marija vucelja the rockefeller university · marija vucelja the rockefeller university ......
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Elasticity and mixing on random graphs
Marija VuceljaThe Rockefeller University
UVa Physics Condensed Matter Seminar, 2013
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• out of equilibrium
• quenched and emergent disorder
• elastic response is typically very heterogeneous
• wide separation of scales between bending and stretching modes
• dissimilar interaction strengths
optical filters
IR lens
CDRW: AgInSbTe
DNA coated colloids
Amorphous solids are ubiquitous
2
Molecular glasses, colloids, granular matter, gels, fibrous networks, semi-flexible networks
Physical properties:
covalent bond energy
hydrogen bond energy
⇡ 100
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3
Outline
• Elasticity of random networks:
• vibrational modes of Laplacian Matrices, Stiffness Matrics
• Mixing on random graphs:
• new kind of Monte Carlo algorithms
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Percolation
• electrical conductivity, diffusion in random
media ... • bonds deposited with
probability p• p>pc finite conductivity• p=pc fractal percolating
cluster
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Rigidity Percolation
• probability p of a spring
Phillips, Thorpe, 1985Guyon, Crapo et al, 1990
M|�Ri = |F i
M stiffness matrix - random and sparse
X
hiji
[(�Ri � �Rj) · nij ]nij = Fi
nij
force balance at each node i
unit vector from i to j
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Floppy network has soft modes (low energy excitations)
�r = 0.2 �r = 0.05
�r ⌘ rc � r
extendedlocal nature
elongate a spring & measure response
over-damped rearrangements at different r:
Soft modes are typically extended!
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Maxwell rigidity criterium
Ncd dimension, N nodes, constrains
Nd�Nc �d(d+ 1)
2' Nd�Nc vibrational number of degrees of freedom
r = 2Nc/N average connectivity
Isostatic network: Nd = Nc ) rc = 2d
Maxwell criteriumrigid r > rcisostaticfloppy
r = rcr < rc
: mechanical stability
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Stiffness matrix
A small change in the displacement of the modes
Mij = �1
2�hijinij ⌦ nij +
1
2�ij
NX
l=1
�hilinil ⌦ nil
M
�E ' 1
2
X
hiji
((�Rj � �Ri) · nij)2 = h�R|M|�Ri
0 0.5 1 1.5 2
0.1
0.2
0.3
0.4
0.5
0.6
t
D(t
)
r = 3.2r = 3.6r = 3.8r = 3.9r = 3.95
0.5 1 1.5 2
0.1
0.2
0.3
0.4
0.5
0.6
frequency t
DO
S D
(t)
r = 4.05r = 4.2r = 4.4r = 4.8
numerics
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Wyart, et al 2005
cut gives:
Density of states of vibrational modes
18 particles confined in a periodic box
D(!) / Ld�1
L�1Ld= const
1024 spheres interacting repulsive harmonic potential, above jamming threshold.
Plato in the density of states
Ld�1 modes
extended modes - approximated by sine waves with frequencies L�1
density of states
Lerner,During and Wyart, 2013
10000 spheres interacting repulsive harmonic potential, above jamming threshold.
Δr
Δr
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Cavity method Mezard, Parisi, Virasoro, 1985
@i neighbors of i extracting vertex i
i
continue
continue
P(xi)
k
P(i)(xk)
P(i,k)(xm)
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Cavity method Mezard, Parisi, Virasoro, 1985
@i neighbors of i
k
P(i)(xk)
•exact on a tree•notice that subtrees independent •generally uncontrolled approximation due to loops •works well for large loops ~logN
assumptions
P(k)(x) =Y
j2@k
P(k)(xj)
P(k,m)(x) =Y
j2@m\k
P(k,m)(xj)
•subgraphs factorize (are independent)
P(k)(xj) = P(k,m)(xj)
•closure
Gaussian ansatz P(k)(xm) ⇠ exp[�ix
2m/2G
(k)mm]
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Diffusion on random graph
@tc(x, t) = r2c(x, t) ! @tc(xi, t) '
1
(�x)2
X
j
Jijc(xj , t)
diffusion on a line
Jij diffusion between i and j nodes on the graphr coordination number
Jij ⌘ �1
rCij + �ij
1
r
NX
k=1
Cik
Cij adjacency matrix
Edwards, Jones, 1976
H(x) =i
2x
T (zI � J)x P(x) = Z�1exp[�H(x)]
density of states of J
Hamiltonian
diagonalizing a matrix “substituted” graph dynamics of fields x or finding the Green’s function G
⇢J(�) =1
N⇡Im
✓@
@zln (det (zI � J))
◆=
�2
N⇡Im
✓@
@zlnZ
◆=
1
⇡NIm (TrG(z))
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Gii =
2
4z �X
j2@i
J2ijG
(i)jj
1 + JijG(i)jj
� Jij
!3
5�1
,
G(k)ii =
2
4z �X
j2@i\k
J2ijG
(i)jj
1 + JimG(i)jj
� Jij
!3
5�1
G =
z � r
r �G()
��1
G() =
z � r � 1
r �G()
��1
F. L. Metz, I. Neri, and D. Bollè, 2010
Cavity Equations
⇢(�) =�(�)(2� r)I[0,2](r)
2+
pr2�(2� �)� (r � 2)2
2�(�� 2)
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Markov Chain Monte Carlo
0 0.5 1 1.5 20
0.5
1
1.5
eigenvalues h
spec
tral d
ensi
ty l(h)
r coordination
number
2 num 2 theory 3 num 3 theory10 num10 theory20 num20 theory
⇢(�) =�(�)(2� r)I[0,2](r)
2+
pr2�(2� �)� (r � 2)2
2�(�� 2)
excellent agreement - theory and numerics
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Cavity equations solution - small heterogeneity
�G�1
�ii= !2I �
X
j2@i
Mij
⇣�G�1
�(i)jj
+Mij
⌘�1Mij � I
�
�G�1
�(k)ii
= !2I �X
j2@i\k
Mij
⇣�G�1
�(i)jj
+Mij
⌘�1Mij � I
�
a = !2 +r � 1
2(1 + ⇠r�1)
a
1� 2a,
A = !2 +r
2(1 + ⇠r)
a
1� 2a
(G�1)ii =
✓Aii Bii
B⇤ii Dii
◆, (G�1)(k)ii =
a(k)ii b(k)ii
b(k)⇤ii d(k)ii
!
a = d,A = d, b = 0, B = 0
Cavity Equations
components
isotropy, simplifies:
⇠s = s�1sX
i=1
cos�s
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!!!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!
!!!!!!!!!!!!
!
!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
"""
"
"
"
""""""""""""""""""""""""""""""""""
"""""""""""""""""""""""""""""
"
"
"
"
"
""""""""""""""""""""""""""
###########################
#
#
#
#
#
#######################################################
#####
#
#
#
#
#
#
##
0.0 0.5 1.0 1.5 2.0 2.50.0
0.5
1.0
1.5
2.0
frequency !
DO
SD!!"
! r " 3.2 num
! r " 3.2 theory
" r " 4.8 num
! r " 4.8 theory
# r " 11 num
! r " 11 theory
D(!)=
Z 1
�1d⇠
rr � 1
⇡e�(r�1)⇠22(r + (r � 1)⇠)
p32!2 � (3� r � (r � 1)⇠ + 4!2)2
⇡!(3(r + (r � 1)⇠)� 4!2)
Vibrational modes D(ω)
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Summary - elasticity
•thermodynamics: What is the role of the coordination number in thermodynamics?
•What distinguishes fragile from strong glasses? •compressed random networks.•cases with stronger spatial inhomogeneities
Angell, 1997
stron
g
Angell plot - strong and fragile glasses
fragil
estron
g
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• energy barriers glassy landscapes
• entropy barriers regions of high probability are separated by narrow paths (small entropy)
• high entropy a large phase space, that is flat in energy
Slow relaxation to a equilibrium, due to:
18
Mixing on random graphs
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set of states
transition matrix
stochastic matrix
If is irreducible the a steady state exists and it is unique
Balance condition
Detailed balance (reversibility):
Detailed balance is sufficient, but not necessary!
Detailed balance
X
x2⌦
[⇡s
(x)T (x, y)� ⇡
s
(y)T (y, x)] = 0
⇡s(x)T (x, y) = ⇡s(y)T (y, x)
⇡
s
(y) =X
x2⌦
⇡
s
(x)T (x, y)
⌦
T (x, y) X
y2⌦
T (x, y) = 1 8x 2 ⌦
T (x, y)
19
⇡
t+1(y) =X
x2⌦
⇡
t(x)T (x, y)
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• How about breaking Detailed balance?
After all if you want your coffee sweet, it is better to stir the sugar,
than to wait for it diffuse around the cup.
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goal: sample with uniform probability from a torus
north east south west
pN = 1�N�1
pE = pW = (2N)�1
pS = 0
N ⇥N
random walker on a torus
mixing time on a torusN
N
pN = pW = pE = pS = 1/4
O(N2)
mixing time on a torus O(N)
O(N) randomize along y-axisrandomize along x-axis O(N)
y
x
Diffusion
Lifting on a torus Chen, Lovasz, Pak 1999
Lifting added advection
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Random walk
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Lifted random walk
Density of visited sites on a torus of 1024 sites, after 1024 steps
high occupancy
low occupancy
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Inverse spectral gap ��1
Autocorrelation of distance from origin r(t)
N2
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Skewed detailed balance
• Create two copies of the system (‘+’ and ‘-’)
• Decompose transition probabilities as
• Compensate the compressibility by introducing transition between copies
⇤
(±,⌥)(x, x)=max
8<
:X
y2⌦
⇣T
(⌥)(x, y)� T
(±)(x, y)
⌘, 0
9=
;
⇡(x)T (+)(x, y) = ⇡(y)T (�)(y, x)
T = T (+) + T (�)
K. S. Turitsyn, M. Chertkov, MV (2008)
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• Extended matrix satisfies balance condition and corresponds to irreversible process:
• Random walk becomes non-Markovian in the original space.
• System copy index is analogous to momentum in physics: diffusive motion turns into ballistic/super-diffusive.
• No complexity overhead for Glauber and other dynamics.
Skewed detailed balance continued
T =
✓T (+) ⇤(+,�)
⇤(�,+) T (�)
◆
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Curie-Weiss Ising model
J > 0 ⇡s1,...,sN = Z�1exp
2
4� J
N
X
k,k0
sksk0
3
5
S =X
k
sk
P (S) ⇠ N !
N+!N�!
exp
✓�JS2
2N
◆
N± =N ± S
2
N-spins ferromagnetic cluster Ising model on a complete graph
Stationary distribution
A state of the system is completely characterized by its global spin (magnetization)
probability distribution of global spin
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Physics of the spin-cluster continued
N !1
J < 1P (S) S = 0
�S ⇠p
N/J
�S ⇠ N3/4
J = 1In the thermodynamic limit
the system undergoes a phase transition at
Away from the transition in the paramagnetic phase
is centered around
and the width of the distribution is estimated by
At the critical point (J=1) the width is
One important consequence of the distribution broadeningis a slowdown observed at the critical point for reversible
MH–Glauber sampling.
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Correlation time of S reversible case
characteristic correlation time of S (measured in the number of Markov chain steps) is estimated as
Trev / (�S)2
the computational overhead associated with the critical slowdown is
⇠p
N
Advantage of using irreversibilityThe irreversible modification of the MH–Glauber algorithm applied to the spin cluster problem achieves complete removal of the critical slowdown.
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Correlation time of S irreversible case
switching from one replica to another the system always go through the S = 0 state, since
The Markovian nature of the algorithm implies that all the trajectories connecting two consequent S = 0 swipes are statistically independent, therefore the correlation time roughly the number of steps in each of these trajectories.
switching + spins in (+) replica(+) to (-)
⇤(�,+)ii = 0 S < 0if
if
Recalling that inside a replica (i.e. in between two consecutive swipes) dynamics of S is strictly monotonous, one estimates
⇤(+,�)ii = 0 S > 0
(-) to (+) switching - spins in (-) replica
Tirr ⇠ �STirr ⇠
pTrev ⌧ Trev
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Numerical verification
Analyzed decay of the pair correlation function, <S(0)S(t)>, with time.
Correlation time was reconstructed by fitting the large time asymptotics with exponential function
for both MH and IMH algorithms we constructed transition matrix corresponding to the random walk in S, calculated spectral gap, Δ, related to the correlation time as,
T = 1/ReΔ
In both tests we analyzed critical point J = 1 and used different values of N ranging from 16 to 4096.
T ⇠ exp(�t/Trev)
T ⇠ exp(�t/Tirr) cos(!t� �)
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Correlation time of (dots)
Inverse spectral gap (crosses)
A square root improvement:
2 4 6 8 10 12 142
4
6
8
10
12
14
16
18
20
log2 N
log 2 T
Irreversible
Reversible
T ⇠ N0.85
T ⇠ N1.43
hS(0)S(t)i
T ⇠ N3/2 ! T ⇠ N3/4
Best case scenario: square root improvement Chen, Lovasz, Pak etc.31
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2d Ising
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Collaborators
Gustavo Düring Pontificia Universidad Catolica de Chile
Matthieu Wyart
NYU
Misha ChertkovLANL, CNLS
Jon MachtaUMass AmherstKonstantin Turitsyn
MIT