Homepage - Vivien Lecomte, LPMA

Lecture at Master 2 - Erasmus Mundus Masters in Complex Systems

Theoretical analysis of complex systems

Homepage of the Master

Date and place

• Location: École Polytechnique, Palaiseau.
  Room: Petite Classe 5, unless otherwise specified.
• Tuesday mornings 9am - 12:30am [with a supplementary lecture on October 9th, 1:30pm - 5pm]
• Starting from October 2nd, for 10 weeks

Recommended reading

• 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

• Lecture 1 (Tuesday, October 2nd, 2012)

  [Pdf of hand-written notes for Part I – pages 1-7]

  • 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.

Homework for next lecture: re-read carefully the notes (pages 1-7 of the pdf) and familiarize yourself with the remaining pages (7-10). Have a look to the notes (see link below) on functional derivatives (a close look on pages 1-3; the rest is mainly illustrative).


• Lecture 2 (Tuesday, October 9th, 2012) Morning

  [Pdf of hand-written notes for Part I – page 7-20]   [Pdf on functional derivatives.]   [Pdf on Gaussian integrals (in french).]

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

• Lecture 3 (Tuesday, October 9th, 2012) Afternoon - 1:30pm to 5pm - Salle n°52, couloir des langues

  [Pdf of hand-written notes for Part II – pages 1-3]

  • Overdamped Langevin dynamics: higher dimensions, correlated noise, non-constant noise. Link to Itō's Lemma.
  • Continuous-time Markov evolution.
  • Operator of evolution and its properties (steady state, spectrum).

[Pdf of the exercice for October 16th] [Directory with the pdf of references.]

• Lecture 4 (Tuesday, October 16th, 2012)

  [Pdf of hand-written notes for Part II – pages 4-6]

  • Description in terms of a jump process.
  • Formal solution using the time-ordered exponential.
  • History-dependent observables.

[Pdf of the correction of homework 1] [Mathematica file for the correction.]

• Lecture 5 (Tuesday, October 23rd, 2012)

  [Pdf of hand-written notes for Part II – pages 6-12]

  • Introductory examples of dynamical phase coexistence (TASEP, kinetically constrained models).
  • Introduction to large deviation functions.
  • Cumulant generating functions.
  • s-ensembles, analogy with Boltzman micro-canonical and canonical thermodynamics.
  • Large deviation functions as largest eigenvalues.

[Pdf version of the Mathematica file.] [Python program for the TASEP in open boundary conditions.]

• Lecture 6 (Tuesday, October 30th, 2012) [Room : Petite Classe 20]

  [Pdf of hand-written notes for Part II – pages 13-17]

  • Population algorithm to estimate large deviation functions: varying population size.
  • An example: the death-and-birth process.

• Lecture 7 (Tuesday, November 6th, 2012)

  [Pdf of hand-written notes for Part III]

  • Population algorithm to estimate large deviation functions: fixed population size.
  • Evolution operator in the Doi-Peliti formalism: the case of lattice gases.

[Correction of homework 2a: Python program for the cloning of the fermionic death-and-birth process (non-constant population).]

• Lecture 8 (Tuesday, November 13th, 2012)

  [Pdf of hand-written notes for Part III]

  • Analogy with quantum mechanics (creation/annihilation operators).
  • Examples: free particles, Kinetically Constrained Models, Zero Range Process, chemical reactions, reaction-diffusion processes.

[Correction of homework 2b: Python program for the cloning of fermionic death-and-birth process, now at constant population.]

• Lecture 9 (Tuesday, November 20th, 2012)

  [Pdf of hand-written notes for Part III] [Pdf of hand-written notes for Part IV]

  • How can one implement exclusion rules?
  • Evolution operator of in terms of quantum spin operators (from SU(2)).
  • Examples: exclusion processes (SSEP, TASEP), Bernoulli equilibrium distribution.
  • Symmetries: conservation of particle number, symmetry by rotation in SU(2).
  • Mapping between equilibrium and non-equilibrium current fluctuations in the Symmetric Exclusion Process.

[Correction of homework 3: Python program for the cloning of bosonic death-and-birth process, at constant population.]

• Lecture 10 (Tuesday, November 27th, 2012)

  [Pdf of hand-written notes for Part IV] [Pdf of hand-written notes for Part V]

  • Coherent states for annihilation/creation and for spin operators.
  • From the evolution operator to the action: path-integral formalism.
  • Large system size limit: gradient expansion.
  • Comparison to the Martin-Siggia-Rose formalism of Langevin dynamics.
  • Continuum limit of exclusion processes: the symmetric exclusion process.
  • Continuum limit of exclusion processes: the Edwards-Wilkinson and the Kardar-Parisi-Zhang classes.
  • Large deviations functions: dynamical phase transitions in exclusion processes, kinetically constrained models (KCMs).

Exam, January 15th, 2012

• Please chose a time-slot on this Doodle page.
• Place: room 0D7, 175 rue du Chevaleret, Paris.
  
  
• Type of presentation:
  • 50-55 minutes in total, split in a 30 minute presentation by you, and a 20-25 minute session of discussion and questions.
  • The 30 minutes presentation should be in the style of short 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
  (As discussed during the lecture, it is easier here to study a SSEP not on a ring but on an isolated segment.)
  
  
  • 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
  (As discussed during the lecture, you can start from the mean-field study of the model on a complete graph.)
  
  
• Note that you are allowed to start from the Python programs for the population algorithms we have discussed during the exercises (see links above).

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