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