Bachelor's Degree (3rd Year) in CPES: Modeling and Numerical Methods in Materials Science—Physics

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.

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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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