Homepage - Vivien Lecomte, LPMA

Lecture at Master 2 - Erasmus Mundus Masters in Complex Systems

Homepage of the Master

Date and place

• Location: Room 393B, Institut Jacques Monod, bâtiment Buffon, 4 rue Marie-Andrée Lagroua Weill-Halle, Paris
  Room 393B is on floor number 3, aisle B, Buffon building.
  When entering, take the wide steps on the right, find an elevator on the wall at right. Go to floor number 3. Take right when exiting the elevator.
• Tuesdays 29/11/2011 and 06/12/2011: 3pm - 6pm
• Wednesdays 30/11/2011 and 07/12/2011: 9am - 12am

Outline of the lecture

• 1st lecture (3pm - 6:15pm, Tuesday, November 29th, 2011)

  [Hand-written lecture notes in pdf]

  • Reminder on continuous-time Markov evolution, jump processes.
  • A simple extension: large deviation functions.
  • Evolution operator in the Doi-Peliti formalism: the case of lattice gases.
  • Analogy with quantum mechanics (creation/annihilation operators).
  • Examples: free particles, Kinetically Constrained Models, Zero Range Process, chemical reactions, reaction-diffusion processes.

• 2nd lecture (9am - 12:15am Wednesday, November 30th, 2011)

  [Hand-written lecture notes in pdf]

  • How can one implement exclusion rules?
  • Evolution operator of in terms of quantum spin operators (from SU(2)).
  • Examples: exclusion processes, Bernoulli equilibrium distribution.
  • Symetries: conservation of particle number, symetry by rotation in SU(2).

• Pause

  • (Re)read the lecture notes carefully.
  • Check that probability is conserved on all the operators seen in the Doi-Peliti formalism, by showing that the total state is a left eigenvector of the operator of evolution.
  • Check that the boundary terms of the evolution operator of exclusion process written in terms of spin operators correspond indeed to the master equation describing the contact with reservoirs.

• 3rd lecture (3pm - 6:15pm, Tuesday, December 6th, 2011)

  [Hand-written lecture notes in pdf]
  • 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.
    (With reminders on Langevin evolution, Fokker-Planck equation.)
  • Continuum limit of exclusion processes: the symmetric exclusion process.

• 4th lecture (9am - 12:15am, Wednesday, December 7th, 2011)

  [Hand-written lecture notes in pdf]
  • 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.
  • Numerical simulations of large deviation functions: cloning algorithms.

• Bibliographical projects

  You have the choice between the following articles
  • Lattice models of nonequilibrium bacterial dynamics
    A G Thompson, J Tailleur, M E Cates and R A Blythe, J. Stat. Mech. (2011) P02029
    [arXiv:1012.0786], link, paper
  • Parts 1-6 on the SSEP of 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
  • Parts 1-5 and 7 on the KMP model of 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
  • 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
  • Parts I, II, IV and V of 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
  • Finite size scaling of the dynamical free-energy in a kinetically constrained model
    Thierry Bodineau, Vivien Lecomte and Cristina Toninelli, submitted to the Journal of Statistical Physics
    [arXiv:1111.6394]

• Numerical projects

  The exercice consists in determining the large deviation function psi(s) of a time extensive observable (like the current of the activity) in simple systems, both by direct diagonalisation of the modified operator of evolution, and by using cloning algorithms in continuous time.
  References are:
  
  
  • Simulating rare events in dynamical processes
    Cristian Giardinà, Jorge Kurchan, Vivien Lecomte, Julien Tailleur, J. Stat. Phys. 145 787 (2011)
    [arXiv:1106.4929], link, paper
  • Simulation of large deviation functions using population dynamics
    Julien Tailleur and Vivien Lecomte, Proceedings of the 10th Granada Seminar on Computational and Statistical Physics, (Granada, 2008), AIP Conf. Proc. 1091 212 (2009)
    [arXiv:0811.1041], link
  • A numerical approach to large deviations in continuous-time
    Vivien Lecomte and Julien Tailleur, J. Stat. Mech. (2007) P03004
    [cond-mat/0612561], link, paper
  
  
  Details on the simulations are given in the [hand-written lecture notes in pdf] of the 4th lecture, or in pages 4-6 of the second article, or in section 2.3 of the third article. Don't hesitate to chose a system that you find physically interesting, and characterize the phase transition by the mean value of the activity -psi'(s), or its second cumulant psi''(s) (in the state s) as a function of s.
  

References

• Lectures will be self-contained (basic familiarity with stochastic processes is needed).
• Here are some references that cover the topics of the lecture: you can browse them to acquaint yourself with the subject, but you don't need to read the articles thoroughly before following 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