Homepage - Vivien Lecomte, LPMA

Lecture at Master 2 - Erasmus Mundus Masters in Complex Systems

Theoretical analysis of complex systems
(1st semester, 2013)

Homepage of the Master

Date and place

• Location: École Polytechnique, Palaiseau.
  Room: Petite Classe 21 on 28/08 and 04/09, and Petite Classe 6 thereafter.
• Usual Dates:Wednesday mornings 9am - 12:30am
• Starting from August 28th 2013, 13 lectures up to December 4th 2013

Prerequisites and Recommended reading

• Prerequisites - mathematical tools

  • Basic linear algebra:
    Finite-dimensional matrices
    Matrix diagonalisation
  • Calculus
    Taylor expansion of functions of one or more variables
    Linear differential equations
    Dirac delta function

• Recommended reading - Physical background

  • Stochastic processes and thermodynamics:
    Random Walks
    Boltzmann distribution
  • More advanced readings
    Legendre transformation and [this article]
    Chapter 1 of Non equilibrium statistical mechanics by Michel Le Bellac

• Going further

  • Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences
    Crispin Gardiner
    Springer
  • Stochastic Processes in Physics and Chemistry
    Nicolaas Godfried van Kampen
    Elsevier

Outline of the lecture

To ask questions on the lectures between classes, you can use the following [Wiki website].

• Lecture 1 (Wednesday, August 28th, 2013)   Room PC 21

  [Pdf of hand-written notes for Part I – pages 1-6]

  • Introduction: the role of fluctuations in the mesoscopic description of complex systems.
  • An archetypal example: random walk and diffusion in 1D.
  • Solution to the diffusion equation
  • Correlation function of the (continuous) Brownian motion and of the white noise.

• Lecture 2 (Wednesday, September 4th, 2013)   Room PC 21

  [Pdf of hand-written notes for Part I – pages 6-10]   [Pdf of hand-written notes for Part Ib – pages 10-13]   [Pdf on functional derivatives]

  • Probability distribution of trajectories for the Brownian motion and the white noise.
  • Extension to the diffusion in a potential; the overdamped Langevin equation, the Fokker-Planck equations.
  • Some tools: delta functions, functional differentiation.

• Mini-project (Due date: Friday, September 20th, 2013)

  [Miniproject]   [Directory with the pdf of references]   [Wiki website for asking questions]

  • Toy-model for the depinning transition: a driven particle in a periodic potential.
  • Analytical results: steady-state and mean velocity.
  • Numerical approach 1: discrete time and space.
  • Numerical approach 2: discrete time and continuous space.
  • Bonus: the case of a particle driven out of equilbrium by its boundaries.

• Lecture 3 (Friday, September 20th, 2013, 9:00-12:00)   Room PC 9

  [Correction of the Miniproject]   [Pdf of hand-written notes for Part Ib – pages 13-16]

  • Correction of the mini-project: analytical part.
  • Detailed balance and reversibility.
  • Density of probability for trajectories.
  • Itō and Stratonovitch.
[Directory with references on the Itō vs Stratonovitch dilemma]

• Lecture 4 (Wednesday, September 25th, 2013)   Room PC 6

  [Mathematica file for the numerics of the Miniproject]   [Pdf of hand-written notes for Part Ib – pages 16-20]   [Pdf on Gaussian integrals (in french)]

  • Backwards Fokker-Planck equation.
  • Reversible Langevin equation: mapping to a quantum evolution in an effective potential.
  • Langevin and Fokker-Planck equation for several correlated degrees of freedom.
  • Langevin equation and density or trajectories for fluctuating fields.

• Lecture 5 (Wednesday, October 2nd, 2013)   Room PC 6

  [Pdf of hand-written notes for Part IIa – pages 1-4]   [Wiki website for asking questions on Homework 4]

  • Correction of Homework 3: Gaussian integrals – a catalogue.
  • Systems with finite or discrete configurations: examples.
  • Continuous-time master equation.
  • Operator formalism; conservation of probability, steady state.
  • Reversibility; mapping to Schrödinger quantum dynamics.

• Lecture 6 (Friday, October 11th, 2013, 9:00-12:30)   Room PC 7

  [Pdf of hand-written notes for Part IIa – pages 4-5]   [Mathematica file for the discrete Brownian walk of Homework 4]

  • Correction of Homework 4: discrete-time Brownian walk and Gaussian integrals; example of evolution operators.
  • Time-ordered exponentials.
  • Probability of tranjectories in configuration space.

• Lecture 7 (Friday, October 18th, 2013, 9:15-12:45)   Room PC 6

  [Pdf of hand-written notes for Part IIa – pages 6-9]   [Python solution for Homework 5, with heap queues]   [Python program to illustrate the scaling of P(K,t)]  

  • Correction of Homework 5: continuous-time Markov chain simulation for birth and death (BaD) process.
  • Time-extensive observables.
  • Large deviation functions: scalings.
  • Cumulant generating functions.
  • Laplace and Legendre transforms.
  • Interpretation of s-ensemble: fixing the average value of time-extensive observables.

• Partial Exam (Wednesday, October 23th, 2013, 10:00-12:00)   Room 67 (couloir des langues)

  [Pdf of the partial exam]

• Lecture 8 (Wednesday, November 6th, 2013)   Room PC 6

  [Pdf of hand-written notes for Part IIb]   [Miniproject 2: cloning algorithm]   [Wiki website for asking questions]   [Miniproject 2: a Python algorithm using heap queues]

  • Correction of partial exam.
  • Correction of miniproject 2.

• Lecture 9 (Wednesday, November 13th, 2013)   Room PC 6

  [Cloning with time mixing, stopped by maximum time] [pdf]   [Cloning, stopped by maximum population] [pdf]   [Cloning, constant population] [pdf]  

  • Correction of miniproject 2.
  • Cloning algorithm: non-constant and constant populations.
  • Heap queues.

• Lecture 10 (Wednesday, November 20th, 2013)   Room PC 6

  [Cloning for bosonic BaD process.] [pdf]   [Mathematica file for Legendre transforms]   [Pdf of hand-written notes for Part III]  

  • Correction of Homework 7 (cloning for bosonic birth and death process).
  • Evolution operators for the large deviations of time-integrals of observables.
  • Doi-Peliti creation and annihilation operators.
  • Evolution operators for creation-annihilation processes on lattices and networks.

• Lecture 11 (Wednesday, November 27th, 2013)   Room PC 6

  [Pdf of hand-written notes for Part IV]

  • Introduction to field theory.
  • Bosonic coherent states: eigenvector properties, representation of the identity.
  • From evolution operators to (time-dependent) numbers: the construction of the action.
  • Example: diffusion process.

• Lecture 12 (Wednesday, December 4th, 2013)   Room PC 6

  [Pdf of hand-written notes for Part V]   [Mathematica file for the dynamical phase transition in the KCM]

  • Going from microscopic to macroscopic: an overview.
  • Martin-Siggia-Rose action for a Langevin equation; comparison to Doi-Peliti action.
  • Example: the dynamical free-energy for the bosonic death and birth process.
  • Example: a Kinetically Constrained Model (KCM) in 1d and on a complete graph.
  • Dynamical phase coexistence, dynamical heterogeneity and non-analyticity of large deviation functions.

• Exam (Wednesday, December 11th, 2013)   Room PC 6

  [Pdf of the exam]   [Pdf of the correction]  

• Project Presentations (Wednesday, December 18th, and Thursday, December 19th, 2013)

Projects (Presentation on December 18th-19th, 2013)

• Place: LPMA, Bâtiment Sophie Germain, avenue de France, 75013 Paris.
  
  
• Type of presentation:
  • 50-55 minutes in total, split in a 40 minute presentation by you, and a 10-15 minute session of discussion and questions.
  • The 40 minutes presentation should be in the style of medium talk, where you present the subject to a naive but scientifically aware audience.
  • You have to present the subject and explain your numerical results.
  • There will be a beamer and a computer (which can read pdf files), and a black or white board at your disposal.
  • Write a report presenting your result. You can be concise and short (5-6 pages are fine), but I would be very glad to read and comment it in case you write a longer report.


• You can either choose a subject on your own, in which you study the large deviation function of a dynamical observable using the cloning algorithm discussed during the lectures, or chose among those possible subjects and references (see also the last lecture):
  • Mapping between equilibrium and non-equilibrium current fluctuation in the SSEP in contact with reservoirs:
  • Current Fluctuations in Systems with Diffusive Dynamics, in and out of Equilibrium
    Vivien Lecomte, Alberto Imparato and Frédéric van Wijland, Prog. Theor. Phys. Supplement 184 276 (2010)
    [arXiv:0911.0564], link, paper
  • Equilibriumlike fluctuations in some boundary-driven open diffusive systems
    Alberto Imparato, Vivien Lecomte and Frédéric van Wijland, Phys. Rev. E 80 011131 (2009)
    [arXiv:0904.1478], link, paper
  
  
  • Dynamical phase transition between clustered and non-clustered profiles in the periodic SSEP on a ring:
  • Inactive dynamical phase of a symmetric exclusion process on a ring
    Vivien Lecomte, Juan P. Garrahan, Frédéric van Wijland, J. Phys. A 45 175001 (2012)
    [arXiv:1203.1600], link, paper
  • Universal cumulants of the current in diffusive systems on a ring
    Cécile Appert-Rolland, Bernard Derrida, Vivien Lecomte and Frédéric van Wijland, Phys. Rev. E 78 021122 (2008)
    [arXiv:0804.2590], link, paper
  
  
  • Dynamical coexistence of phase between active and inactive regions in a KCM:
  • First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
    Juan P. Garrahan, Robert L. Jack, Vivien Lecomte, Estelle Pitard, Kristina van Duijvendijk and Frédéric van Wijland, J. Phys. A 42 075007 (2009)
    [arXiv:0810.5298], link, paper
  • Finite size scaling of the dynamical free-energy in a kinetically constrained model
    Thierry Bodineau, Vivien Lecomte and Cristina Toninelli, J. Stat. Phys. 147 1 (2012)
    [arXiv:1111.6394], link, paper
  
  
• Note that you are allowed to start from the Python programs for the population algorithms we will discuss during the exercises.

References

• Here are some references that cover the topics of the lecture
• H. Hilhorst, lecture notes on statistical physics (in french), chapter 1-10
• L. Peliti, J. Phys. France 46 1469 (1985)
• J. Tailleur, J. Kurchan and V. Lecomte, J. Phys. A 41 505001 (2008), part 1-5
• H.G. Solari, J. Math. Phys. 28 1097 (1987)
• V. Lecomte, A. Imparato and F. van Wijland, Prog. Theor. Phys. Supplement 184 276 (2010)
• Pdfs files of the references can be found at this place