Connecting microscopic simulations with kinetically constrained models
  of glasses
Connecting microscopic simulations with kinetically constrained models of glasses
Matthew T. Downton and Malcolm P. Kennett
Physics Department, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada
(Dated: August 23, 2021)
Kinetically constrained spin models are known to exhibit dynamical behavior mimicking that of
glass forming systems. They are often understood as coarse-grained models of glass formers, in
terms of some “mobility” field. The identity of this “mobility” field has remained elusive due to the
lack of coarse-graining procedures to obtain these models from a more microscopic point of view.
Here we exhibit a scheme to map the dynamics of a two-dimensional soft disc glass former onto a
kinetically constrained spin model, providing an attempt at bridging these two approaches.
PACS numbers: 05.10.-a, 61.20.Gy, 61.43.Fs, 64.70.Pf
I. INTRODUCTION
The origin of the onset of ultra-slow dynamics in glassy
systems, and in particular, glass-forming liquids, remains
a murky subject, with many competing ideas and tanta-
lizing clues as to underlying causes, despite years of ef-
fort by a large community of researchers [1]. Recently
it has become increasingly clear that dynamical hetero-
geneities, regions of atypically fast dynamics that are lo-
calized in space and time, are intimately connected to the
phenomenon of glassiness [2, 3, 4, 5, 6], becoming increas-
ingly important at lower temperature scales towards and
below the glass transition temperature Tg [7, 8, 9, 10].
Early ideas about heterogeneous dynamics focused on
the idea of co-operatively rearranging regions which grow
with decreasing temperature [11]. Currently, molecular
dynamics (MD) simulations of supercooled liquids allow
much greater access to the microscopic details of this het-
erogeneity [12, 13, 14]. This has included the observation
of “caging” of particles, and string-like excitations that
allow particles to escape these cages [9, 15, 16, 17], which
has been confirmed in experiments on colloidal glasses
[18]. However, MD simulations have the drawback that
it is difficult to reach the low temperatures and long times
characteristic of the glassy phase.
An alternative approach to reach low temperatures
and long times, but without a microscopic foundation,
is to study simple models of glassiness, kinetically con-
strained models (KCMs) [19, 20, 21, 22, 23, 24], such
as the Fredrickson-Andersen (FA) model [20] or the East
model [25], or variations such as the North-East model
[23], which mimic the constrained dynamics of real glassy
systems but have trivial thermodynamics. These may be
viewed as effective models for glasses, in terms of some
coarse-grained degree of freedom often labelled “spins”,
also termed a “mobility field” by some authors [23]. In
Fredrickson and Andersen’s original work, they posited
that the degrees of freedom may be high and low den-
sity regions, related to earlier suggestions by Angell and
Rao [26]. Despite these appealing physical pictures, it
has not been particularly clear to what physical quan-
tity this “mobility field” corresponds. If KCMs are truly
effective models of glassy behavior then it should be pos-
sible to make a connection between some set of degrees of
freedom, in a MD simulation, for instance, and a KCM.
In this paper we propose a specific coarse-graining pro-
cedure to explore whether a link can be made between
MD simulations and KCMs of glassiness. Previous work
in this direction found evidence of dynamic facilitation in
MD simulations [10, 27], however, there was no attempt
to map the dynamic facilitation onto a KCM. We use
an approach directly related to the idea of a “mobility”
field, using the local mean-square displacement (MSD)
in a suitably defined box to define a spin variable. Re-
gions with large average MSD correspond to “up” spins
and those with low average MSD correspond to “down”
spins. We give specific details of our procedure below.
We investigate the time and length-scale dependence of
this coarse-graining. The two characteristic time scales
are the beta relaxation time scale, tβ (corresponding
physically to the time for relaxation within a cage) and
the longer alpha relaxation time scale, tα, which corre-
sponds to the time-scale on which structural relaxation
of cages occurs. We find that evidence of dynamic facil-
itation becomes much stronger at longer times of order
tα, than at earlier times of order tβ . We also study the
effect of changing the size of the coarse-graining box, l,
in space and consider values 0.02 ≤ l/L . 0.25, where
L is the system size. With appropriate choices of time
and lengthscales, we find a clear mapping from our MD
simulations onto a KCM similar to the 1-spin facilitated
FA model. This is not what one might naively expect.
Since we study a fragile glass-former, we expect to find a
KCMwhich exhibits super-Arrhenius relaxation, whereas
the 1-spin facilitated FA model has Arrhenius relaxation
– the super-Arrhenius growth of timescales is absorbed
into the coarse graining time, which is most effective at
capturing kinetically constrained behaviour when it is of
order tα contrary to expectations based on the idea of a
mobility field [20, 23].
The demonstration of a coarse-graining procedure to
translate from a microscopic model to a coarse-grained
KCM for glasses can shed light on the following: it pro-
vides a physical interpretation to the “mobility field”; it
can give a stronger theoretical justification for the use
of KCMs to study glassy dynamics; and may open the
door to further exploration of the link between micro-
scopic models and long-time features of dynamics, i.e.
http://arxiv.org/abs/0704.1497v3
answering the question: for a given interparticle interac-
tion potential, how will the dynamics of the glass behave?
This paper is structured as follows: in Sec. II we give
details of our molecular dynamics simulations, in Sec. III
we describe our coarse-graining procedure and present
the results of coarse-graining MD simulations, and then
in Sec. IV we discuss our results.
II. MOLECULAR DYNAMICS SIMULATIONS
We study the dynamics of a well-characterized glass
former: the binary soft disc model with a potential of
the form ǫ
in two dimensions [13, 15, 28]. This
mixture of discs with size ratio 1:1.4 inhibits crystalliza-
tion upon cooling. We choose a 75:25 ratio of small to
large particles and cool the system with the density fixed
at ρ = 0.85σ−2
. All temperatures and lengths are quoted
in the standard reduced units of the Lennard-Jones po-
tential using the small disc diameter σ11, as a length
scale. This mixture has the same glassy characteristics
as the model previously documented in Ref. [28]. Post-
equilibration calculations are performed using the re-
cently introduced iso-configurational (IC) ensemble [29].
Our results are qualitatively similar if we follow a single
trajectory with no IC averaging, but IC averaging gives
smoother trends as a function of temperature.
FIG. 1: Averaged mean square displacement from IC simula-
tions of length τe at T = 0.360. On this time scale there is
movement throughout the cell.Not all of the the larger groups
grow into regions of large displacements at longer timescales.
We show the corresponding spin configuration, with up spins
represented as black squares.
In Fig. 1 we show the MSD for each particle averaged
over 500 independent trajectories started from the
same particle configuration in a N = 1600 particle
system. Each trajectory is evolved for a time τe, where
τe is the time that it takes for the self-intermediate
scattering function for the small particles Fs(k, t) =
i 〈sin (k |ri(t)− ri(0)|) / (k |ri(t)− ri(0)|)〉 (note
that the form we use has already been averaged over
angle) to decay by a factor of 1/e – this is roughly
tα. The k value chosen is that of the first peak of the
static structure factor. It is clear that there are regions
of much higher MSD than the average, and that these
regions are reasonably widely spaced. The important
question for mapping the dynamics to a KCM is how
such regions influence the behavior of their neighbors.
The MSD in the IC ensemble simulations can be seen
as a measurement of the propensity for a particle to move
based on the initial configuration. As noted by Widmer-
Cooper and Harrowell, each trajectory within the ensem-
ble does not reproduce the same dynamics [29]. The final
propensity is therefore the composite of a set of trajec-
tories that is determined solely by the initial particle po-
sitions. We can follow the change of the propensity in
time by following a single trajectory and performing IC
simulations separated by a time τs (which we mostly take
to be ∼ τe). The KCM that we determine is one that is
obtained from an IC average over 500 trajectories.
III. COARSE GRAINING PROCEDURE
There are two parts to our coarse-graining procedure.
First, we identify the spins that enter in the KCM (using
the results of the MD simulations). Second we infer the
dynamics of these spins. Specificly, we construct a model
which has a Hamiltonian
si, (1)
where si is a “spin” variable on a site i ∈ [1, NS] (where
NS is the number of sites in the spin model) for which
up (si = 1) corresponds to an active region and down
(si = −1) corresponds to an inactive region, and J is
some (yet to be determined) energy scale. The first part
of the coarse-graining procedure is to find a way to deter-
mine the separation of regions into up and down spins.
In general one might also consider terms in the Hamil-
tonian related to spin-spin interactions [24], but in their
simplest forms, KCMs are usually taken to have the form
in Eq. (1). This implies that at high temperatures there
are no static correlations, as one expects in a liquid. The
model Eq. (1) has no interactions and any glassy phe-
nomenology must come from the dynamical rules that
govern how spins flip. These dynamical rules are usually
stated in the form that the probability of a spin flipping
is dependent on the state of its neighbors [20]. To be
more precise, we can note that Glauber rates for flipping
a spin i are given by [19, 24]:
wi(s) = fi(s)
n↓, si = 1
n↑, si = −1
which respects detailed balance, and the concentration of
up spins (with n↑ + n↓ = 1) is
1 + eJ/T
, (3)
and s = (s1, . . . , sNS ). We determine the function fi(s)
assuming that it has the form f(m), where m is the num-
ber of up spins on sites neighboring site i, similarly to the
formulation of the FA model [20]. We analyze the data
from the MD simulations to determine f(m). It is desir-
able that the results be relatively insensitive to the pa-
rameters entering the coarse-graining procedure, which
is what we find.
Our coarse-graining procedure to determine spins and
sites is as follows: we perform a set of simulations to give
n ≃ 100 timesteps in the IC ensemble. The timesteps are
chosen to be tβ , 0.6τe, and τe to check the coarse-graining
in time. We find that the fitting form τe =
works well over the entire temperature range we consider
(T = 0.36 to T = 0.96), although for T ≤ 0.48 the form
τe = 0.025e
(1.1T )
works equally well. We take each of
the snapshots of IC averaged particle configurations and
coarse-grain in space, by dividing the sample into (l/L)2
boxes lying on a square lattice [30], so that each particle
is assigned to a box. We take l to be small and fixed
to l = 2 most of the time (this appears to be roughly
the length-scale of the cage-breaking process, and also
of the order of the dynamic correlation length, ξ [28])
and assume l to be temperature and time independent.
Since we have at most 1600 particles in our system, when
we go to large coarse graining lengths (l ≥ 8), we start
to get close to the system size and finite size effects are
important, i.e. l/L & 0.2.
We associate a spin with each box, either up or down
depending on whether the MSD per particle in the box
is larger or smaller than some cutoff. We adjust the cut-
off so that n↑ takes its equilibrium value, Eq. (3). This
leaves the freedom to choose the energy scale J . A seem-
ingly natural energy/temperature scale associated with
glassy dynamics appears to be that where the relaxation
time for the small particles starts to stretch more quickly
and there is a marked onset of dynamic heterogeneity;
roughly T ∼ 0.5. This is also the temperature where the
scaling between diffusion and relaxation times changes
[28], motivating us to choose J = 0.5 (in the same units
as T ). However, we check that our results are robust un-
der varying this choice (see Fig. 2). In general we find
that if J & 0.3 the probabilities of spin flips that we
identify are identical.
Now, one of the assumptions that underlies writing
down Eq. (1) is that there are no static correlations.
Given that the spins we consider are themselves defined
through dynamics, albeit within a single coarse-graining
time, it is difficult to be certain that one can extract
truly static correlations. Nevertheless, we checked for
static correlations in the spin model defined above and
find that at high temperatures there are no static corre-
lations, whereas for temperatures below about T = 0.45,
there are “static” correlations on a lengthscale of up to
two lattice spacings (at T = 0.36), that appears to be
growing with decreasing temperature. This is in accord
with the expectation that as the spins diffuse, they lead
to local relaxations that appear as a static correlation in a
single snapshot, but can be resolved, for example, by the
dynamic four-point susceptibility that has been discussed
extensively in the literature [31, 32, 33, 34, 35, 36, 37, 38].
It is also possible that extra terms involving spin-spin in-
teractions should perhaps be included in the model as
these can also lead to static correlations, but the rela-
tively short correlation length, and the existence of dy-
namic correlations as discussed above suggests that we
can ignore these interactions as a first approximation. We
shall proceed under this assumption of non-interacting
spins and discuss some consequences of interactions that
appear to give small corrections to our non-interacting
results.
0 0.5 1 1.5 2
FIG. 2: P (m) at m = 0, 1, 2, 3, and 4, for spin flips down
to up as J is varied, with a coarse-graining timescale of τe at
T = 0.36 with l = 2.
In Figs. 2 and 3 we show some of our checks on the
coarse-graining procedure. All of these are at T = 0.36.
In particular, in Fig. 2 we show how P (m), the prob-
ability of a spin flip between consecutive time steps
changes with variations in J at fixed l = 2 and T for
spin flips from down to up. [At a fixed temperature,
P (m) ∝ f(m).] In Fig. 3 we show how P (m) changes
with variations in l at fixed J and T for spin flips with
up to down. Comparable results are found for the spin
flips not shown. Statistical error bars are comparable to
the size of the symbols.
We have thus defined our spins. Now we must under-
stand their dynamics. To do this, we ask the question,
for an up or down spin with m nearest neighbors that
are up spins, what is the probability that it will flip in
a given time-step? This is the way that the classic FA
model is posed. We display the function f(m) as a func-
tion of temperature in Figs. 4 and 5 for coarse graining
0 2 4 6 8 10
FIG. 3: P (m) as a function of m for spin flips up to down as
l is varied, with a coarse-graining timescale of τe at T = 0.36
with J = 0.5.
times of tβ and τe respectively.
In Figs. 4a) and 5a) we consider m = 0, 1, and 2,
whilst for clarity, in Figs. 4b) and 5b) we show m = 3
and m = 4. The behaviour between these two coarse-
graining times is quite distinct. For a coarse-graining
time of tβ there is no strong tendency towards kineti-
cally constrained dynamics, whereas for a coarse graining
time of τe there are quite strong indications. For a coarse
graining time of τe, at low temperatures (T . 0.5), f(0),
f(1) and f(2) are distinctly different, and there is qual-
itative agreement between f(m) determined from either
up to down spin flips or down to up spin flips. Similar
results are seen for f(3) and f(4).
Detailed balance implies that f(m) determined from
either type of spin flip should be the same if the system
is described precisely by a model of the type in Eq. (1).
In Figs. 4 and 5 there are small but clear differences be-
tween f(m) determined from the two types of spin flips.
We believe that there are two sources for this discrepancy.
Probably most important is the presence of spin-spin in-
teractions which are not accounted for in Eq. (1). Such
interactions mean that f(m) does not depend solely on
m, although at low temperatures it is clear that m is
the most important variable controlling the behaviour
of f(m). Secondly, the magnitude of the discrepancy
between rates varies from temperature to temperature
point. This is likely to be from biases that are forced on
us by computational restrictions. In principle, one would
like to equilibrate a large number of independent parti-
cle configurations, then perform an IC average for each
initial condition to determine f(m). In practice, it is
computationally expensive to equilibrate at low temper-
atures, so we equilibrate one particle configuration and
then use this to generate subsequent particle configura-
tions from one member of the IC ensemble. This intro-
duces a sampling bias that is likely to contribute to the
discrepancy between the rates. We note that even for a
single trajectory, there are signatures of kinetically con-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(0) ud
f(0) du
f(1) ud
f(1) du
f(2) ud
f(2) du
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(3) ud
f(3) du
f(4) ud
f(4) du
FIG. 4: f(m) as a function of temperature from both up to
down and down to up spin flips a) m = 0, 1, 2; b) m = 3, 4
with coarse graining time tβ
strained dynamics, but cleaner results are obtained from
our IC averaging, and we expect averages over more in-
dependent particle configurations to yield more precise
results. Despite these caveats, Fig. 5 clearly illustrates
that the coarse-graining procedure we have devised gives
strong evidence of kinetically constrained dynamics, and
appears to give a mapping of a microscopic simulation to
a KCM.
The signatures of kinetically constrained behavior only
start to show up in the coarse grained spin model for
T . 0.5. which is around the temperature at which
the relaxation time for small particles increases quickly
with decreasing temperature, and is also around where
the Stokes-Einstein relation breaks down. An important
point to note about KCMs is that they are expected to
give the best account of glassy dynamics when n↑ ≪ 1. In
the temperature range we consider, this is only approx-
imately true, for instance, at T = 0.36, n↑ ≃ 0.2 (with
J = 0.5, although for larger J , n↑ is considerably smaller,
and the f(m) we have determined are unchanged). The
requirement of a vanishing number of up spins suggests
that to extract a specific KCM from our data, one should
consider the limit that T → 0. Figure 5 certainly suggests
that as T → 0, f(0) → 0, but it is harder to determine
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(0) ud
f(0) du
f(1) ud
f(1) du
f(2) ud
f(2) du
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(3) ud
f(3) du
f(4) ud
f(4) du
FIG. 5: f(m) as a function of temperature from both up to
down and down to up spin flips a) m = 0, 1, 2; b) m = 3, 4
with coarse graining time τe
the fate of f(m) for m 6= 0. However, our results suggest
a KCM that might apply in the limit T → 0 defined by
(with α constant)
f(m) =
0, m = 0
α, m ≥ 1.
The most important feature of the model that is realised,
whether it is exactly as in Eq. (4) or not, is that it is
of the one-spin facilitated type [19]. It appears unlikely
from Fig. 5 that f(1) → 0 as T → 0 as would be required
for a two-spin facilitated KCM. We have verified with
Monte Carlo simulations that this model gives timescales
that diverge in an Arrhenius fashion at low temperatures.
However, the fact that the timestep in the KCM is also
strongly temperature dependent (through τe) leads to a
fragile behaviour of timescales when expressed in terms
of the time units of the MD simulations, i.e. τ ∼ eA/T
for some constant A.
IV. DISCUSSION
There are several goals in trying to map MD simula-
tions of a glass former onto a KCM. The first is making
a connection between microscopic particle motions and
some effective theory of glassy dynamics. A second is to
determine the long-time dynamics of a given glass for-
mer at very low temperatures where MD simulations are
ineffective, say through Monte Carlo simulations of the
KCM, or in some cases, analytic calculations.
The mapping of MD simulations to a KCM that we
have achieved is not what one might naively expect.
From the discussions in the literature [20, 23], the expec-
tation would be that for coarse-graining on some small
lengthscale (as we do), and on a timescale much less
than tα or τe, one obtains a KCM which has within it
the physics of the alpha relaxation time. Our attempts
along these lines are shown in Fig. 4, which clearly in-
dicates that this expected behaviour does not hold. It
is only when one coarse-grains on timescales of order tα
that we start to see effective kinetic constraints emerg-
ing. In order to get fragile glass like behaviour, as seen
in the MD simulations, from the constant coarse-graining
time scenario, one would require the KCM obtained from
the coarse-graining procedure to be multi-spin facilitated,
since single-spin facilitated models of the type found in
Eq. (4) are known to have activated dynamics [19]. This
suggests that there may be alternative coarse graining
schemes that capture a connection between MD simula-
tions and KCM. Nevertheless, to the best of our knowl-
edge, we have demonstrated the first effective mapping of
MD simulations onto a KCM. The “spins” of our model
correspond to regions of high or low MSD per particle
and hence are in the spirit of the “mobility field”[23].
Given that the mapping we exhibit does not allow us
to obtain the fragile glass behaviour of the original model
from our KCM, it is interesting to ask what physics the
spins that we map to are sensitive to. We believe they
are sensitive to slow structural relaxation that proceeds
on time scales even longer than tα. This appears to be
consistent with recent work by Szamel and Flenner [39]
that suggests that continuing relaxations beyond tα lead
to the timescale for the onset of Fickian diffusion to be an
order of magnitude longer than tα and grow faster than
tα at low temperatures. The same authors also proposed
a non-gaussian parameter which differs from that con-
ventionally used in studies of glass formers [40]. This
non-gaussian parameter has a peak at later times than
the conventional one, at timescales of order tα, where it
appears that heterogeneity in the distribution of parti-
cle displacements is maximal. This might be related to
why a “spin” definition based on mean square displace-
ment, as we use here, is most sensitive to physics on
the timescale of tα. This suggests that in order to make
a mapping to a KCM more in line with naive expecta-
tions, one should use quantities that have maximum con-
trast on timescales which are considerably shorter than
tα. Such an approach may be a viable way to construct
effective theories of glassy dynamics in this and in other
systems. We have demonstrated a particular numerical
coarse-graining procedure, and we hope that our results
may help point the way to analytic approaches to con-
nect microscopic models of glasses to KCMs, and further
insight into the glass problem.
V. ACKNOWLEDGEMENTS
Calculations were performed on Westgrid. The au-
thors thank Horacio Castillo, Claudio Chamon, and Mike
Plischke for discussions, and David Reichman for critical
comments on the manuscript. The authors also thank
the anonymous referees for their comments. This work
was supported by NSERC.
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