Bifurcation and fluctuations in jamming transitions
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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
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
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)