Dynamic relaxation of a liquid cavity under amorphous boundary conditions.

Abstract

The growth of cooperatively rearranging regions was invoked long ago by Adam and Gibbs to explain the slowing down of glass-forming liquids. The lack of knowledge about the nature of the growing order, though, complicates the definition of an appropriate correlation function. One option is the point-to-set (PTS) correlation function, which measures the spatial span of the influence of amorphous boundary conditions on a confined system. By using a swap Monte Carlo algorithm we measure the equilibration time of a liquid droplet bounded by amorphous boundary conditions in a model glass-former at low temperature, and we show that the cavity relaxation time increases with the size of the droplet, saturating to the bulk value when the droplet outgrows the point-to-set correlation length. This fact supports the idea that the point-to-set correlation length is the natural size of the cooperatively rearranging regions. On the other hand, the cavity relaxation time computed by a standard, nonswap dynamics, has the opposite behavior, showing a very steep increase when the cavity size is decreased. We try to reconcile this difference by discussing the possible hybridization between mode-coupling theory and activated processes, and by introducing a new kind of amorphous boundary conditions, inspired by the concept of frozen external state as an alternative to the commonly used frozen external configuration.

Full text

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arXiv:1006.3746v2 [cond-mat.dis-nn] 6 Dec 2010
Finite-size scaling under amorphous boundary conditions unveils cooperative
rearrangement in glass-forming liquids
Andrea Cavagna,1,2Tom´ as S. Grigera,3,4and Paolo Verrocchio5,1,6, ∗
1Istituto Sistemi Complessi (ISC-CNR), UOS Sapienza, Via dei Taurini 19, 00185 Roma, Italy
2Dipartimento di Fisica, Universit` a “Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy
3Instituto de Investigaciones Fisicoqu´ ımicas Te´ oricas y Aplicadas (INIFTA) and Departamento de F´ ısica,
Facultad de Ciencias Exactas, Universidad Nacional de La Plata, c.c. 16, suc. 4, 1900 La Plata, Argentina
4CCT La Plata, Consejo Nacional de Investigaciones Cient´ ıficas y T´ ecnicas, Argentina
5Dipartimento di Fisica, Universit` a di Trento, via Sommarive 14, 38050 Povo, Trento, Italy
6Instituto de Biocomputaci´ on y F´ ısica de Sistemas Complejos (BIFI), Spain
The growth of cooperatively rearranging regions was invoked long ago by Adam and Gibbs to
explain the slowing down of glass-forming liquids.
though, turned them into one of the longest-sought chimeras in condensed matter physics. The lack
of knowledge about the nature of the growing order complicates the definition of appropriate cor-
relation functions. One option is the point-to-set correlation function, which shows that the spatial
rule of amorphous boundaries grows when lowering the temperature. By measuring the equilibration
time of a liquid droplet bounded by amorphous boundary conditions in a low temperature model
glass-former, we show that the point-to-set correlation length is naturally identified with the size of
the cooperatively rearranging regions. The equilibration time increases with the size of the droplet
and saturates to the bulk value when the droplet outgrows the point-to-set correlation length. Our
results can be interpreted within the Random First Order theory of the glass transition, provided
that the surface tension between different amorphous states is allowed to fluctuate.
The absence of experimental confirmations,
PACS numbers:61.43.Fs, 62.10.+s,64.60.My
It is common wisdom that the spectacular slowing
down of supercooled liquids at low temperature is caused
by the growth of a correlation length of some sort. The
underlying idea is that of cooperativity: at lower temper-
atures, larger regions (termed cooperatively rearranging
region) must move together in order to fully relax [1].
Unfortunately, the standard tools used in critical phe-
nomena to detect a growing correlation length fail in
glass-forming liquids, as it is not at all clear a priori
what the order parameter should be. No obvious do-
main or structure can be observed in a low temperature
liquid to distinguish it from a high temperature one. If
order is growing in glass-formers, it must be some sort of
amorphous order, and the corresponding order parameter
must be nonstandard.
Recently, some progress has been achieved by using
amorphous boundary conditions (ABC) [2–4]. The idea
goes as follows [2]. Consider a low temperature equi-
librium configuration of a liquid and freeze all particles
outside a certain region. This region (or cavity) is then
let free to evolve and thermalize, subject to the pinning
field produced by all the frozen particles surrounding it.
Clearly, the smaller the region the stronger the effect of
the pinning field, hence keeping the region in a very re-
stricted portion of its own phase space. The idea, then,
is to check how large the region must be to emancipate
from the boundary conditions, i.e. to regain ergodicity
and thermalize into a state different from the sorround-
ing one. The advantage of this method is that the sys-
tem chooses its own definition of ‘order’ by means of the
amorphous boundary conditions, and we do not need to
have any a priori knowledge of the nature of such order.
Practically speaking, the procedure amounts to measure,
as a function of the size R of the region, the correla-
tion between the original region’s configuration (that of
the frozen surrounding) and that achieved after the re-
gion has equilibrated subject to the amorphous boundary
conditions. This quantity is called point-to-set correla-
tion function [5, 6], q(R), and it has shown an interest-
ing feature [3, 4]: its decay length-scale, ξs, increases on
lowering T. Regions smaller than ξs cannot relax com-
pletely, even given infinite time, due to the presence of
the pinning ABC.
Even though it seems quite natural to interpret the
point-to-set lengthscale ξs as the bona fide correlation
length of the system, the numerical evidence gathered
up to now does not prove unambigously that ξs is the
size of the cooperatively rearranging regions (CRRs), i.e.
the lenghtscale relevant for the dynamics. In particular,
we do not know how long it takes for a cavity that can
relax to actually do so. In fact, although a CRR cannot
be smaller than ξs, a region of this size could still relax
differently from the bulk (as is the case for instance in
the square plaquette model (SPM) [7]). Another critical
issue is whether the cavity equilibration time approaches
the bulk equilibration time from above or from below.
Theories based on kinetic constraints, where the slow-
down is ascribed to the low density of mobile regions
(termed excitations or defects) [8, 9], seem to imply that
the bulk equilibration time should be approached from
above (i.e. small systems relax more slowly). This kind
of behaviour has in fact been observed for the equilibra-

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2
0
0.2
0.4
0.6
0.8
1
10 100 1000
t
10000 100000
C(t)
R
0.98
1.68
2.27
2.87
3.62
4.57
7.25
10.95
bulk
FIG. 1: Dynamical correlation function C(t;R) for a few rep-
resentative sizes R at T = 0.89TMC. It is also shown (solid
line) the decay of the dynamic correlation function for the case
with periodic boundary conditions in a bulk system (N = 4096
particles).
tion time of a cavity with frozen amorphous boundary
conditions in the SPM, a model whose dynamics is in-
deed ruled by diffusing defects [7]. On the other hand,
according to a cooperative view of the glass transition,
and in particular to RFOT [10–13], one expects the cav-
ity equilibration time to have the opposite behaviour:
systems smaller than the cooperative lengthscale have to
relax collectively, so that large systems should relax more
slowly than small systems. To clarify these issues we an-
alyze here the dynamic relaxation of a cavity subject to
amorphous boundary conditions.
We perform Monte Carlo simulations of a 3-d soft-
spheres system [26]. In particular, we analyze the con-
nected auto-correlation function of the overlap q(t;R).
The overlap measures the correlation between the run-
ning configuration and a reference one at t = 0 [27]. The
asymptotic value of the overlap, q(R) ≡ ?q(t → ∞;R)?,
averaged over many realizations of the boundary condi-
tions, is the point-to-set correlation function [2–4, 6, 13].
The dynamical fluctuations of the overlap are measured
by the following connected correlation function [28],
C(t;R) =?(q(t0+ t;R) − q(R))(q(t0;R) − q(R))?
?
We have studied the cavity at five temperatures, of which
the lowest one is below the mode-coupling temperature
TMCof the system. Fig. 1 shows the dynamical correla-
tion function C(t;R) at various values of R for our lowest
temperature T = 0.89TMC. We stress that for those val-
ues of R such that the order parameter q(R) ?= 0, ergod-
icity is broken [4]. In this case the connected correlation
function (1) describes the equilibrium dynamics within a
local region of the cavity’s phase space, which does not
(q(t0;R) − q(R))2?
.
(1)
1
10
100
1000
10000
100000
1 2 3 4 5 6
R
7 8 9 10 11
τ
ξs
ABC
bulk
ψ=1
FIG. 2: Equilibration time of a region subject to frozen amor-
phous boundary conditions, as a function of the region’s size
R. The overlap is computed only at the centre of the sphere.
The kink between the growth and the saturated regime oc-
curs at R close to ξs = 3.82 ± 0.46, which is the point-to-set
lengthscale for T = 0.89TMC, as measured in [4]. Also shown
are the curves τ(R) ∼ eRψwith ψ = 1 (dotted line) and the
bulk equilibration time (dashed line).
coincide with genuine structural relaxation of the cav-
ity. In Fig. 2 we report the equilibration time τ(R) for
T = 0.89TMC [29] . Three features of this curve stand
out: i) the equilibration time saturates for large R to a
value independent of the cavity size; ii) the equilibration
time grows with R, so that saturation occurs from be-
low; iii) growth and saturation are separated by a rather
sharp kink at a well-defined value of R. The first fact
is quite intuitive: the effect of the boundary conditions
is expected to fade away for large R, so that τ(R) must
eventually reach its bulk value, which is exactly what
it does with a very sharp kink. The remarkable point
is that the kink occurs at R ∼ ξs, i.e. at the point-to-
set correlation length measured in Ref. [4]. For R < ξs
the whole region is correlated, since the overwhelming ef-
fect of the amorphous border breaks the ergodicity. For
R > ξs, just as the effect of the border fades away, the
region is able to decorrelate by breaking up into smaller
correlated sub-parts. In this regime relaxation factorises.
In Fig. 3 we plot logτ(R) (normalized to its bulk value)
for all the temperatures studied.
haviour is the same: growth, kink, and saturation. More-
over, we see that the kink moves to larger R upon lower-
ing the temperature, as ξsbecomes larger at smaller T.
In the inset of Fig. 3 we report as a function of tempera-
ture the value R where τ(R) has the kink and the point-
to-set correlation length ξsobtained in [4]. Although it
is hard to say anything certain about their exact T de-
pendence, the two lengthscales seem to track each other
quite nicely. We therefore conclude that the point-to-set
lengthscale ξs has indeed a dynamical meaning, as the
typical size of a CRR.
For each T the be-

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3
0.0001
0.001
0.01
0.1
1
1 1.5 2 2.5 3 3.5 4 4.5
τ(R)/τBULK
R
2
3
4
0.2 0.3 0.4 0.5
ξ
T
τ(R)
q(R)
FIG. 3: Equilibration time τ, normalized to its bulk value,
vs.
R for all temperatures.
0.89TMC,1.1TMC,1.5TMC,2.1TMC.
static correlation length is compared to the position of the
kink of τ(R) as a function of T.
From right to left:
Inset:
T =
the point-to-set
According to random first-order theory (RFOT),
whether or not a region of radius R relaxes depends on
the balance between the surface tension Y that devel-
ops when that region actually rerranges and the config-
urational entropy Σ unleashed by the rearrangement: if
Y > TΣRd−θ(d is the dimension of the system, θ is
the surface tension —or stiffness— exponent) the sur-
face cost is larger than the entropic gain and the region
does not rearrange. On the other hand, if Y < TΣRd−θ
the entropic gain outweighs the surface energy cost and
the region has a thermodynamic advantage to rearrange.
The minimal rearranging size where entropy and surface
tension balance, ξ = (Y/TΣ)1/(d−θ), is the mosaic corre-
lation length of RFOT.
Therefore, within RFOT a cavity with amorphous
boundary conditions of radius R < ξ has broken ergod-
icity, and it can only explore the state imposed by the
boundary conditions [2]. In this regime the equilibration
time is the time needed to explore that one state, which
is roughly equal to the β-relaxation time, τ(R) ∼ τβ(we
neglect in this analysis a possible dependence of τβ on
R due to the extended nature of the excitations related
to β-relaxation [14]). For R > ξ, instead, rearrangement
occurs and ergodicity of the cavity is restored. In this
regime the region is larger than the minimal rearranging
size, so that relaxation factorizes: different subregions of
size ξ will rearrange independently from each other, and
the equilibration time will be equal to its bulk value, i.e.
τ(R) ∼ τ0exp(ξψ/T), where τ0 is an Arrhenius prefac-
tor and ψ is the exponent regulating the barrier growth.
Hence, this sharp RFOT description, within which the
surface tension has just one value, Y , predicts a step-like
jump of τ(R) at R = ξ, from τβup to τ0exp(ξψ/T). This
is clearly not what we observe.
However, it has been pointed our that for the typical
temperatures and sizes studied in simulations and exper-
iments both surface tension [4] and configurational en-
tropy [11] fluctuations are relevant. At the practical level
disantangling the two effects is hard, and given that large
surface tension fluctuations have been reported [15, 16],
a generalized version of RFOT that incorporates only
surface tension fluctuations seems reasonable [4]. If the
surface tension fluctuates so must the equilibration time,
τ(R,Y ) ≈
?τβ
τ0exp
Y > TΣRd−θ
Y < TΣRd−θ.
?
1
T(Y/TΣ)
ψ
d−θ?
(2)
Let us write the surface tension distribution as P(Y/Yc),
where Yc is no longer the unique value of the surface
tension, but rather a scale. In this case we can still define
a typical mosaic correlation length, ξ = (Yc/TΣ)1/(d−θ)
[4]. The macrosopic equilibration time will be given by
an average over Y of the time in (2),
τ(R) = τβ
?∞
?TΣRd−θ
TΣRd−θP(Y/Yc)dY +
+τ0
0
P(Y/Yc) exp
?1
T(Y/TΣ)
ψ
d−θ
?
dY.
(3)
The first term in (3) corresponds to regions surrounded
by large surface tension, which do not rearrange, and it
is negligible (it equals at most τβ). The second term
corresponds to the low surface tension regions that do
rearrange, and at low temperatures this term is large.
Clearly, if P(Y ) = δ(Y − Yc) we recover the step-like
behaviour of τ(R) described above. If, on the other hand,
P(Y ) is broad, then the result is nontrivial. At low T the
second integral in (3) is dominated by the exponential
and we can use the saddle-point approximation. There
are two regimes: A small R regime, for TΣRd−θ< Yc,
where the saddle-point coincides with the right edge of
the integration domain, YSP= TΣRd−θ. In this case we
have,
τ(R) ∼ τ0P(r) eRψ/T
,TΣRd−θ< Yc, (4)
where r ≡ (R/ξ)d−θis a sort of renormalized radius.
The large R regime, on the other hand, is defined by
TΣRd−θ> Yc. In this regime the saddle point is an R-
independent value close to the peak of the distribution
[30]. This gives the expected saturation of the equilibra-
tion time for large sizes,
τ(R) ∼ τ0P(1) eξψ/T
,TΣRd−θ> Yc.(5)
The final result is:
T logτ(R) ∼
?
Rψ
ξψ
for R < ξ ,
for R > ξ ,
(6)

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4
which is exactly the behaviour we find (Fig. 2). This
agreement between soft RFOT and numerical results
seems to indicate that the correlation length ξsthat we
extract both from the point-to-set correlation function
and from the equilibration time, is the same as the mo-
saic correlation length ξ of RFOT. This lengthscale is
the typical size of the cooperatively rearranging regions,
which dominate activated dynamics at low temperatures.
Note that through (6) the growth of logτ(R) for R < ξ
can be used to estimate the value of the barrier exponent
ψ. Former investigations [16, 21] suggest ψ = 1, so in
Fig. 2 we show the best fit with this value of the ex-
ponent, which seems compatible with the data. On the
other hand, a nonlinear fit for R < ξsgives ψ = 1.41(2).
Hence, the value ψ = 1.5 proposed in Ref. [10] is equally
compatible with the data.
The fact that the cavity equilibration time under amor-
phous boundary conditions grows with R is, as we have
seen, natural in the context of cooperative rearrange-
ment, but somewhat harder to explain in the context
of theories based on diffusing defects. If the slowdown is
caused by a low density of defects, one would expect the
equilibration time to decrease for increasing size of the
region, and thus to saturate from above to the bulk value.
This is exactly what happens in the SPM model, where
defects play a major role in the relaxation of the system
[7]. Given our results, it seems that in the quest for the
main relaxation mechanism in low temperatures glass-
forming liquids, activated cooperative rearrangement is
the most natural candidate for the job.
We thank L. Berthier, G. Biroli, J.-P. Bouchaud,
C. Cammarota, L. Cugliandolo, S. Franz, J.P. Garrahan,
G. Gradenigo, R.L. Jack, A. Heuer, W. Kob, M. Mezard,
G. Parisi, G. Tarjus, M. Wyart and F. Zamponi for sev-
eral important remarks, and ECT* and CINECA for
computer time. The work of TSG was supported in part
by grants from ANPCyT, CONICET, and UNLP (Ar-
gentina).PV has been partly supported through Re-
search Contract Nos.FIS2009-12648-C03-01,FIS2008-
01323 (MICINN, Spain).
Appendix A: The dynamical decay of the overlap
We use the correlation function C(t;R) to emphasize
the role of q(R) as an order parameter. However, the
overlap q(t,R) is itself a time correlation function, and
its time decay is in fact sufficient to characterize the
relaxation dynamics of the system. In Supplementary
Fig. 4 we show the time decay of the overlap q(t,R) at
various values of R, computed by using a standard Mon-
tecarlo dynamics (without swap) to make the compar-
ison with other studies more straightforward. Definin-
ing τ(R) through the decay of the connected overlap,
q(t,R)−q(R), gives results similar to those obtained from
C(t;R). In particular we see from Supplementary Fig. 4
a very clear growth of τ(R) with R.
It is interesting to compare Supplementary Fig. 4 with
Fig. 8 of Ref. [7], where a cavity with amorphous bound-
ary conditions was studied in a system whose dynamics
is dominated by defects (SPM). In both cases the asymp-
totic overlap q(R) decreases with increasing R. However,
the behaviour of the equilibration time, i.e. the time em-
ployed by q(t,R) to reach its asymptotic value q(R), is
exactly the opposite: in the defect-ruled model studied
in [7] the equilibration time decreases when R increases,
while in the glass-former studied here the equilibration
time increases when R increases. The comparison be-
tween these two figures is one of the starkest visualiza-
tions of the difference between the two cases.
Appendix B: Comparison with finite-size scaling
under periodic boundary conditions
As we have seen, we find that the equilibration time of
the cavity with amorphous boundary conditions reaches
its bulk value from below. Several finite-size scaling stud-
ies of similar glass-forming liquids under periodic bound-
ary conditions (PBC) [22–24] find an equilration time
approaching its infinite-volume limit from above (even
though an approach from below has been reported also
with PBC for the same model at T = 0.89TMC in
Ref. [25], where the auto-correlation time of energy was
addressed). The reader could wonder about this differ-
ence. First of all, we note that the behaviour we find has
nothing to do with the fact that we use a swap Monte
Carlo dynamics. We have already shown in Supplemen-
tary Fig. 4 how the time defined by the standard overlap
without swap increases with R. Here, in Supplementary
Fig. 5, we show the correlation function C(t;R) calcu-
lated without swap dynamics, and again we find an un-
ambiguous increase of the equilibration time with the size
of the cavity. Hence, the qualitative feature of the ap-
proach from below has nothing to do with the specific
dynamics.
Secondly, we note that in the case of PBC the finite size
effects on the equilibration time are considerably weaker
than in the ABC case. In our system the time increases
by more than three orders of magnitude, for an increase
of R by a factor 4. In the PBC study of [24] the equili-
bration time changes by barely one order of magnitude
by changing the linear size L by a similar factor 4. Com-
parable results are obtained in [22]. Hence, it seems that
the physical origin of the effect of varying the system size
on the relaxation time is different for PBC and ABC.
In fact, we believe that PBC and ABC are two physi-
cally different cases. Consider a low T liquid. As we have
explained before, in the case of ABC, when R < ξs the
region simply does not decorrelate: the order parameter
q(t;R) does not decay to zero even for large times, and
ergodicity of the region is broken. Under these conditions

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5
the equilibration time that one measures from the con-
nected correlation function, i.e. the time needed to relax
to the nonzero asymptotic value q(R), is quite short: it is
the relaxation time within a single state. This is certainly
not the case for a PBC system, where (at equilibrium)
ergodicity is not broken: the correlation function must
always go to zero and, if T is small, it will take a long
time to do so, even for relatively small L.
In terms of phase space, we can rephrase this argument
in the following way. For ABC the cavity with R < ξsis
confined within a finite portion of its own phase space.
This means that, not only the volume of the phase space
decreases when R decreases, but also that the fraction of
phases space accessible to the cavity decreases with R.
Indeed, this fraction is proportional to 1 − q(R). In the
case of PBC, when we decrease L the size of the phase
space decreases (as for the cavity), but the fraction of
accessible phase space is always 1. Hence, the physical
set-up of ABC and PBC is radically different.
Even though we cannot exclude that in some temper-
ature and size regimes PBC and ABC may give similar
results (for example, it seems reasonable that very small
PBC systems —few particles— should relax quickly),
from what we have explained above we do not expect
PBC and ABC to always give the same qualitative be-
haviour of the equilibration time vs. size.
Appendix C: Speculations about the high
temperature behaviour
It has been argued in [13] that at temperatures higher
than the mode-coupling temperature TMC, where the dy-
namics is dominated by unstable directions in the phase
space, rather than by activated barrier crossing, τ(R)
should show the opposite trend, approaching its bulk
value from above. The idea is that in a smaller cavity
the frozen boundary conditions stabilize unstable sad-
dles (which play the role of defects), thus increasing the
non-activated relaxation time. On the other hand, the
activated relaxation time goes the opposite way.
If this picture is true, we may speculate that the mea-
sured relaxation time is the smallest of the two, so that
the most sizeble effect of the crossover between the two
mechanisms around TMCshould be a non-monotonic be-
haviour of τ(R): the relaxation time should grow with
R, reach a maximum, and slope down to its bulk value
for larger R.
To unveil such behaviour (which is probably a rather
weak feature of τ(R) anyway), one should use a dynam-
ics sensitive to the crossover between the high-T non-
activated mechanisms and the low-T activated one. How-
ever, the price one pays by doing that is to mix the
two mechanisms, thus making the determination of the
lengthscale where τ(R) saturates to its bulk value, quite
blurred. Here, on the other hand, we wanted to focus
on the determination of ξsfrom the saturation of τ(R).
The swap Montecarlo dynamics that we used, by heavily
suppressing energy barriers, is intrinsically insensitive to
the crossover around TMC[25] and therefore it is ideal for
the task. We are blind to any non-monotonous behaviour
of τ(R), but we are rewarded by a quite sharp determi-
nation of ξs, which was our main point in this work.
Clearly, though, the analysis of the crossover between
non-activated to activated dynamics from the point of
view of the lengthscale, is something that deserves fur-
ther investigation.
Appendix D: An unexpected inequality
In order to have a finite bulk equilibration time, we
need the second integral in equation 3 of the article to
be finite for R → ∞. Therefore P(Y/Yc) must decay
sufficiently fast to suppress the Arrhenius factor. For this
reason, YSP∼ Yc. If we make the reasonable assumption,
P(Y/Yc) ∼ e−(Y/Yc)ν,Y ≫ 1, (7)
we must have,
ν ≥
ψ
d − θ.
(8)
Since the distribution P(Y ) implies an equivalent distri-
bution of the rearranging regions’ size, P(R), inequal-
ity (8) means that P(R) must decay fast enough to sup-
press the growth of the equilibration times for large R.
This is reasonable. In [4] it was shown that the expo-
nent ν is related to the anomaly exponent ζ that rules
the nonexponential decay of the point-to-set correlation
function q(R),
q(R) ∼ e−(R/ξ)ζ, (9)
with
ζ = ν(d − θ),ζ ≥ 1.(10)
where θ is the surface tension exponent. This leaves us
with the inequality
ζ ≥ ψ.(11)
On increasing the temperature the anomaly ζ must go to
1, as the point-to-set correlation function q(R) becomes
a pure exponential [4]. If ψ is temperature-independent,
relation (11) then implies,
ψ ≤ 1. (12)
We note that the value ψ ∼ 1 previously reported in [16]
satisfies (12). Of course, if we allow ψ to depend on T
(as ζ does), then there would be no reason for(12) to be
valid in general, whereas(11) would still hold.

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0
0.2
0.4
0.6
0.8
1
1 10 100 1000 10000 100000 1e+06
qc(t)
time
R=1.06
R=1.68
R=2.88
R=4.57
FIG. 4: Time dependence of the overlap q(t;R) calculated
with a standard Montecarlo dynamics (no swap), at T =
1.55TMC, for various vlaues of R. The growth of the equi-
libration time τ(R) with R is unambiguous.
0
0.2
0.4
0.6
0.8
1
10 100 1000 10000
C(t)
t
R
1.68
2.27
2.87
3.62
4.57
5.76
7.25
FIG. 5: Correlation function C(R,t) calculated with standard
Monte Carlo dynamics (no swap), at T = 1.1TMC, for various
vlaues of R. The growth of the equilibration time τ(R) with
R is clear also in this case.
∗Electronic address: paolo.verrocchio@unitn.it
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[26] Our glass-former is a 3-d soft-spheres binary mixture [17]
with parameters as in Ref. [4]. Simulations were done
with Metropolis Monte Carlo with particle swaps [18]
to equilibrate at low enough temperatures, but we also
ran some non-swap simulations to check that our results
did not depend on the algorithm (see Appendices A and
B for the results). The mode-coupling temperature [19]
for this system is TMC =0.226 [20]. Our largest system
has N =16384 particles in a box of length L =25.4. We
ran simulations at T =0.482,0.350,0.246,0.214,0.202, first
with periodic boundary conditions to generate a set of
equilibrium configurations. We then run the amorphous
boundary simulations by picking an equilibrium configu-
ration and artificially freezing all particles but those oc-
cupying a spherical cavity of radius R =1.06, 1.68, 1.92,
2.12, 2.28, 2.61, 2.87, 3.29, 3.62, 4.15, 4.57, 5.75, 7.2,
9.14, and 10.95. We use at least 16 samples for each tem-
perature and cavity size.
[27] The sphere is partioned in small cubic boxes where ni is
the number of particles in box i. The side ℓ of the cells
is such that ni = {0,1}. We focus [4] on the overlap at
the centre of the sphere q(t;R) ≡
where the sum runs over all boxes within a small volume
v at the center of the sphere and Ni is the number of
boxes. To minimize statistical uncertainty without losing
the local nature we choose Ni = v/ℓ3= 125. The over-
1
ℓ3Ni
?
i∈vni(t)ni(0),

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7
lap of two identical configurations is 1, while for totally
uncorrelated configurations q = q0 = ℓ3= 0.062876.
[28] see appendix A for a different correlation function
[29] We have estimated the equilibration time τ(R) by fit-
ting the correlation function C(t;R) to an exponential
form exp(−t/τ) in the long time tail (C < 0.1). Differ-
ent choices, such as the integrated time τ ≡?dt′C(t′;R)
or C(τ;R) = e−1give similar results.
[30] see appendix D for details.