Metastable states in the East model and
plaquette spin models
Robert Jack (University of Bath)
Warwick meeting, Jun 2014
Thanks to:!
Juan Garrahan (Nottingham)
Ludovic Berthier (Montpellier)
Outline
− Part I : Metastable states in the East model!
− Part II : Plaquette models,
Interpreting point-to-set measurements
and overlap fluctuations, using metastable
state arguments
vaporization near Tg. This means that OTP’s landscape is very heterogeneous. The basins sampled at high temperature allow relaxation by
surmounting low barriers involving the rearrangement of a small
number of molecules. The very large activation energy at T ≈ Tg, on
the other hand, corresponds to the cooperative rearrangement of
many molecules. These differences between strong and fragile behaviour imply a corresponding topographic distinction between the two
Metastable states
-7.05
0.0
Figure 6 Mean inherent struct
sized Lennard–Jones atoms, a
from which the inherent structu
dynamics simulations at consta
temperatures for 256 Lennard–
are of type B. The Lennard–Jon
$BB"0.88, $AB"0.8, and %AA
temperature, energy and time a
$AA(m/%AA )1/2, respectively, with
were performed at a density of
and 3.33&10#6. When T > 1
entire energy landscape, and th
are shallow. Under these condi
activation energy for structural
and T ≈ 0.45, the activation en
‘landscape-influenced’, and th
strongly temperature-depende
separating sampled adjacent e
(calculations not shown). This i
rare jumps
over distan
and execute
Stillinger,
2001]
crossover between landscapecorresponds closely with the m
from refs 70 and 72.)
“Basins” / “minima” / “valleys” on a free energy
landscape
Transition
states
Potential energy
Basin
Ideal glass
Coordinates
Crystal
[Debenedetti
Figure 5 Schematic illustration of an energy landscape. The x-axis represents all
configurational coordinates. (Adapted from ref. 44.)
0.
Regions of configuration space with fast “intra-state”
motion and
264slow transitions between states
© 2001 Macmillan Magazines Ltd
Counting states
[Kurchan and Levine, J Phys A, 2011]
Regions of configuration space with fast “intra-state”
motion and slow transitions between states
States with lifetime ⌧ , take a time tobs with 1 ⌧ tobs ⌧ ⌧
Averaging over a time period tobs :
same as Boltzmann averaging within the state
Diversity (entropy) of averaged properties:
… gives number (and properties) of metastable states!
… including configurational entropy (complexity)
2
II.
East model
MODEL, METHODS, AND METASTABLE
STATES
A.
(a)
1
0.8
Model
Spins ni = 0, 1, with i = 1 . . . N .
The East model [17, 18] consists of L binary spins
ni = 0, 1, with i = 1 . . . L, and periodic boundaries.
i with
1
We refer to spins with ni = 1 as ‘up’ and those
ni = 0 as ‘down’. The notation C = (n1 , n2 , . . . , nL ) indicates a configuration of the system. The key feature
of the model is that spin i may flip only if spin i 1 is
up. If this kinetic constraint [19, 20] is satisfied, spin i
E flips from 0 to 1 with rate c and from 1 to 0 with rate
(a)
1 c. We take1c/(1 c) = e
where is the inverse
temperature, so small c corresponds to low temperature.
The rates for0.8
spin flips obey detailed balance, so that
the probability
ofPconfiguration C at equilibrium is sim0.6
0
i ni /Z, where Z = (1 + e
pins
ply p (C)C(t)
= e
)L is the
=5
ies.
partition function.
one has hni i = c, and
=4
0.4 At equilibrium,
with
3
the regime of interest is= small
c (low temperature).
2 of p0 (C), the kinetic conin- Despite the0.2trivial =form
= 1 to complex co-operative dyure
straint in this model leads
0 -2c. In 0particular,
1 is
2
4
6
8 time
10
namics at small
the
relaxation
10
10
10
10
10
10
10
in of
i the model diverges in a super-Arrhenius fashion, as
t
2
/(2 ln 2)
rate
⌧0 ⇠ e (b)
[18, 33]. The rapid increase in relaxation
2
= 0 1a where
erse
time is illustrated in bFig.
we
show
b
=
1
hn
(t)n
(0)i
hn
i
i
i
1
ure.
b=2 i
2 i hn 2i2
hni (0)i
hat
hni (t)n
hni ii b = 3
i
C(t)
=
,
(1)
b=0
imC(t)
c(1 c)
the
0.95
and
Spin i can flip only if n
Flip rates c and 1
C(t) =
C(t) 0.6
= 1.
=5
=4
=3
=2
=1
0.4
0.2
c gives hn
2 i i =0 c =
-2
10
(b)
10
1
01+e2
10
4
10
t
b=0
1
.
b=1
10
10
8
10
10
b=2
b=3
b=0
C(t)
6
0.95
b=1
=5
=4
0.9
-2
10
10
0
b=3
b=2
2
10
b=4
4
10
t
10
6
10
8
10
FIG. 1: Correlation functiontimes
C(t) in the t
East
model.
Characteristic
c
b ⇡
10
1
(b
)
2
(a) The
inverse temperature is varied from = 1 to = 5, showing
a dramatic increase in relaxation time. (b) At the lowest
[RLJ, Phys
E, 2013]
temperatures, plateaus are apparent
during Rev
the early
stages
B.
Metastable states
0.1
↵
ciated with a profile C , holds very
East
model. As discussed by Sollich
0.01
metastable states, we perform a time averalso [34]), motion on a time scale t
0.001
0
50
pins, defining
stricted
to domains
of size100
2b 1 , with
[RLJ, Phys Rev E, 2013]
b=1 1
immediately to the right of a long-liv
3
Z t
1
1 b
b
(b
) within suc
0
0
c
⌧
t
⌧
c
then
spins
0.1
2
n
=
(1/t)
dt
n
(t
),
(2)
Characteristic
times
t
⇡
c
i
b
[ it
=5]
tories
of
the
East
model,
starting
from a particular initial
1/2
2
hnit i0ic
h nit iic
flip
many
times
within
timeprocesses
t, while
condition that helps to reveal the physical
at
0.01
and l
work.likely
Fig. 2to
shows
spins hnof
flipthe
attime-averaged
all. The result
this
it iic
Initial condition:
11..
1.1.
1..1
1...1
1....1
eraged spin1...profile
their variances h( nit )2 iic , where the subscript “ic” indiisthat
that
nit !was
hntaken
c if aspin
i is with
i i = with
b=0 1
cates0.001
the
average
fixed
initial
con0
50
100
Sollich
and
[18]
used
the
term
dition
(in
contrast
to Evans
other averages
in
this work
which
b
=
2
1
C t = 0.1
(n1t , n2t , . . . , nLt ).
(3) are conventional averages over equilibrium trajectories).
describe these mobile domains. A m
We emphasise the following threeb points:
0.1
lifetime
of order c can be identifie
0.01
rvation of [14] is that if the system has
• All of the hnit iic are either close to 1 or of order
1/2
position
superdomains.
0.01
ates indexed
by
↵
=
1,
2,
.
.
.
,
then
each
c ⌧ 1. Also,oftheits
standard
deviation h( nit )2 iic ⌧
0.001
↵50
1
0
1 In
for all
spins. ofThus,
for
thisthe
initial
condition
ated with a profile C . Further, if100t is much
terms
Fig.
1b,
limit
c
and for each of these times t, one almost certainly
0.001
b=1 1
0
50
100within a
he intrastate relaxation times, but small
sponds
to
choosing
a
finds a profile
C t that is close
totime
a reference
profile
↵
b = 3C 1 = hn i . Each reference profile (one for each
it icfunction.
t
0.1
heir lifetimes,
then the observed profile C t
relation
However, when
↵
value of t shown) can therefore be identified with a
rely be close to one of the C . Hence, the
sults
to state
gainthat
information
about
0.1
metastable
is stable on a time
scale t.me
Time averaging…
0.01
0.001
0
=2 1
[ b=
5]
0.1
Initial condition:
50
hnit iic
100
1/2
h n2it iic
•0.01
For b 1, any spins with hnit iic ⇡ 1 are separated
by at least 2b 1 spins with hnit i ⌧ 1. If two up
spins are closer than this in the initial condition
0.001
tories
of the
starting
from
then
rightmost
themmodel,
is within
the mobile
0 the
50ofEast
100
domaincondition
associated with
leftmost
one. Inthe
thatp
thati the
helps
to reveal
case, the
rightmost
(nj ) tends
to flip many
work.
Fig. spin
2 shows
the time-averag
times and njt converges to a value2 close to c.
Time-averages
(almost)
determined
bytheir
initial
condition…
1...
11..
1.1.
1..1
1...1
1....1
variances
h( nit ) iic , where the
0.01
B.
Metastable states
↵
0.1
ciated with a profile C , holds very
East
model. As discussed by Sollich
0.01
metastable states, we perform a time averalso [34]), motion on a time scale t
0.001
0
50
pins, defining
stricted
to domains
of size100
2b 1 , with
[RLJ, Phys Rev E, 2013]
b=1 1
immediately to the right of a long-liv
Z t
1
1 b
b
(b
) within suc
0
0
c
⌧
t
⌧
c
then
spins
0.1
2
nit = (1/t)
dt ni (t ),
(2)
Characteristic times tb ⇡ c
flip many times within time t, while
0
0.01
likely
to flip at all. The result of this l
1
eraged
ñi =spin
⇥(nprofile
i
is that nit ! hni i = c if spin i is with
2 ) = 0, 1
0.001
0
50
100
Sollich
and
Evans
[18]
used
the
term
b
=
2
1
C t = (n1t , n2t , . . . , nLt ).
(3)
describe these mobile domains. A m
Measure entropy of ñi
b
0.1
lifetime
of
order
c
can be identifie
rvation of [14] is that if the system has
byindexed
pattern-counting
`
position
of its superdomains.
0.01
ates
by ↵ = 1, 2, .at
. . ,length
then each
↵
ated with a profile C . Further, if t is much
In terms of Fig. 1b, the limit c1
0.001
P
0
50
100within a
he intrastate
relaxation
times,
but
small
sponds
to
choosing
a
time
S` = B` p(B` ) ln(1/p(B` )) b = 3 1
heir lifetimes, then the observed profile C t
relation function. However, when
↵
rely be close to one of the C . Hence, the
sults
to gain information about me
0.1
Counting…
B` : state of (ñi , . . . , ñi+`
1)
0.01
0.001
0
50
100
i
E.g., B`=4 = (1, 0, 1, 0) almost never occurs for b
2
Counting…
[RLJ, Phys Rev E, 2013]
Convenient to plot S` S` 1
... converges for large ` to entropy per site
[
(a)
0.09
b=0
b=1
b=2
b=3
b=4
theory
Sl - Sl-1
0.08
0
=4]
5
10
15
20
(a)
⇤
b
Length scale
`
⇡
2
g (r)
1
t
0.005
For b = 0, expect
ñ = n,
‘static’ result
Well-defined complexity
. . . for states of00 lifetime
2 tb
l
Theory: up spins are never closer than `⇤(b), 0.03
[ =5]
(b)
otherwise randomly distributed. . .
G4(r,t)
b=0
. . 0.04
. “hard rods”
0.02
b=1
4
r
Hard rod states
[RLJ, Phys Rev E, 2013]
1...
11..
1x..
111.
1.1.
1x1x..
1cx.
1.11.
1x1x1x
1cxx1c
1ccxx
1 : ni ⇡ 1
⇡0
FIG.0 :3:ni Schematic
1.1.1.
b=0
b=1
b=2
T
patc
a p
(ñit
eval
b=3
c : ni ⇡ c ± O(c )
1/2
x
:
n
⇡
c
±
O(c
illustration of
i time evolution) among
3/2
metastable states. Each box represents a state, labelled by
its characteristic local density profile, and a value of b that
Evolution
to
“longer
rods”,
some
b deterministic and
indicates its lifetime (of order c ). To describe the profile
some
steps…
we stochastic
use a notation
suggested by Fig. 2, based on the relative
2 1/2
sizes of hnit iic and h nit iic . We distinguish four cases: (i) a
whe
C˜t , a
in C˜
patc
A
one
profi
Part I summary
Time scale separation means metastable states are welldefined…!
Kurchan-Levine method allows counting of metastable
states (of given lifetime)…!
[Kurchan and Levine, J Phys A, 2011]
In the East model, these states correspond with those of
“hard rod” systems…
... at least for times tobs ⌧ ⌧↵ (that is, `⇤ ⌧ 1/c)
… diverging length scale coupled with diverging time scale
[RLJ, Phys Rev E, 2013]
Triangular Plaquette Model
J. Chem. Phys. 123, 164508 !2005"
[Newman and Moore 1999, Garrahan and Newman 2000, Garrahan 2002]
barriers
only in
Ising spins si = ±1
ty of its
on vertices of triangular lattice
Energy
1
2
X
N
2
X
s1 s2 s3 =
nµ
ven in a E =
µ square and rhomboid droplets !L
4 showing relation between
FIG. 9. Sketch
caging
1
#11=is0,
marked
by a filled
=
6#.
The
spins
are
on
the
intersections
of
grids;
µ
labels
“plaquettes”,
n
=
(1
s
s
s
)
1
µ
1
2
3
ed than
2
circle. The & symbol marks the plaquette interaction q24 = #24#25#34. Rotaof Ref.
tions of 120° leave both Hamiltonian and triangular lattices invariant. HowAt equilibrium,
n
are
uncorrelated
(as
in
KCMs)
µ
ets with
ever, note that the boundaries of the droplets are not all equivalent, since the
rhombus shape is not invariant under this rotation.
pin-spin
Glauber dynamics (single spin flips). . .
entropy,
2
c/T
ir bulk
Relaxation time ⌧ ⇠ e
, as in East model
H = − !1/2# & #xy#x+1,y#x,y+1 ,
!31#
TPM: spin correlations
J. Chem. Phys. 123, 164508 !2005"
[Newman and Moore 1999, Garrahan and Newman 2000, Garrahan 2002]
barriers
only in
Ising spins si = ±1
ty of its
on vertices of triangular lattice
hsi i = 0,
hsi sj i = 0
ven in a
Selected
three-point
and
higher
correlations
are
non-zero,
FIG. 9. Sketch showing relation between square and rhomboid droplets !L
caging
log 2/ log 3
Correlation
length
⇠
⇠
c
= 6#. The spins are on the intersections of grids; #11 is marked by a filled
ed than
circle. The & symbol marks the plaquette interaction q24 = #24#25#34. Rotaof Ref.
tions of 120° leave both Hamiltonian and triangular lattices invariant. Howets with
note that the states.
boundaries. .of the droplets are not all equivalent, since the
Manyever,
metastable
rhombus shape is not invariant under this rotation.
pin-spin
Assuming complexity ⇡ simple entropy,
entropy,
S` ⇡ `2 c ln c + 2` ln 2
[Cammarota and Biroli, EPL, 2012]
ir bulk
H = − !1/2# & #xy#x+1,y#x,y+1 ,
!31#
approached at the Kauzmann temperature.
DOI: 10.1103/PhysRevLett.86.5526
The striking universality of relaxation dynamics in supercooled liquids has remained intriguing for decades. In
addition to the overall dramatic slowing of transport as the
glass transition is approached, one finds the emergence of
a strongly nonexponential approach to equilibrium when
a supercooled liquid is perturbed [1]. This contrasts with
the behavior of chemically simple liquids at higher temperatures, where only a single time scale for a highly exponential structural relaxation is usually encountered for
times beyond the vibrational time scales. Both the range
of time scales and the typical magnitude of the relaxation
time require explanation.
Several theoretical threads lead to the notion that the
universal behavior of supercooled liquids arises from
proximity to an underlying random first order transition
[2–6] which is found in mean field theories of spin glass
without reflection symmetry [7–9], and in mode coupling
[10,11] and density functional [12–14] approaches to
the structural glass transition. This picture explains both
the breakdown of simple collisional theories of transport
that apply to high temperature liquids at a characteristic
temperature TA and the impending entropy crisis of supercooled liquids first discovered by Simon and brilliantly
emphasized by Kauzmann [15] at the temperature TK .
Furthermore the scenario suggests that the finite range
of the underlying forces modifies mean field behavior
between TA and TK in a way that leads to an intricate
“mosaic” structure of a glassy fluid in which mesoscopic
local regions are each in an aperiodic minimum but are
separated by more mobile domain walls which are strained
and quitepfar from local minima structures, as shown in
Fig. 1 [16]. Relaxation of the elements of the mosaic,
reconfiguring to other low energy structures, leads to the
[RLJ and
slow relaxation, and a scaling treatment
of theGarrahan,
most proba-
PACS numbers: 64.70.Pf
Relaxation in supercooled liquids is well approximated
by the stretched exponential or Kohlrausch-Williams-Watts
b
(KWW) formula f!t" ! e2!t#tKWW " . A study conducted
by Böhmer et al. [18] on over sixty glass formers at about
Tg shows b and D are strongly correlated. The smallest
b is found for the most fragile liquids. The heterogeneity
of time scales suggests possible heterogeneity in space.
The existence of spatial heterogeneity received support
from recent computer simulations [19–21], although these
are at temperatures near TA where the clusters are small,
KCM or mosaic?
Can think of TPM as a kinetically-constrained model
… or as a “mosaic” made up of metastable states
BUT:
No domain walls,
No surface tension between
states
“Nucleation” dynamics
among states:
barrier scales with log `
(not ` as in CNT)
FIG. 1. An illustration of the “mosaic structure” of supercooled liquids. The mosaic pieces are not necessarily the same
size due to the fluctuations in the driving force, configurational
[Xia
andfrom
Wolynes,
PRLconfigura2001]
entropy. The system
escapes
a local metastable
tion by an activated process equivalent to forming a liquidlike
droplet inside a mosaic element. For droplets with size much
smaller2005]
than the mosaic (as shown with the small circle), the
JCP,
droplet shape and its surface energy cost are well described by
“Cavity” point-to-set
To measure “mosaic” tile size…
Immobilise system outside a “cavity” of size `
What is the smallest `
for which the cavity has more than 1 state available?
[Bouchaud and Biroli, J Chem Phys, 2004]
For the TPM: this size is ⇠mos ⇠ c
log 2/ log 3
[RLJ and Garrahan, J Chem Phys, 2005]
Relaxation on length scale ⇠mos
determines bulk relaxation time
(consistent with rigorous bound)
[Montanari and Semerjian, J Stat Phys, 2006]
fi = 1
(a) f = 0.12
pi = 1
pi = −1
(b) f = 0.18
(c) f = 0.25
Random pinning
Random pinning: from an equilibrium configuration C0 ,
‘pin’ (immobilise) each spin w/prob f
ODELSAllow
WITH .model
..
to relax to configuration
C,
PHYSICAL
REVIEW E 85, 021120 (20
measure overlap hq(C, C0 )i = hsi s0i i
0
Also,
measure
p
=
hs
s
i
i fiii=
C1
β=3
(a) f = 0.12
(e)
(d)
104
105
pi = 1
f = 0.10
(d)
Finite length scale ⇠ ⇠ (1/c)
(b) f = 0.18
f = 0.15
(e)
pi = −1
(f)
(c) f = 0.25
f(f)= 0.20
[RLJ and Berthier, PRE 2012]
IN MODELS WITH . . .
PHYSICAL REVIEW E 85, 021120 (2012)
TPM “mosaic”
fi = 1
β=3
(a) f = 0.12
(e)
(d)
103
104
105
pi = 1
f = 0.10
(d)
(b) f = 0.18
f = 0.15
(e)
pi = −1
(f)
(c) f = 0.25
f(f)= 0.20
[RLJ and Berthier, PRE 2012]
FIG. 3. scale
(Color⇠ online)
of the propensities pi in the
Finite length
⇠ (1/c)Snapshots
(from numerics)
long-time limit. (a)–(c) SPM at β = 3, varying
f
.
At
this
temperature
log 2/ log 3
Natural
∗ mosaic length scale ⇠mos ⇠ (1/c)
∗
3 for which
f
f ≈ 0.18. (d)–(f) TPM at fβ ==0.10
f = 0.15≈ 0.15. Pinned
f = 0.20 spins
are blackidea:
while pinning
unpinnedreduces
spins are number
color coded
dark blue
(dark
Mean-field
of with
available
states,
FIG.pale
3. (Color
online)
Snapshots
ofgray)
the propensities
p0
the
i in
≈
1,
and
blue
or
green
(light
for
p
≈
and
gray)
for
p
i
i
with fixed surface long-time
tension:
diverging
length
scale
(at
fixed
c)
limit. (a)–(c) SPM at β = 3, varying f . At this temperature
On
increasing
f
,
the
spins
become
polarized
pi ≈ −1, respectively.
[Cammarota
and Biroli,
EPL 2012]
∗
∗
f ≈ 0.18. (d)–(f) TPM at β = 3 for which f ≈ 0.15. Pinned spins
memost
The correlations
associated
with
the??
so that
pi ≈ 1 inarespatial
black cases.
while
unpinned
spins
are
color coded
with dark
blue
(dark
3 Coupling
4
5 between
domains
too
weak
for
this,
in
TPM
10
10
10
∗ pale blue or green (light gray) for p ≈ 0 and
susgray) for p
i ≈ 1,fand
i
, consistent with χ being maximal.
propensity are strongest
near
Overlap fluctuations
For fixed (quenched) C0 , bias C to lie near C0 ,
✏N q(C,C0 )
with biasing probability e
Allow model to relax to configuration C,
measure overlap hq(C, C0 )i = hsi s0i i
RANDOM PINNING IN GLASSY SPIN MODELS WITH . . .
random pinning…
average overlap
quenched
biased
overlap…
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(a)
1
β=3
0.8
eps=0
0.01
0.05
0.07
0.09
0.11
0.13
0.15
1e+00
1e+01
C(t)
0.6
0.4
f = 0.00
0.2
1e+02
1e+03
1e+04
t
[RLJ and Garrahan, unpublished]
1e+05
0
100
14
0.05
0.10
0.15
0.20
101
102
t
103
104
105
[RLJ and Berthier, PRE 2012]
Overlap fluctuations
For two configurations C, C0 , bias C to lie near C0 ,
with biasing probability e✏N q(C,C0 )
(annealed case)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
annealed
biased overlap
(annealed)…
quenched
biased overlap
(quenched)
average overlap
average overlap
First order phase transition at non-zero ✏⇤
eps=0
0.01
0.05
0.07
0.08
0.09
0.10
0.11
1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06
t
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
eps=0
0.01
0.05
0.07
0.09
0.11
0.13
0.15
1e+00
1e+01
1e+02
1e+03
1e+04
1e+05
t
[Garrahan PRE 2014, RLJ and Garrahan, unpublished]
Overlap fluctuations
[RLJ and Garrahan, unpublished]
Why are the quenched/annealed cases so different?
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
annealed
biased overlap
(annealed)…
quenched
biased overlap
(quenched)
average overlap
average overlap
Defect density hnµ i ⇡ c2
eps=0
0.01
0.05
0.07
0.08
0.09
0.10
0.11
1e-01 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
eps=0
0.01
0.05
0.07
0.09
0.11
0.13
0.15
1e+00
1e+01
1e+02
1e+03
1e+04
1e+05
t
t
Defect density hnµ i ⇡ c
Annealed case: system is sampling non-typical metastable states
Dimensionality?
[RLJ and Garrahan, unpublished]
Are two-dimensional models special?
If surface tension between metastable states exists,
can relate pinned systems to random-field Ising magnets…
(hence no phase transitions in 2d)
In TPM, no surface tension, so this argument does not apply…
preliminary results for 3d generalisation of TPM
… consistent with 1st order transition with annealed coupling,
no transition for quenched coupling,
(as in 2d)
Dynamics and metastability
Alternate probe of non-typical metastable states:!
bias trajectories of a system using time-integrated quantities
Let C be a “configuration”.
Consider trajectories C(t) of length tobs (sample paths)
Bias probabilities as:
Prob[C(t); s] = Prob[C(t); 0]
e
stobs k[C(t)]
Z(s)
k[C(t)]: time-averaged measurement along trajectory
[ Ruelle, Gallavotti-Cohen, Evans, Derrida, Lebowitz, Gaspard, Maes, etc ]
Dynamics and metastability
Bias probabilities as:
Prob[C(t); s] = Prob[C(t); 0]
e
stobs k[C(t)]
Z(s)
Effect of field s is to enhance probabilities of
long-lived states with small k
This can lead to “dynamical phase transitions”:
singular responses to field s
[ Garrahan et al 2007, Hedges et al 2009, … ]
System arrives in non-typical metastable states. . .
(cf annealed overlap calculation)
Summary
KCMs (and the TPM) have well-defined metastable states
(with a range of lifetimes…)!
Random pinning and quenched overlap calculations reveal
correlations among “mosaic tiles” but (apparently) no phase
transitions…!
No evidence for surface tension between states in TPM,
[Relevant for “nucleation” arguments about dynamics,
could be a scenario for atomistic systems(?)]!
Annealed overlap and s-ensemble calculations probe nontypical metastable states and do lead to phase transitions in
KCMs, TPM, …