Effective theories of the glass
transition:
thermodynamic versus
dynamic approaches
Gilles Tarjus
(LPTMC, CNRS/Univ. Paris 6)
+ G. Biroli (IPhT, Saclay), C. Cammarota (Rome),
M. Tarzia (LPTMC)
Warwick 2014
1
Outline
• In search of an effective theory and an appropriate
order parameter for the glass transition: dynamic or
thermodynamic?
• Contrasting the dynamical facilitation and the
landscape approaches. Enlarging the phase diagram.
• Mean-field dynamical transitions.
• Beyond mean-field: Constraints and pinning fields.
2
The glass transition:
What makes the problem interesting...
There are hints that glass formation involves
✴ some form of universality
✴ some form of collective/cooperative
behavior
Yet, of an unusual kind...
➡Search for effective theories of glass-forming
liquids
➡What is the appropriate local order
parameter? A thermodynamic/structural or a
purely kinetic approach?
Some theoretical ingredients
•Frustration: The energy of a system cannot be minimized by
simultaneously minimizing all the local interactions.
=> Multiplicity of low-energy (‶metastable″) states.
•Thermal activation in a rugged (free) energy landscape:
Presence of an exponentially large number of metastable states that
may trap the system.
=> Relaxation slowdown is associated with thermally
activated escape from metastable states.
•Dynamical facilitation: Mobility triggers mobility in nearby regions.
=> Spatial correlations in the dynamics.
Different ways to incorporate the ingredients in a general
theory!
Local order parameter(s)
•Local structural order: Observables characterizing the locally
preferred molecular arrangement in the liquid, if present (e.g., related
to bond-orientational order).
=> e.g., poly-tetrahedral/icosahedral in metallic glasses.
• Similarity or ‶overlap″ between configurations: Measures of the
similarity between two equilibrium configurations of the liquid.
High overlap => in the same state (‶localized″)
Low overlap => in different states (‶delocalized″)
•Local mobility field: Mobility or activity defined by following the
dynamics in small volumes of space over short periods of time.
=> Easier at low T where mobility is localized and scarce.
Which is the best starting point
to describe glassy liquids?
Weak constraints from comparison to
experimental data...
With the help of (unavoidable ?) adjustable parameters,
several theories fit the same data equally well
Configurational entropy/RFOT
(Wolynes et al.)
Facilitation (Garrahan-Chandler)
log(viscosity or time) vs Tg/T
Frustration (Kivelson-GT)
Focus on:
Dynamical facilitation & KCM’s
(rarefaction of dynamical paths)
vs
free-energy landscape approach
(rarefaction of metastable states)
Contrasting the predictions by
enlarging the parameter space
The configurational entropy/landscape
scenario as the mean-field theory of
glass-forming liquids
[Wolynes, Kirkpatrick, Thirumalai, 80’s + Parisi-Mezard-Franz +many!!!]
large number
of metastable
states
• Exponentially
(1)
(2)
(3)
s
=
effective potential
=
F[ !(r)]
sc the liquid
s c(configurational
c entropy)
that may trap
between two temperatures, a dynamical transition
N1
N2
N3
at Td and a “random first-order
transition”
at TK.
states
states
states
• Order parameter = overlap between equilibrium
configurations => sbuild
a
Landau
functional
(i)
kB Log(Ni )/ Vdrop
~
c
through the replica formalism.
•
•
cut in !( r ) space
FIG. 1: Appearance of an exponentially large number of
Found
in mean-field spin glasses, in mean-field-like
FIG. 2: The typical energy lands
meta-stable configurations and the possibility for a system to
phase as a function of density (
realize these configurations
gives(HNC,
rise to the
configurational
approximations
of liquids
DFT),
in the hard-sphere
fluid
in to des
potential,
which
is
used
entropy. The latter serves as the driving force for the structhe nonergodic2013-14]
glassy states and
theture
limit
of in
infinite
dimension
change
the glassyspatial
phase. The
energy cost to[Kurchan-Zamponi-Parisi,
match the
panel).
boundaries between the different density configurations gives
riseof
to fluctuations
the energy barriers.
fluctuations
of the configuraRole
in The
finite
dimensions
????????????
tional entropy give rise to the energy barrier fluctuations.
dom hard sphere packing29 , in
e constraint forces are finite, but the transition
e at non-zero s. This result means that the
Thewith
importance
of adding
fields/constraints...
system
finite short-ranged
forces
of interat exhibits long relaxation times and dynamic
This is a way to enlarge the phase diagram, possibly generating
without equilibrium
thermodynamic
transitions
bona fide transitions.
ng theory
random
first-order
theory
[3]
=> [5]
Oneand
can check
qualitative
behavior
and universality
classes
ical non-equilibrium
glass
transition.
The sowithout recourse to
adjustable
parameters.
For instance: In KCM’s (dynamical facilitation), add a source s
conjugate to the mobility field
T
(b)
T
Criticality
Active
Inactive
Inactive
Criticality
s
s
neric space-time
phase
diagram
KCMs
[9].constraint,
There isElmatad
a firstet al., PNAS, 2010]
Non-equilibrium
phase
diagramfor
T versus
s [soft
y that occupies the s = 0 axis, separating an active fluid phase
The importance of adding fields/constraints...
Similar spirit yet quite different: Adding equilibrium/
thermodynamic constraints induces new phase transitions and
critical points in the mean-field theory of glass-forming liquids
and structural glasses (random-first order transition scenario)
0.09
Field ε conjugate to the overlap
versus T
[Franz-Parisi, PRL 1997]
εc,Tc
T versus the concentration of
randomly pinned particles
[Cammarota-Biroli, PNAS 2012]
0.75
0.08
0.07
(co,To)
0.7
0.05
T
Coupling !
0.06
0.04
0.65
0.03
Td
0.01
‘
f
cK(T)
0.02
0
cd (T)
cd(T)
0.6
0.6
0.62
0.64
0.66
0.68
0.7
Temperature T
0.72
0.74
0.76
0.78
TK
0
0.05
c
0.1
0.15
FIG. 1: Phase diagram for the spherical 3-spin model with spins pinned from an equilibrium con
Contrasting dynamical-facilitation and
configurational-entropy scenarios
Diagram T vs field ε conjugate to the overlap (sketch from
mean-field theory): 3 regions of interest I, II, III
T
Tc, εc
LOW
Td
I
: Spinodal(s)
III
MCT
TK
RFOT
0
: First-order transition
II
Tc, εc : Terminal critical point
Td
HIGH
OVERLAP
: Dynamical (MCT) singularity
TK : Thermodynamical RFOT
ε
Contrasting dynamical-facilitation and
landscape scenarios
Obstacles for a direct comparison:
• Dynamical facilitation and KCM’s have trivial thermodynamics.
• The landscape/configurational-entropy approach is essentially
formulated at the mean-field level.
=> Introduce a mean-field limit in the dynamical facilitation
models.
=> Map back KCM’s on models without kinetic constraints and
with nontrivial static (multi-spin) interactions.
=> Include fluctuations in the landscape approach.
Contrasting ...: Region I near the (meanfield) dynamical glass transition
• Quasi-equilibrium description of the beta-regime near the MCT
transition (Td): Critical phenomenon can be described as a cubic field
theory in a random source (RFIM spinodal, dsup=8) for the fluctuations of
the overlap between configurations [Franz, Parisi and coll., 2011-2013]
• KCM’s on Bethe lattice and regular random graphs [a mean-field limit for
dynamical facilitation]: The transition has the same character as MCT
(discontinuous, relations between exponents, etc).
• Yet, the order parameters for the quasi-equilibrium description are very
different: ‶nonlocal mapping″ [Foini et al, J. Stat. Mech. 2012, Franz-Sellitto, J.
Stat. Mech., 2013]
• Beyond mean-field: seemingly very different behaviors....
+ Digression on KCM’s and bootstrap percolation on hyperbolic lattices
[Sausset et al., J. Stat. Phys., 2010]
Contrasting ...: Region II near the terminal
critical point in a nonzero applied source
coupled to the overlap
Mean-field RFOT-like phase diagram
T
Td
approach: Add fluctuations to the
mean-field description -> Biroli,
Cammarota, Tarzia, GT, PRL, 2014
Tc, εc
II
• For the dynamical-facilitation
TK
0
• For the landscape/thermodynamic
ε
approach: Consider spin models
with plaquette interactions (duality
with KCM’s) -> Garrahan, PRE
2014
The overlap between configurations as an
order parameter: Sketch of formalism (I)
• Liquid (N,V,T): Hamiltonian H[rN], with rN = atomic configuration.
Pick an equilibrium reference configuration r0N and define the
microscopic overlap between rN and r0N as
N
j
i
�
r
+
r
0
N
N
j
i
q̂x [r , r0 ] =
f (|r − r0 |)δ(x −
)
2
i,j=1
where f (r) = 0 for r > a with a << atomic diameter
• Define an overlap field p(x) and its associated effective hamiltonian:
�
N
dr
N
N
N
−βH[rN ]
S[p; r0 ] = − log
δ[p − q̂[r , r0 ]] e
N!
• The correlation functions are generated by the functional W[ε; r0N]:
W [�; r0 N ] = − log
�
Dp e−S[p;r0
N
]+
�
x
�(x)p(x)
Sketch of formalism (II)
• One is interested in the cumulants of W[ε; r0N], S[p; r0N], etc, when
averaged over r0N
=> The reference configuration r0N plays the role of a ‶quenched
disorder″.
• Goal: To derive an effective theory for the overlap field p(x) =>
approximate form for S[p; r0N] or its cumulants by integrating out
short-distance fluctuations (through liquid-state theory, general
Landau-like arguments, etc).
• To generate the cumulants: Introduce replicas raN, a=1,...,n, each
coupled to a distinct field εa
=> Now, one can define replicated overlap fields pa(x) between raN
and r0N and overlap fields qab(x) between raN and rbN.
Sketch of formalism (III)
• Generically, the effective hamiltonian for the overlap fields pa(x) is
obtained from
e−Srep [{pa }] = exp(−
n
�
a=1
S[pa ; r0 N ]) ∝
� �
ab�=
Dqab e−S[{pa ,qab }]
0.2
where S[{pa,qab}] is obtained by integrating
out short-distance fluctuations (through
liquid-state theory, general Landau-like
arguments, etc) and at the Landau level
for pa=qab =Q has the typical form
illustrated here:
0.15
0.1
Td>T> TK
0.05
0
-0.05
0
Q
• Crucial: The fields pa can be critical (near the terminal critical
point) but never the fields qab due to the external field provided by
the coupling with the pa’s !!!!!!
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
FIG. 7. The annealed potential VA (q) as a function of q for p = 4 and various values of T , in o
from top to bottom. For p = 4 one has Tc = 0.503 and Ts = 0.544. The curves are normalized
IV. REAL GLASSES
A. The Model
We have tested the ideas presented in this note for binary fluids. Although we did no
of the phase transition line, we found evidence for a first order transition in presence of
in the annealed and in the quenched case. The model we consider is the following. We
of different sizes. Half of the particles are of type A, half of type B and the interaction a
! " σ(i) + σ(k) #12
,
H(x) =
rthier
Relevance of the 2-state, first-order-like,
effective description after integrating out
short-distance fluctuations
Université Montpellier 2, Montpellier, France
1, 2013)
rational entropy in supercooled liquids, based
ize a system in a single metastable state. We
show that the temperature dependence of the
broad temperature range. This allows a new
time to configurational entropy. Finally we
onfigurational entropy.
Evidence from finite-size computer simulations
=> Landau-like potential with cubic term
ε
8
1.6
procedure
in a similar problem). Indeed, in the umbrella sam1
7
pling approach, one needs to compute p(q) to obtain the over1.4
1
500
lap
potential
[W
(q)
=
−
log
P(q)],
which
is
much
costlier
in
6
0.8
N = 64N
250
time and less precise than only recording the central point of
1.2
124
5
the distribution and performing a line integral with it.
62
1
4
0.6
We run simulations at 0.3 ≤ ! ≤ 0.57 at 21 values of q
0.555 0.56 0.565 0.57
T =between
13
φ
evenly spaced
0 and 1. In addition, we consider five
3
(100
0.4 for N = 500 7and 50 for N = 250 for 0.54 ≤ ! ≤ 0.57) re2
alizations of each experiment, and results presented here are
1
averaged over all these samples. The set up is the follow0.2
0
ing. We start with a thermalization of each ofΣthe two repli0.3
0.5
0.55
cas. We consider thermalizations of "0 elementary MC steps
-1
0.4
0.53
0.57
0
(EMCSs)[39],
defining EMCS as N attempts at ordinary individual random particle moves. Only once the initial config0
0.2
0.4
0.6
0.8
1
q
0 are thermalized,
0.2
0.4we run0.6
0.8 simulations
1 at
urations
the tethered
QIn order to ensure the thermalfixed q using the weight (11).
FIG. 1: (Color online) Derivative of the replica potential, Ŵ % (q) ob0.8
ization,
we
consider
two
alternative
experiments.
On
the
one
Effective ‶annealed″ potential (left, Berthier,tained
PREas2013)
and its
derivative
(right,
(9)
at
different
packing
fractions.
Inset: Phase diagram
hand, we move sequentially from q = 1 to 0 in steps of 0.05,
(# , ! ) extracted
fromplotted
the Maxwell
construction
different system
PRE 2014)
forprocedure.
a glass-forming
liquid
model
versus
the for
overlap
and Parisi-Soane,
on the other, we consider
the reverse
At each
sizes. The errors are computed using the jackknife method.
0.6
value
q we remaintemperatures
using thesystem
latter size
"int = 0.1"0 EMCS,
atofdifferent
forand
a small
0.05"0 EMCS in the analysis. We completely avoid hystereW’(q)
βV (Q)
3
Results: Universality class
• If present, the terminal critical point is in the universality class of
the RFIM (random field Ising model). Effective (disordered)
hamiltonian for the overlap fluctuations ϕ(x)=p(x)-pc (up to ϕ4):
� �
�
r3
r4
r2
2
3
4
S[φ] =
c[∂φ(x)] + φ(x) + φ(x) + φ(x)
2
3!
4!
x
� �
�
δr3 (x)
δr2 (x)
2
3
+
− h(x)φ(x) +
φ(x) +
φ(x)
2
3!
x
2
h(x) random field (most relevant).Expressions for the parameters
• When one proceeds instead to an ‶annealed calculation″ (just two
coupled replicas): universality class of pure Ising model (no
quenched disorder)
[Confirmed via a different method by Franz-Parisi: J. Stat. Mech.,2014]
Results: Existence of a
critical point in D=3?
One can evaluate the effective parameters of the RFIM description
for specific models:
The transition exists if the ratio of the effective disorder strength
to the effective interaction (surface tension) is small enough =>
accounts for long-distance fluctuations.
From simulations of the 3D RFIM, the critical ratio ≃ 2.3.
=> The Wolynes et al Landau theory for the fragile glass-forming
liquid o-TP: Ratio ≃ 0.94, the transition seems to survive.
=> The disordered p-spin model with p=3, Ratio ≃ 4.48, no
transition [cf. also Cammarota et al, PRB 2013]
ated thermodynamic glass transition in the uncoupled system. We
nt interpretation is possible. We consider the triangular plaquette
m which displays (East model-like) glassy dynamics in the absence
w that when two replicas are coupled there is a curve of equilibrium
ses of small and large overlap, in the temperature-coupling plane
an exact temperature/coupling duality of the system) which ends
n the limit of vanishing coupling the finite temperature transition
system is in the disordered phase at all temperatures. We discuss
Annealed computation (2 coupled replicas)
lations in light of this result.
The terminal critical point in plaquette spin
models
for the 2D triangular
plaquette spin model (Garrahan, 2014)
2
3] have studied
ties of (reasonself-dual
line
two copies (or
1.5
e studies have
low
sition between
overlap
critical
T 1
ween the repliendpoint
be first-order
high
ure (or density,
0.5
overlap
coupling plane
h a behaviour
0
0
0.5
1
1.5
2
heoretical pre!
transition thenumerical Critical
reendpoint
universality
of thetriangular
2D 4-state
FIG.
1. Exact =
phase
diagram of class
two coupled
plar the existence
models.
The(ν=2/3,
temperature,
T , and
�, are
modelquette
≠ 2Dglass
Ising
model
α=2/3
vscoupling,
ν=1, α=0).
ss state in the
in units of the energy constant of the model, J. The full line
No
finite-T
transition
when
ε=0.
issue, since a
is a curve of first-order transitions from a phase with low over-
Potts
Contrasting ...: Region III near the meanfield static transition (RFOT)
• Interesting per se and to compare with the dynamic facilitation
picture: in the latter, the presence of localized defects destroys
any finite-T transition in the absence of constraints.
• Difficulty for deriving an effective theory for the overlap with a
reference configuration: All overlap fields are now affected by
the presence of static point-to-set correlations (that precisely
diverge in mean-field at the RFOT TK) => more tricky to integrate
out the qab’s for fixed pa’s.
• Work in progress: indication for a RFIM with a renormalized
effective external field (configurational entropy) and
antiferromagnetic interactions on top of the ferromagnetic ones.
Conclusion
• Contrasting the dynamical facilitation and the landscape
approaches => Enlarging the phase diagram by applying
fields/constraints.
• Transitions in the enlarged diagram give info on the
structure of the configurations and metastable states.
Describable by a 2-state Ising theory or not? Consequences
for the nature of the defects?
• Need more systematic investigation in the dynamical-
facilitation approach: quenched case, 3D, more models...
• More easily testable in simulations of liquid models.
24