# Aging in thermal active glasses

Giulia Janzen  Liesbeth M. C. Janssen<EMAIL_ADDRESS>Department of
Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eindhoven, The Netherlands

###### Abstract

It is well established that glassy materials can undergo aging, i.e., their
properties gradually change over time. There is rapidly growing evidence that
dense active and living systems also exhibit many features of glassy behavior,
but it is still largely unknown how physical aging is manifested in such
active glassy materials. Our goal is to explore whether active and passive
thermal glasses age in fundamentally different ways. To address this, we
numerically study the aging dynamics following a quench from high to low
temperature for two-dimensional passive and active Brownian model glass-
formers. We find that aging in active thermal glasses is governed by a time-
dependent competition between thermal and active effects, with an effective
temperature that explicitly evolves with the age of the material. Moreover,
unlike passive aging phenomenology, we find that the degree of dynamic
heterogeneity in active aging systems is relatively small and remarkably
constant with age. We conclude that the often-invoked mapping between an
active system and a passive one with a higher effective temperature rigorously
breaks down upon aging, and that the aging dynamics of thermal active glasses
differs in several distinct ways from both the passive and athermal active
case.

Introduction – Glasses are disordered solids that exhibit extremely slow
relaxation dynamics Biroli and Garrahan (2013); Debenedetti and Stillinger
(2001); Langer (2014). An important hallmark of glasses is that they undergo
physical aging, i.e., the behavior of the material depends explicitly on its
age Hodge (1995); Berthier and Biroli (2009); Lunkenheimer  _et al._ (2005);
Zhao  _et al._ (2013). In supercooled liquids and glasses, aging dynamics is
often studied by applying a temperature quench toward a lower temperature;
after such a quench, the structural relaxation time tends to increase with the
waiting time (or ’age’ of the material) $t_{w}$ Kob and Barrat (1997); Foffi
_et al._ (2004). In general, this aging behavior can be regarded as an out-of-
equilibrium phenomenon whereby the material seeks to recover equilibrium at
the new temperature Struik (1977); Hutchinson (1995); Barrat and Kob (1999);
Kob and Barrat (2000). More specifically, physical aging of thermal glasses is
qualitatively understood as the gradual approach toward a lower-lying energy
state on the energy landscape Debenedetti and Stillinger (2001).

The recent advent of active matter, i.e., systems whose constituent particles
can move autonomously through the consumption of energy, is fueling renewed
interest in the study of glassy dynamics Janssen (2019); Berthier  _et al._
(2019). Both theory and simulations Szamel  _et al._ (2015); Szamel (2016);
Feng and Hou (2017); Liluashvili  _et al._ (2017); Szamel (2019); Henkes  _et
al._ (2011); Ni  _et al._ (2013); Berthier and Kurchan (2013); Berthier
(2014); Szamel  _et al._ (2015); Flenner  _et al._ (2016); Berthier  _et al._
(2017), as well as recent experiments of synthetic systems Klongvessa  _et
al._ (2019a, b) and living cells Zhou  _et al._ (2009); Parry  _et al._
(2014); Garcia  _et al._ (2015); Bi  _et al._ (2016); Nishizawa  _et al._
(2017); Angelini  _et al._ (2011), have established that dense active matter
can show remarkable similarities with conventional glassy phenomenology. For
example, slow structural relaxation and ultimate kinetic arrest, varying
degrees of fragility, Stokes-Einstein violation, and dynamic heterogeneity
have all been observed in disordered active systems Ediger (2000); Berthier
and Biroli (2011); Binder and Kob (2011); Berthier  _et al._ (2005); Richert
(2002); Ediger and Harrowell (2012); Janssen (2019). A popular model for such
active glassy dynamics is the active Brownian particle (ABP) model, which
combines thermal diffusive motion with a constant self-propulsion speed for
each particle Romanczuk  _et al._ (2012); Ramaswamy (2017, 2010); Lindner and
Nicola (2008); Löwen (2020); Sprenger  _et al._ (2020). The prevalent
qualitative picture of ABP glassy dynamics is that – at least in steady-state
conditions – the presence of active forces generally facilitates cage-breaking
events at high densities Ni  _et al._ (2013). This culminates in overall
faster relaxation dynamics, and hence ABPs are often mapped onto an effective
passive system with a higher effective temperature Howse  _et al._ (2007);
Bechinger  _et al._ (2016).

In contrast to many other hallmarks of glassy dynamics, the manifestation of
physical aging in active glasses is still largely unexplored. A priori it is
not clear how an active glass will age, as the conventional picture of aging,
i.e., the material’s tendency to reach equilibrium via minimization of the
energy, cannot hold in an active non-Hamiltonian system that is inherently far
from equilibrium Vicsek and Zafeiris (2012); Marchetti  _et al._ (2013); Speck
(2016); De Groot and Mazur (2013); Lebon  _et al._ (2008). Moreover, since
aging, activity, and glass formation all constitute different types of non-
equilibrium phenomena, one might expect a complex interplay of various dynamic
processes within such a system. The limited number of active aging studies
reported thus far have focused only on athermal systems Janssen  _et al._
(2017); Mandal and Sollich (2020), and hence it is still unclear to what
extent physical aging in thermal glasses becomes fundamentally different when
introducing activity.

In this work, we explore how the role of temperature – the key parameter in
most conventional (passive) aging studies –, affects the aging dynamics of
thermal active glasses following a temperature quench. We find that the active
aging dynamics of ABPs is governed by a non-trivial and time-dependent
competition between thermal and active effects, which generally precludes the
mapping of the active system onto an effective passive one. In particular,
activity leads to a relative speed-up in structural relaxation as time
progresses, while the degree of dynamic heterogeneity remains remarkably
constant with age. We attribute these findings to activity-enhanced cage
breaking – a reverberation of the well-known steady-state ABP dynamics Ni  _et
al._ (2013). Overall, we conclude that the aging behavior of thermal active
glasses is distinct from both the passive and athermal active case.

Simulation details – We study a two-dimensional (2D) binary mixture of thermal
ABPs. The overdamped equations of motion for each particle $i$ are given by

$\displaystyle\gamma\,\dot{\bm{r}}_{i}=\sum_{i\neq
j=1}^{N}\bm{f}_{ij}+f\,\bm{n}_{i}+\sqrt{2D_{T}}\,\bm{\eta}$ (1)
$\displaystyle\dot{\theta}_{i}=\sqrt{2D_{r}}\,\eta_{\theta}$ (2)

where $\bm{r}_{i}=(x_{i},y_{i})$ represents the particle’s spatial
coordinates, $\theta_{i}$ the rotational coordinate, and the dots denote the
time derivative. The thermal noise is represented by independent white noise
stochastic processes $\bm{\eta}=(\eta_{x},\eta_{y})$ with zero mean and
variance $2k_{B}T/\gamma\delta(t-t^{\prime})$, where $k_{B}$ is the Boltzmann
constant, $T$ the temperature, and $\gamma$ a friction coefficient. In Eq. 2,
$\eta_{\theta}$ is a Gaussian stochastic process with zero mean and variance
$2D_{r}\delta(t-t^{\prime})$. The translational and rotational diffusion
constants are denoted as $D_{T}$ and $D_{r}$, respectively, and we can further
define the ABP persistence time as $\tau_{r}=D_{r}^{-1}$. Each ABP has a
constant self-propulsion speed $f/\gamma$ along a direction
$\bm{n}_{i}=(\cos{\theta}_{i},\sin{\theta}_{i})$; if the magnitude of the
active force, $f$, is zero, Eq. 1 reduces to the equation of motion for a
passive Brownian particle. Lastly, $\bm{f}_{ij}=-\nabla_{i}V(r_{ij})$ is the
interaction force between particles $i$ and $j$, where
$r_{ij}=\left|\textbf{r}_{i}-\textbf{r}_{j}\right|$. For $V$ we use a Lennard-
Jones potential

$V(r_{ij})=4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right]$

with a cutoff distance $r_{ij}=2.5\sigma_{ij}$. In order to prevent
crystallization we use the parameters of the 2D binary Kob-Andersen mixture
Kob and Andersen (1995): $A=65\%$, $B=35\%$, $\epsilon_{AA}=1$,
$\epsilon_{BB}=0.5\epsilon_{AA}$, $\epsilon_{AB}=1.5\epsilon_{AA}$,
$\sigma_{AA}=1$, $\sigma_{BB}=0.88\sigma_{AA}$ and
$\sigma_{AB}=0.8\sigma_{AA}$. We set the density to $\rho=1.2$, the number of
particles to $N=10\,000$ and $D_{T}=\gamma=1$. Results are in reduced units,
where $\sigma_{AA}$, $\epsilon_{AA}$,
$\frac{\sigma_{AA}^{2}\gamma}{\epsilon_{AA}}$, and
$\frac{\epsilon_{AA}}{k_{B}}$ are the units of length, energy, time, and
temperature, respectively. The Brownian dynamics simulations were performed by
integrating the equations of motion using the Euler-Maruyama method with a
step size $\delta t=10^{-4}$.

In order to reach the glass transition temperature $T_{g}$, we slowly decrease
the temperature of the liquid starting from $T=1$. To reach the steady state,
for each temperature $T$, we let the system equilibrate for a time larger than
the structural relaxation time before collecting data. To study the aging
behavior we prepare $100$ independent configurations and we let them
equilibrate at the initial temperature $T_{i}=1$. After this equilibration
process we apply a quench to the final temperature $T_{q}\lesssim T_{g}$ and
then we follow the evolution in time at constant temperature ($T=T_{q}$). In
the passive case we use quenching temperatures between $T_{q}=0.25$ and
$T_{q}=0.4$, while for the active system ($f=0.5$, $D_{r}=0.1,\,1,\,10$) we
typically use $T_{q}=0.25$; we have chosen these values to admit comparisons
between passive and active systems at both the same absolute temperature, and
at the same quasi-effective temperature in steady state. Finally, while here
we consider the aging dynamics following a temperature quench, it may be
expected that a qualitatively similar aging behavior applies when using an
activity quench. We have verified this similarity for a few dynamical
properties, but we will focus solely on the temperature-quench dynamics in the
remainder of this work.

Steady state dynamics – Before discussing the aging dynamics, we first
consider the steady-state dynamics as a benchmark. In Fig. 1(a) we show the
temperature dependence of the $\alpha$ relaxation time for both a passive
($f=0$) and active ($f=0.5,\tau_{r}=D_{r}=1$) system. We find that at any
given temperature below $T\approx 0.8$, the active system always tends to
relax faster than the passive reference case, in agreement with literature
Mandal  _et al._ (2016); Ni  _et al._ (2013); Berthier  _et al._ (2019). In
particular, the temperature at which $\tau_{\alpha}$ reaches a value of
$10^{3}$ is $T\approx 0.4$ for the passive case, and $T\approx 0.3$ for the
active case. To estimate the glass transition temperature more precisely, we
have also fitted both data sets with a power law
$\tau_{\alpha}=\tau_{0}\,(T-T_{c})^{-\Gamma}$, yielding $T_{c}=0.36$ and
$T_{c}=0.23$ for the passive and active mixture, respectively. Note that,
despite the high densities in both systems, the observed differences in
$T_{c}$ are fairly close to the expected effective temperature differences in
the dilute limit Bechinger  _et al._ (2016): For a single ABP with activity
parameters $f=0.5$ and $D_{r}=1$, a temperature of $T=0.25$ would correspond
to a ’passive’ effective temperature of $T_{eff}=0.375$. In the following, we
will compare steady-state active and passive systems at different temperatures
such that the relaxation times are of the same order of magnitude.

Figure 1: (a): Temperature dependence of the structural relaxation time
$\tau_{\alpha}$ for a passive and active system in steady state. In the
passive case ($f=0$, red dots), we report $\tau_{\alpha}$ for
$T=1,0.8,0.7,0.6,0.5,0.45,0.4$. In the active case ($f=0.5$ and
$\tau_{r}=D_{r}=1$, blue stars), we plot $T=1,0.8,0.7,0.6,0.5,0.4,0.35,0.3$.
Lines represent a fit to the power law
$\tau_{\alpha}=\tau_{0}\,(T-T_{c})^{-\Gamma}$. (b): Non-Gaussian parameter
$\alpha_{2}(t)$ as a function of time $t$ in steady state. The red line
corresponds to the passive system at $T=0.4$ and the blue dashed line to the
active system at $T=0.3$. The red dot and blue star indicate the respective
peaks of $\alpha_{2}(t)$. The inset shows the corresponding probability
distributions $P(n)$ of the size $n$ of clusters of mobile particles around
the peak of $\alpha_{2}(t)$.

To quantify the degree of dynamic heterogeneity in passive and active glassy
systems, we first benchmark the non-Gaussian parameter in the steady state Kob
_et al._ (1997); Wisitsorasak and Wolynes (2014). Figure 1(b) shows the time
dependence of $\alpha_{2}(t)$ at two representative temperatures $T=0.3$
($f=0.5$, $\tau_{r}=D_{r}=1$) and $T=0.4$ ($f=0$). It can be seen that the
$\alpha_{2}(t)$ of the active system starts to deviate from the passive case
at a time $t\sim 1$. This time scale can readily be understood in terms of the
ABP persistence time $\tau_{r}=D_{r}^{-1}=1$, i.e., the characteristic time
scale at which the self propulsion force is expected to reorient and become
prominently visible in the dynamics Zöttl and Stark (2016). At intermediate
times (t $>\tau_{r}$), both the passive and active system exhibit a peak in
$\alpha_{2}(t)$, defining the time scale at which the Van Hove function has
the strongest deviations from Gaussianity Fernandez-Rodriguez  _et al._
(2020). In the passive system this peak is higher, meaning that the active
system is less heterogeneous.

To further study spatial correlations between the most mobile particles, one
can compute the cluster size distribution Kob  _et al._ (1997); Weeks  _et
al._ (2000). Here we identify mobile particles as those that move more than a
distance $\Delta r=0.2$ in a time interval $\Delta t$. Two mobile particles
belong to the same cluster if their distance is less than the average radius
of the nearest neighbor shell $r_{c}$ (with $r_{c}$ determined from the radial
distribution function). The cluster size distribution of such mobile particles
thus gives direct insight into the real-space size of dynamically
heterogeneous regions. The probability distribution $P(n)$ of clusters of size
$n$ around the peak position is shown in the inset of Fig. 1. In the passive
case, the probability to find a bigger cluster is higher than in an active
system. Therefore, we can conclude that the passive system is indeed more
heterogeneous than the steady-state active system. Moreover, for both systems
we have also verified that $P(n)$ is dominated by small clusters at short and
long times, while bigger clusters can be found at intermediate time scales
Gebremichael  _et al._ (2001).

Figure 2: Mean-squared displacements as a function of time $t$ at $T_{q}=0.25$
for (a) $t_{w}=0$ and (b) $t_{w}=10$. The solid curves correspond to active
systems with $f=0.5$ and different persistence times $\tau_{r}$; the dashed
red line corresponds to a passive system. The inset of panel (a) shows the
waiting-time dependence of the $\alpha$ relaxation time for different
$\tau_{r}$; dashed lines represent a fit to the power law
$\tau_{\alpha}\propto t_{w}^{\delta}$. For $\tau_{r}=0.1$ we obtain a fitted
exponent of $\delta=0.52$, while for $\tau_{r}=10$ we find $\delta=0.24$. In
panel (b) the arrows indicate which active system yields the highest MSD at
the corresponding time $t$ (colors are the same as in panel a).

Aging is governed by a competition between thermal and active effects – To
characterize the aging behavior, we first compare an active and passive
thermal system quenched to the same temperature $T_{q}$. Figure 2 shows
representative mean-squared displacements (MSDs) as a function of time $t$ for
two waiting times $t_{w}$ and various persistence times $\tau_{r}$ at
$T_{q}=0.25$. It can be seen that the MSD of the active thermal system is
always equal to or higher than the MSD of the passive reference system; we
have verified that this is true for all persistence times and waiting times
studied (up to $t_{w}=1000$). To analyze the active aging dynamics in more
detail, we distinguish between three regimes: a short-time regime
$t\ll\tau_{r}$, a long-time regime $t\gg\tau_{r}$, and a non-trivial
intermediate-time window $t\sim\tau_{r}$ separating the short- and long-time
dynamics. In the short-time regime, we find that the aged MSD of the active
system is virtually indistinguishable from that of the passive reference
system at the same $T_{q}$. This finding, which holds for all considered
values of $\tau_{r}$ and $t_{w}$ and is also consistent with the short-time
MSD of ideal ABPs in steady state Bechinger  _et al._ (2016), thus suggests
that the short-time dynamics of an active thermal glass is fully governed by
thermal effects, regardless of the material’s activity and age.

In the opposite limit of $t\gg\tau_{r}$, we observe a notable effect of the
activity: The long-time dynamics is significantly sped up by the presence of
active forces. This relative speedup depends monotonically on the persistence
time, i.e. a larger $\tau_{r}$ (smaller $D_{r}$) yields faster dynamics. We
also note that for the smallest persistence time considered here,
$\tau_{r}=0.1$, the active system’s MSD deviates only marginally from that of
the passive reference system. This result holds regardless of the waiting
time, and can be understood by the fact that a small $\tau_{r}$ causes
particles to undergo fast reorientation, rendering the self-propulsion term
less effective Liluashvili  _et al._ (2017); Mandal and Sollich (2020).
Overall, the above findings lead to our first main conclusion: The time-
dependent aging dynamics of an active thermal glass is governed by a
competition between thermal and active effects. Specifically, an active glass
initially behaves very similar to a passive glass at the same temperature, but
its long-time dynamics is controlled by activity. Note that Mandal and Sollich
Mandal and Sollich (2020) also reported an activity-dominated long-time regime
for athermal systems (referred to as ’ADA’); however, in contrast to Ref.
Mandal and Sollich (2020), the short-time aging dynamics in our case is
inherently governed by the temperature.

To further quantify the long-time relaxation, we have also extracted the
$\alpha$ relaxation times as a function of waiting time (inset of Fig. 2(a)).
This waiting-time dependence follows a power law $\tau_{\alpha}\propto
t_{w}^{\delta}$ – analogous to simple aging in passive systems Kob and Barrat
(1997) and in athermal active systems Mandal and Sollich (2020) – with an
exponent $\delta$ that decreases monotonically with $\tau_{r}$. Thus, as the
system gets older, the active long-time dynamics will become increasingly
faster compared to its passive analogue at the same $T_{q}$.

At intermediate times $t$, i.e. in the crossover regime between the
temperature-dominated (short-time) and activity-dominated (long-time) regimes,
the roles of activity and age become more complex. In particular we find that
the ordering of MSD curves now depends on both $\tau_{r}$ and $t_{w}$. Careful
inspection of Fig. 2 reveals that for $t_{w}=0$ (panel a) a larger $\tau_{r}$
always leads to a higher MSD, but for longer waiting times ($t_{w}=10$, panel
b) this trend is violated. Specifically, for $t_{w}=10$, we find that when
$t\sim 10^{-1}$ the active system with $\tau_{r}=0.1$ is the fastest (pink
arrow in Fig. 2(b)), but when $t\sim 1$ the system with $\tau_{r}=1$ becomes
faster (blue arrow). Finally, when $t\gtrsim 10$ the system with $\tau_{r}=10$
yields the fastest dynamics (green arrow) and we recover the scenario of the
long-time limit. We have verified that a similar picture also applies for
other finite waiting times (see supplementary material). We can rationalize
this finding by noting that the persistence time $\tau_{r}$ is the intrinsic
time scale of the active system, and the self-propulsion term can only start
to become effective when the relevant time scale becomes comparable to
$\tau_{r}$. In steady-state conditions this merely requires $t\sim\tau_{r}$,
but in the case of aging the active system also needs to be sufficiently old
($t_{w}\gtrsim\tau_{r}$) to observe this effect.

Figure 3: Structural relaxation times $\tau_{\alpha}$ as a function of waiting
time $t_{w}$ for initial temperature $T_{i}=1$. The respective quenching
temperatures for the passive ($f=0$) and active ($f=0.5,\tau_{r}=D_{r}=1$)
system are (a) $T_{q}=0.375$ and $T_{q}=0.25$, and (b) $T_{q}=0.35$ and
$T_{q}=0.25$. The dashed lines represent a fit to the power law
$\tau_{\alpha}\propto t_{w}^{\delta}$.

The ’effective temperature’ changes with age – We now seek to establish
whether the long-time aging dynamics of an active glass can be mapped onto a
passive system with a different effective temperature. To this end, we fix
$\tau_{r}=D_{r}=1$ and compare the active system to its passive counterpart at
a higher quenching temperature $T_{q}$. Figure 3 shows representative $\alpha$
relaxation times as a function of waiting time $t_{w}$ for two quenching
temperatures $T_{q}$; the values of $T_{q}$ for the active and passive system
are chosen such that the relaxation times in steady state are of the same
order of magnitude. There are two important findings we infer from these data.
First, the $\alpha$ relaxation times always follow a power law as a function
of age, i.e. $\tau_{\alpha}\propto t_{w}^{\delta}$. The fitted exponents
$\delta$ are also in good agreement with the values reported in Ref. Mandal
and Sollich (2020) for athermal systems (see also supplemental material).
Second, we find that for small $t_{w}$ the relaxation dynamics of the active
system is generally slower than the passive reference system, whereas for
large $t_{w}$ the relaxation of thermal ABPs becomes faster. Specifically, the
active system at $T_{q}=0.25$ surpasses the passive curve for $T_{q}=0.35$
around $t_{w}\sim 10$ (Fig. 3(a)), while for the passive system with
$T_{q}=0.375$ the crossing point lies near $t_{w}\sim 100$ (Fig. 3(b)). Hence,
the simple picture of ABPs as hot colloids with a fixed effective temperature
Cugliandolo  _et al._ (2019) breaks down during aging.

The above finding is in stark contrast with the athermal results of Ref.
Mandal and Sollich (2020), which showed virtually identical aging dynamics for
active and passive glasses in the same parameter range
($\tau_{r}=1,T_{q}=0.37$). Our work thus implies that the addition of thermal
noise in an active glassy system can change the aging dynamics profoundly, and
leads to our second main conclusion: There is no rigorous mapping possible for
the aging behavior of a thermal active system onto a passive system at a
higher temperature, since the active system will always have a faster
relaxation after a certain $t_{w}$. That is, the effective temperature of a
thermal active glass evolves, and increases, with age.

To study the role of the initial temperature $T_{i}$, we have also tested the
relation $F^{(T_{i,1})}(k,t_{w},t+t_{w})\sim
F^{(T_{i,2})}(k,t_{w}+t_{(2,1)},t+t_{w}+t_{(2,1)})$, for the intermediate
scattering functions, with $t_{(2,1)}$ a fit parameter. This mapping has been
previously reported for passive thermal systems Kob and Barrat (1997), and
here we find that it also holds for our active thermal samples, at least for
$T_{i,1}=1$ and $T_{i,2}=5$ (see supplementary material). We can rationalize
this relation by considering that, as the system goes from $T_{i,2}$ to
$T_{q}$, it will visit the configuration at $T_{i,2}>T_{i,1}>T_{q}$, so
$t_{(2,1)}$ is the time needed for the system starting at $T_{i,2}$ to reach
the same relaxation as the one found by a system equilibrated at $T_{i,1}$ at
a fixed $t_{w}$ Kob and Barrat (1997). Overall, we can thus conclude that the
effect of $T_{i}$ on the aging relaxation dynamics is qualitatively similar
for thermal active and passive systems.

Figure 4: Non-Gaussian parameters as a function of time $t$ for waiting times
$t_{w}=1,10,50,80$. (a) Passive system at quenching temperature $T_{q}=0.4$.
(b) Active system ($f=0.5$, $\tau_{r}=D_{r}=1$) at $T_{q}=0.3.$

Non-Gaussianity and dynamic heterogeneity remain constant with age – We now
turn to an aspect of glassy dynamics that has hitherto remained unexplored for
aging active systems, namely dynamic heterogeneity. Figure 4 shows how the
non-Gaussian parameters $\alpha_{2}(t)$ for a passive and active thermal
system, quenched to $T_{q}=0.4$ and $T_{q}=0.3$ respectively, evolve with
$t_{w}$ as the system attempts to recover its steady-state behavior (cf. Fig.
1(b)). In the passive case (Fig. 4(a)), we find that for small $t_{w}$ the
peak height of $\alpha_{2}(t_{w},t+t_{w})$ is below the steady-state value of
$\alpha_{2}\approx 1.2$, thus indicating that the passive relaxation dynamics
is initially less heterogeneous than in steady state; as the system becomes
older, the peak height grows and reaches the passive steady-state maximum
after $t_{w}\approx 80$. This gradual increase in heterogeneity is, however,
strikingly different from the behavior we see in our active systems. Figure
4(b) shows that in the active case the peak height of the non-Gaussian
parameter remains remarkably constant with age, and is in fact always
consistent with its steady-state value ($\alpha_{2}\approx 1.1$). Thus, the
degree of dynamic heterogeneity is manifestly age-independent for active
thermal glass-formers. The only significant change with $t_{w}$ for our active
system is that the peak position of $\alpha_{2}(t_{w},t+t_{w})$ shifts in time
to ultimately recover the active steady-state dynamics. We have verified for
both the active and passive case that the same trend can be found for other
values of $T_{q}$.

Finally, let us examine the non-Gaussian parameters and cluster size
distributions during aging for different temperatures. Figure 5 compares
$\alpha_{2}(t_{w},t+t_{w})$ for an active system at $T_{q}=0.25$ with a
passive system at the same quenching temperature and at $T_{q}=0.375$. Here
again we can identify a time-dependent competition between thermal and active
effects: For sufficiently small $t_{w}$ (panel a) and small $t<\tau_{r}$
($\tau_{r}=1$), the non-Gaussian parameters for the active and passive samples
at $T_{q}=0.25$ are very similar, but as time progresses the effect of
activity becomes more prominent. To see this effect, let us first consider
young glasses with $t_{w}=1$ (Fig. 5(a)). At intermediate time scales
$t\gtrsim\tau_{r}$, the peak of $\alpha_{2}(t_{w},t+t_{w})$ is somewhat lower
for the active system (Fig. 5(a)), implying that the active thermal sample
undergoes less heterogeneous relaxation than its passive counterpart. However,
compared to an equally young but higher-temperature passive system
($T_{q}=0.375$), the active glass is slightly more dynamically heterogeneous,
at least at the time scale where $\alpha_{2}(t)$ peaks. These differences are
also reflected in the corresponding cluster size distributions (inset panel
a).

Importantly, as the waiting time increases ((Fig. 5(b)), we find that the
active system at $T_{q}=0.25$ exhibits a significantly lower peak than both
the $T_{q}=0.25$ and $T_{q}=0.375$ passive sample. This is in fact a direct
consequence of the trend reported in Fig. 4: In the active case the peak of
the non-Gaussian parameter remains approximately constant with age, while in
the passive case it grows. The measured size distributions of mobile clusters
(inset panel b) also confirm that, for $t_{w}=100$, the largest correlated
clusters are found for the passive system. Moreover, we have verified that in
the passive case, regardless of $t_{w}$, the particle cluster size grows with
decreasing temperature, consistent with the scenario in steady state Donati
_et al._ (1999). Overall, the above results unambiguously establish that, in
terms of the microscopic relaxation dynamics, there is no simple mapping
possible between an aging active thermal glass and a passive system with a
different, but constant, effective temperature. We argue that the less
pronounced, and approximately age-independent, degree of dynamic heterogeneity
in active thermal glass-formers stems from the fact that the self-propulsion
term enables particles to escape more easily, and more autonomously, from
their local cages Ni  _et al._ (2013); this not only induces faster overall
relaxation, but it also curtails the need for strongly cooperative, i.e.,
heterogeneous, relaxation dynamics.

Figure 5: Non-Gaussian parameters as a function of time $t$ for (a) $t_{w}=1$
and (b) $t_{w}=100$. The active system ($f=0.5$, $\tau_{r}=D_{r}=1$) is
quenched to $T_{q}=0.25$, whereas the passive system is quenched to
$T_{q}=0.25$ or $T_{q}=0.375$. The insets show the respective cluster size
distributions $P(n)$ around the corresponding peaks of
$\alpha_{2}(t_{w},t+t_{w})$.

Conclusions – In summary, our work reveals that thermal active glasses share
some non-trivial aspects of aging phenomenology with athermal active and
passive thermal glasses, but there are also several fundamental differences.
Notably, the aging relaxation dynamics of active systems is governed by a
time-dependent competition between thermal and active effects, with the ABP
persistence time controlling both the time scale and magnitude of the
activity-enhanced speedup in dynamics. From our results it has not been
possible to map the aging behavior of an active system onto a passive system
at a different temperature; instead, we find that the manifested effective
temperature changes both with time and with age. Moreover, for an active
thermal system, the dynamics is generally less heterogeneous than in the
passive case, and the degree of non-Gaussianity remains remarkably constant
with age. We hypothesize that both the relative speedup in dynamics and the
relatively weak dynamic heterogeneity in aged active glass-formers is due to
activity-enhanced cage breaking. Finally, while we have focused on the aging
dynamics following a temperature quench, we have verified that a similar
phenomenology applies when using an activity-quench protocol (see
supplementary material). In particular, at intermediate persistence times
($\tau_{r}\sim 1$), quenching the temperature at fixed activity or quenching
the activity at fixed temperature is essentially the same Mandal and Sollich
(2020).

Acknowledgments – We thank Vincent Debets and Kees Storm for their critical
reading of the manuscript. This work has been financially supported by the
Dutch Research Council (NWO) through a Physics Projectruimte grant.

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