Bifurcation and fluctuations in jamming transitions

1
Bifurcation and fluctuationsin jamming
transitions

• University of Tokyo
• Shin-ichi Sasa
• (in collaboration with Mami Iwata)
• 08/08/29_at_Lorentz center

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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
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MCT transition
Eg. Spherical p-spin glass model
µ supplementary variable to satisfy the
spherical constraint
Stationary regime
Equilibrium state with T
The relaxation time diverges as
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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
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• These developments clearly show that the first
  stage already ends (when I decide to start this
  research.. )
• What is the research in the next stage ? Not
  necessary?

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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
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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
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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

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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, k3)
light particle particle with contact number
less than k (say, k3)
K-core maximally connected region of heavy
particles
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K-core percolation
transition from non-existence to existence
of infinitely large k-core in the limit n ? 8
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 k4 d2 ?)
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K-core problem (dynamics)
Time evolution (decimation process)

• (i) Choose a particle with a constant rate
  a(1)
• 
  (for each particle)
• (ii) If the particle is light, it is removed.
• If the particle is heavy, nothing is done

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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
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K-core problem in a random graph
n number of vertices m number of edges
Initial state
particle ? vertex connection ? edge
Time evolution

• Choose a vertex with a constant rate a(1)
• 
  (for each vertex)
• (ii) If the vertex is light,
• all edges incident to the vertex are
  deleted

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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 (k3)
discontinuous transition !
Chalupa, Leath, Reich, 1979
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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
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Fluctuation of relaxation events
Dynamical heterogenity in jamming systems
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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 in a
class of systems undergoing a saddle-node
bifurcation under the influence of noise.
Iwata and Sasa, arXiv0808.0766
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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

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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
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Master equation (transition table)
..
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Master equation (transition rate)
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Langevin equation
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Deterministic equation
density of light vertices
initial condition
? z as one of dynamical variables
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Bifurcation
Conserved quantities
Transformation of variables
?
marginal saddle
The k-core percolation in a random graph
is exactly given as a saddle-node bifurcation !!

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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)

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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
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Simplest example
Saddle-node bifurcation
Stable fixed point
Potential
Marginal saddle
Mean field spinodal point
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Basic idea
transient
small deviation
special solution
? Goldstone mode associated with
time-traslational symmetry
divergent fluctuations of
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Fluctuations of ?
Poisson distribution of ? for ? gtgt 1
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Determination of scaling forms
A Langevin equation valid near the marginal
saddle
Scaling form
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Fluctuation of trajectories
Gaussian integration of ?
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Numerical observations
Square Symbol direct simulation of k-core
percolation with n8192
Red Langevin equation with T3/16384
Blue Langevin equation with T1/2097152
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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

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MCT equation
Exact equation for the time-correlation function
for the Spherical p-spin glass model
(stationary regime)
Attach Graph
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Singular perturbation I
Step (0)
Step (1)
Multiple-time analysis
dilation symmetry
We fix D1 as the special solution A
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Singular perturbation II
Step (2)
different ?
Derive small ? in a perturbation method
Determine ? and ?
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Variational formulation
The variational equation is equivalent to the
MCT equation
Substitute
into the variational equation
The solvability condition determines
and the value of ?
? can be solved (formally) under the solvability
condition
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Analysis of Fluctuation Idea
MCT equation
fluctuation of ? and ?(t)
divergent part
Determine the divergence of fluctuation intensity
of ?
? Goldstone mode associated with the dilation
symmetry
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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

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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
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• APPENDIX

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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
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Spatially extended systems II
Characterize fluctuations leading to
smearing the MF calculation

The Goldstone mode is massless in the limit e ?
0
Existence of activation process mass
generation of this mode
? slope of the effective potential of ?
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Spatially extended systems III
Seek for simple finite-dimensional models
related to jamming transitions in granular
systems
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Simplest example
Saddle-node bifurcation
Stable fixed point
Potential
Marginal saddle
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Question
trajectory
transient
small deviation
special solution
-- Instanton analysis
-- difficulty the interaction between the
transient part and ?
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Fictitious time evolution
a stochastic bistable reaction diffusion system
s-stochastic evolution for
(e.g. Kink-dynamics in pattern
formation problems)
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Result