1 – Introduction et motivations

2 – Intégrales de chemin en mécanique quantique

3 – Compléments sur dérivée fonctionnelle et intégrales gaussiennes

4- Intégrales de chemin gaussiennes - Déterminants fonctionnels

5- Fonctionnelle génératrice et développements perturbatifs

6 – Fonctions de corrélation

7- Indications sur l'intégrale de chemin en théorie quantique des champs

8 – États cohérents et représentation holomorphe

I. Free Quantum Fields (Review)

The Klein–Gordon field, Canonical commutation relations, Propagators, Generating functionals

II. Interacting Quantum Fields

The S-matrix, Scattering amplitudes and cross sections, The LSZ reduction formula

III. Perturbation Theory

The interaction picture, Dyson series, Correlation functions, The Gell-Mann–Low formula, Feynman rules, Applications and examples

IV. Quantum Electrodynamics

Spinor fields: spin (1/2), Vector fields: spin (1), Interactions (QED), Electron–electron scattering

V. Renormalisation

Electron self-energy, Dimensional regularisation, Renormalisation, One-loop divergences in QED, Higher-order corrections and renormalisability

VI. Ward–Takahashi Identities

Functional integrals for scalar and fermionic fields, The effective action, Ward–Takahashi identities

• Molecular Dynamics Simulations for the exploration of emergent dynamic properties
  • TD: Integrating Newton’s equations of motion for continuous potentials: The secret behind the Verlet algorithm
  • Liouville formulation of symplectic and multiple-timestep integrators
  • Langevin dynamics
  • Thermodynamic averages, fluctuations and transformations between ensembles
  • Linear response and Green-Kubo relations for transport coefficients
  • TD: MD simulations of Lennard-Jones liquids
  • Non-equilibrium Molecular Dynamics
•  Monte Carlo Simulations for the exploration of emergent static properties
  • Exact enumeration of small systems: Reweighting and the exact evaluation of partition functions
  • Monte Carlo simulations
    • Simple Sampling: Statistical errors and the limits of reweighting
    • Importance Sampling: The Metropolis algorithm; Statistical errors, dynamical correlations and the power of well-designed trial moves: Glauber vs. Kawasaki vs Cluster moves for spin systems; Simulating polymers; Quenched disorder: spin glasses; Other ensembles.
    • TD: Monte Carlo simulations of the Ising model
• Free energies
  • Widom insertion and thermodynamic integration
    • TD: free energy of a methane molecule in water from TI (L. Monticelli)
  • Advanced techniques:
    • TD: The density of states of the Ising model from Multihistogram analysis/WHAM
    • TD: Protein binding energies from Umbrella Sampling and Steered Molecular Dynamics (L. Monticelli)
• Exploring Rare events
  • (sampling) transition states and paths
  • Kinetic Monte Carlo

I. General Relativity

A. The geometry of space-timeB. The Einstein equations

C. Basic GR solutions

D. Hamiltonian formulation: Einstein equations as evolution equations

E. Black Holes

II. Cosmology

A. Standard cosmological model

B. Hot Big-Bang and primordial cosmic plasma

C. Structure formation

Ondes nonlinéaires.
- Notions sur les équations de solitons. Equation de Korteweg-de Vries; compétitions entre nonlinéarité et dispersion.  Existence et interactions d’ondes solitaires.
 - Equation de Burgers. compétition entre nonlinéarité et dissipation.
 - Notions sur les systèmes de réaction diffusion. Propagation de fronts. Croissance d’un milieu stable dans un milieu métastable ou dans un milieu instable. Notions sur les milieux excitables.
 - Instabilité des trains d’onde. Développements multi-échelles et équations d’amplitude.
 - Equation de Schrödinger nonlinéaire. 


Morphogénèse:
- panorama de la morphogénèse dans les systèmes hors équilibre:  physique, hydrodynamique, biologie, etc.
- Etude de stabilité linéaire dans les systèmes étendus
- Patterns et leurs instabilités secondaires
- Coarsening et lois d'échelle
- Chaos spatiotemporel

Lecture 1: Sums of random variables
Statistics of sums of random variables, central limit theorem and its generalization to broadly distributed random variables. Examples of correlated or non-identically distributed random variables. Large deviations and their relevance in statistical physics.
Application of sums of random variables to equilibrium disordered systems, example of the Random Energy Model.

Lecture 2: Stochastic processes
Markov jump processes, master equation, detailed balance. Random walk on a lattice. Diffusive limit and diffusion equation. Continuous time random walks and anomalous diffusion.
Statistics of the first return time of a random walk.
Relaxation to equilibrium, increase of entropy, expansion over the eigenvectors of the Markov operator. Non-stationary dynamics at large time and aging phenomenon.

Lecture 3: Langevin and Fokker-Planck equations
Langevin equation, definition and physical motivation. Detailed balance.
Fokker-Planck equation. Derivation from a biased random walk on a lattice. Kramers-Moyal expansion. Stationary solution without flux (equilibrium) and with flux. Physical examples.
Relaxation to equilibrium and Ornstein-Uhlenbeck process.
Langevin equation with multiplicative noise, Ito and Stratonovitch discretization rules. Corresponding Fokker-Planck equation. Wong-Zakai theorem. Stochastic on-off intermittency as an example of multiplicative noise. Stochastic calculus and Ito's lemma.

Lecture 4: Response to an external field
Linear response close to equilibrium. Response function and Kubo formula. Time-correlation function and fluctuation-dissipation relation. Microreversibility and symmetry of response functions. Kramers-Krönig relation.
Statistics of trajectories and fluctuation relation far from equilibrium. Comparison of a trajectory with the time-reversed one. Statistics over trajectories. Fluctuation relation for discrete-time stochastic processes. Jarzinsky and Crooks relations.

Lecture 5: Examples of N-body distributions in stationary nonequilibrium states
One-dimensional lattice particle models, Zero-Range Process, Asymmetric Simple Exclusion Process. Matrix Product Ansatz solution.
Tapping dynamics for granular matter: Edwards postulate for the N-body distribution.
Approximate N-body distribution for interacting active particles with random self-propulsion forces.

Lecture 6: Effective description in terms of one-body problem
Dynamics with persistent interactions, approximation in terms of a non-linear Fokker-Planck equation with a self-consistent force field.
Collisional dynamics in the dilute regime, kinetic theory for the one-body distribution.
Principle of the derivation of continuum equations for the slow modes.

Part I — Passive soft matter:

(1) Suspensions — microhydrodynamics, Stokes flow, Brownian motion and statistical description;

(2) Dense and jammed systems — concentrated suspensions, emulsions, foams and granular materials, yield-stress rheology (Bingham, Herschel–Bulkley), elastoplastic and soft-glassy-rheology models;

(3) Polymer and biological networks — polymer elasticity, gelation and sol–gel transition, rubber elasticity, swelling (Flory–Rehner), viscoelasticity, fracture and toughening of gels.

Part II — Active soft matter:

(4) Active suspensions — microswimmer hydrodynamics, pushers/pullers, kinetic theory, active stress and rheology, motility-induced phase separation (MIPS);

(5) From foams to tissues — vertex model, cell shape and rigidity transition, T1 rearrangements;

(6) Active solids and gels — active polar/nematic gel theory, active poroelasticity, spontaneous flow, oscillations and buckling instabilities.

Each lecture is paired with a tutorial session and a research article used as the basis of the final oral exam.

Ce cours utilise une pédagogie par projet. Il s'agira de mener un projet de modélisation en géophysique interne ou planétologie afin de répondre à une question scientifique. Le projet est construit de A à Z par l'étudiant avec l'équipe pédagogique en soutien. La première séance fixe le cadre organisationnel et scientifique avec quelques bases et les enseignants proposent quelques idées de projets. Les séances qui suivront serviront à accompagner les étudiants dans l'avancée de leur projet, que ce soit sur les concepts, les expériences/calculs ou l'acquisition d'outils techniques. L’UE est évaluée par un rapport et une soutenance début janvier. L’accent est mis sur la démarche scientifique plutôt que sur les résultats, forcément limités sur une courte durée. 

1. Introduction: phenomenology of phase transitions, quantum phase transitions and non-equilibrium phase transitions
2. The Ising model : transfer matrix, spontaneous symmetry breaking, Mermin-Wagner theorem

3. Mean-field theory: variational approach, Ginzburg-Landau theory, Ginzburg-Landau criterion for the fluctuations
3. Phenomenological description of scale invariance : scaling forms and scaling laws
4. Real-space renormalization: Block spins and renormalization group, fixed points and critical exponents, renormalization of the 2D Ising model
5. Momentum-space renormalization: Wilson, Polchinski and Wetterich equations, equilibrium and non-equilibrium formalism.

Introduction / outils théoriques
Cinématique, règles de Feynman
QED
Amplitudes d’hélicité, Renormalisation, Effets NLO
Structure du proton
Fonctions de structure, Modèle des partons
Interaction faible
Courants chargés, structure chirale, courant neutres
Physique de la saveur
Matrice CKM, violation de CP
Modèle standard
Unification électrofaible, Higgs
Physique des neutrinos
Oscillation de neutrinos, masse des neutrinos
Astrophysique et cosmologie
Multi-messagers, ondes gravitationnelles (exp.)

Outils et méthodes
Détection
Interaction particule/Matière, trajectographes, calorimètre, etc..
Outils d’analyse
Probabilité, Statistique 

Part 1:  Conformal Symmetry
- the conformal group in arbitrary dimensions
- correlation functions in conformal field theories
- two-dimensional conformal theories and the Virasoro algebra

Part 2:  Supersymmetry
- Coleman-Mandula theorem and superalgebras
- superspaces
- representation theory: supermultiplets
- supersymmetric field theories

1. Introduction à la mécanique des fluides en général et à la turbulence en particulier
Phénoménologie et rappel de l'équation de transport d'impulsion (équations de Navier-Stokes). Présentation de quelques écoulements turbulents traditionnels (Canal, Convection de Rayleigh-Bénard, la turbulence homogène et isotrope). Introduction des échelles caractéristiques de la turbulence.

2. Approches empiriques en turbulence
Méthodes expérimentales (fil chaud et imagerie de particules) et numériques (la simulation numérique directe des équations de Navier-Stokes).

3. Une description probabiliste de la turbulence
Quelques notions sur les champs aléatoires: symétries, spectres de puissance, fonctions de corrélation. Définition de la turbulence homogène et isotrope.

4. Transport turbulent de quantité de mouvement et de chaleur
Décomposition et tenseur de Reynolds. Viscosité turbulente et modèle de longueur de mélange. Application au cas du canal. Dynamique de la température et nombre de Richardson.

5. Bilan d'énergie
Exemples de la convection de Rayleigh-Bénard et des écoulement cisaillés avec parois. Définition de la couche limite (laminaire et turbulente).

6. Turbulence homogène et isotrope
Théories de Kolmogorov. Lois de répartition spectrale de l'énergie, la cascade d'énergie vers les petites échelles. Phénomènes de transport d'énergie à travers les échelles : Relation de Kármán–Howarth et approches analytiques. Étude et modélisation du phénomène d'intermittence. Le mécanisme d'étirement de la vorticité. Invariance d'échelle – Fractalité et Multifractalité. Ouverture sur une approche stochastique de la turbulence.

Bibliographie
A first cours in turbulence, Tennekes and Lumley, MIT Press.
Turbulent Flows, S. Pope, CUP.
Turbulence, U. Frisch, CUP.

Week 1 — Open quantum systems. Density matrices, partial trace, purification, entropy. The real qubit, T₁, T₂, thermal occupation, two-level-system defects. Quantum channels, Kraus operators, the Lindblad master equation.

Week 2 — Measurement and phase space. POVMs, stochastic master equation, quantum trajectories, feedback, feedforward, heralding. Phase-space representations, Wigner, Husimi, Glauber–Sudarshan, characteristic functions, Gaussian states.

Week 3 — Linear systems, amplifiers, detectors. Coupled linear systems, cooperativity, mediated coupling, adiabatic elimination, reservoir engineering. Parametric oscillators and amplifiers, quantum limits on amplification, photon counting, homodyne and heterodyne detection, efficiency, added noise, noise temperature, mode matching.

Week 4 — Quantum metrology. Standard quantum limit, single- and two-mode squeezing, SU(1,1) interferometry, squeezed light in gravitational-wave detectors. Fisher information, Cramér–Rao bounds, atomic clocks, quantum advantage in sensing.

Week 5 — Resources and error correction. Entanglement, discord, contextuality, magic, state of the art in many-body entanglement and quantum simulation. Quantum error correction, key theorems, surface code, bosonic codes, quantum LDPC codes, cryogenics and scaling bottlenecks across platforms.

Week 6 — Quantum computing. Key quantum algorithms, quantum advantage in computing, out-of-time-order correlators as a path to practical quantum advantage.

1) Introduction to the complexity of the quantum many-body problem (1h);

2) Exact diagonalization and its limitations (3h);

3) Variational approaches; the Monte Carlo method, and variational Monte Carlo for physics and chemistry (4h);

4) Improving on the variational approach: diffusion Monte Carlo, fixed-node approximation for strongly interacting fermions (4h); 

5) Density-matrix approaches and tensor-network Ansätze for lattice quantum systems and beyond (4h);

6) Path-integral Monte Carlo methods for systems in continuous space: applications to quantum fluids and solids (4h); 

7) Path-integral Monte Carlo methods (and variants) for systems on lattice: applications to condensed matter, statistical physics, etc. (4h). 

The TD sessions will be hands-on, aimed at programming simple instances of some of the methods presented in the course. 

Ce cours utilise une pédagogie par projet. Il s'agira de mener un projet de modélisation en géophysique interne ou planétologie afin de répondre à une question scientifique. Le projet est construit de A à Z par l'étudiant avec l'équipe pédagogique en soutien. La première séance fixe le cadre organisationnel et scientifique avec quelques bases et les enseignants proposent quelques idées de projets. Les séances qui suivront serviront à accompagner les étudiants dans l'avancée de leur projet, que ce soit sur les concepts, les expériences/calculs ou l'acquisition d'outils techniques. L’UE est évaluée par un rapport et une soutenance début janvier. L’accent est mis sur la démarche scientifique plutôt que sur les résultats, forcément limités sur une courte durée. 

1. Introduction to the General Circulation of the Atmosphere and Ocean
2. Fundamental Equations and Phenomena
   - Boussinesq approximation, primitive equations, shallow water equations, quasi-geostrophic approximation
   - Rossby waves, gravity waves, barotropic instability
3. Atmospheric dynamics
   - baroclinic instability and mid-latitude dynamics
   - tropical dynamics: Hadley cell, convection and tropical waves
4. Oceanic dynamics
   - wind-driven circulation
   - thermohaline circulation
5. Predictability

I. Overview: What is string theory?

II. Classical (bosonic) string

III. Quantization of bosonic string

IV. Compactification

1) Atmosphere: Earth’s energy balance, radiative transfer, greenhouse effect, emissions and climate projection scenarios, etc.

2) Ocean: Ocean water mass structure and distribution, horizontal circulation, meridional overturning, ocean heat storage and vertical mixing, global ocean monitoring, etc.

3) Sea ice: Formation and properties, ice-albedo feedback, dense water formation, observed changes and future projections, human and geopolitical impacts, etc.

4) Ice sheets: Polar ice sheets in the coupled climate system, ice-sheet instabilities, satellite monitoring of ice sheets, sea-level change, etc.

5) Climate: Climate variability and dynamics, modes of natural variability, detection of anthropogenic signals, climate models, etc.

I.- Introduction to elasticity of continuous media

II.- Thin elastic plates

II.1.- Introduction: differential geometry of a surface, equations

II.2.- Buckling: roll-like patterns, growth-induced patterns

II.3.- Singularities: focussing of energy around points (developable cones) and lines (stretching ridges)

II.4.- Patterns from singularities and foldable structures

 
III.- Fracture Mechanics

III.1.- Linear Elastic Fracture Mechanics

III.2.- Quasi-static crack propagation: instabilities and crack-induced patterns

III.3.- Fast fracture: dynamics, instabilities.

III.4.- Application to friction

1. The relation between large deviation theory and thermodynamical potentials
 
2. Computation of free energy and entropy functions for some basic problems in statistical physics

3. The relation between large deviation theory and thermodynamical potentials

4. Computation of free energy and entropy functions for some basic problems in statistical physics

5. Gallavoti, Cohen and Evans fluctuation theorems, Crooks and Jarzynski inequality and fluctuation theorem, some of the major developments of statistical mechanics basis during the last two decades

6. The relation between kinetic theory, large deviation theory, and the irreversibility paradox

7. The use of large deviation theory for dynamical system theory (finite time Lyapunov exponents)
 
8. Large deviation theory and disordered systems

9. propagation, transport) and turbulence (self-organization, turbulent transport)

10. The study of multistability phenomena in a multitude of physical applications (magnetic systems, physical chemistry, polymers, turbulence, and so on)

11. Application of large deviation theory for a model of Jupiter's Great Red Spot, and several applications to climate dynamics.)

- The nature of GW and its historical evolution
- Sources of GW
- Specific characteristics of GW: what we can read from the GW signal
- The interferometric detectors of GW
- Some challenges in the design of GW detectors
- The future of GW Astronomy

1- Introduction: what is active matter?

2- Self propelled bodies and persistent random walkers: statistics of active-particle trajectories.

2- Active systems in a background fluid: description of microswimmers.
 
3- Active Matter thermodynamics: we will address questions such as, can we a define a temperature, a pressure for those systems?
 
4- Continuum description of active matter.

I. From Classical to Mesoscopic Transport

II. Ballistic Quantum Transport and the Landauer Formalism

III. Disorder and Quantum Interference

IV. Quantum Transport and Interaction Effects

V. Frontiers of Contemporary Quantum Transport