M2 iPCS: Out of Equilibrium Statistical Mechanics, Classical and Quantum
Presentation of the lecture series
Exam: TBA. All hand-written documents are allowed. No electronic devices allowed (not even the cell phone).
Lectures on Wednesdays (room 348A) and Thursdays (room 054A) starting Feb 2nd, 2022 (02/02, 03/02, 09/02, 10/02, 16/02, 17/02, 23/02, 24/02, 02/03, 03/03, 09/03, 10/03, 16/03, 17/03) (location: Condorcet bldg, 4 rue Elsa Morante, 75013 Paris).
Some useful problems to work your way through the lectures (feel free to ask for help if you get stuck).
Some fully written out solutions by the 2017, 2018, 2019, 2020 , 2021 (and soon the 2022) graduates:
Differential calculus likes Stratonovich discretization (by Bruno V.); A solvable master equation and dynamical complexity (by Martin M.); Glauber dynamics and the FDT (by Kevin B.); Kawasaki dynamics (by Sebastian G.); Triplet annihilation (by Bruno V.); Playing around with stochastic calculus (by Mallory D.); Recipe for a Gaussian white noise (by Aigars L.); Quantum formulation of classical stochastic dynamics (by Ludovico C.); Path statistics, Crooks and Jarzynski theorems (by Arthur A.) or the same exercise whose solution is written up differently (by Andrea P.); Critical Dynamics of Model C (by Julien H.); Dean-Kawasaki Equation (by Marc B.); Dean-Kawasaki dynamics from a microscopic approach (Gianluca B.); Persistence in pair annihilation (Brieuc B.); Macroscopic Fluctuation Theory of a weakly asymmetric exclusion process (Toni M); Lotka–Volterra prey-predator population dynamics (Beatriz A); Differential caluclus likes Stratonovich discretization (Alberto D., notes of the 04/02/2021 presentation); How "natural" is Stratonovich calculus? (Guillaume C.); Current distribution in a TASEP with periodic boundary conditions (F); On a family of equilibrium Langevin processes (Marie S. P. & Pierre H.); Particle in contact with a thermostat (Yanis B.); Thermally driven coupled oscillators (Pierre S.).
Outline
Chapter 1 Methods of stochastic dynamics
A review and some new things. Master Equation/Fokker-Planck and Langevin, and path integrals (Janssen-De Dominicis). Introducing trajectories and histories. Large deviations, Gallavotti-Cohen, Onsager and Green-Kubo relations. Gallavotti-Cohen, Onsager and Green-Kubo relations. Reversibility vs irreversibility. Roughly 3 lectures.
References
Van Kampen, Stochastics processes in physics and chemistry
Risken, The Fokker-Planck equation
Gardiner, Stochastic methods
Chapter 2 Critical Dynamics
Dynamics in the 1d Ising model and quantum spin chains. Persistence, coarsening. Halperin and Hohenberg classification. Critical dynamics and the concept of dynamical scaling. Roughly 2 lectures.
References
Täuber, Critical dynamics: a field theory approach to equilibrium and non-equilibrium scaling behavior
Hohenberg and Halperin, Theory of dynamic critical phenomena
Chapter 3 Driven Systems
Diffusive systems display long range correlations. Macroscopic Fluctuation Theory. Phase transitions are possible in d=1 (the open TASEP as a model for traffic). A taste of the matrix ansatz solution for the open TASEP by Derrida, Evans and Hakim. The failure of mean-field and superdiffusive dynamics. Phase transitions in DDS. The Katz-Lebowitz-Spohn standard model. Effect of a drive on a critical point. The difficulty of building up a phenomenological coarse-grained description. Roughly 4 lectures.
References
Zia and Schmittmann, Statistical Mechanics of Driven Diffusive Systems
Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, Macroscopic Fluctuation Theory
Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process
Chapter 4 Population Dynamics
Connections between a master equation and a many-body quantum problem. The Doi-Peliti path integral formulation. Application of the Doi-Peliti formalism to pair annihilation with one or two species. Segregation phenomena and reaction zones. Introduction to epidemic models (SIR and percolation, SIS). Epidemics: SIS, and the Contact Process. Front propagation into an unstable state. Networks as growth processes. Networks as growth processes. Epidemics on networks and the vanishing of the epidemic threshold. Preys and Predators. Roughly 4 lectures.
Chapter 5 Roughly 1 lecture. Choose from
Chapter 5 Active Matter One particle models. Signatures of the departure from Eqilibrium. Coarse-grained descriptions of active matter.
Chapter 5 Glasses
Chapter 5 Grains
Chapter 5 The Quantum Langevin Equation Coupling to a thermostat in a quantum system.
Roll-out
Chapter 1 Methods of stochastic dynamics
02/02: How do we get to a master equation (separation of time scales). Deriving the probability of a full trajectory. The continuum version of a master equation and its truncation into a Fokker-Planck equation (diffusive limit, separation of length scales). And we started discussing stochastic processes and what a Langevin equation is.
A good training exercise is # 7.
03/02: Langevin equation and stochastic calculus. The subtleties of discretization. Multiplicative noise and applications to real physics. Path integrals formulation (Onsager-Machlup and Janssen-De Dominicis formulations).
Suggested exercises: # 2,3 & 4.
09/02: Being in or out of equilibrium. Quantifying the distance to equilibrium. Gallavotti-Cohen and Evans-Searles theorems arbitrarily far from equilibrium. Recovering the FDT and Green-Kubo relations close to equilibrium.
Suggested exercices: # 1, 6 & 24.
Chapter 2 Critical Dynamics
10/02: Dynamics of the one-dimensional spin chain à la Glauber. Relaxation time. Persistence. A reminder on the Landau-Ginzburg theory of phase transitions and the first steps into the dynamics. Phenomenological construction of the Stochastic Partial Differential Equation of Model A according to the Hohenberg & Halperin classification.
Suggested exercises: # 8 &9.
16/02: Model B, Model C, etc, and the importance of symmetries and invariances. Dynamical scaling. The dynamical exponent z, and the critical initial slip.
Suggested exercises: # 5, 10, 11 &20.
Chapter 3 Driven Systems
16/02: Boundary-driven diffusive systems. The example of the Simple Symmetric Exclusion Process. Closed, open and in contact with particle reservoirs. Condition for equilibrium (equality of the chemical potentials). Density profile.
Suggested exercises: # 14
17/02: The emergence of weak but long range correlations. Effective Coulomb like interactions. The SSEP viewed as a member of a large family of systems described by the so-called Macroscopic Fluctuation Theory. And finally introducing the TASEP.
Suggested exercises: # 21.
23/02: The mean-field solution to the TASEP. A few hints of the exact solution based on the Derrida matrix ansatz. Hydrodynamic description in terms of a noisy Burgers equation and the emergence of anomalous (superdiffusive) scaling in low space dimensions.
24/02: Driven diffisve systems, and the effect of a drive on a critical point.
Chapter 4 Population dynamics
24/02: Pair annihilation. The existence of a scale invariant regime and the importance of fluctuations. Reaction-diffusion equation. The failure of a Langevin stochastic PDE. Master equation approach.
02/03: A session devoted to solving problems 14 and 20.
03/03: Follow up on the master equation approach to diffusing and reacting particles. The so-called Doi-Peliti formalism. Back to the original annihilation problem. More on pair annihilation with two-species, and the power of scaling arguments to understand segregation phenomena.
Suggested exercises: # 9, 13 & 17.
09/03: Epidemic modeling. The SIR model and percolation. The SIS model and its variations.
10/03: Superdiffusion. Finite life time of the absorbing state. Front propagation into an unstable state.
Suggested exercises: # 15 (pretty hard, but interesting) & 26.
16/03: Epidemics on networks, and networks as a population dynamics problem. Preys and predators.
17/03: Collective phenomena in Active Matter.