids, J.-L. Barrat & J.-P. Hansen Capillarity and wetting phenomena : drops, bubbles, pearls, waves, P. G. de Gennes, F. Brochard-Wyart & D. Quéré, Springer. Theory of Simple Liquids, J.-P. Hansen & I. R. McDonald, Elsevier. Liquides: solutions, dispersions, emulsions, gels, B. Cabane & S. Henon, Belin.

Septembre - Octobre - Novembre - Décembre.

PARIS

PARIS

Introduction à l'élasticité
Théorie de l’élasticité
Ondes élastiques
Flexion faible des poutres
Flambage d’une poutre (buckling).

PARIS

Pre?sentation ge?ne?rale
Milieux granulaires confine?s
Transports granulaires
Les effets du gaz interstitiel
Me?lange et se?gre?gation.

PARIS

Le cours “Les bases de l’hydrodynamique” traite dans une approche physique les aspects spécifiques à la mécanique des fluides :
– lois de conservation,
– régimes d’écoulement,
– écoulements de fluides “parfaits”,
– écoulements de fluides “réels”,
– couches limites laminaires,
– ondes de surface,
– diffusion de la chaleur et convection thermique,
– applications aux instabilités hydrodynamiques.

PARIS

Interactions moléculaires de base
Interactions électrostatiques, Interactions de polarisation mutuelle
Les solutions
solutions de petites molécules, solubilité/démixion
Solutions de macromolécules, bon/mauvais solvant, solvant theta
Solutions aqueuses
Tensio-actifs et micellisation
Structure des tensio-actifs, critère d’association
Concentration micellaire critique
Les différentes phases micellaires
Dispersions et émulsions
Définition
Métastabilité
Interactions entre particules
Tension de surface
Pression de disjonction – Constante de Hamaker
Approche mécanique/thermodynamique
Loi de Laplace
Mouillage
Paramètre d’étalement
Longueur capillaire
Mouillage statique
Mouillage dynamique
Interfaces complexes : rôle de la rugosité, de la porosité et de l’élasticité.
Hydrodynamique aux interfaces
Approximation de lubrification
Cas des films minces
Mouillage forcée
Démouillage.

Septembre - Octobre - Novembre - Décembre.

PARIS

Rhe?ologie : ge?ne?ralite?s
De?termination pratique du comportement des mate?riaux : rhe?ome?trie
Introduction aux fluides complexes
Rhe?ologie des fluides complexes.

Septembre - Octobre - Novembre - Décembre.

PARIS

A.L. Fameau (L'oréal), J.F. Argilliers (IFPEN), G. Cuvelier (Agroparistech).

Chapitre 1 : Système colloïdaux: introduction
Caractéristiques: système colloïdal
Mouvement brownien versus gravité
Surfaces spécifiques
Forces entre particules au sein d’une suspension colloïdale
Notion de double couche électrostatique et théorie DLVO
Potentiel zeta, point isoélectrique
Chapitre 2 : Les émulsions
Généralités, les grands types d’émulsions, applications
Métastabilité
Caractérisation des émulsions
Définition et propriétés des tensio-actifs
Mécanismes de déstabilisation d’une émulsion
Chapitre 3 : Les mousses
Généralités, distribution du liquide
Caractéristiques, propriétés, applications
Formation de la mousse
Les 3 étapes dans la vie d’une mousse (murissement, persistance…)
Rôle de la formulation.

Septembre - Octobre - Novembre - Décembre.

PARIS

Le but de ce cours est de pouvoir effectuer des calculs d’écoulement de fluides, d’abord simples puis complexes, dans des milieux poreux modèles et naturels. Les fluides seront essentiellement des fluides dits “colloidaux” avec phase continue huileuse ou acqueuse. Les domaines industriels d’application sont nombreux. Parmi ceux-ci, la problématique “injectivité & productivité des puits” fournira l’essentiel des supports du cours.

I- Méthodes de caractérisation des roches

II- Perméabilité, écoulement monophasique

– En capillaire: loi de Poiseuille

– En milieux poreux: loi de Darcy

– Relation Porosité/perméabilité (Kozeny-Carman)

III- Ecoulement de fluides

– Traceur, dispersion

– Ecoulement de fluides colloïdaux (transport, transfert aux interfaces et dépôt)

IV- Effet sur les propriétés pétrophysiques

– Réduction de perméabilité, de porosité

– Exemples d’application: injectivité/productivité des puits.

The heterogeneous inter-relations between the components of complex systems are frequently represented via graphs or networks; examples range from protein-protein interactions in biology, over social networks connecting human beings up to large-scale technological networks like the internet or power grids. A huge variety of methods have been developed over the last years to reconstruct networks from observational data, to analyse a network’s structure, or to understand its implications on processes taking place on top of these networks. Many of these methods have been inspired by the statistical physics of disordered systems. My lecture series will cover four major topics:
(i) models of networks (random graphs, small-world networks, scale-free networks, network ensembles),
(ii) characterisation of networks (degree distribution, centrality measures, k-core, community detection),
(iii) models on networks (Ising model, optimisation over networks, epidemic spreading models),
(iv) inference of networks (correlation networks, Gaussian networks, inverse statistical physics).
The focus of the course will be to give a solid and coherent methodological and theoretical basis, which can be used across a broad range of applications.

Networks: An Introduction, M. Newman, Oxford University Press. Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics, A. K. Hartmann & M. Weigt, Wiley-VCH.

Janvier - Février - Mars.

PARIS

N. Sator.

Economics deals with human activities, such as production, distribution and consumption of wealth. However mainstream economic theory has been driven by general physical concepts, like equilibrium, with the aim of providing scientific basis for its forecasts. Furthermore, economic data stem from the interaction of a large number of agents and therefore call for the tools and ideas of statistical physics.
In this course, some basic economic and financial concepts will be first introduced and discussed (no prerequisite is required), in particular the underlying hypothesis which constitute the cornerstones of modern financial theory. We will then investigate the modeling of financial markets from a physics perspective, by using empirical analysis and by comparing the real-life financial data properties with those predicted by models. From the evolution of stock prices as a random walk to the option pricing by the famous Black-Scholes model, the aim of this course is to discuss the validity of modern financial theory and to show to what extent statistical physics allows for a better understanding of financial markets.

An Introduction to Econophysics: Correlations and Complexity in Finance, R.N. Mantegna and H.E. Stanley, Cambridge University Press The statistical Mechanics of Financial Markets, J. Voit, Springer. The Mathematics of Financial Derivatives: A student introduction, P. Wilmott, S. Howison and J. Dewynne, Cambridge University Press.

Janvier - Février - Mars.

PARIS

A. Decelle.

This course will be an introduction to both theoretical and practical aspects of modern Machine Learning. Several problems will be studied from unsupervised learning (with a particular enphasis on generative models such as community detection in graphs, or neural networks), to supervised methods of deep learning. We will pay a special attention to the theoretical basis of these methods, using applied mathematics and statistical physics tools, and to their applications to real problems. Students will have the opportunity to learn to code most of the algorithms seen during the lectures and apply them on real datasets.

Pattern Recognition And Machine Learning, C. Bishop, Springer-Verlag New York Inc Information, Physics, and Computation, M. Mézard and A. Montanari, OUP Oxford.

Janvier - Février - Mars.

PARIS

D. Ullmo.

The socio-economic world represents an example of a complex system, some aspects of which can be analyzed in terms that are actually quite familiar to physicists. The purpose of this course is to give an introduction to some basic concepts and technical tools making possible for students to enter this field at the interface between physics and social sciences. This field is actually relatively vast. Rather than attempting to cover all of its facets, this class focuses on two aspects which are ubiquitous in social sciences, namely optimization problems (with or without uncertainties), and game theory.

Lecture on macroeconomics, O. Jean Blanchard & S. Fischer, MIT Press. Game Theory, M. Maschler, E. Solan & S. Zamir, Cambridge University Press.

Janvier - Février - Mars.

PARIS

Building from the stochastic modeling of physical systems, we address collective effects specific to the nonequilibrium nature of the dynamics, mostly in classical systems, but also in quantum ones. Our walk begins with a landmark result in nonequilibrium statistical mechanics that tells us that fluctuations even far from equilibrium, possess intriguing statistical properties inherited from microscopic reversibility. Relaxation towards equilibrium, possibly in the vicinity of a critical point, and then the build-up of correlations in driven systems (with the possibility of phase transitions down to one space dimension) make up the first half of the lectures. We then embark in the modeling of population dynamics (from chemical reagents to living matter) and conclude the lecture series with phenomena and modeling specific to the nonequilibrium quantum realm. Along the lectures, we map these faces to names and physical phenomena. They’ll all be familiar to the audience by the end of the course.

Stochastic processes in physics and chemistry, N. van Kampen, Elsevier. Critical dynamics, U. C. Täuber, Cambridge University Press.

Janvier - Février - Mars.

PARIS