M2PCS: Nonequilibrium and Active Systems 2023-2024
Exam: Tuesday January Xth, 2024. 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).
Lectures on Tuesdays 14:00 - 16:00 (05/09, 12/09, 19/09, 26/09, 03/10, 10/10, 17/10, 24/10, 14/11, 21/11, 28/11, 05/12, 12/12). Location: Sophie Germain bldg, room 2017.
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
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).
Chapter 2 is continued. 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.
We 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.
During the tutorials, we looked at 2.1 (how to build a Gaussian white noise) and 2.3 that covers the basics of Brownian motion, plus a small bit on exponential functions of Brownian motion.
The solutions to the chapter 1 and chapter 2 exercises can be found here.
In the lecture 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).
Path integrals are on the menu. That's the third manner in which one can actually describe a stochastic process (on par with the Fokker-Planck equation and the Langevin equation). It is neither superior nor inferior in any qualitative way. It's simply more convenient in some given frameworks.
Lecture 6 - 10/10/2023
Exercises from chapter 3 on stochastic processes. The solutions can be found here.
The Fokker-Planck equation (chapter 4) is on the agenda. We established a one-to-one correspondence between a Langevin equation and a Fokker-Planck equation. The proof relied on Itô's calculus.
We finish our considerations on the Fokker-Planck equation with a few words on first passage problems and on stochastic thermodynamics. We also prove the fluctuation-dissipation theorem, a remarkable connection holding in equilibrium only connecting the spontaneous decay of equilibrium correlations to the system's response to an infinitesimal perturbation.
Lecture 9 - 14/11/2023
Exercises on the Fokker-Planck equation (chapter 4). Solutions can be found here.
We will enter the chapter on ratchets.
Molecular motors
Active particles
Lecture 13 - 12/12/2023
Exercises to wrap the first semester up. Solutions can be found here.
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