# Master 2 courses in Physics - Physics Complex Systems track

The Master's degree Physics Complex Systems is a research-oriented track in fundamental physics, aimed at the study of complex systems through statistical physics in equilibrium and out of equilibrium, field theory, stochastic processes, dynamical systems, non-linear physics, inference techniques and numerical simulation. The objects of study are in particular complex networks, active matter, the physics/social science interface, "soft" matter, biological systems, complexity in quantum physics, or questions arising from "big data" or "machine learning" issues.

## 2nd year Master courses (M2) taught in English

### Semester 1

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

Stochastic phenomena, the description of which involves randomness, are ubiquitous in physics, chemistry, biology and beyond (information theory, computer science, economy etc.) The goal is to equip students with the essential tools for their modeling and understanding. The course starts with a probability bootstrap with emphasis on large deviations, establishing a bridge with statistical physics approaches. Markov processes, Langevin and Fokker-Planck equations are then presented, together with the stochastic differential equation framework and stochastic calculus (Stratonovich versus Ito-Döblin rules). The linear-response theory features, found along the way, will be put in a broader perspective. The second half of the course is devoted to fluctuation theorems, first passage properties, and key results in stochastic thermodynamics. Follows an introduction to martingales and functionals of Brownian motion, including the Feynman-Kac corres.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 6 ECTS - English Level: B2 (no test required)**

**Brief Description**

Machine learning (ML) is one of the most dynamic and exciting areas in modern datadriven research. Built upon inspiration from fields as different as statistics, computer science, neurosciences and physics, it allows for automatic learning from complex large-scale datasets. The lectures aim at introducing the core concepts and algorithms of ML in a way easily understood by physicists, both in the setting of supervised learning (linear and logistic regression, ensemble methods, deep neural networks…) and unsupervised learning (dimensional reduction, clustering, generative modelling…). The course is accompanied by Python Jupiter Notebooks, which allow for testing the main algorithms presented in the lectures, and introduces some highly used ML Python packages to the students.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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 illustrateand 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.).

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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 nanometerscale 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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

In this lecture, we explore transport properties of quantum systems whose dimensions are smaller than the characteristic phase coherence length. Below this length scale, the quantum nature of the electronic wavefunction shows up. In such systems, the complex interplay between disorder, Coulomb interaction and dimensionality results in original behaviours. By using Landauer formalism, we show how quantum statistics and electronic interferences affect the transport and break the classical laws for electricity. We see also how non-equilibrium current fluctuations are related to the fundamental properties of these systems. Using Green’s function formalism, we study how the interplay between electronic interferences and disorder leads to universal conductance fluctuations, weak localization and even to persistent currents at equilibrium. Finally, we investigate the interplay between such a coherent system and a superconductor, which yields new fundamental transport mechanisms.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

Classical electromagnetism before quantum electrodynamics: Maxwell’s equations in Fourier space, transverse and longitudinal modes, energy and momentum, gauges choices: Coulomb and radiation.

Schrödinger’s equation in the presence of a classical electromagnetic field, perturbation theory and Fermi’s Golden Rule, excitation rates.

The Fock space of the quantum electromagnetic field, photons, the interaction Hamiltonian between matter and the electromagnetic field.

Some properties of the quantum electromagnetic field: energy, momentum and angular momentum; vacuum fluctuations and the Casimir effect.

Some remarkable states of the quantum electromagnetic field: coherent states and squeezed states, classical limit of quantum electromagnetism.

The interaction between photons and matter: absorption and emission of a single photon by an atom, spontaneous emission, laser, the Weisskopf-Wigner model.

If time allows: entangled photons and the violation of Bell’s inequalities.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 1st Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

The objective of this course 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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

### Semester 2

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

In these lectures, I will provide useful analytical tools typically employed in the derivation of statistical properties of the eigenvalues of various classes of random matrices. I will start from a characterization of matrix models with real spectrum and discuss in particular: i) Ensembles with independent entries; ,ii) Ensembles that have rotational invariance. Then, I will enter into more detail performing rigorous computations based also on the replica method. In the last lectures, I will focus on exactly solvable mean-field models with applications to spin glasses, non-convex optimization problems, and theoretical ecology.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

Information technology and nanotechnologies urge physics and chemistry to interpenetrate with different fields of mathematical and information sciences. Thermodynamic structure in the Langevin stochastic process to which this lecturer has contributed is just one example. Nowadays, we need to understand the essential scientific concepts from different viewpoints, since specific methods for specific subjects become soon obsolete. To meet such requirements, the present course aims at grasping the important concepts of advanced stochastics by multiple approaches, statistical vs individual, microscopic vs coarse-grained, forward vs inverse, etc. Being accompanied by weekly preparatory reading material and after-course exercises, the lectures bring workable knowledges to both who knows already and who learns for the first time.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

As soon as space is organized, there is a need for transport. This is the case in systems as different as cities or cell interior. In spite of these differences, transport can be described by the same families of models, inspired from those developped for fluids, and corresponding to different description levels: microscopic, mesoscopic, or macroscopic.

In this course, we shall study the properties of the different types of models, see how conservation laws constrain the form of model equations, and discuss which models are more appropriate for given applications. We shall also see how the competition for space that can be described through game theory.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

This course provides an introduction to the theory and the practice of inference and learning from data. I discuss the different kinds of learning from data: supervised, unsupervised and reinforcement learning. As a simple example of supervised learning I discuss the « perceptrons » and « support vector machines ». I also discuss in a lesser detail multilayer neural networks and « deep learning ». The problem of data clustering is addressed in detail and used to illustrate unsupervised learning. I discuss applications from computer revised learning. I discuss applications from computer science (hand writing recognition, matrix completion), biology (inverse Ising models for protein folding), and associative memory (Hopfield model). The course includes 3 tutorials to implement the discussed algorithms.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

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 BlackScholes 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.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)

**2nd year Master - 2nd Semester - 3 ECTS - English Level: B2 (no test required)**

**Brief Description**

This course is a follow up to Martin Weigt’s « Computational science » class. The class will consist in lectures and lab sessions where we will code some algorithms from scratch, and apply them to textbook or real-life datasets. We will cover the basic building blocks of modern ML, emphasizing the mathematical origins of the algorithmic recipes used in learning algorithms, which allows to make sense out of the diversity of algorithms in use. In the same time, key methodological points will be outlined and illustrated on concrete examples (datasets and tasks). Typical success and pitfalls of statistical learning and their interpretation by statistical physics methods will be provided. Overall, we will cover the vocabulary used in ML litterature, so as to become autonomous when confronted with new methods. Depending on the appetite of the class, we will spend more or less time on Deep Learning methods.

**Contact**

Dominique Mouhanna (dominique.mouhanna@sorbonne-universite.fr)