# M2ICFP Advanced Statistical Mechanics

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

Lectures on Fridays (room 418 C) : 02/09, 09/09, 16/09, 23/09 in Jussieu (bât B, 3rd floor, Room #11), 30/09 (22.23.317), 07/10 (54.55.201), 14/10 (22.23.317), 21/10 (22.23.317), 28/10 (22.23.317)**,** **25****/****11 (24.25.101)****,** 02/12 (54.55.125), 09/12(54.55.125), 16/12 (54.55.125) . *Fri**days Nov 4, 11 and 18 are without lectures/tutorials; Dec 16 is for tutorials only.*

**Tutorials**

Some useful prerequisites (by E. Trizac, M. Lenz & FvW) and the companion solutions.

Sheet 7

**Outline of the lectures**

**Chapter 1 ****Statistical dynamics and Markov processes**

**02/09/2022** After motivating the ubiquity of a master equation, we have covered its basic mathelmatical properties. We have also determined the probability of a time realization of the process and we have used this to construct a trajectory-dependent observable that seems to have flavors in common with entropy (as we know it from thermodynamics).

**09/09/2022** The adjoint master equation is related to first passage questions, and the mean first passage-time is a solution of an inhomogeneous equation involving the adjoint operator.

**Chapter 2 ****Stochastic processes**

**09/09/2022** We now focus on master equations for continuous degrees of freedom. The master equation is expanded in a power series of increasing derivatives and each coefficient has a simple meaning. This is a moment of the infinitesimal jump the process executes over an infinitesimal time window. This takes us to Fokker-Planck equations (the diffusive limit of a continuous master equation) and then to a representation of a process described by a Fokker-Planck equation in terms of a random process. This is the Langevin equation. A Langevin equation conceals a number of mathematical difficulties that we have reviewed. These are related to the requirement to specify an underlying discretization scheme whenever one writes a Langevin equation.

**16/09/2022** Differential calculus must be adapted when dealing with functions that depend on a Brownian motion. The chain rule is modified (Ito's lemma). Problems emerge when a Langevin equation has multiplicative noise, which is actually a pretty common situation. Finally, we introduced a path integral representation of stochastic processes.

**23/09/2022** After a clean derivation of a what a path integral really is we have seen how to manipulate these objects without caring too much about mathematical rigor. We learnt about Onsager Machlup and Janssen De Dominicis formulations. We also had a taste of how to build an effective evolution equation for an open quantum system in contact with a thermostat. The quantum Langevin equation was approached by the Caldeira-Leggett model in which the bath is made of quantum harmonic oscillators. The derivation parallels the classical one, but the mathematics involves operators, rather than real numbers, which was to be expected.

**Chapter 3 Time reversal and its consequences**

**23/09/2022** We defined equilibrium by means of a zero entropy production condition. Using the freshly derived path integral formulation, we realized that entropy production was, up to the inverse temperature, the work provided to the system by the nonconservative forces acting on it.

**30/09/2022** For equilibrium dynamics we pointed to a connection to quantum mechanics. For a system weakly driven out of equilibrium, we have seen that the response of a physical observable to a small drive is, up to the temperature as a prefactor, the relaxation rate of an equilibrium correlation. We also investigated the fate of entropy production arbitrarily far from equilibrium (the Gallavotti-Cohen theorem). We'll get back to that next week.

And a Quizz.

**07/10/2022 **We drew some consequences of the Gallavotti-Cohen theorem when the systems considered are weakly driven out of equilibrium, thereby making contact with linear response. We recovered the Onsager reciprocity relations along with the Green-Kubo & Yamamoto-Zwanzig fluctuation relations.

**Chapter ****4 Metastability and rare events**

**07/10/2022** When a system evolves in a non convex potential landscape with deep enough wells and valleys, it may have a hard time probing the whole landscape. We discussed the typical time it takes a system endowed with equilibrium dynamics to hop over a potential barrier (this is the Arrhenius formula for the Kramers escape time). We hinted that things may get complex in higher dimension than one. However, the relevant one-dimensional degree of freedom that is relevant for the description of the dynamics was taken for granted (the so-called reaction coordinate in chemical physics).

**14/10/2022** We have seen that the reaction currents were related to the excited states of an extended evolution operator, which could be used to focus on long-live metastable states. We have also superficially described the phenomenon of stochastic resonance by which a small forcing exerted on a bistable system gets amplified by joining forces with stochastic noise.

**Chapter ****5 The mean-field approximation**

**14/10/2022** The ubiquity of the mean-field approximation, whether in condensed matter, high energy physics or in complex systems, makes it a central element of theoretical approaches. We have reviewed known facts on the statics of an Ising model and we have endowed the mean-field Ising model (defined on a fully connected lattice) with dynamical evolution rules, respecting the detailed balance condition. We have also expressed the evolution operator of the system in terms of quantum spins. We are left with exploring the properties of this evolution operator, which we shall do next week.

**21/10/2022** More on the mean-field dynamics of the Ising model (accelerating the dynamics, changing dynamical scaling). An introduction to the statics of liquids in infinite dimensions. We argued why mean-field was way more difficult to construct for systems not living on a lattice (such as particles in a fluid) and we introduced the idea that the graph of effectively interacting particles in infinitely many space dimensions was tree-like.

**28/10/2022** We continued our analysis of the dynamics of liquids in infinite dimension. This led us to effective equations of motion for a single tagged particle, of for two tagged particles. We realized that in this dynamical version of mean-field the proper self consistent quantity to determine was not a parameter, but a full function of time (a memory kernel). We finally explained that solving this mean-field version of fluids told us that at low temperature or high density the dynamics would simply freeze. We superficially mentioned the counter-intuitive possibility of reaching a similar result based on the statics only. The last section of our presentation of mean-field was devoted to the McKean-Vlasov equation and to an application to the Brownian motion of some matrix elements. We found that the stationary (equilibrium) distribution of the corresponding eigenvalues was rather simple (this is the Wigner semi-circle law).

**Chapter 6 Exactly solvable models**

**28/10/2022** An introduction to exactly solvable models, by the example. A model for traffic.

**25/11/2022** We assumed that there was some interesting physics in the boundary-driven TASEP and a mean-field analysis indeed told us that there were phase transitions in the stationary state, in spite of the TASEP being a one-dimensional model. To make sure that mean-field was not playing tricks on us, we solved the TASEP with a matrix ansatz. Our next goal is to diagonalize the XXZ chain with a Bethe ansatz.

**02/12/2022** XXZ quantum spin chain and the Bethe ansatz.

**Chapter ****7 Critical dynamics and mesoscopic descriptions**

**02****/1****2****/2022** The Hohenberg & Halperin classification, and the renormalization group for dynamic critical penomena.

**09/12/2022** Driven diffusive systems and active matter. And a Quizz...

**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

Schütz, Exactly solvable models for many-body systems far from equilibrium

De Dominicis & Giardina, Random fields and spin glasses: a field theory approach

Sekimoto, Stochastic energetics