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Computation of transition trajectories and rare events in nonequilibrium statistical physics Lyon, June 11th June 15th, 2012 Universality of large deviations in diffusive systems Frdric van Wijland Laboratoire Matire et Systmes

  1. Computation of transition trajectories and rare events in nonequilibrium statistical physics Lyon, June 11th – June 15th, 2012

  2. Universality of large deviations in diffusive systems Frédéric van Wijland Laboratoire Matière et Systèmes Complexes Université Paris Diderot – Paris 7 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  3. In collaboration with Cécile Appert, Orsay Bernard Derrida, Paris Juan P Garrahan, Nottingham Alberto Imparato, Århus Vivien Lecomte, Paris 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  4. Outline Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  5. Outline Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  6. Diffusive systems : example 1 Simple symmetric exclusion process 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  7. Diffusive systems : example 1 Simple symmetric exclusion process 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  8. Diffusive systems Coarse grained description Space scale of the system size Time scale of the diffusiion time Make occupation numbers a smoothly varying field : 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  9. Diffusive systems : equilibrium statics In equilibrium, the probability to observe a given profile is 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  10. Diffusive systems : dynamics Continuity equation where, on phenomenological grounds 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  11. Diffusive systems : dynamics Continuity equation where, on phenomenological grounds 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  12. Diffusive systems : dynamics Continuity equation where, on phenomenological grounds 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  13. Diffusive systems : dynamics Green – Kubo relation 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  14. Diffusive systems : example 2 Independent oscillators (Giardiná, Kurchan, Redig version of the Kipnis-Marchioro-Presutti model) 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  15. Diffusive systems : example 2 Independent oscillators 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  16. Diffusive systems : example 2 Independent oscillators with a local conservation law for the energy 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  17. Diffusive systems : example 2 Connection to a thermostat 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  18. Diffusive systems : example 2 Local energy conservation 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  19. Diffusive systems : example 2 Working at macroscopic scales 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  20. Diffusive systems : example 2 Equilibrium Consistent with naive continuum limit from the dynamics 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  21. Diffusive systems : summary Ingredients : Smooth variations over space and time scales Diffusive dynamics characterized by Noise is vanishingly small (note the ) 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  22. Outline Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  23. Questions about rare events Fluctuations of the current (particles, energy) 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  24. Questions about rare events Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  25. Questions about rare events Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response Asking the question reveals the structure 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  26. Questions about rare events Fluctuations of the current (particles, energy) Transport coefficients, linear response Burnett coefficients, non-linear response Asking the question reveals the structure Evans-Morris-Cohen-Searles-Gallavotti-Cohen- Kurchan-Lebowitz-Spohn-Maes fluctuation theorem Deep understanding of the kinetic foundations of the second law 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  27. Questions about rare events Equilibrium statistical mechanics/thermodynamics Partial view, ignoring dynamics Nonequilibrium : quest for generic properties Mimick the approach from the statics (ideas from Ruelle & Bowen, 1970's). 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  28. Questions about rare events Equilibrium statistical mechanics/thermodynamics Partial view, ignoring dynamics Nonequilibrium : quest for generic properties Mimick the approach from the statics (ideas from Ruelle & Bowen, 1970's). 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  29. Questions about rare events For a time realization with a prescribed current (or whatever else is of physical interest), what does the system look like ? In or out of equilibrium 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  30. Questions about rare events Extensive constraints : Time and space integrated current : Time and space integrated activity (traffic for C. Maes) 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  31. Questions about rare events Generating functions (canonical approach) : Dynamical versions of a partition function and of an intensive free energy. 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  32. Outline Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  33. WKB-like technique Langevin equation for Equivalent to a path-integral formulation 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  34. WKB-like technique Action encodes dynamics + reweighting. 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  35. WKB-like technique Action encodes dynamics + reweighting. 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  36. WKB-like technique Partition function dominated by the saddle 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  37. WKB-like technique : credits Macroscopic fluctuation theory : Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim Lebowitz, Speer, Derrida, Bodineau See the lecture by Bernard Derrida 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  38. WKB-like technique : recipe Step 1: Find the saddle : 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  39. WKB-like technique : recipe Step 2: Evaluate action at the saddle Step 3 : Integrate out fluctuations around the saddle and get 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  40. Outline Diffusive systems, with examples Questions about rare events Technical know-how Message 1 : universality of large deviations Message 2 : existence of phase transitions 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  41. Universality of large deviations Consider the integrated current large deviations 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  42. Universality of large deviations : equilibrium With periodic boundary conditions: Saddle is stationary and homogeneous at 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  43. Universality of large deviations : equilibrium With periodic boundary conditions: where the universal scaling function has a branch cut along the real axis for 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  44. Universality of large deviations : equilibrium For the SSEP, hence Burnett coefficients are infinite. 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

  45. Universality of large deviations : equilibrium For the KMP, the argument of the scaling function Can hit the singularity : phase transition. 13/06/2012 F. van Wijland - MSC/Paris 7 - ENS Lyon, 2012, Large deviations

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