Bifurcation and fluctuationsin jamming transitions

University of TokyoShin-ichi Sasa

(in collaboration with Mami Iwata)08/08/29@Lorentz center

MotivationToward a new theoretical method for analyzing “dynamical fluctuations” in Jamming transitions

PROBLEM: derive such statistical quantities from a probability distribution of trajectories for given mathematical models

TARGET: Discontinuous transition of the expectation value of a time dependent quantity, , 　 accompanying with its critical fluctuations)(t

MCT transition Eg. Spherical p-spin glass model

)0()(1

)(1

i

N

ii sts

Nt

321

321

3211

iiiNiiiiii sssJH

3p

)2/(! 12 pNpJ

N

ii Ns

1

2

iii

i ss

H

dt

ds

)()(2

3 2

0sstds

TT s

t

t

N Stationary regime

4

6 dTT

μ: supplementary variable to satisfy the spherical constraint

0t Equilibrium state with T

tt for 0)( The relaxation time diverges as )( dTT tft for 0)( )( dTT

Theoretical study on fluctuation of

Effective action for the composite operator

Response of 　　　 to a perturbation

hsssJH iiiNiiiiii

321

321

3211

Franz and Parisi, J. Phys. :Condense. Matter (2000)

ht)(

Response of 　　　 to a perturbation ht)(

i

N

iiiii

Niiiiii shsssJH

11

321

321

321

Biroli , Bouchaud, Miyazaki, Reichman, PRL, (2006)

Biroli and Bouchaud, EPL, (2004)

)(2

1log

2

1)( pI2

10 trtr

spatially extended systems

spatially extended systems

Cornwall, Jackiw,Tomboulis,PRD, 1974

4

3

• 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

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

Simpler mathematical description of the divergence simple story for coexistence of discontinuous transition and critical fluctuation

What we did recently

- (Exactly analyzable) many-body model exhibiting discontinuous transition with critical fluctuations

We analyzed theoretically the dynamics of K-core percolation in a random graph

-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

• 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

n hard spheres are uniformly distributedin a sufficiently wide box

compress

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)

(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

Time evolution （ decimation process)

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

(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

n: number of vertices m: number of edges

Initial state:

Time evolution:

particle vertex; connection edge

k-core percolation point

nn

mR fixed in the limit;

control parameter

All vertices are isolated

A k-core remains

cRR cRR

density of heavy vertex whose degree is at least (k=3)h

discontinuous transition !

RcR

)( th

Chalupa, Leath, Reich, 1979

Relaxation behavior

)(th

h

t

h

t

density of heavy vertex whose degree is at least k(=3) at time t

4096 ; nRR c 03.005.007.0

Red

Green

Blue

Green and blue represent samples of trajectories

03.0

Fluctuation of relaxation events

22 hhn

tmaximum becomes )( when timethe: t

)(

0 RRc

～ Dynamical heterogenity in jamming systems

Our resultsThe 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.

and

Iwata and Sasa, arXiv:0808.0766

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

Master equation (preliminaries)

: the number of vertices with r-edgesrv

),,,( 210 vvvw

: the number of edges

The number of edges of a heavy vertex obeys a Poisson distribution

zrr ez

rzQq

!)(

1

3222

rrrqhv

rr qhv / )3( rthe law of large numbers

Markov process of w Pittel, Spencer, Wormald, 1997

tP during ' :)|'( wwww

3

1r

rq

4r

r zrq z: important parameter

Master equation (transition table)

jww

……..

)0,2,2,1(1

)1,0,1,1(2

)1,1,2,1(3

)0,1,1,1(4

)1,2,3,2(5

)2,0,2,2(6

)0,1,2,2(7

)1,1,2,2(8

)3,2,1,2(9

)1,1,1,2(10

)2,1,1,2(11

)1,0,1,2(12

)0,0,1,2(13

)1,0,1,2(14

),,,( 210 vvvw

Master equation (transition rate)

jww

Langevin equationn/w

Deterministic equation

initial condition

21 2 s density of light vertices

2

t

),,,( from determined is 2103 z as one of dynamical variables

BifurcationConserved quantities

Transformation of variables

→

cRR cRR cRR

)(2 zRzzt )1()( zeez zz Rz 2)0(

The k-core percolation in a random graph is exactly given as a saddle-node bifurcation !!

/21 zJ )(/2 zQhJ

z z z

czz2 bat cRR

cz

4r

r zrq

marginal saddle

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

Langevin equation of z :

the simplest Langevin equation associated with a SN bifurcation:

Fluctuation of relaxation trajectories of z

22 )()()( tztzntz

*t )( *tz*at peak a has )( tttz

0 , cRR

The perturbative calculation wrt the nonlinearity seems quite hard even for

nT /1

fix :1)0(

Simplest example

Saddle-node bifurcation

Potential Stable fixed point

Marginal saddle

fix :1)0(

nT /1

Mean field spinodal point

Basic idea

)()()()( tztztztz cBu

)( )( tztz cu

)( 0 tzu

transient small deviation special solution

(t) and ofn fluctuatio

)0()0( zzB

)( )( tztz cB

cRR

)( ofn fluctuatio tz

divergent fluctuations of t

z

cz)(tzB

)( tzu

θ: Goldstone mode associated with time-traslationalsymmetry

Fluctuations of θsaddle marginal thefrom exit time :

)( ** /11

/' nfn

22 n

)( ** /12

/' nfn

* cn

*for ' n

*for ' n

1/'2/' ** 0 Poisson distribution of θfor θ 　 >> 1

2/1'2 bat

czz)()( 2/12/1 tt

Determination of scaling forms

n

dbat 2

czz

)()( 3/13/1 tnnt

A Langevin equation valid near the marginal saddle

)(2)0()( tdt

3/1/' * 2/3*

)( 3/21

3/1 nfn )( 3/22

3/5 nfn

)( ** /11

/' nfn

0

22 2)(2

exp1

])([ bn

dbadt

d

n

Z t

0Scaling form:

2/5'

Fluctuation of trajectories

)()( 03/2 nOnO

2/1*

t 2/5*)( tz*at peak a has )( tttz

2

)( 2

1)(

n

eZ

p )()( 03/2 nOnO

)()( 05/2 nOnO

Gaussian integration of 　 θ

Numerical observations

Red: Langevin equation with T=3/16384

Blue: Langevin equation with T=1/2097152

Square Symbol: direct simulation of k-core percolation with n=8192

5.08 5.21.0

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)()(2

0sstdsg s

t

t 1)0(

cgg tt 0)(

cgg tft 0)( )(

ttGtGft as 0)( ; )()(

Exact equation for the time-correlation function for the Spherical p-spin glass model (stationary regime)

)1(2 fgff

Attach Graph

4cg

)3/(2Tg ,3 2p

5g

4g

3g

f2/1cf

Singular perturbation I

0 )0( cgg

)()( 0 cftGt

)()()( 0 AtGt t

cgg 1for )(0 tCttG a

2))1((2)21( aa

Step (0)

Step (1)

later determined be will0Multiple-time analysis

0)(')(42 2

0

2 sAsAdsAA

cfA )0(

solution:)( solution :)( AA 1 )( b

c DfA 2))1((2)21( bb

We fix D=1 as the special solution A

dilation symmetry

Singular perturbation II

)()()()( 0 tAtGt

yet determinednot are )( and ,, t

Step (2) small )0( cgg

|) (|)( **0 cftAtG

ba

b

t

*

t

Derive small ρ in a perturbation method

tlog

)(0 tG

)( tA different λ

Determine λ 　 and 　 ζ

Variational formulation

0

0 0)(),( sstdsM

)()(0 tAt

ba 2

1

2

1

0

2 );(2

1)( tdtFI

)()()()( 0 tAtGt

)()();( 2

0sstdsgtF s

t

t The variational equation is equivalent to the MCT equation

0

)()(),( tBsstdsM

),(),( tsMstM

The solvability condition determines and the value of λ

)2/(1*

at ρ 　 can be solved (formally)under the solvability condition

Substitute into the variational equation

Analysis of Fluctuation: Idea

)()()()( 0 tAtGt

)()()()( ttzztztz ucB

)(.)( NeconstP

fluctuation of λ 　 and ρ(t)

divergent part

Determine the divergence of fluctuation intensity of λ

0)(

t

MCT equation

　 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

K-core percolation with finite dimension

KCM in a random graphSN-bifurcation

Bifurcation analysis of MCT transition

Fluctuation of

Fluctuation of (Spherical p-spin glass)

Spatially extendedsystems

Granular systems

spatially extended systems

APPENDIX

Spatially extended systems I

2/3* * cd

Analyze diffusively coupled dynamical elements exhibiting a SN bifurcation under the influence of noise

Ginzburg criteria c 4/1 RR

near a marginal saddle

Schwartz, Liu, Chayes,EPL, 2006

Curie-Wise theory

Ginzburg-Landau theory = diffusively coupled dynamical systems undergoing pitch-folk bifurcation under the influence of noise

Pitch-fork bifurcation

n

dbat 2

),(),( 2/14/12/1 txtx

but, be careful for c RR

Binder, 1973

Spatially extended systems IICharacterize 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

Saddle-node bifurcation

Potential Stable fixed point

Marginal saddle

fix :1)0(

nT /1

Question trajectory

),;( TP

)(1)()()( * tttt B special solution transient small deviation

)( 1)(* tt

)( 0* t

)0()0( B

)( 1)( ttB

t

-- Instanton analysis

-- difficulty: the interaction between the transient part and θ

Fictitious time evolution

s-stochastic evolution for

VF T ,

a stochastic bistable reaction diffusion system

),;( TP

(e.g. Kink-dynamics in pattern formation problems)

Result