Bifurcation and fluctuations
in jamming transitions
University of Tokyo
Shin-ichi Sasa
(in collaboration with Mami Iwata)
08/08/29@Lorentz center
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,
 (t ) , accompanying with its critical fluctuations
PROBLEM: derive such statistical quantities from
a probability distribution of trajectories for given
mathematical models
MCT transition
Eg. Spherical p-spin glass model
H 
 J i1i2i3 si1 si2 si3
1i1 i2 i3  N
dsi
H

 si   i
dt
si
1
 (t ) 
N
N 
N
 s (t )s (0)
i 1
t 0
i
N
p3
 si  N
2
J   p!/(2N
2
p 1
)
i 1
μ: supplementary variable to
satisfy the spherical constraint
i
Equilibrium state with T
3
 t  T 
2T

t
0
ds 2 (t  s )  s ( s )
Stationary regime
T  Td 
 (t )  0 for t  
The relaxation time diverges as
T  Td
 (t )  f  ( 0) for t  
6
4
(T  Td ) 
Theoretical study on fluctuation of
Response of
 (t)
h
1i1 i2 i3  N
Response of
H 
 (t)
4
to a perturbation
 J i1i2i3 si1 si2 si3  h
H 
h
 J i1i2i3 si1 si2 si3   hi si
1i1 i2 i3  N
Franz and Parisi,
J. Phys. :Condense. Matter (2000)
3
to a perturbation
N
i 1
Biroli , Bouchaud, Miyazaki, Reichman,
PRL, (2006)
spatially extended systems
Effective action for the composite operator
Biroli and Bouchaud, EPL, (2004)
1
1
1
( )   tr log   tr0   2 pI ( )
2
2


spatially extended systems
Cornwall, Jackiw,Tomboulis,
PRD, 1974
• 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?
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
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
Outline of my talk
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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
Example
compress
n hard spheres are uniformly distributed
in a sufficiently wide box
parameter : volume fraction
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
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 ?)
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
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
K-core problem in a random graph
Initial state:
n: number of vertices
m: number of edges
particle  vertex; connection  edge
Time evolution:
(i) 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
k-core percolation point
n 
m
R
n
fixed in the limit;
control parameter
R  Rc
R  Rc
All vertices are isolated
A k-core remains
h
density of heavy vertex
whose degree is at least (k=3)
h (t  )
discontinuous transition !
Chalupa, Leath, Reich, 1979
Rc
R
Relaxation behavior
h (t )
density of heavy vertex
whose degree is at least k(=3) at time t
h
h
t
t
R  Rc   ; n  4096
Red
Green
Blue
  0.03
  0.05
  0.07
Green and blue represent
samples of trajectories
  0.03
Fluctuation of relaxation events

  n h

2
 h
2


  Rc  R  0
 
t

 ( )   
 : the time when  (t ) becomes maximum
～Dynamical heterogenity in jamming systems
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
 and are calculated theoretically
as one example in a class of systems undergoing
a saddle-node bifurcation under the influence of noise.
Iwata and Sasa, arXiv:0808.0766
Outline of my talk
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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
Master equation (preliminaries)

vr
: the
number of edges
: the number of vertices with r-edges
w  (  , v0 , v1 , v2 )
Markov process of w
Pittel, Spencer, Wormald, 1997
P(w ' | w ) : w  w ' during t
The number of edges of a heavy vertex obeys a Poisson distribution

q
r 3
r
1
1
qr 
z r e z
Q( z )r!

2     2v2  h rqr
z: important parameter
vr / h  qr (r  3)
the law of large numbers
r 3

 rq
r 4
r
z
Master equation (transition table)
w  (  , v0 , v1 , v2 )
w  w  j
 10  (2,1,1,1)
 5  (2,3,2,1)
1  (1,2,2,0)
 6  (2,2,0,2)
 2  (1,1,0,1)
11  (2,1,1,2)
12  (2,1,0,1)
 7  (2,2,1,0)
 3  (1,2,1,1)
 4  (1,1,1,0)
……..
 13  (2,1,0,0)
 8  (2,2,1,1)
 9  (2,1,2,3)
14  (2,1,0,1)
Master equation (transition rate)
w  w  j
Langevin equation
 w/n
Deterministic equation
s  1  22
density of light vertices
2

initial condition

3 is determined from ( , 0 , 1 , 2 )
 z as one of dynamical variables
t
Bifurcation
J1  z / 
Conserved quantities
2
J 2  h / Q( z )

 rq
r 4
r
z
Transformation of variables
→
 t z   z  2 R( z )
 t  a  b 2
z  zc  
( z )  e  z (e z  1  z ) z (0)  2 R
R  Rc  
R  Rc
R  Rc
R  Rc
marginal
saddle
zc
z
The k-core percolation in a random graph
is exactly given as a saddle-node bifurcation !!
z
z
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)
Question
Fluctuation of relaxation trajectories of z

 z (t )  n z(t ) 2  z(t )
 z (t ) has a peak at t  t*
2

R  Rc   ,   0
t*  

 z (t* )   
Langevin equation of z :
The perturbative calculation wrt the nonlinearity seems quite hard even for
the simplest Langevin equation associated with a SN bifurcation:
 (0)  1 : fix
T  1/ n
Simplest example
 (0)  1 : fix
T  1/ n
Saddle-node bifurcation
Stable fixed point
Potential
Marginal saddle
Mean field spinodal point
Basic idea
z (t )  zu (t   )  z B (t )  zc   (t )
special solution
transient
zB (0)  z(0)
zu (t )  zc (t  )
zu  0 (t  )
z B (t )  zc (t  )
R  Rc
fluctuatio n of  and  (t)
fluctuatio n of z (t )
z
small deviation
θ: Goldstone mode
associated with time-traslational
symmetry
zB (t )
zc
zu (t   )

t
divergent fluctuations of

Fluctuations of θ
 : exit time from the marginal saddle

  n 
2

  n '/ f1 (n1/  )
*
  n
 '/ *
*
1 / *
f 2 (n
 '  1/ 2
 ' / *  2 ' / *  1
)
2

nc  
 *
    ' for n   
  
 '
z  zc  
 t  a  b 2
 0
for n  
*
 *
 (t )   1/ 2 ( 1/ 2t )
Poisson distribution of θ
for θ >> 1
Determination of scaling forms
A Langevin equation valid near the marginal saddle
z  zc  
 (t ) (0)  2d (t )
d
 t  a  b 2  
n
 n
1
([ ])  exp 
Z
 2d
Scaling form:
  n '/ f1 (n1/  )
*
*


0
d


dt ( t  a  b 2 ) 2  2b  
n


 (t )  n
1/ 3
( n
 ' / *  1 / 3
  n1/ 3 f1 (n2 / 3 )
1/ 3
t)   0
*  3/ 2
 ' 5/ 2
  n 5 / 3 f 2 (n 2 / 3 )
Fluctuation of trajectories
p( ) 
1
e
Z
n
(   ) 2
2 
O(n 2 / 3 )    O(n0 )
Gaussian integration of θ
O(n 2 / 3 )    O(n0 )
O(n 2 / 5 )    O(n0 )
 z (t ) has a peak at t  t* t*   1/ 2
 z (t* )   5 / 2
Numerical observations
  8 0.5

  0.1 2.5



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
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
MCT equation
t
 t    g  ds 2 (t  s) s (s)
0
 ( 0)  1
Exact equation for the time-correlation function for
the Spherical p-spin glass model (stationary regime)
p  3, g  3/(2T 2 )
 (t )  f   G(t ) ; G(t )  0 as t  
f   gf  (1  f  )
2
g  gc
gc  4
 (t )  0 t  
g 5
Attach Graph
g  gc
g4
g 3
 (t )  f  ( 0) t  
fc  1/ 2
f
Singular perturbation I
Step (0)
g  g c  (t )  G0 (t )  f c
G0 (t )  Ct  a for t  1
(1  2a)  2((1  a)) 2
Step (1)
g  g c   (  0)   0
 (t )  G0 (t )  A( )
  t
  0 will be determined later
Multiple-time analysis

 A  2 A2  4 ds A2 (  s) A' ( s)  0
A(0)  f c
0
A( ) : solution  A( ) : solution
A( )  f c  D b   1
We fix D=1 as the special solution A
dilation symmetry
(1  2b)  2((1  b)) 2
Singular perturbation II
Step (2)
g  g c   (  0)
 small
 (t )  G0 (t )  A( )   (t )
  t
 ,  , and  (t ) are not determined yet

G0 (t )
t*  
G0 (t* ) | A(  t* )  f c |

b
a b
A(  t )
different λ
log t
Derive small ρ in a perturbation method
Determine λ and ζ
Variational formulation
t
F (t; )   t    g  ds 2 (t  s) s (s)
0
I ( ) 
1 
dtF 2 (t ;  )

0
2
The variational equation is equivalent to
the MCT equation
 (t )  G0 (t )  A( )   (t )
Substitute
into the variational equation

 dsM (t, s)  (s)  B(t )

 dsM (t, s) (s)  0
0
0
0
M (t , s)  M ( s, t )
The solvability condition determines
ρ can be solved (formally)
under the solvability condition
 0 (t )    A(  t )
 
1
1

2a 2b
t*   1 /(2 a )
and the value of λ
Analysis of Fluctuation: Idea
P( )  const.e  N ( )

0
(t )
MCT equation
 (t )  G0 (t )  A( )   (t )
fluctuation of λ and ρ(t)
divergent part
z (t )  z B (t )  z c  zu (t   )   (t )
Determine the divergence of fluctuation intensity of λ
λ: Goldstone mode associated with the dilation symmetry
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
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
APPENDIX
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
d
2
 t  a  b   

near a marginal saddle
n
 ( x, t )   1/ 2 ( 1/ 4 x,  1/ 2t )
Binder, 1973
 *  3 / 2 d c   *
Ginzburg criteria
Schwartz, Liu, Chayes,
EPL, 2006
  1 / 4 R  Rc
but, be careful
for R  Rc
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 this mode
 slope of the effective potential of θ
Spatially extended systems III
Seek for simple finite-dimensional models
related to jamming transitions in granular systems
Simplest example
 (0)  1 : fix
T  1/ n
Saddle-node bifurcation
Stable fixed point
Marginal saddle
Potential
Question
trajectory
 (t )  * (t   )  B (t ) 1   (t )
special solution
transient
* (t )  1 (t  )
B (0)   (0)
*  0 (t  )
small deviation
B (t )  1 (t  )

P( ;  , T )
-- Instanton analysis
-- difficulty: the interaction between the transient part and θ
t

Fictitious time evolution
a stochastic bistable reaction diffusion system
F ,T  V
s-stochastic evolution for

(e.g. Kink-dynamics
in pattern formation problems)
P( ;  , T )
Result