Bifurcation and fluctuations in jamming transitions

1. Bifurcation and fluctuationsinjamming transitions University of Tokyo Shin-ichi Sasa (in collaboration with Mami Iwata) 08/08/29@Lorentz center

2. Motivation Toward a new theoretical method for analyzing “dynamical fluctuations” in Jamming transitions TARGET: Discontinuous transition of the expectation value of a time dependent quantity, ,accompanying with its critical fluctuations PROBLEM: derive such statistical quantities from a probability distribution of trajectories for given mathematical models

3. MCT transition Eg. Spherical p-spin glass model μ: supplementary variable to satisfy the spherical constraint Equilibrium state with T Stationary regime The relaxation time diverges as

4. Theoretical study on fluctuation of Response of to a perturbation Franz and Parisi, J. Phys. :Condense. Matter (2000) Response of to a perturbation Biroli , Bouchaud, Miyazaki, Reichman, PRL, (2006) spatially extended systems Effective action for the composite operator spatially extended systems Biroli and Bouchaud, EPL, (2004) Cornwall, Jackiw,Tomboulis, PRD, 1974

5. These developments clearly show that the first stagealready ends (when I decide to start this research….. ) • What is the research in the next stage ? Not necessary?

6. Questions Simpler mathematical description of the divergence simple story for coexistence of discontinuous transition and critical fluctuation Classification of systems exhibiting discontinuous transition with critical fluctuations (in dynamics) other class which MCT is not applied to ? jamming in granular systems ? Systematic analysis of fluctuations Description of non-perturbative fluctuations leading to smearing in finite dimensional systems

7. What we did recently We analyzed theoretically the dynamics of K-core percolation in a random graph - (Exactly analyzable) many-body model exhibiting discontinuous transition with critical fluctuations -The transition = saddle-node bifurcation (not MCT transition) We devised a new theoretical method for describing divergent fluctuations near a SN bifurcation - Fluctuation of “exit time” from a plateau regime We applied the new idea to a MCT transition

8. Outline of my talk • Introduction • Dynamics of K-core percolation (10) • K-core percolation = SN bifurcation (10) • Fluctuations near a SN bifurcation (10) • Analysis of MCT equation (10) • Concluding remarks (2) • Appendix

9. Example compress parameter : volume fraction n hard spheres are uniformly distributed in a sufficiently wide box heavy particle : particle with contact number at least k (say, k=3) light particle : particle with contact number less than k (say, k=3) K-core = maximally connected region of heavy particles

10. K-core percolation transition from “non-existence’’ to “existence” of infinitely large k-core in the limit n  ∞ with respect to the change in the volume fraction --- Bethe lattice : Chalupa, Leath, Reich, 1979 --- finite dimensional lattice: still under investigation (see Parisi and Rizzo, 2008) --- finite dimensional off-lattice: no study ? Seems interesting. (How about k=4 d=2 ?)

11. K-core problem (dynamics) Time evolution （decimation process) (i) Choose a particle with a constant rate α(=1) (for each particle) (ii) If the particle is light, it is removed. If the particle is heavy, nothing is done

12. Slow dynamics near the percolation It takes much time for a large core to vanish !  slow dynamics arise when particles are prepared in a dense manner.  characterize the type of slow dynamics. glassy behavior or not ? Study the simplest case: dynamics of k-core percolation in a random graph

13. K-core problem in a random graph Initial state: n: number of vertices m: number of edges particle  vertex; connection  edge Time evolution: • Choose a vertex with a constant rate α(=1) • (for each vertex) • (ii) If the vertex is light, • all edges incident to the vertex are deleted

14. k-core percolation point fixed in the limit; control parameter All vertices are isolated A k-core remains density of heavy vertex whose degree is at least (k=3) discontinuous transition ! Chalupa, Leath, Reich, 1979

15. Relaxation behavior density of heavy vertex whose degree is at least k(=3) at time t Red Green and blue represent samples of trajectories Green Blue

16. Fluctuation of relaxation events ～Dynamical heterogenity in jamming systems

17. Our results The k-core percolation point is exactly given as the saddle-node bifurcation point in a dynamical system that describes a dynamical behavior. The exponents are calculated theoretically as one example ina class of systems undergoing a saddle-node bifurcation under the influence of noise. Iwata and Sasa, arXiv:0808.0766

18. Outline of my talk Introduction Dynamics of K-core percolation K-core percolation = SN bifurcation(10) Fluctuations near a SN bifurcation (10) Analysis of MCT equation (10) Concluding remarks 2 Appendix

19. Master equation (preliminaries) : the number of edges : the number of vertices with r-edges Markov process of w Pittel, Spencer, Wormald, 1997 The number of edges of a heavy vertex obeys a Poisson distribution z: important parameter the law of large numbers

20. Master equation (transition table) ……..

21. Master equation (transition rate)

22. Langevin equation

23. Deterministic equation density of light vertices initial condition  z as one of dynamical variables

24. Bifurcation Conserved quantities Transformation of variables → marginal saddle The k-core percolation in a random graph is exactly given as a saddle-node bifurcation !!

25. Outline of my talk Introduction Dynamics of K-core percolation K-core percolation = SN bifurcation Fluctuations near a SN bifurcation (10) Analysis of MCT equation (10) Concluding remarks (2)

26. Question Fluctuation of relaxation trajectories of z Langevin equation of z : The perturbative calculation wrt the nonlinearity seems quite hard even for the simplest Langevin equation associated with a SN bifurcation:

27. Simplest example Saddle-node bifurcation Stable fixed point Potential Marginal saddle Mean field spinodal point

28. Basic idea special solution transient small deviation θ: Goldstone mode associated with time-traslational symmetry divergent fluctuations of

29. Fluctuations of θ Poisson distribution of θ for θ>> 1

30. Determination of scaling forms A Langevin equation valid near the marginal saddle Scaling form:

31. Fluctuation of trajectories Gaussian integration of θ

32. Numerical observations Square Symbol: direct simulation of k-core percolation with n=8192 Red: Langevin equation with T=3/16384 Blue: Langevin equation with T=1/2097152

33. Outline of my talk Introduction Dynamics of K-core percolation K-core percolation = SN bifurcation Fluctuations near a SN bifurcation Analysis of MCT equation (10) Concluding remarks (2) Appendix

34. MCT equation Exact equation for the time-correlation function for the Spherical p-spin glass model (stationary regime) Attach Graph

35. Singular perturbation I Step (0) Step (1) Multiple-time analysis dilation symmetry We fix D=1 as the special solution A

36. Singular perturbation II Step (2) different λ Derive small ρ in a perturbation method Determine λandζ

37. Variational formulation The variational equation is equivalent to the MCT equation Substitute into the variational equation The solvability condition determines and the value of λ ρcan be solved (formally) under the solvability condition

38. Analysis of Fluctuation: Idea MCT equation fluctuation of λand ρ(t) divergent part Determine the divergence of fluctuation intensity of λ λ:Goldstone mode associated with the dilation symmetry

39. Outline of my talk Introduction Dynamics of K-core percolation K-core percolation = SN bifurcation Fluctuations near a SN bifurcation Analysis of MCT equation Concluding remarks Appendix

40. Summary and perspective K-core percolation in a random graph KCM in a random graph SN-bifurcation K-core percolation with finite dimension Fluctuation of Spatially extended systems Bifurcation analysis of MCT transition Granular systems Fluctuation of (Spherical p-spin glass) spatially extended systems

41. APPENDIX

42. Spatially extended systems I Curie-Wise theory Pitch-fork bifurcation Ginzburg-Landau theory = diffusively coupled dynamical systems undergoing pitch-folk bifurcation under the influence of noise Analyze diffusively coupled dynamical elements exhibiting a SN bifurcation under the influence of noise near a marginal saddle Schwartz, Liu, Chayes, EPL, 2006 Binder, 1973 Ginzburg criteria but, be careful for

43. Spatially extended systems II Characterize fluctuations leading to smearing the MF calculation The Goldstone mode is massless in the limit ε 0 Existence of activation process = mass generation of thismode  slope of the effective potential of θ

44. Spatially extended systems III Seek for simple finite-dimensional models related to jamming transitions in granular systems

45. Simplest example Saddle-node bifurcation Stable fixed point Potential Marginal saddle

46. Question trajectory transient small deviation special solution -- Instanton analysis -- difficulty: the interaction between the transient part and θ

47. Fictitious time evolution a stochastic bistable reaction diffusion system s-stochastic evolution for (e.g. Kink-dynamics in pattern formation problems)

48. Result