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 have proceeded with the construction of path integrals, thus building one more tool to deal with stochastic processes. This has allowed us to define and determine the probability of a full trajectory. Given this path probability, we have defined an equilibrium as one in which the probability of a path is identical to the probability of a time reversed path (a useful quantity is the Kullback-Leibler divergence between these to probabilities). In the exercise session, we have covered exercises 3.1, 3.2 and 3.3.
Lecture notes for week 3 are available here and solutions to the exercises of chapter 3 can be found here.
Week 4 - 03/12 & 04/12/2024
The Fokker-Planck equation is the last face of stochastic processes we shall need for our developments. Instead of characterizing an instantaneous path (as in a stochastic differential equation) or the probability of a path, the Fokker-Planck equation tells us about the probability to find the process at a given point in space and at a given time. Time-reversal translates into a nice property of the probability evolution operator (it maps to a Hermitian operator). The spectrum of the evolution operator need not be real (except for equilibrium dynamics) and it contains the hierarchy of relaxation rates of the process. An important development of week 4 is the so-called "stochastic energetics" interpretation of systems evolving according to Langevin equations, which promotes (and establishes) the basic principles of thermodynamics (macroscopic systems in their stationary equilibrium state, non fluctuations) downto the scale of small and fluctuating systems.
Lecture notes for week 4 are available here.
Week 5 - 10/12 & 11/12/2024
In the last section of the Fokker-Planck chapter we illustrated how, even if simple to formulate, optimal transport converts into difficult mathematics. We also discussed the emergence, only out of equilibrium, of the ratchet effect.
Solutions to chapter 4 are here, and the lecture notes for week 4 are here.
Week 6 - 17/12 & 18/12/2024
We have introduced the notion of active particles. The run-and-tumble motion of a bacterium such as E. Coli was illustrated with characteristic numbers. Whether bacteria or synthetic colloids, active particles are modeled by means of a few simple ingredients (force with non Gaussian statistics and a simple exponential decay of their correlations) that lead to remarkable features. One of these is the ability to be attracted to repulsive obstacle, which, when translated in a many body language, is responsible for the possibility of phase separation in the ttotal absence of attractive forces.
Wrap up of the lecture notes can be found here, and the corresponding exercises there.
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