# Exploring the Relationship Between Softness and Excess Entropy in Glass-
forming Systems

Ian R. Graham  Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104, USA  Paulo E. Arratia  Department of
Mechanical Engineering and Applied Mechanics, University of Pennsylvania
Philadelphia, PA 19104, USA  Robert A. Riggleman  Department of Chemical and
Biological Engineering, University of Pennsylvania Philadelphia, PA 19104, USA

###### Abstract

We explore the relationship between a machine-learned structural quantity
(softness) and excess entropy in simulations of supercooled liquids. Excess
entropy is known to scale well the dynamical properties of liquids, but this
quasi-universal scaling is known to breakdown in the supercooled and glassy
regimes. Using numerical simulations, we test whether a local form of the
excess entropy can lead to predictions that derive from softness, which has
been shown to correlate well with the tendency for individual particles to
rearrange. To that end, we explore leveraging softness to compute excess
entropy in the traditional fashion over softness groupings. Our results show
that by computing the excess entropy over softness-binned groupings, we can
build a strong quantitative relationship between the rearrangement barriers
across the explored systems.

##  I Introduction

Despite their importance to numerous technologies, the properties of
disordered (amorphous) materials far from equilibrium continue to elude
comprehensive understanding (Stillinger and Debenedetti, 2013; Charbonneau et
al., 2017; Berthier and Biroli, 2011). Disordered materials made of
particulates can span a wide range of length scales, and examples include
metallic glasses, nanoparticle packings, colloidal suspensions, foams &
emulsions, and granular materials Cubuk et al. (2017); van Hecke (2009);
Jerolmack and Daniels (2019). Our knowledge of these materials is primitive
compared to our understanding of crystalline solids, where symmetry and order
guide detailed theories Torquato (2018); Liu and Nagel (2010). In disordered
systems, the dynamics are known to be strongly heterogeneous and can vary by
orders of magnitude even for a supercooled liquids at equilibrium. For many
glass-forming systems, the dynamics exhibit a temperature dependence where the
activation energy grows upon cooling. Thus, relating the evolution of the
material microstructure (at the constituent level) to the material’s dynamics
(and bulk response) is quite difficult. Much work has been devoted to
developing structural indicators of glassy dynamics in disordered systems,
with different degrees of success (Richard et al., 2020; Galloway et al.,
2020; Dyre, 2018; Bonnecaze et al., 2020). If one is interested in developing
thermodynamically consistent coarse-grained and constitutive models,
identifying the relevant structural parameters and their connection to
thermodynamic quantities is a critical step (Öttinger et al., 2005; Öttinger,
2007).

In recent years, machine learning (ML) and data science techniques have
matured to a point that they are ubiquitous in research and industry. Past
studies have successfully applied ML models to predict local dynamical
properties of glassy systems more accurately than traditional quantities such
as local potential energy and free volume, for example (Tong and Xu, 2014;
Riggleman et al., 2007; Widmer-Cooper and Harrowell, 2006a; Richard et al.,
2020). These ML models range in complexity from relatively simple linear
support vector machines (SVMs) which operate on vectors of pre-selected
structural features for each particle, to convolutional and graph models that
are able to extract more intricate spatial correlations within the material
and thus lead to improved prediction accuracy on similar tests (Schoenholz et
al., 2016, 2017; Bapst et al., 2020; Fan and Ma, 2021). A notable ML indicator
developed to characterize structural defects and predict rearrangements in
disordered packings is the quantity known as _softness_ Schoenholz et al.
(2016); Cubuk et al. (2016, 2017). This quantity was one of the first models
developed for glasses and, though relatively simple in its construction from
local radial distribution functions, is still able to provide good predictive
power (Schoenholz et al., 2016, 2017).

Although softness may be more interpretable compared to other machine-learning
based models, particles of similar softness still possess different local
structures, complicating its overall physical meaning. For systems in which
particles are dominated by isotropic interactions, softness can be constructed
as a weighted sum of the radial distribution function, where the weights are
determined by the SVM. While these weights do inform us to a degree that
certain features in the pair correlation function (notably the first and
second peaks of $g(r)$) are critical to local dynamics, it remains unknown how
the general form of the radial weights may be related to a priori knowledge of
the particles and their interactions. On the other hand, excess entropy, a
quantity known to scale with the dynamical properties of simple liquids, can
be similarly constructed as a simple function of local particle coordinates
where deviations in the radial distribution function from that of the ideal
gas (a flat distribution) are penalized (Dyre, 2018; Bell et al., 2020;
Bonnecaze et al., 2020; Ma et al., 2019; Galloway et al., 2020). Due to the
similar construction of these quantities, it is tempting to consider whether
there are any connections between them. This would lead to an improved
understanding of softness, for example. However, one difficulty of such a
comparison is that softness is defined on a particle basis to strongly
correlate with particle-level dynamics, while excess entropy is usually
defined for an entire configuration and is associated with system-average
dynamics. In this work, we formulate a comparison between these quantities
(i.e., excess entropy and softness) using two separate approaches. First, we
define a local form of excess entropy that takes in a Gaussian-smeared, coarse
radial distribution function (similar to softness) and compare the performance
of this measure relative to softness. Next, we utilize softness as an
intermediary tool so to compute excess entropy in a more traditional fashion.
Essentially, we treat particles of similar softness as comprising their own
ensembles, allowing us to compute ensemble average quantities (like excess
entropy) over softness-grouped subsystems. Our results suggest interesting
future directions for combining equilibrium tools with machine-learned
quantities in out-of-equilibrium disordered systems.

##  II Methods

###  II.1 Simulation Details

Numerical simulations comprise sets of equilibrated supercooled states of
bidisperse, Lennard-Jones character generated using HOOMD-blue (Anderson et
al., 2020). Configurations are composed of 32,768 particles in a 3D periodic
box and a Nosé-Hoover thermostat is used to integrate the dynamics with a time
step of $10^{-3}$ in the NVT ensemble. We employ a Lennard-Jones potential
with a modifiable well-width parameter $\Delta$ to control the level of caging
in the system; the standard Lennard-Jones definition is obtained when
$\Delta=0.0$. The potential is defined as

$\begin{split}\cr
V_{ij}(r)=\begin{cases}4\epsilon_{ij}\left[\left(\frac{\sigma^{\prime}_{ij}}{r-\Delta}\right)^{12}-\left(\frac{\sigma^{\prime}_{ij}}{r-\Delta}\right)^{6}\right]&r\leq
2.5\sigma_{ij}\\\ 0&r>2.5\sigma_{ij},\end{cases}\end{split}$ (1)

where $\sigma^{\prime}_{ij}=\sigma_{ij}(1-\Delta/{2^{1/6}})$ is defined to
keep the minimum at the same position as $\Delta$ varies, $r$ is the pair
distance between particles, $\epsilon_{ij}$ is the interaction energy scale
between species $i$ and $j$, and $\sigma_{ij}$ is the interaction length
scale. The parameter $\Delta$ is varied from 0.0 to 0.4, and $\epsilon_{ij}$
and $\sigma_{ij}$ are set in accordance with a standard 80:20 Kob-Anderson
type mixture ($\epsilon_{AA}=1.0$, $\epsilon_{AB}=1.5$, $\epsilon_{BB}=0.5$,
$\sigma_{AA}=1.0$, $\sigma_{AB}=0.8$, $\sigma_{BB}=0.88$). The functional
forms are shown in Figure 1a for different choices of $\Delta$. Depending upon
the value of $\Delta$ used, packing fraction $\rho$ is varied between 1.2 and
1.12 to minimize the rapid increase in system pressure $p$ as the well width
is decreased. We explored the $(\Delta,\rho)$ pairings [$(0.0,1.2)$,
$(0.1,1.18)$, $(0.2,1.16)$, $(0.3,1.14)$, $(0.4,1.12)$] . Simulations at
$\Delta=0.5$ were also produced, but they were found to readily crystallize in
the supercooled regime; they are omitted from the analysis. To ensure
reproducibility, three randomly seeded replicas were generated for each
system. All stated Lennard-Jones units are with respect to the A-species of
the standard Kob-Anderson mixture (i.e., $\Delta=0$). Configurations are
initialized on a cubic lattice with random placements of particle species and
thermalized well into the liquid regime for each system. The systems are then
swept through the analyzed temperatures into the supercooled regime, waiting
at most 20 $\tau_{\alpha}$ (where $\tau_{\alpha}$ is the alpha relaxation
time) at the highest temperature in the supercooled regime and 8
$\tau_{\alpha}$ for the coldest temperature before continuing with the
temperate quench. We use the self-intermediate scattering function
($F(\bm{Q},t)=\frac{1}{N}\sum_{j=1}^{N}\langle\exp[i\bm{Q}\cdot(\bm{r}_{j}(0)-\bm{r}_{j}(t))]\rangle$)
to estimate $\tau_{\alpha}$ by calculating the time required for $F(\bm{Q},t)$
to drop below $\frac{1}{e}$. In all systems we use $|\bm{Q}|=7.14$, the
wavenumber coinciding with the distance to the first peak of $g(r)$.
Furthermore, we apply FIRE minimization as a post-processing step to obtain
the inherent structures at each sampled time for final analysis; this step is
not strictly necessary to obtain the relationships we find, but serves to
remove thermal fluctuations from the analysis (Bitzek et al., 2006). In all
parts of the analysis, we explore only the dynamics of A-species particles. We
made substantial use of the _freud_ , _signac_ , and _signac-flow_ python
packages to perform post-processing analysis and to manage data & job
workflows (Ramasubramani et al., 2020, 2018; Adorf et al., 2018).

###  II.2 Softness

As briefly discussed above, softness ($S$) is a machine-learned quantity
trained on examples of particles undergoing rearrangement. Here, we closely
follow the techniques employed in previous works to construct $S$ in our
thermal system (Schoenholz et al., 2017, 2016). We assess whether or not a
particle participated in a rearrangement by examining $p_{hop}$. $p_{hop}$ is
defined as

$p_{hop}(i,t)=\sqrt{\langle(x_{i}-\langle
x_{i}\rangle_{B})^{2}\rangle_{A}\langle(x_{i}-\langle
x_{i}\rangle_{A})^{2}\rangle_{B}}.$ (2)

where $x_{i}$ is the position of the particle $i$ in the simulation, $A$ and
$B$ are time intervals defined as $A=[t-t_{R}/2,t]$ and $B=[t,t+t_{R}/2]$, and
$t_{R}$ is the time window used. A fairly coarse period is used between frame
dumps of $1\tau_{A}$, where $\tau_{A}$ is the Lennard Jones time unit in
reference to the A species of the $\Delta=0.0$ system, and we use a time
window $t_{R}=10\tau_{A}$. To categorize rearrangements, two cutoffs are
defined: $p_{H}=0.05$ and $p_{S}=0.2$. Particles are deemed _soft_ if
$p_{hop}>p_{S}$ and _hard_ if $p_{hop}<p_{H}$. We further restrict our dataset
by only processing the peaks and troughs of $p_{hop}$, and asserting that
rearrangement events are separated by $10\tau_{A}$, and non-rearrangements by
$80\tau_{A}$. In case any events are too close, the more prominent peak (or
deepest trough) is selected.

Structure functions, $\mathcal{G}_{K}(\mu_{i})$, are calculated from a
Gaussian-smeared radial distribution function by the following equation

$\mathcal{G}_{K}(\mu_{i})=\sum_{j\in\\{K\\}}{e^{\frac{(r_{j}-\mu_{i})^{2}}{2{\Delta\mu}^{2}}}}$
(3)

where $\mu_{i}$ is the radial distance of the $i^{\textrm{th}}$ structure
function, $\Delta\mu$ is variance of the gaussian, the label $K$ selects for
particle species A or B, and $r_{j}$ is the radial distance to the particle
$j$. $\mu_{i}$ varies linearly from 0.4 to 3.0 with a step size of
$\Delta\mu=0.1$ (Behler and Parrinello, 2007). With these structure functions,
we proceed to train a LinearSVM (using the _scikit-learn_ package(Pedregosa et
al., )) to classify our soft and hard particles. The final hyperplanes are
insensitive to the random seeds relating to the SVM optimization and subset of
data used. The LinearSVM works by finding an optimal hyperplane to separate
our classes in the high-dimensional space in which our structure functions
reside. Note that we apply a pre-processing step to our data that shifts and
scales the structure functions such that within each index pair ($K,\mu_{i}$)
our data has zero mean and unit variance. After fitting the data to our model,
we can extract the decision function which is the signed distance to the
hyperplane and use this measure as our softness $S$. We find that the
accuracy, measured as $P(R|S>0)$, is between 73-80% for the coldest
temperatures measured in each system, though this accuracy decreases to 40-50%
at our highest temperatures due to the inherent thermal stocasticity of the
liquid state.

###  II.3 Excess Entropy

Excess entropy, $s^{(2)}$, is classically defined as the difference in
configurational entropy between the ideal gas and the system of interest
(Dyre, 2018). It can be calculated by a variety of means, including
thermodynamic integration (Bell et al., 2020). Here, we utilize the simplified
and approximate calculation based upon the species-dependent radial
distribution function $g_{K}(r)$ as

$s^{(2)}=-2\pi\sum_{K}\rho_{K}\int(g_{K}(r)\log\\{g_{K}(r)\\}-g_{K}(r)+1)r^{2}dr$
(4)

in 3D. $\rho_{K}$ is the density of species K in the sample and $r$ is the
radial distance. We additionally construct a per-particle version of $s^{(2)}$
in close analogy to softness. The local $s^{(2)}$ utilizes the same structure
functions as softness, but these inputs are rescaled by the appropriate
spherical measure to be transformed back into a coarse, smeared
$g_{K}(r_{i})$.

$g_{K}(r_{i})=(4\pi\rho_{K}^{*}r^{2}_{i})^{-1}\mathcal{G}_{K}(r_{i})$ (5)

where $\rho_{K}^{*}$ is an effective local density calculated by integrating
the RDF for each species, here computed with a sphere of radius $r=3.0$. From
this RDF, we can apply a simple midpoint integration to obtain

$s^{(2)}=-2\pi\sum_{K}\rho_{K}^{*}\sum_{i}^{i_{max}}{[g(r_{i})\log\\{g(r_{i})\\}-g(r_{i})+1]r_{i}^{2}L(r_{i}-\eta)\Delta
r}$ (6)

where $\Delta r$ is the bin width, and $L(r_{i}-\eta)$ is the inverse logistic
function centered at a distance of $\eta$ away. We applied a gentle
optimization pass, finding $\eta=2.0$ to adequately dampen fluctuations at
distances at and beyond the second peak of $g(r)$. A few aspects of the form
we have chosen here substantially improve the performance of this measure.
Notably, respecting the bidispersity of the systems and using the local
species density $\rho_{K}^{*}$, as opposed to the average system density.

##  III Results

###  III.1 Comparison of softness and local excess entropy

Figure 1: (a) Pair potentials used in this work. Marginal variation in the
Lennard-Jones well width, controlled here by the parameter $\Delta$, leads to
significant dynamical changes. (b) Example snapshot of our standard Lennard-
Jones Kob-Anderson mixture at $k_{B}T=0.45$, $\Delta=0.0$, produced with the
_fresnel_ python library. (c) The total radial distribution functions (RDFs)
for the 5 systems we consider. Higher $\Delta$ values lead to a sharpening of
RDF peaks, in turn leading to more pronounced caging and slower dynamics at a
given temperature. (d) Diffusion coefficients for A species particles measured
in both the liquid and supercooled regimes.

Here, we examine simulations of a Lennard-Jones Kob-Anderson mixture (Kob and
Andersen, 1994). We explore four other variations of this system with a
modified well width parameter $\Delta$, ranging from 0.0 to 0.4, as shown in
Figure 1a. Although the modifications to the potential appear subtle, this
translates into considerably stronger caging effects within the system and
more brittle response overall as $\Delta$ is increased (Lin and Riggleman,
2019). Figures 1b,c show an example snapshot of our $\Delta=0.0$ system at
$k_{B}T=0.45$ and samples of $g(r)$ at similar depths into the supercooled
regime for each system. The increased caging behaviour is reflected in the
increased height of the $g(r)$ peaks when $\Delta$ is increased. Figure 1d
shows the diffusion coefficients for these five systems as a function of
inverse temperature; particle diffusivity decreases considerably as $\Delta$
is increased for same $T$.

The dynamic properties of liquids, such as diffusion, at high temperatures are
expected to exhibit an Arrhenius temperature dependence, implying that a
characteristic energy describes the temperature dependence of the diffusivity
or viscosity. In this regime details of individual particle microstructure are
unimportant, and all particles of the same type obey similar dynamics. At
short time scales, particles travel ballistically as they move through their
local neighborhood, but the short mean free path of the liquid ensures that
these trajectories are quickly redirected, and diffusive behavior soon
follows. As the system is cooled, it reaches a temperature where the Arrhenius
trend begins to break down, known as the crossover temperature. For
temperatures below the crossover, details of the individual particle
environments become more important as particles spend more time in any given
local configuration. This coincides with the emergence of stretched
exponential behaviour in the self intermediate scattering and overlap
functions, which is understood to result from an underlying wide distribution
of relaxation rates within the system. This emerging importance of the
specific microstructure, and resultant wide distribution of relaxation times,
greatly complicates the prediction of dynamics in these systems. It becomes
mostly hopeless to make first-principles predictions of these behaviours from
information of the potential and macroscopic observables alone.

In Figure 2 we illustrate how the reduced diffusion coefficient, calculated as
$D_{A}^{\ast}=\rho^{\frac{1}{3}}\sqrt{\frac{m}{kT}}D_{A}$, correlates with
both the average softness ($S$) and excess entropy ($s^{(2)}$) of the systems.
The prefactor of the reduced diffusion coefficient originates from Enskog
theory, i.e. the kinetics of a dense hard-particle gas (Dzugutov, 1996). As
expected, we observe monotonic behavior in both quantities, but they differ
qualitatively in both the liquid and supercooled regimes. In the liquid state,
the average $S$ varies little. This is consistent with the understanding that
softness is well correlated with the effective energy barrier of particles,
which should be constant in the liquid regime. As the temperature is decreased
below the onset temperature and the average values of $S$ take on a more
substantial temperature dependence where $S$ decreases with decreasing $T$,
the relationship between $D_{A}^{\ast}$ and softness changes to a nearly
exponential relationship, though the relation is clearly stronger than a
simple exponential at the lowest temperatures considered. While these trends
are qualitatively similar across the different systems, it remains difficult
to relate them quantitatively due to the construction of $S$. This is because
$S$, obtained here as a signed distance from an SVM hyperplane where all
dimensions have been rescaled to unit variance over the data distribution, has
poorly defined units. And it is a non-trivial issue how one would rescale and
shift these dimensions such that particles of a given softness correspond to,
for example, the same (reduced) diffusion coefficient from first principles.

For the case of excess entropy in Figure 2b, we see expected behaviour where
in the liquid state the pair approximation of excess entropy does of fair job
of estimating the reduced diffusion coefficients. However, the trends
quantitatively differ once temperatures drop below onset. Application of the
3-body term in the excess entropy or its estimation using thermodynamic
integration could perhaps improve the collapse in the liquid state, however we
wish to limit our analysis to predictors that only require a single (or small
number) of configurations. Unfortunately, both quantities, $S$ and $s^{(2)}$,
fall short as universal predictors of bulk dynamics in the supercooled regime.

Figure 2: Reduced diffusion coefficients of the A species as a function of
both the average softness (a) and excess entropy (b).
$D_{A}^{*}=\rho^{\frac{1}{3}}\sqrt{\frac{m}{kT}}D_{A}$. The behaviour of these
systems as softness varies, while qualitatively quite similar, fails to be
agree quantitatively. The results regarding excess entropy though depend upon
the regime. In the liquid state the excess entropy yields good agreement in
the reduced diffusion coefficient across systems, though this breaks down
considerably below the crossover temperature.

Following previous analyses of softness, an Arrhenius-like relationship can be
found as a function of temperature within the supercooled regime when tracking
particles at constant $S$ across temperature. In Figure 3a,c we show this
stratification for the probability of rearrangement in two of our systems
binned by softness. In each system, we take the 5th and 95th percentile bounds
of softness, and then use five equally spaced bins to aggregate our
populations. Furthermore, these Arrhenius fits can be interpreted as relating
to an Eyring-like equation $P_{R}(S)=\exp(\Sigma(S)-\Delta E(S)/T)$, where
$\Sigma$ is the activation entropy, $\Delta E$ is the activation energy, and
$T$ is the simulation temperature. When viewed this way, effective energetic
and entropic barriers can be extracted from the fits, providing a connection
between glassy local structures and the characteristic energy barriers
governing the system’s dynamics.

Figure 3: Probability of rearrangement against inverse temperature across the
systems analyzed. Each system is separated into 5 equally spaced bins of
softness, and at each temperature point we utilize a cutoff in $p_{hop}$ to
measure the occurrences of local rearrangements. In subplots (a) and (c)
softness is used to group particles with similar dynamics, leading to
Arrhenius behaviour within each strata. In (b) and (d) we perform the same
analysis, but instead using the local excess entropy as our structural proxy
for dynamics. Both quantities are able to reproduce Arrhenius behaviour within
the strata, though softness is more effective in capturing the range of
dynamical behaviour within the system.

While these results regarding softness are promising, one may wonder if the
qualities that we find surprising here (the Arrhenius-like behavior and
separation of relaxation rates) are hard to find using other measures of local
packing. To explore this issue, we perform a similar analysis using our
locally-defined excess entropy, as shown in Figure 3b,d. Although the quantity
$s^{(2)}$ is not designed specifically to identify rearrangements, one still
finds that the local excess entropy exhibits properties similar to those of
$S$. For example, one still obtain fairly good Arrhenius fits to the strata
with $s^{(2)}$, though the separation of these strata is not as large as with
$S$. Depending upon the system, we observe at most an order of magnitude
spread in the probability of rearrangement when looking at the coldest
temperature sample when applying excess entropy. Softness, on the other hand,
is capable of extracting an additional order of magnitude or more in this
spread.

Next, we directly assess whether excess entropy and softness are correlated
with each other. We do so by calculating the Pearson correlation coefficient
$\rho_{S,s^{(2)}}=\tfrac{cov(S,s^{(2)})}{\sigma_{S}\sigma_{s^{(2)}}}$ between
$S$ and $s^{(2)}$ as a function of the parameter $\Delta$. Across our systems
we find a moderate correlation, as shown in the inset to Figure 4a. We find an
increase in the correlation between softness and excess entropy as $\Delta$
increases, which appears to plateau at $\approx 0.5$ (4a, inset). We believe
that this is because the tighter well width is increasing the importance of
structural entropy on dynamics, though this effect is limited. Our local
excess entropy is successful in picking out some important features of
softness, but ultimately falls short. We can understand this through the
difference in the construction of these quantities. Excess entropy is directly
penalizing fluctuations of the RDF relative to that of the ideal gas, i.e.
$g(r)=1.0$. What seems to be crucial to softness’ power is it’s knowledge of
the potential which it encodes transitively through the average $g(r)$ of a
system and local fluctuations around $g(r)$ observed during the training
process.

Figure 4: (a) 2D histogram of softness and local excess entropy at a particle
level for the $\Delta=0.0$ system at $k_{B}T=0.47$. (inset) Pearson
correlation as a function of $\Delta$. We find a moderate correlation across
the systems, which upon closer examination improves with increasing $\Delta$,
likely due to the sharpening of the underlying RDF that enters the
calculation. (b) Cross-over temperatures estimated from the Arrhenius fits in
the supercooled regime using softness and excess entropy. A simple IQR outlier
rejection method is applied to remove poor estimations that occur due to
nearly co-linear Arrhenius fits. Good agreement is found between the estimates
using the two quantities.

Past studies were able to use dynamics (through dynamical heterogeneities) or
structure (through softness) to estimate the crossover temperature into the
supercooled regime(Keys et al., 2011; Schoenholz et al., 2016) where the
dynamics first become non-Arrhenius. In Figure 4b we show the crossover
temperatures estimated from the Arrhenius fits of softness and local excess
entropy. Surprisingly, the crossover temperatures obtained through the two
quantities agree relatively well. This is interesting to note, since
traditional methods of extracting a crossover temperature require numerous
measurements of long time dynamical quantities such as the diffusivity,
viscosity, or alpha relaxation time. The distinct advantage that we hold by
using excess entropy is that it requires no pre-training on data. Thus local
excess entropy appears to be good and useful structural quantity to determine
when the supercooled regime has been entered. We believe that estimating
cross-over temperature is possible from $s^{(2)}$ because the structure
captured by excess entropy is correlated with short-time dynamical
heterogeneities in this regime.

One important observation here is that the Arrhenius trend found by softness
may not be as unique as previous studies suggest. Instead, the observance of
the Arrhenius trend in the supercooled regime may simply hinge on the quantity
being at most weakly correlated with rearrangement probability. This also
raises concerns about how to validate the performance of softness in general.
Within thermal simulations, we have the freedom to use techniques such as the
isoconfigurational ensemble and score our structural quantities by their
correlation with propensity or other dynamical quantities averaged over the
ensemble (Widmer-Cooper and Harrowell, 2006b; Bapst et al., 2020). But in
athermal sheared systems, where the rearrangement dynamics are binary in
nature, it is difficult to frame softness’ effectiveness as a correlation to
average per-particle dynamics.

To summarize this section, we find that softness and local excess entropy,
although moderately correlated, possess clear differences in their performance
that affect their application as local indicators of rearrangement. Softness
is clearly more suited to this task, but the moderate correlation suggests
that there may be an underlying correspondence between these quantities. It
remains that in cases where it is difficult or impossible to compute softness,
the local excess entropy may make a fair proxy of dynamical behaviour and as a
means to estimate the crossover temperature.

###  III.2 Building ensembles with softness

A broad distribution of relaxation rates is a hallmark of systems that exhibit
glassy dynamics (Palmer et al., 1984). However, this feature greatly
complicates the analysis of these systems with tools from statistical
mechanics and thermodynamics. Thus, it is challenging to relate local
structure to the relaxation rates of glassy systems, as is evident from the
unsatisfying performance of purpose-built quantities discussed above. The
concept of machine-learning softness appears to be a fair solution to this
issue in simple supercooled liquids, as recent works has demonstrated a clear
connection between softness and relaxation rates in glassy systems(Schoenholz
et al., 2016, 2017). This should be understood to be one of softness’ key
contributions, that is, as a framework to infer dynamics from structure. And
since these dynamics follow Arrhenius laws for a given softness, we can obtain
information regarding local rearrangement barriers. But while softness
possesses these valuable traits, it still cannot explain the dynamical
behavior of systems in a way that connects to prior intuition.

Figure 5: (a) Radial distribution functions of inherent structures of the
$\Delta=0.0$, $k_{B}T=0.49$ system when particles are grouped by softness.
Lower-softness inherent structures tend to exhibit enhanced densities of
particles at peaks in the RDF and suppression of those within the troughs. (b)
Relationship between excess entropy and softness computed through RDFs of
particle groupings with a given softness. The resulting relationship is not
quite linear, but monotonically increasing.

In order gain a deeper insight into the connection between thermodynamics and
structure in glassy systems, we propose leveraging the predictive power of
machine-learned indicators, like _softness_ , to partition our systems into
sub-groupings that posses similar structure and dynamics. This can be done by
simply grouping particles by their softness. Once these groupings are made, we
posit that it is appropriate to compute thermodynamic quantities, like the
excess entropy, in the traditional way by averaging over configurations, and
that the results of these computations are qualitatively relatable to the
actual dynamics. In this way, we propose thinking about the supercooled liquid
not as possessing a homogeneous entropy, but as a number of distinct liquid
states characterized by a spatially varying entropy, similar to the idea of
“entropic droplets” from the random first-order transition theory (Kirkpatrick
and Wolynes, 1987).

After grouping particles by softness, it is straightforward to construct the
radial distribution functions from the inherent structure states. In Figure 5a
we show examples of the A-A and A-B radial distribution functions (RDFs)
calculated for the inherent structure configurations for particles of
different softness. Particles of lower softness possess expected features,
with neighbors more highly concentrated at peaks and deficient at troughs. We
find that the inherent structure RDFs for the softness groupings are
insensitive to the choice of temperature in the supercooled regime. These RDFs
are used to compute $S$ and $s^{(2)}$ and extract a trend between softness and
excess entropy of the particles binned by softness, as shown in Figure 5b.
Using the excess entropy at a given softness from Figure 5b and the
energy/entropy barriers extracted from Figures 3a and 3c, we can look for a
relationship between the excess entropy and the barriers impeding
rearrangements; we show this comparison in Figure 6. Surprisingly, we find
that the local entropic barriers of the systems, rescaled by the log-
probability of rearrangement at $T_{A}$ ($\log[P_{R}(T_{A})]$), appear to
collapse the data as a function of the pair excess entropy. Interestingly,
this relationship here appears linear, in contrast to the relationship between
softness and rearrangement barriers, which often posses a slight quadratic
character.

Figure 6: (a) Entropic and (b) energetic barriers as a function of excess
entropy. We select similar temperatures relative to onset in each system,
though the softness-grouped excess entropies of the inherent structure
quenched states is insensitive to temperature. We find a simple rescaling by
$\log[P_{R}(T_{A})]$ and $T_{A}$ is sufficient to collapse the trends.

Figure 6b shows a collapse in the energetic barriers by additionally rescaling
by the onset temperature $T_{A}$ found in each system. Although this is
somewhat unexpected, it follows from the fact that 1) we observe a collapse in
the entropic barriers and 2) that the softness strata approximately intersect
at the onset temperature $T_{A}$. Though a little algebra we find that

$P_{R}(T_{A})=\exp\left(\Sigma-\frac{\Delta E}{T_{A}}\right)$ (7)
$\frac{\Delta E}{\log[P_{R}(T_{A})]T_{A}}=\frac{\Sigma}{\log[P_{R}(T_{A})]}-1$
(8)

The required scaling with $\log[P_{R}(T_{A})]$ appears to be an effect of our
universal criterion for rearrangements in these systems, $p_{hop}>0.2$. The
results here are curious. The microscopic dynamics of the supercooled liquid,
a system that has long been difficult to analyze using the tools of
statistical mechanics and thermodynamics, appears to be well described by a
thermodynamic structural measure, excess entropy, over these softness-grouped
ensembles. Even if we take a leap and interpret $s^{(2)}$ here as a
microscopic measure of configurational entropy in these systems, it is
surprising that this also well predicts the barrier heights as the potential
is modified. The only other critical information to forming the observed
collapse is the onset temperature, which is straightforward to measure.

###  III.3 Discussion

Our first set of results suggest that quantities such as the local excess
entropy are able to recover many of the notable features attributed to
softness, e.g., Arrhenius trends in the supercooled regime and an ability to
infer the onset temperature for supercooled dynamics. While the overall
performance (in terms of predicting particle rearrangements) of excess entropy
is not as robust as softness, it does surprisingly well for a quantity that
has not been aggressively optimized. Though the simple application of excess
entropy as a local, non-averaged estimate is indeed somewhat naive, and
information of the interactions must be somehow incorporated (even indirectly)
to elevate the local estimate of excess entropy to at least the level of
softness. A more intricate exploration of the transition states may help to
reveal these connections between the local inherent structures and the
barriers to rearrangement. Particularly, if one considers ensembles of
transition states with the same energy barrier height, what average behavior
emerges for the per-particle energies with distance? Is it possible that we
could intuit this from a combination of $U(r)$ and its derivatives?

Furthermore, our analysis using softness-grouped ensembles exposes a
remarkably simple connection between excess entropy and the rearrangement
barriers within supercooled liquids. Even though the supercooled state is
fraught with many complications due to its rich behavior and metastable
nature, there appears to be an optimal partitioning of particles that allows
us to make connections to equilibrium thermodynamics. Unfortunately, however,
effectively obtaining these groups at the moment requires either dynamical
sampling techniques like isoconfigurational ensembles or machine-learned
quantities that pose superior correlations with rearrangement dynamics.
Quantities derived from low-frequency modes may also be good candidates for
this type of aggregation (Tong and Xu, 2014). The framework presented here is
a possible path for framing thermodynamics in supercooled liquids, and it may
help us to better understand the physical principles responsible for the
emergence of glassy behavior at the onset temperature.

###  III.4 Acknowledgements

This work is funded by University of Pennsylvania’s MRSEC NSF-DMR-1720530.

## References

  * Adorf et al. [2018] Carl S. Adorf, Paul M. Dodd, Vyas Ramasubramani, and Sharon C. Glotzer.  Simple data and workflow management with the signac framework.  _Computational Materials Science_ , 146:220–229, April 2018\.  ISSN 0927-0256.  doi: 10.1016/j.commatsci.2018.01.035.  URL https://www.sciencedirect.com/science/article/pii/S0927025618300429.
  * Anderson et al. [2020] Joshua A. Anderson, Jens Glaser, and Sharon C. Glotzer.  HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations.  _Computational Materials Science_ , 173:109363, February 2020.  ISSN 09270256.  doi: 10.1016/j.commatsci.2019.109363.  URL https://linkinghub.elsevier.com/retrieve/pii/S0927025619306627.
  * Bapst et al. [2020] V. Bapst, T. Keck, A. Grabska-Barwińska, C. Donner, E. D. Cubuk, S. S. Schoenholz, A. Obika, A. W. R. Nelson, T. Back, D. Hassabis, and P. Kohli.  Unveiling the predictive power of static structure in glassy systems.  _Nature Physics_ , 16(4):448–454, April 2020\.  ISSN 1745-2473, 1745-2481.  doi: 10.1038/s41567-020-0842-8.  URL http://www.nature.com/articles/s41567-020-0842-8.
  * Behler and Parrinello [2007] Jörg Behler and Michele Parrinello.  Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces.  _Physical Review Letters_ , 98(14):146401, April 2007.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.98.146401.  URL https://link.aps.org/doi/10.1103/PhysRevLett.98.146401.
  * Bell et al. [2020] Ian H. Bell, Jeppe C. Dyre, and Trond S. Ingebrigtsen.  Excess-entropy scaling in supercooled binary mixtures.  _Nature Communications_ , 11(1):4300, August 2020\.  ISSN 2041-1723.  doi: 10.1038/s41467-020-17948-1.  URL https://www.nature.com/articles/s41467-020-17948-1.  Number: 1 Publisher: Nature Publishing Group.
  * Berthier and Biroli [2011] Ludovic Berthier and Giulio Biroli.  Theoretical perspective on the glass transition and amorphous materials.  _Reviews of Modern Physics_ , 83(2):587–645, June 2011.  ISSN 0034-6861, 1539-0756.  doi: 10.1103/RevModPhys.83.587.  URL https://link.aps.org/doi/10.1103/RevModPhys.83.587.
  * Bitzek et al. [2006] Erik Bitzek, Pekka Koskinen, Franz Gähler, Michael Moseler, and Peter Gumbsch.  Structural Relaxation Made Simple.  _Physical Review Letters_ , 97(17):170201, October 2006.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.97.170201.  URL https://link.aps.org/doi/10.1103/PhysRevLett.97.170201.
  * Bonnecaze et al. [2020] Roger T. Bonnecaze, Fardin Khabaz, Lavanya Mohan, and Michel Cloitre.  Excess entropy scaling for soft particle glasses.  _Journal of Rheology_ , 64(2):423–431, March 2020\.  ISSN 0148-6055, 1520-8516.  doi: 10.1122/1.5133852.  URL http://sor.scitation.org/doi/10.1122/1.5133852.
  * Charbonneau et al. [2017] Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi, Pierfrancesco Urbani, and Francesco Zamponi.  Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions.  _Annual Review of Condensed Matter Physics_ , 8(1):265–288, March 2017.  ISSN 1947-5454, 1947-5462.  doi: 10.1146/annurev-conmatphys-031016-025334.  URL https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-031016-025334.
  * Cubuk et al. [2017] E. D. Cubuk, R. J. S. Ivancic, S. S. Schoenholz, D. J. Strickland, A. Basu, Z. S. Davidson, J. Fontaine, J. L. Hor, Y.-R. Huang, Y. Jiang, N. C. Keim, K. D. Koshigan, J. A. Lefever, T. Liu, X.-G. Ma, D. J. Magagnosc, E. Morrow, C. P. Ortiz, J. M. Rieser, A. Shavit, T. Still, Y. Xu, Y. Zhang, K. N. Nordstrom, P. E. Arratia, R. W. Carpick, D. J. Durian, Z. Fakhraai, D. J. Jerolmack, Daeyeon Lee, Ju Li, R. Riggleman, K. T. Turner, A. G. Yodh, D. S. Gianola, and Andrea J. Liu.  Structure-property relationships from universal signatures of plasticity in disordered solids.  _Science_ , 358(6366):1033–1037, November 2017\.  ISSN 0036-8075, 1095-9203.  doi: 10.1126/science.aai8830.  URL https://www.sciencemag.org/lookup/doi/10.1126/science.aai8830.
  * Cubuk et al. [2016] Ekin D. Cubuk, Samuel S. Schoenholz, Efthimios Kaxiras, and Andrea J. Liu.  Structural Properties of Defects in Glassy Liquids.  _The Journal of Physical Chemistry B_ , 120(26):6139–6146, July 2016.  ISSN 1520-6106, 1520-5207.  doi: 10.1021/acs.jpcb.6b02144.  URL https://pubs.acs.org/doi/10.1021/acs.jpcb.6b02144.
  * Dyre [2018] Jeppe C. Dyre.  Perspective: Excess-entropy scaling.  _The Journal of Chemical Physics_ , 149(21):210901, December 2018.  ISSN 0021-9606, 1089-7690.  doi: 10.1063/1.5055064.  URL http://aip.scitation.org/doi/10.1063/1.5055064.
  * Dzugutov [1996] Mikhail Dzugutov.  A universal scaling law for atomic diffusion in condensed matter.  _Nature_ , 381(6578):137–139, May 1996.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/381137a0.  URL http://www.nature.com/articles/381137a0.
  * Fan and Ma [2021] Zhao Fan and Evan Ma.  Predicting orientation-dependent plastic susceptibility from static structure in amorphous solids via deep learning.  _Nature Communications_ , 12(1):1506, December 2021.  ISSN 2041-1723.  doi: 10.1038/s41467-021-21806-z.  URL http://www.nature.com/articles/s41467-021-21806-z.
  * Galloway et al. [2020] K. Lawrence Galloway, Xiaoguang Ma, Nathan C. Keim, Douglas J. Jerolmack, Arjun G. Yodh, and Paulo E. Arratia.  Scaling of relaxation and excess entropy in plastically deformed amorphous solids.  _Proceedings of the National Academy of Sciences_ , 117(22):11887–11893, June 2020.  ISSN 0027-8424, 1091-6490.  doi: 10.1073/pnas.2000698117.  URL http://www.pnas.org/lookup/doi/10.1073/pnas.2000698117.
  * Jerolmack and Daniels [2019] Douglas J. Jerolmack and Karen E. Daniels.  Viewing earth’s surface as a soft-matter landscape.  _Nat. Rev. Phys._ , 1:716–730, 2019.  doi: 10.1038/s42254-019-0111-x.  URL https://doi.org/10.1038/s42254-019-0111-x.
  * Keys et al. [2011] Aaron S. Keys, Lester O. Hedges, Juan P. Garrahan, Sharon C. Glotzer, and David Chandler.  Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids.  _Physical Review X_ , 1(2):021013, November 2011\.  ISSN 2160-3308.  doi: 10.1103/PhysRevX.1.021013.  URL https://link.aps.org/doi/10.1103/PhysRevX.1.021013.
  * Kirkpatrick and Wolynes [1987] T. R. Kirkpatrick and P. G. Wolynes.  Stable and metastable states in mean-field Potts and structural glasses.  _Physical Review B_ , 36(16):8552–8564, December 1987.  ISSN 0163-1829.  doi: 10.1103/PhysRevB.36.8552.  URL https://link.aps.org/doi/10.1103/PhysRevB.36.8552.
  * Kob and Andersen [1994] Walter Kob and Hans C. Andersen.  Scaling Behavior in the $\beta$ -Relaxation Regime of a Supercooled Lennard-Jones Mixture.  _Physical Review Letters_ , 73(10):1376–1379, September 1994.  ISSN 0031-9007.  doi: 10.1103/PhysRevLett.73.1376.  URL https://link.aps.org/doi/10.1103/PhysRevLett.73.1376.
  * Lin and Riggleman [2019] Emily Y. Lin and Robert A. Riggleman.  Distinguishing failure modes in oligomeric polymer nanopillars.  _Soft Matter_ , 15(32):6589–6595, 2019.  ISSN 1744-683X, 1744-6848.  doi: 10.1039/C9SM00699K.  URL http://xlink.rsc.org/?DOI=C9SM00699K.
  * Liu and Nagel [2010] Andrea J. Liu and Sidney R. Nagel.  The jamming transition and the marginally jammed solid.  _Annual Review of Condensed Matter Physics_ , 1(1):347–369, 2010.  doi: 10.1146/annurev-conmatphys-070909-104045.
  * Ma et al. [2019] Xiaoguang Ma, Jiachen Liu, Yikang Zhang, Piotr Habdas, and A. G. Yodh.  Excess entropy and long-time diffusion in colloidal fluids with short-range interparticle attraction.  _The Journal of Chemical Physics_ , 150(14):144907, April 2019.  ISSN 0021-9606, 1089-7690.  doi: 10.1063/1.5091564.  URL http://aip.scitation.org/doi/10.1063/1.5091564.
  * Öttinger [2007] Hans Christian Öttinger.  Systematic Coarse Graining: “Four Lessons and A Caveat” from Nonequilibrium Statistical Mechanics.  _MRS Bulletin_ , 32(11):936–940, November 2007\.  ISSN 0883-7694, 1938-1425.  doi: 10.1557/mrs2007.191.  URL http://link.springer.com/10.1557/mrs2007.191.
  * Öttinger et al. [2005] Hans Christian Öttinger, Hans Christian Öttinger, and Hans Christian Öttinger.  _Beyond Equilibrium Thermodynamics_.  John Wiley & Sons, Incorporated, Newy York, UNITED STATES, 2005.  ISBN 978-0-471-72791-0.  URL http://ebookcentral.proquest.com/lib/upenn-ebooks/detail.action?docID=228475.
  * Palmer et al. [1984] R. G. Palmer, D. L. Stein, E. Abrahams, and P. W. Anderson.  Models of Hierarchically Constrained Dynamics for Glassy Relaxation.  _Physical Review Letters_ , 53(10):958–961, September 1984.  ISSN 0031-9007.  doi: 10.1103/PhysRevLett.53.958.  URL https://link.aps.org/doi/10.1103/PhysRevLett.53.958.
  * [26] Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, and David Cournapeau.  Scikit-learn: Machine Learning in Python.  _MACHINE LEARNING IN PYTHON_ , page 6.
  * Ramasubramani et al. [2018] Vyas Ramasubramani, Carl Adorf, Paul Dodd, Bradley Dice, and Sharon Glotzer.  signac: A Python framework for data and workflow management.  pages 152–159, Austin, Texas, 2018.  doi: 10.25080/Majora-4af1f417-016.  URL https://conference.scipy.org/proceedings/scipy2018/vyas_ramasubramani.html.
  * Ramasubramani et al. [2020] Vyas Ramasubramani, Bradley D. Dice, Eric S. Harper, Matthew P. Spellings, Joshua A. Anderson, and Sharon C. Glotzer.  freud: A software suite for high throughput analysis of particle simulation data.  _Computer Physics Communications_ , 254:107275, September 2020.  ISSN 0010-4655.  doi: 10.1016/j.cpc.2020.107275.  URL https://www.sciencedirect.com/science/article/pii/S0010465520300916.
  * Richard et al. [2020] D. Richard, M. Ozawa, S. Patinet, E. Stanifer, B. Shang, S. A. Ridout, B. Xu, G. Zhang, P. K. Morse, J.-L. Barrat, L. Berthier, M. L. Falk, P. Guan, A. J. Liu, K. Martens, S. Sastry, D. Vandembroucq, E. Lerner, and M. L. Manning.  Predicting plasticity in disordered solids from structural indicators.  _Physical Review Materials_ , 4(11):113609, November 2020.  ISSN 2475-9953.  doi: 10.1103/PhysRevMaterials.4.113609.  URL https://link.aps.org/doi/10.1103/PhysRevMaterials.4.113609.
  * Riggleman et al. [2007] Robert A. Riggleman, Jack F. Douglas, and Juan J. de Pablo.  Tuning polymer melt fragility with antiplasticizer additives.  _The Journal of Chemical Physics_ , 126(23):234903, June 2007.  ISSN 0021-9606, 1089-7690.  doi: 10.1063/1.2742382.  URL http://aip.scitation.org/doi/10.1063/1.2742382.
  * Schoenholz et al. [2016] S. S. Schoenholz, E. D. Cubuk, D. M. Sussman, E. Kaxiras, and A. J. Liu.  A structural approach to relaxation in glassy liquids.  _Nature Physics_ , 12(5):469–471, May 2016.  ISSN 1745-2473, 1745-2481.  doi: 10.1038/nphys3644.  URL http://www.nature.com/articles/nphys3644.
  * Schoenholz et al. [2017] Samuel S. Schoenholz, Ekin D. Cubuk, Efthimios Kaxiras, and Andrea J. Liu.  Relationship between local structure and relaxation in out-of-equilibrium glassy systems.  _Proceedings of the National Academy of Sciences_ , 114(2):263–267, January 2017.  ISSN 0027-8424, 1091-6490.  doi: 10.1073/pnas.1610204114.  URL http://www.pnas.org/lookup/doi/10.1073/pnas.1610204114.
  * Stillinger and Debenedetti [2013] Frank H. Stillinger and Pablo G. Debenedetti.  Glass Transition Thermodynamics and Kinetics.  _Annual Review of Condensed Matter Physics_ , 4(1):263–285, April 2013.  ISSN 1947-5454, 1947-5462.  doi: 10.1146/annurev-conmatphys-030212-184329.  URL https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-030212-184329.
  * Tong and Xu [2014] Hua Tong and Ning Xu.  Order parameter for structural heterogeneity in disordered solids.  _Physical Review E_ , 90(1):010401, July 2014\.  ISSN 1539-3755, 1550-2376.  doi: 10.1103/PhysRevE.90.010401.  URL https://link.aps.org/doi/10.1103/PhysRevE.90.010401.
  * Torquato [2018] S. Torquato.  Perspective: Basic understanding of condensed phases of matter via packing models.  _The Journal of Chemical Physics_ , 149(2):020901, 2018.  doi: 10.1063/1.5036657.  URL https://doi.org/10.1063/1.5036657.
  * van Hecke [2009] M van Hecke.  Jamming of soft particles: geometry, mechanics, scaling and isostaticity.  _J. Phys. Condens._ , 22(3):033101, dec 2009.  doi: 10.1088/0953-8984/22/3/033101.  URL https://doi.org/10.1088%2F0953-8984%2F22%2F3%2F033101.
  * Widmer-Cooper and Harrowell [2006a] Asaph Widmer-Cooper and Peter Harrowell.  Free volume cannot explain the spatial heterogeneity of Debye–Waller factors in a glass-forming binary alloy.  _Journal of Non-Crystalline Solids_ , 352(42-49):5098–5102, November 2006a.  ISSN 00223093.  doi: 10.1016/j.jnoncrysol.2006.01.136.  URL https://linkinghub.elsevier.com/retrieve/pii/S0022309306008994.
  * Widmer-Cooper and Harrowell [2006b] Asaph Widmer-Cooper and Peter Harrowell.  Predicting the Long-Time Dynamic Heterogeneity in a Supercooled Liquid on the Basis of Short-Time Heterogeneities.  _Physical Review Letters_ , 96(18):185701, May 2006b.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.96.185701.  URL https://link.aps.org/doi/10.1103/PhysRevLett.96.185701.