The student is expected to acquire an overview of various computational aspects of complex systems, and to develop an understanding of several interdisciplinary applications

Module 1: Nonlinear Physics [B. Andreotti] (COMPULSORY MODULE) The physics of the so-called “complex” systems is based on a multitude of coexisting dynamic mechanisms, on a multiplicity of spatial or temporal scales, or on dynamics that are violently out of equilibrium. In this context, the linearization of problems, which makes powerful analytical mathematical tools available, becomes inoperative. Non-linear differential equations, for which there is no longer any basis for functional decomposition, since there is no longer any principle of superposition, require specific problem-solving techniques, either theoretically or experimentally. Nonlinear physics has no particular object of study. It brings together a culture and a set of theoretical and experimental tools that can be used in all disciplinary fields. For this reason, researchers with this background can be found in many different communities: biophysics, soft matter, hydrodynamics, economics, condensed matter, social theory, geosciences, atmospheric sciences, optics, acoustics, non-equilibrium statistical physics, etc. For this reason, there is no fixed corpus associated with ‘Nonlinear Physics’. I have chosen to approach it from the point of view of hydrodynamics, elasticity and soft matter, i.e. by working on partial differential equations, to complement the rest of the curriculum. We willl start with… linear problems. Then we will consider weakly nonlinear developments and bifurcations. Then we will see the fundamental aspects of asymptotic calculus, singularities, self-similar solutions and their emergence. Finally, we will discuss the physics of dynamical systems, chaos and turbulence. This outline, organized according to mathematical tools, will be associated with a second layer organising the visit of various modern problems in continuum mechanics. Module 2: Advanced nonlinear physics [B.Andreotti] In 1917, the biologist and mathematician d’Arcy Thomson published a book that has inspired many contemporary scientists: On growth and form. The question posed is the mechanistic origin of forms. This question, which requires the articulation of different disciplines, lies at the heart of a branch of non-linear physics that is concerned with the natural sciences: morphogenesis. It’s a question that remains extremely topical in biophysics for understanding the origin of life and its diversity. In the advanced non-linear physics course, we will take up this question a century after d’Arcy Thomson. We’ll start by looking at some of the big questions on the existing Universe. How are planets formed? What forms the Earth’s relief? How can we describe volcanoes, mountains, rivers, hot and cold deserts and their sand and ice dunes? How are clouds formed? Why don’t most clouds cause rain? How do cyclones form? How does water circulate in the soil? How do animals move? How do living micro-organisms work? How do plants and animal embryos take shape? What is the dynamics of populations? Where does global warming and the ongoing collapse of life come from? And much more. Module 3: Disordered systems [P. Urbani] Most physical systems in nature can be considered as disordered in the sense that either the relevant dynamical degrees of freedom interact heterogeneously, or because they are subjected to random forces or fields or because they are frozen into amorphous/glassy structures. In all these cases, disorder gives rise to a complex and rich phenomenology characterized for example by metastability, the emergence of rough energy landscapes, slow dynamics and avalanche-like responses. The study of disordered systems has become important not only because they are central in statistical and condensed matter physics but also because the theoretical concepts developed to study them have important ramifications in a large variety of fields including computer science, optimization, statistics, high-dimensional inference and learning problems. This course will introduce a broad class of classical disordered systems and will discuss the theoretical fundamental concepts and methods to understand their equilibrium and out of equilibrium physics. A particular attention will be devoted to discuss how such concepts and methods can be apply beyond physics. Module 4: Nonequilibrium and active systems [F. Van Wijland] 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 calculus, 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 5: Statistical field theory and soft matter [J.-B. Fournier] Statistical field theory (SFT) applies ordinarily to systems close to a critical phase transition, where large-scale fluctuations are observed. However, soft matter (polymers, biological membranes, interfaces, liquid crystals, etc.) often exhibits strong thermal fluctuations because larges deformations have excitations energies close to kT. As the number of symmetries in nature is limited, these systems are described by the same SFT tools as critical systems. They thus exhibit critical exponents and scaling laws. For instance, the partition function of a self-avoiding walk or polymer is a path integral that can be mapped to an O(n) model in the replica n→0 limit and studied with the renormalization group techniques. Thus polymers share the same exponents as magnets. In the O(n) models, the field are vectors, but what if they become tensorial? We may use again our SFT skills and end up describing the liquid crystal ordering that lies within your computer screen. This is what physics is about: describing a large variety of phenomena with a limited number of concepts and tools. Module 6: Biophysics [A. Callan-Jones] To motivate how physics can give insight into — and also get insight from — living systems, consider how an organism develops. Yes, genetics is important, but how a homogeneous embryo breaks symmetry to become a developed being involves collective motions from the sub-cellular to the organism scale. As a result, modern biophysics borrows and has helped to advance concepts from nonlinear physics, soft matter, and non-equilibrium statistical physics. In this course we will focus on cell (and some tissue) biophysics, but will also delve into how organisms respond and adapt to their surroundings. We will emphasize throughout how intrinsically driven chemical reactions and molecular changes can give rise to out of equilibrium macroscopic responses. The course will broken down into five chapters, as illustrated in the figure. Module 7: Introduction to quantum and out of equilibrium dynamics [L. Cugliandolo] In these lectures we will introduce a selection of theoretical methods used to study the out of equilibrium dynamics of quantum complex systems. We will start by recalling some essential features of quantum mechanics. Then we will enter the discussion of modern topics such as: 1- the definition and effects of a quantum bath, 2- the modelization and treatment of quenched randomness, 3- the description of chaos in quantum systems, 4- the dynamics following quenches. At all stages a parallel with the classical counterparts will be made. Module 8: Quantum information [A. Keller] The objective of this module is to introduce some basic theoretical tools to understand what is meant by “quantum information and computation”. After recalling the formalism of quantum mechanics, we will introduce the quantum circuit model, give some simple examples of quantum algorithms and present an introduction to the problem of quantum error corrections. Depending on the time remaining, we will try to address actual questions related to the experimental proof of quantum computational supremacy with noisy intermediate-scale quantum device.

The module of “Nonlinear physics” is compulsory, while the other 3 modules can be chosen by the student among the list specified here above

Reading and study materials can be found at https://physics-complex-systems.fr/cursus.html

Exam: Written test;

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.