Module 1: Advanced nonlinear Physics (3 ECTS - Nicolas Pavloff) 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. Module 2: Advanced Statistical Mechanics (3 ECTS - L. Cugliandolo) This course deals with advanced statistical physics topics and techniques. Most of the themes treated are subjects of current research. For example, I revisit the conditions under which Gibbs-Boltzmann equilibrium establishes, or the impossibility of reaching equilibrium due to conservation laws, long-range interactions, frustration or quenched disorder. In all these cases I present alternatives to Gibbs-Boltzmann equilibrium and the special features brought about in static descriptions of some concrete problems. Module 3: Nonequilibrium and active systems (3 ECTS - J. Tailleur) Statistical mechanics has brought a fundamental change of paradigm in physics: rather than solving complex dynamics (e.g. Newton or Schroedinger equations), the study of matter can now be done using static, probabilistic approaches, provided the system under study is in (thermal) equilibrium. As statistical mechanics progresses towards new area of research (biophysics, geophysics, driven systems), new frameworks are needed to reproduce the successes of equilibrium statistical mechanics. In these lectures, we will study the new tools which have been developed over the past few decades to study non-equilibrium systems. The first part of the lectures will be dedicated to study these tools in the context of relaxations towards thermal equilibrium (derivation of Langevin equation, Ito calculous, Fokker-Planck equation & operator, Master equation). In the second part, we will illustrate and apply these tools to study a research field which has attracted a lot of interest recently: active matter. This field encompasses systems in which individual units are able, at the microscopic scale, to convert energy stored in the environment to self-propel (bacteria, active colloids, vibrated granular media, etc.). Module 4: Numerical simulations (3 ECTS - P. Viot) Six decades after the first simulation in physics, simulations became a tool that have invaded all fields of sciences. The lecture starts by an overview of basic methods of molecular dynamics and Monte-Carlo simulations. In a second part, we consider various observables that are available in simulations. A third part is dedicated to the investigation of phase transitions by implementing finite size analysis, reweighting method, as well as several advanced Monte-Carlo methods (tempering, Wang-Landau, cluster algorithm). In a fourth part, we consider small systems with Brownian dynamics and the relevant role of fluctuations in the framework of large deviation function, fluctuation theorems and stochastic thermodynamics. In the last part, we propose an introduction to non-equilibrium simulations by considering some paradigmatic models and simulation methods for non Hamiltonian systems. Module 5: Statistical physics of simple and complex fluids (3 ECTS - M. Durand) 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. Module 6: Biophysics (3 ECTS - M. Lenz) 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. Module 7: Quantum field theory (3 ECTS - J. Serreau) The course aims at introducing the basic notions of quantum field theory (QFT) as a successful unification of quantum mechanics (QM) and special relativity (SR). We identify the deep issues of this unification rooted in the very principles of QM and SR and we show how these are resolved by QFT. On our way, we discuss some great historical successes, such as the prediction of antiparticles, the Casimir effect, and others. We also briefly discuss the path integral quantization and make link with the tools of statistical field theory. Module 8: Mathematical Tools (3 ECTS - G. Roux) The course is based on miscellaneous small chapters, built from examples. The goal is to recall and introduce useful mathematical tools with hands on. The concepts and techniques introduced are useful for other lectures and for future everyday work. We try to treat classical examples as well as some taken from the physics literature. Since we cannot cover all possible subjects, the objective is rather to train and manipulate mathematics, in particular since exact solutions are appealing and indispensable for benchmarking numerical tools.

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4 modules to be chosen from the following table of 8

Module 1: Advanced nonlinear Physics Linear and nonlinear waves, G. B. Whitham, Wiley-Blackwell. Physique des solitons, M. Peyrard & T. Dauxois, EDP sciences. Module 3: Nonequilibrium and active systems The Fokker-Planck Equation. H. Risken, Springer Stochastic Methods – A Handbook for the Natural and Social Sciences, C. Gardiner, Springer. Stochastic Processes in Physics and Chemistry, H. Risken, Elsevier. Module 4: Numerical simulations Understanding Molecular Simulation–From Algorithms to Applications. D. Frenkel & B. Smit, Academic Press. Stochastic Energetics. K. Sekimoto, Springer. A Guide to Monte Carlo Simulations in Statistical Physics. D. P. Landau & K. Binder, Cambridge University Press. Module 5: Statistical physics of simple and complex fluids 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. Module 6: Biophysics 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). Module 7: Quantum field theory Quantum Field Theory, L. H. Ryder, Cambridge University Press. An Introduction to Quantum Field Theory, M. E. Peskin & D. V. Schröder, Westview Press. The Quantum Theory of Fields Vol 1: Foundations, S. Weinberg, Cambridge University Press. Module 8: Mathematical Tools Physics and Mathematical Tools: Methods and Examples, A. Alastuey, M. Clusel, M. Magro & P. Pujol, Word Scientific. Mathematical Methods for Physicists, T. L. Chow, Cambridge University Press. Group Theory in a Nutshell for Physicists, A. Zee, Princeton University Press.

Modalità di esame: Prova scritta (in aula);

Exam: Written test;

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Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.