# M2PCS: Nonequilibrium and Active Systems

**Exam: TBA. **All hand-written documents are allowed. No electronic devices allowed (not even the cell phone).

Lectures on Tuesdays (room 304 A) 13:45 - 15:45 : 06/09, 13/09, 20/09, 27/09, 04/10, 11/10, 18/10, 25/10, 08/11, 22/12, 29/12, 06/12, 13/12) (location: Condorcet bldg, 4 rue Elsa Morante, 75013 Paris).

**Tutorials (sheet posted ****a week ahead of time)**

Sheet 1 - 20/09/2022 (and a few hints for the solutions of the exercises)

Sheet 2 - 11/10/2022 (and a few hints for the solutions of the exercises)

Sheet 3 - 08/11/2022 (and a few hints for the solutions of the exercises)

Sheet 4 - 06/12/2022

**Outline of the lectures**

**Chapter 1 ****Equilibrium statistical mechanics**

06/09 What we know, and what we don't know. What works. Liouville equation, microcanonical ensemble, canonical ensemble.

**Chapter 2 ****The Langevin equation**

06/09 A colloid in a bath of water molecules: questions of interest (no interest whatsoever in the water molecules). The hope of describing the bath by a linear combination of two contributions, one that is deterministic and colloid-position-dependen, and one that is random and Gaussian.

13/09 Eliminination the degrees of freedom of the bath in the particular model where the bath is a set of harmonic oscillators. Generalized Langevin equation: a memory kernel expressing viscous damping, and a noise term which has Gaussian statistics. Its variance is given by the same kernel as in the friction contribution (further comments on that "coincidence" will be given later). Limits of interest: Markov limit (the time-scale of the memory of the kernels is short wrt other relevant time scales) & Overdamped limit (the inertial time scale is short wrt other relevant time scales). A simple system to study: a colloid in an optical trap.

20/09 Tutorial 1 on the Langevin equation and how to manipulate it.

**Chapter 3 Stochastic calculus**

27/09 We began by explaining where the mathematical problem lies with Brownian motion. In a Langevin equation the derivative of the position is at best ill-defined and the whole lecture was devoted to understanding how to properly understand what it means, and to manipulating it without making errors. This led to Ito's lemma and to a few applications to illustrate the mathematical dangers of careless dealing with a nondifferentiable as if it was smooth.

04/10 We learnt about path integrals which we immediately put to use to properly define what equilibrium is. We also applied path integration to the Kramers' escape problem.

**Chapter ****4**** ****The Fokker-Planck equation**

04/10 We used Ito calculus to derive an evolution equation for the probability to observe a signal evolving through a Langevin equation in a given position at a given time.

18/10 We explored some properties of a Fokker-Planck equation (especially in equilibrium) and we have solved it in a few simple cases. We have also discussed similarities and differences with quantum mechanics.

25/10 We returned to the notion of being in, or out of equilibrium, and how this reflects on the Fokker-Planck equation (there is an underlying Hermitian operator), or on some specific relations between correlation functions and the response of a system to a weak nonequilibrium drive (he famous fluctuation-dissipation theorem).

**Chapter ****5 Thermal Ratchets, Stochastic Engines**

25/10 We projected ourselves 200 years back in time and realized that Carnot was right: thermal baths with unequal temperatures are necessary to extract work out of a system. And we were stunned by Feynman's ratchet and pawl's device, and left with the assignment to find the (big) hole in the reasoning leading to the possible of extracting work with a single thermal bath.

22/11 We examined the minimal nonequilibrium ingredients necessary to trigger the net motion of a particle in a potential.

**Chapter ****6 Molecular Motors**

22/11 We looked at some concrete numbers for molecular motors and discussed the additional, chemical, ingredient, that comes in addition to the physics discussed in chapter 5.

29/11 Motors & Active particles

06/12 Active Particles

**Some lecture notes (sketchy, probably riddled with typos) **

Lecture 1 - 06/09/2022

Lecture 2 - 13/09/2022

Lecture 3 - 27/09/2022

Lecture 4 - 04/10/2022

Lecture 5 - 18/10/2022

Lecture 6 - 25/10/2022

Lecture 7 - 22/11/2022

Lecture 8 - 29/12/2022

Lecture 9 - 13/12/2022

**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