B.A. in CPES: Modeling and Numerical Methods in Materials Science

Atomic Physics: Periodic Table, Lewis and VSEPR Models
# Electron Configuration
# Periodic Table
# Description of Molecular Chemical Entities: Lewis and VSEPR

Introduction to Organic Chemistry and Basic Reaction Mechanisms
# Nomenclature
# Stereochemistry
# Fundamentals of Reactivity in Organic Chemistry
# Aliphatic Nucleophilic Substitutions
# β-Eliminations
# Electrophilic Addition to Alkenes
# Nucleophilic Addition: The Example of Organomagnesium Compounds

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First-Year Mathematics Curriculum at Lycée Joffre

# Set-Theoretical Reasoning and Vocabulary

# Supplementary Materials on Algebraic Calculus, Elementary Analysis, and Geometry

# Census

# Real Analysis: Sequences, Limits, and Continuity; Differentiability; Calculation of Antiderivatives and Integrals.

# Complex Numbers

# Differential Equations

# Linear Algebra: Vectors, Linear Mappings, and Matrices; Bases and Dimension,

# Asymptotic Analysis

# Introduction to Statistics

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The curriculum below covers the two physics courses in the first year of the CPES program at Lycée Joffre

Geometric Optics

# Laws of Geometric Optics
- Light: source models and spectra.
- Descartes' laws, plane diopter

#
Lenses - Spherical Diopter, Spherical Mirror - Spherical, Thin Lenses
- Conjugation Relations

# Optical Instruments
- The Eye and Corrective Eyeglasses
- Astronomical Telescope or Galilean Telescope
- Microscope

Stitch Mechanics, Vibrations

# Point Kinematics
- Coordinate Systems
- Position, Velocity, Acceleration
- Classical Motion (Constant Acceleration, Circular Motion)
- Chronophotography

# Laws of Dynamics

# Energy Approach

# Oscillators

Electrostatics, Magnetostatics

# Electric charge, electric field

# Electrostatic potential, potential energy

# Gauss's Theorem: Statement and Examples

# Current element, magnetostatic field

# Circulation and Flux of the Magnetic Field

# Electrostatic Dipoles

# Motion of a Charged Particle in an Electrostatic or Magnetostatic Field

Electrokinetics

# Current and voltage, charge and current densities, the law of conservation. Mesh and node theorems

# Simple dipoles and models.

# RC, RL, and RLC Circuits, Filtering.

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This modern language course focuses on the written and spoken use of English, with a particular emphasis on scientific and socioeconomic contexts.

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The first-year curriculum is as follows:

# Introduction to Contemporary Issues

# Introduction to Economics: An Introduction to Geopolitics

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The curriculum below covers the two first-year computer science courses at Lycée Joffre:

Algorithms and Programming 1: Basic Concepts in Algorithms and Programming:
• The concept of a problem, problem instances, complexity, termination, and validity proofs
• Algorithms and iterative processing
• Proofs: correctness, termination, complexity
• Linear data structures: arrays, linked lists, stacks, queues
• Sorting: insertion, selection, merge sort, other sorting examples
• Introduction to recursive processing (binary search, etc.)
• Programming in Python

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Using Computer Systems: Introduction to the Main Concepts of Computer Systems:
• Practical proficiency with a Linux environment
• Knowledge of the main Linux commands
• Understanding documentation pages (man)
• Sessions, files, directories, access permissions
• Programs, processes, commands, and scripting languages
• Console (terminal) vs. graphical execution modes
• Familiarization with shells, writing a few scripts

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Introduction to Numerical Analysis 

# Basics, Challenges, and Limitations; Representation of Numbers in Computers

# Additional Concepts for Numerical Analysis

# Numerical Solution of Nonlinear Equations

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Instruction takes the form of projects led by pairs of students, supplemented along the way by online sessions that allow students to apply concepts from physics and chemistry on a computer.

Depending on the needs, a few more traditional classes are scheduled to supplement the students' training.

Tentative schedule:

1. Preliminaries: Visualization of scalar and vector fields: gradient, divergence, rotator, Laplacian, contour lines, surface and volume integrals, numerical solutions to point mechanics problems

2. Supervised projects offered to students: These projects involve the collection, processing, and interpretation of data. The general theme we have chosen is to train students to understand the concepts of “statistical field theory,” that is, the concept of probability applied to functions. This theme is ubiquitous in modern science: image processing and analysis, AI, statistical physics, dynamical systems and stochastic processes, sequencing, epidemiology, economics, observational cosmology..., and the combination of an approach that integrates numerical methods, mathematical ideas, and modeling seems to emerge naturally (examples of projects, list provided for illustrative purposes only: sand-grain statistics, the Poisson and Gaussian distributions, connection to the Beer-Lambert law via the concept of effective cross-section; observational data in cosmology: matter/radiation distribution on the scale of the universe; epidemiological statistics and dynamics; microscopy and analysis of Brownian motion; etc.)

3. Additional Topics in Chemistry: Slater’s effective theory for atoms (effective screening potential), molecular orbitals, fragment approach

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General Knowledge (Epistemology and Philosophy):

# History of Ideas

# Epistemology of Science and Digital Technology

#Argumentation and Communication

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The curriculum below covers the two physics courses in the first year of the CPES program at Lycée Joffre

Geometric Optics

# Laws of Geometric Optics
- Light: source models and spectra.
- Descartes' laws, plane diopter

#
Lenses - Spherical Diopter, Spherical Mirror - Spherical, Thin Lenses
- Conjugation Relations

# Optical Instruments
- The Eye and Corrective Eyeglasses
- Astronomical Telescope or Galilean Telescope
- Microscope

Stitch Mechanics, Vibrations

# Point Kinematics
- Coordinate Systems
- Position, Velocity, Acceleration
- Classical Motion (Constant Acceleration, Circular Motion)
- Chronophotography

# Laws of Dynamics

# Energy Approach

# Oscillators

Electrostatics, Magnetostatics

# Electric charge, electric field

# Electrostatic potential, potential energy

# Gauss's Theorem: Statement and Examples

# Current element, magnetostatic field

# Circulation and Flux of the Magnetic Field

# Electrostatic Dipoles

# Motion of a Charged Particle in an Electrostatic or Magnetostatic Field

Electrokinetics

# Current and voltage, charge and current densities, the law of conservation. Mesh and node theorems

# Simple dipoles and models.

# RC, RL, and RLC Circuits, Filtering.

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Program Overview:

– Probability Vocabulary

– Random variables and probability distributions

– Expectation and variance

– Applications

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Thermodynamics
# Fundamentals: state functions, thermodynamic equilibrium, changes of state
# Kinetic theory of gases
# First law
# Thermochemistry
# Second law
# Thermal machines (simple).

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The first-year curriculum is as follows:

# Introduction to Contemporary Issues

# Introduction to Economics

# Introduction to Geopolitics

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This course is an introduction to the research process: it includes an immersion in a research laboratory (meeting with researchers, learning about research projects) and a literature review designed to help students articulate their understanding of the research process, as well as the objectives and expected and hoped-for outcomes of a project.

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First-Year Mathematics Curriculum at Lycée Joffre

# Set-Theoretical Reasoning and Vocabulary

# Supplementary Materials on Algebraic Calculus, Elementary Analysis, and Geometry

# Census

# Real Analysis: Sequences, Limits, and Continuity; Differentiability; Calculation of Antiderivatives and Integrals.

# Complex Numbers

# Differential Equations

# Linear Algebra: Vectors, Linear Mappings, and Matrices; Bases and Dimension,

# Asymptotic Analysis

# Introduction to Statistics

View the full page

Instruction takes the form of projects led by pairs of students, supplemented along the way by online sessions that allow students to apply concepts from physics and chemistry on a computer.

Depending on the needs, a few more traditional classes are scheduled to supplement the students' training.

Tentative schedule:

1. Preliminaries: Visualization of scalar and vector fields: gradient, divergence, rotator, Laplacian, contour lines, surface and volume integrals, numerical solutions to point mechanics problems

2. Supervised projects offered to students: These projects involve the collection, processing, and interpretation of data. The general theme we have chosen is to train students to understand the concepts of “statistical field theory,” that is, the concept of probability applied to functions. This theme is ubiquitous in modern science: image processing and analysis, AI, statistical physics, dynamical systems and stochastic processes, sequencing, epidemiology, economics, observational cosmology..., and the combination of an approach that integrates numerical methods, mathematical ideas, and modeling seems to emerge naturally (examples of projects, list provided for illustrative purposes only: sand-grain statistics, the Poisson and Gaussian distributions, connection to the Beer-Lambert law via the concept of effective cross-section; observational data in cosmology: matter/radiation distribution on the scale of the universe; epidemiological statistics and dynamics; microscopy and analysis of Brownian motion; etc.)

3. Additional Topics in Chemistry: Slater’s effective theory for atoms (effective screening potential), molecular orbitals, fragment approach

View the full page

The curriculum below covers the two first-year computer science courses at Lycée Joffre:

Algorithms and Programming 1: Basic Concepts in Algorithms and Programming:
• The concept of a problem, problem instances, complexity, termination, and validity proofs
• Algorithms and iterative processing
• Proofs: correctness, termination, complexity
• Linear data structures: arrays, linked lists, stacks, queues
• Sorting: insertion, selection, merge sort, other sorting examples
• Introduction to recursive processing (binary search, etc.)
• Programming in Python

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VBA Programming in Excel: Developing VBA Macros in Excel:

# Working with Macros in Excel

# Applying the Basics of VBA in Excel

# Debugging VBA Macros in Excel

# Writing Custom Functions

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General Knowledge (Epistemology and Philosophy):

# History of Ideas

# Epistemology of Science and Digital Technology

#Argumentation and Communication

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This modern language course focuses on the written and oral use of English, with a particular emphasis on scientific and socioeconomic contexts.

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Second-year mathematics curriculum, portion taught at Lycée Joffre:

# Review of Linear Algebra and Additional Concepts in Euclidean Geometry

# Advanced Probability

# Further Analysis of Functions of a Single Variable

# Functions of Several Variables

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The oscillator is a fundamental concept in physics: matter is often modeled as a collection of oscillators (harmonic or otherwise) that interact with one another and with the external environment. The environment acts on the matter through a wave, such as an acoustic or electromagnetic wave. This allows us to lay the theoretical foundations for problems involving radiation-matter interaction and thus to develop one of the fundamental tools for the study of matter (in the broadest sense): spectroscopy.

Spectroscopy is, in fact, the fundamental tool for studying the physical properties of the objects around us, such as molecules, crystals, stars, and galaxies. These properties are deduced either from their spontaneous emission or from their response to external excitation. For example, we measure the absorption, reflection, and transmission properties of applied electromagnetic radiation (visible, infrared, X-rays, neutrons, etc.). The response to this radiation then provides a way to identify the various types of oscillators that make up the medium under study.

 In short, the study of the physical environments around us relies on two fundamental theoretical tools: oscillators and waves, which are precisely the subject of this course.  

The approach taken here is a step-by-step progression, starting with the harmonic oscillator, then moving on to coupled oscillators, and finally to waves analyzed within the framework of discrete systems: infinite and then finite coupled oscillators with different boundary conditions.

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Advanced Topics in Linear Algebra, Reduction Theory, and Applications 

# Additional Concepts in Linear Algebra: The Concept of Direct Sum

# Determinants

# Reduction: Diagonalization and its applications to the study of discrete and continuous dynamical systems in physics, chemistry, biology, and economics.

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This course continues to build students' skills in applied mathematics, with a focus on numerical methods.

Schedule (subject to change depending on the year and the time available):

# Matrix Numerical Analysis

# digital integration

# Numerical Solution of Differential Equations

# Introduction to the Qualitative Analysis of Nonlinear Differential Equations

.

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This module supplements and formalizes the thermodynamic concepts introduced in the Thermodynamics 1 course unit, exploring several aspects in greater depth: thermodynamic potentials defined using Legendre transformations, the thermodynamics of open systems, phase transitions in pure substances, and irreversible processes, with forays into the microscopic level to provide an overview of the physical foundations of the theory.

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This course unit is taken only by students in the Materials Science track of the CPES program. It includes a mathematics section and a chemistry section, focusing on the concept of symmetry groups (of a chemical molecule) and its application to the determination of molecular orbitals through the linear combination of atomic orbitals.

Tentative schedule:

# Introduction to Group Theory, Finite Groups, Symmetry Groups of Molecules

# Linear representations of groups, character theory, and applications to molecular orbitals or the vibrational modes of a molecule

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The 10th-grade computer science course offered at the high school focuses on advanced concepts in algorithms and programming:

# Tree Data Structures: Binary Trees, Heaps, ABR, Priority Queues

# Sorting into piles, lower bound on the sort

# Graph Structures: Representations (Adjacency Matrices, Edge Lists, Neighbor Lists)

# Basic Algorithms (Connectivity, Breadth-First and Depth-First Search, Topological Sorting)

# Distance Calculation (Dijkstra, Heap Implementation)

# Introduction to Object-Oriented Programming (encapsulation, late binding, classes, methods, etc.)

# Python Programming

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Curriculum taught at Lycée Joffre during the 2nd year

# Solid Mechanics (statics, dynamics—not torsional forces)

# Electromagnetism in a Vacuum

# Diffraction optics, Fourier optics

# Waves and Radiation, Wave Scattering.

# Kinetics and Thermochemistry, Introduction to Electrochemistry

# Organic Chemistry: Some Reaction Mechanisms Well-Suited for Quantum Modeling Using Boundary Orbitals

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This course unit complements the computer science curriculum by introducing students to the study of information systems and databases (designing processes within an information system and managing relational databases):

# Information Systems: Introduction to the Entity-Relationship Model, the Relational Model, and Business Process Modeling (Conceptual Business Process Model, Organizational Business Process Model)

# Databases: Creating, Manipulating, and Querying Relational Databases.

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As in the first year, this course consists of supervised projects involving data collection, processing, and interpretation: the approach combines numerical methods, mathematical concepts, and modeling.

The topics of study may change from year to year in order to keep students engaged.

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This modern language course focuses on the written and spoken use of English, with a particular emphasis on speaking.

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General Knowledge (Epistemology and Philosophy):

# History of Ideas

# Epistemology of Science and Digital Technology

#Argumentation and Communication

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General Knowledge (Epistemology and Philosophy):

# History of Ideas

# Epistemology of Science and Digital Technology

#Argumentation and Communication

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Introduction to the Use of Data Mining Methods:

# Motivations, Areas of Application

# Data mining tasks (classification, estimation, prediction, etc.)

# Data Processing (Pre- and Post-Processing)

# Supervised learning (Bayesian method, k-nearest neighbors, neural networks, etc.)

# Unsupervised Learning (k-means)

# Evaluation of Methods

# Data mining software (R, Scikit, Weka, etc.)

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Second-year mathematics curriculum, portion taught at Lycée Joffre:

# Review of Linear Algebra and Additional Concepts in Euclidean Geometry

# Advanced Probability

# Further Analysis of Functions of a Single Variable

# Functions of Several Variables

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Curriculum taught at Lycée Joffre during the 2nd year

# Solid Mechanics (statics, dynamics—not torsional forces)

# Electromagnetism in a Vacuum

# Diffraction optics, Fourier optics

# Waves and Radiation, Wave Scattering.

# Kinetics and Thermochemistry, Introduction to Electrochemistry

# Organic Chemistry: Some Reaction Mechanisms Well-Suited for Quantum Modeling Using Boundary Orbitals

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The first part of this course presents the fundamentals of quantum chemistry for chemists and physical chemists. It begins by reviewing the principles of quantum mechanics and its governing equation, the Schrödinger equation. Solving the Schrödinger equation in simple cases and the concepts of wave functions and quantization are presented and illustrated using simple examples. The hydrogen atom is then studied.

The course also examines approximation methods used to determine the properties of complex systems where the Schrödinger equation cannot be solved directly. The effect of spin on the electronic properties of atoms and molecules will also be discussed.

The second part of this course focuses on the quantum description of molecular properties and reactivity. The qualitative construction of molecular orbitals using symmetry properties will be introduced, and the connection between molecular orbital diagrams and chemical bonding will be explained. The relationship between molecular geometry and electronic structure will be discussed. This course will then examine the Hückel method, which is used to obtain molecular orbital diagrams for π systems. The classical concepts of conjugation, delocalization, donor and acceptor character, and aromaticity will thus be studied within this framework. Bound orbital theory is used to explain molecular reactivity (cycloadditions, electrocyclization) and molecular geometries.

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This course continues to provide students with training in applied mathematics, particularly in the use of statistical tools:

# Concepts of Populations and Random Samples.

# Estimators.

#Mean Square Error and Bias-Variance Decomposition.

# Empirical estimators of the mean and variance. Construction of estimators using the method of moments. Maximum likelihood.
Interval estimation.

# Hypothesis testing: null hypothesis, alternative hypothesis. Type I and Type II errors. Test power. Test statistic, rejection region. Examples of classic tests.

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This course, similar in concept to the first-year project, is an introduction to the research process that goes deeper than the first-year course: it includes an immersion in a research laboratory, a literature review, and the completion of a project involving the study of a physical or chemical context, the collection of experimental or observational data, and the implementation of a modeling and/or numerical simulation approach.

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This module provides an introduction to the process of using computer tools in physics: it involves analyzing a phenomenon, idealizing or modeling it, and then studying it on a computer. The critical interpretation of the results is also part of this process. The examples covered are chosen to relate to other current topics in the course.

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This course covers the theoretical tools used to describe distributed and conserved quantities, such as density, energy, or charge density.

The first introductory chapter presents the general conservation equation and the vector analysis tools needed to understand it, as well as the various fields in which this equation is used (virtually all of physics, but also chemistry, economics, and more...).

The following chapters are primarily devoted to the case with one spatial dimension and one temporal dimension (1+1 D). Chapter 2 addresses convective transport and presents the characteristic method for solving the corresponding first-order quasi-linear partial differential equation. Diffusive transport is then studied in Chapter 3, along with the heat equation and the Fourier method. Finally, Chapter 4, devoted to reaction-diffusion equations, considers the case of convective and diffusive transport using the Smoluchowsky equation, and derives the famous Einstein equation for diffusion.

The course includes theoretical and computational tutorials covering topics such as road traffic and queueing models.

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The 10th-grade computer science course offered at the high school focuses on advanced concepts in algorithms and programming:

# Tree Data Structures: Binary Trees, Heaps, ABR, Priority Queues

# Sorting into piles, lower bound on the sort

# Graph Structures: Representations (Adjacency Matrices, Edge Lists, Neighbor Lists)

# Basic Algorithms (Connectivity, Breadth-First and Depth-First Search, Topological Sorting)

# Distance Calculation (Dijkstra, Heap Implementation)

# Introduction to Object-Oriented Programming (encapsulation, late binding, classes, methods, etc.)

# Python Programming

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As in the first year, this course consists of supervised projects involving data collection, processing, and interpretation: the approach combines numerical methods, mathematical concepts, and modeling.

The topics of study may change from year to year in order to keep students engaged.

View the full page

This modern language course focuses on the written and spoken use of English, with a particular emphasis on speaking.

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Thermodynamics: Microscopic and Macroscopic Aspects

Thermodynamics is the tool of choice for studying matter on a macroscopic scale. In particular, when it comes to chemical reactions, it allows us to predict the direction in which they proceed and their equilibrium state. In the early years of the bachelor’s program, the focus is on describing the principles of thermodynamics and their direct application to chemistry in the case of simple, single-phase equilibrium reactions or reactions between homogeneous phases. This course will expand on this knowledge in two directions.

First, we will generalize this macroscopic thermodynamic descriptive framework to more complex systems, such as interfacial systems where surface tension plays a role, or non-uniform phases where the composition is not the same everywhere due to an external field. We will also study phase transitions and equilibrium displacements.

Next, we will examine the connection to the microscopic world, where matter is described at the atomic scale. We will show that the evolution predicted by thermodynamics is statistical in nature, with the equilibrium state corresponding to the most probable macroscopic state given the constraints imposed on the system. This will allow us to deduce the macroscopic thermodynamic properties of a physicochemical system from its microscopic description.

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The first part of the module will introduce metals and alloys through the lens of crystallography (from the “ideal” crystalline solid to defects and solid solutions), followed by a second part devoted to their characterization by X-ray diffraction, and the final part will address their synthesis using their solid-liquid binary phase diagram (description and construction) and the various solid-state transformations.

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This course builds on the mathematics covered in the first year (L1) and the^first semester of the second year (L2). It will introduce the mathematical tools that physicists need in the areas of integration theory, functional transformations, complex variables, and distributions.

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This module provides an introduction to the concepts and methods of statistical physics for systems at equilibrium, using a bottom-up approach: starting with examples and then deriving the general principles. It draws heavily on the course by Harvey Gould and Jan Tobochnik. The final chapter of the course offers a historical introduction to the development of the theory of Brownian motion.

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This course aims to introduce the fundamentals of physical hydrodynamics. The kinematic aspects are covered first: Eulerian and Lagrangian formalisms, analysis of the motion of a fluid volume element, introduction to the stream and potential velocity functions, and applications to various types of flows. In the next section on fluid dynamics, we derive Euler’s equation and Bernoulli’s equation for the flow of ideal fluids, followed by the Navier-Stokes equation describing the flow of Newtonian viscous fluids. This section will lead us to define the stress tensor as well as the Reynolds number, which allows us to determine whether a flow is laminar or turbulent. The course concludes with an introduction to the mechanics of deformable solids: displacement field, strain tensor, and deformation tensor.

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In this course, we will build upon the basic concepts previously covered in Quantum Mechanics in Semester 5. The course is organized around the following main topics: extension of the wave mechanics formalism, angular momentum theory, the hydrogen atom, perturbations, and an introduction to relativistic quantum mechanics.

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Thermodynamics: Microscopic and Macroscopic Aspects

Thermodynamics is the tool of choice for studying matter on a macroscopic scale. In particular, when it comes to chemical reactions, it allows us to predict the direction in which they proceed and their equilibrium state. In the early years of the bachelor’s program, the focus is on describing the principles of thermodynamics and their direct application to chemistry in the case of simple, single-phase equilibrium reactions or reactions between homogeneous phases. This course will expand on this knowledge in two directions.

First, we will generalize this macroscopic thermodynamic descriptive framework to more complex systems, such as interfacial systems where surface tension plays a role, or non-uniform phases where the composition is not the same everywhere due to an external field. We will also study phase transitions and equilibrium displacements.

Next, we will examine the connection to the microscopic world, where matter is described at the atomic scale. We will show that the evolution predicted by thermodynamics is statistical in nature, with the equilibrium state corresponding to the most probable macroscopic state given the constraints imposed on the system. This will allow us to deduce the macroscopic thermodynamic properties of a physicochemical system from its microscopic description.

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The first part of the module will introduce metals and alloys through the lens of crystallography (from the “ideal” crystalline solid to defects and solid solutions), followed by a second part devoted to their characterization by X-ray diffraction, and the final part will address their synthesis using their solid-liquid binary phase diagram (description and construction) and the various solid-state transformations.

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This course builds on the mathematics covered in the first year (L1) and the^first semester of the second year (L2). It will introduce the mathematical tools that physicists need in the areas of integration theory, functional transformations, complex variables, and distributions.

View the full page

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This module provides an introduction to the concepts and methods of statistical physics for systems at equilibrium, using a bottom-up approach: starting with examples and then deriving the general principles. It draws heavily on the course by Harvey Gould and Jan Tobochnik. The final chapter of the course offers a historical introduction to the development of the theory of Brownian motion.

View the full page

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This course aims to introduce the fundamentals of physical hydrodynamics. The kinematic aspects are covered first: Eulerian and Lagrangian formalisms, analysis of the motion of a fluid volume element, introduction to the stream and potential velocity functions, and applications to various types of flows. In the next section on fluid dynamics, we derive Euler’s equation and Bernoulli’s equation for the flow of ideal fluids, followed by the Navier-Stokes equation describing the flow of Newtonian viscous fluids. This section will lead us to define the stress tensor as well as the Reynolds number, which allows us to determine whether a flow is laminar or turbulent. The course concludes with an introduction to the mechanics of deformable solids: displacement field, strain tensor, and deformation tensor.

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In this course, we will build upon the basic concepts previously covered in Quantum Mechanics in Semester 5. The course is organized around the following main topics: extension of the wave mechanics formalism, angular momentum theory, the hydrogen atom, perturbations, and an introduction to relativistic quantum mechanics.

View the full page

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Thermodynamics: Microscopic and Macroscopic Aspects

Thermodynamics is the tool of choice for studying matter on a macroscopic scale. In particular, when it comes to chemical reactions, it allows us to predict the direction in which they proceed and their equilibrium state. In the early years of the bachelor’s program, the focus is on describing the principles of thermodynamics and their direct application to chemistry in the case of simple, single-phase equilibrium reactions or reactions between homogeneous phases. This course will expand on this knowledge in two directions.

First, we will generalize this macroscopic thermodynamic descriptive framework to more complex systems, such as interfacial systems where surface tension plays a role, or non-uniform phases where the composition is not the same everywhere due to an external field. We will also study phase transitions and equilibrium displacements.

Next, we will examine the connection to the microscopic world, where matter is described at the atomic scale. We will show that the evolution predicted by thermodynamics is statistical in nature, with the equilibrium state corresponding to the most probable macroscopic state given the constraints imposed on the system. This will allow us to deduce the macroscopic thermodynamic properties of a physicochemical system from its microscopic description.

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This course builds on the mathematics covered in the first year (L1) and the^first semester of the second year (L2). It will introduce the mathematical tools that physicists need in the areas of integration theory, functional transformations, complex variables, and distributions.

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This unit is a natural continuation of the units on classical Newtonian mechanics.

In the first part of this unit, we cover classical mechanics, starting with the principle of least action and arriving at two new formulations: the Lagrangian formalism and the Hamiltonian formalism. We study the connection between physical symmetries and conservation laws (E. Noether’s theorem) and introduce Poisson brackets, which allow us to write the classical laws of time evolution of physical quantities in a form that already foreshadows those of quantum mechanics.

In the second part of the course, beginning with an examination of the experimental limits of classical mechanics, a new theory of mechanics is introduced: quantum mechanics. This theory is conceptually entirely different from previous classical theories, based on a description of physical phenomena in terms of probabilities and is therefore no longer deterministic. This is a radical paradigm shift that revolutionized physics in the last century and has enabled a deeper understanding of the physical world, with fundamental and practical implications that have radically changed human life (atomic physics, chemistry, nuclear energy, transistors, and lasers, to name just a few). 

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This module provides an introduction to the concepts and methods of statistical physics for systems at equilibrium, using a bottom-up approach: starting with examples and then deriving the general principles. It draws heavily on the course by Harvey Gould and Jan Tobochnik. The final chapter of the course offers a historical introduction to the development of the theory of Brownian motion.

View the full page

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This course aims to introduce the fundamentals of physical hydrodynamics. The kinematic aspects are covered first: Eulerian and Lagrangian formalisms, analysis of the motion of a fluid volume element, introduction to the stream and potential velocity functions, and applications to various types of flows. In the next section on fluid dynamics, we derive Euler’s equation and Bernoulli’s equation for the flow of ideal fluids, followed by the Navier-Stokes equation describing the flow of Newtonian viscous fluids. This section will lead us to define the stress tensor as well as the Reynolds number, which allows us to determine whether a flow is laminar or turbulent. The course concludes with an introduction to the mechanics of deformable solids: displacement field, strain tensor, and deformation tensor.

View the full page

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In this course, we will build upon the basic concepts previously covered in Quantum Mechanics in Semester 5. The course is organized around the following main topics: extension of the wave mechanics formalism, angular momentum theory, the hydrogen atom, perturbations, and an introduction to relativistic quantum mechanics.

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