M2PCS: Nonequilibrium and Active Systems 2024-2025
Exam: Tuesday January 7th, 2025 (Université Paris Cité, from 09:00 through 12:00 in Sophie Germain bldg, Room #0011). All hand-written documents are allowed. No electronic devices allowed (not even the cell phone). No printed document allowed (no lecture notes, no tutorials, no solutions ; printout of notes taken on a tablet on demand a few das before).
Lectures on Tuesdays 08:30-10:30 (UPC, Condorcet bldg, Room #050A, 26/11,03/12, 10/12, 17/12) and Wednesdays (SU, 27/11,04/12, 11/12, 18/12).
Outline of the lectures
Chapter 1 Equilibrium statistical mechanics
Chapter 2 The Langevin equation
Chapter 3 Stochastic calculus
Chapter 4 The Fokker-Planck equation
Chapter 5 Thermal Ratchets, Stochastic Engines
Chapter 6 Molecular Motors
Chapter 7 Active Particles
Week 1 - 12 & 13/11/2024
Chapter 1 was covered (a reminder on equilibrium statistical mechanics) and the core ideas behind the description of a system by means of a Langevin equation (chapter 2) were introduced. The key ingredient is the separation of time and length scales between the degrees of freedom of interest, and those of the environment (aka bath or reservoir). A solvable model leads to a Langevin equation by an explicit integration of the degrees of freedom of the bath. This shows the trade off between loss of information (on the bath degrees of freedom) and memory. The Markov limit is then discussed, and the overdamped limit is also introduced. Wealso entered Chapter 3. We defined a Gaussian process as the limiting process of a large-dimensional Gaussian variable. We specifically considered the so-called Gaussian white noise.
Lecture notes for week 1 available here.
Week 2 - 19 & 20/11/2024
On Tuesday morning, have worked out three exercises (2.1, 2.3 and 2.4, solutions can be found here). Lectures-wise, we learnt about stochastic calculus, namely how to manipulate functions that are nowhere differentiable as though they actually were. This has led for instance to the celebrated Ito's lemma. Difficulties arise whenever a Langevin equation with multiplicative noise shows up (either directly, or as a by-product when considering a nonlinear function of a signal evolving via an additive Langevin equation).
Lecture notes for week 2 are available here.
Week 3 - 26 & 27/11/2024
We shall proceed with the construction of path integrals. In the exercise session, for those that may want to a have a look ahead of Wednesday, we shall cover exercises 3.1, 3.2 and 3.3.
References
Van Kampen, Stochastics processes in physics and chemistry
Risken, The Fokker-Planck equation
Gardiner, Stochastic methods
Täuber, Critical Dynamics A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior
Sekimoto, Stochastic energetics
Dorfman, An introduction to chaos in nonequilibrium statistical mechanics