# Extremely Persistent Dense Active Fluids

Grzegorz Szamel  Department of Chemistry, Colorado State University, Fort
Collins, Colorado 80523, USA  Elijah Flenner  Department of Chemistry,
Colorado State University, Fort Collins, Colorado 80523, USA

###### Abstract

We examine the dependence of the dynamics of three-dimensional active fluids
on persistence time $\tau_{p}$ and average self-propulsion force $f$. In the
large persistence time limit many properties of these fluids become
$\tau_{p}$-independent. These properties include the mean squared velocity,
the self-intermediate scattering function, the shear-stress correlation
function and the low-shear-rate viscosity. We find that for a given $f$ in the
large $\tau_{p}$ limit the mean squared displacement is independent of the
persistence time for times shorter than $\tau_{p}$ and the long-time self-
diffusion coefficient is proportional to the persistence time. For a large
range of self-propulsion forces the large persistence time limits of many
properties depend on $f$ as power laws.

Particles that use energy from their environment to perform persistent motion,
i.e. self-propelled or active particles, behave in surprising and interesting
ways Marchetti2013 ; Elgeti2015 ; Bechinger2016 . Recently, novel intermittent
dynamics was identified in extremely persistent dense homogeneous two-
dimensional systems Mandal2020 ; Mandal2021 ; Keta2022 ; Keta2023 . It was
shown that these systems evolve through sequences of mechanical equilibria in
which self-propulsion forces balance interparticle interactions. Here we
examine extremely persistent dense homogeneous three-dimensional active fluids
in which the interparticle interactions never manage to balance the self-
propulsion forces. Many properties of these fluids become $\tau_{p}$
independent and scale with the root-mean-squared self-propulsion force $f$ as
power laws.

We recall that the phase space of active systems is much larger than that of
passive ones. At a minimum, one has to specify the average strength of active
forces and their persistence time in addition to the set of parameters
characterizing the corresponding passive system. If one considers athermal
active systems, this results in a three-dimensional control parameter space.
Thus, when comparing results of diverse studies, one needs to specify the path
in the parameter space that one is following.

Early studies of dense homogeneous active systems focused on the glassy
dynamics and the active glass transition Berthier2013 ; Ni2013 ; Berthier2014
; Szamel2015 ; Mandal2016 ; Flenner2016 ; Berthier2017 ; Klongvessa2019 ;
Berthier2019 ; Janssen2019 . These studies considered a limited range of
persistence times and often examined the behavior of the systems at constant
active temperature $T_{a}$ that characterizes the long-time motion of an
isolated active particle. At constant $T_{a}$, with increasing $\tau_{p}$ the
strength of active forces decreases and dense active systems typically
glassify, see Fig. 2c of Ref. Keta2022  for a recent example. Thus, to
investigate the effects of extremely persistent active forces it is common to
fix their strength while increasing their persistence time Mandal2020 ;
Mandal2021 ; Keta2023 .

Recent simulational studies of dense two-dimensional active systems
demonstrated that, for large persistence times, there is a new phase between
fluid and glass phases with intermittent dynamics Mandal2020 ; Mandal2021 ;
Keta2023 . Importantly, in the large $\tau_{p}$ limit the relaxation happens
on the time scale of the persistence time, and the mean-square displacement
and the two-point overlap function exhibit well-defined limits when plotted
versus time rescaled by the persistence time Mandal2021 . This observation
suggests that the dynamics at large persistence times may be studied by
assuming that the active and interparticles forces converge to a force-
balanced state for times much less than the persistence time and that all the
rearrangements happen on the time-scale of the persistence time. This approach
is termed activity-driven dynamics Mandal2021 ; MandalSollich2020 .

A recent study that used the activity-driven dynamics algorithm Keta2022
discovered very interesting extreme persistence time limit dynamics of two-
dimensional active systems. Displacement distributions were found to be non-
Gaussian and to exhibit fat exponential tails. An intermediate-time plateau in
the mean-square displacement was absent. Instead, a region scaling with time
as $t^{\beta}$ with $\beta\approx 1.6$ for times less than the persistence
time was identified. The complex intermittent dynamics resembled that found in
zero-temperature driven amorphous solids, but with some important differences.

Here we focus on extremely persistent three-dimensional dense homogeneous
active fluids. Like Keta et al. Keta2022 ; Keta2023 , we use $\tau_{p}$ and
$f$ as our control parameters. In the systems we investigated, for large
enough $\tau_{p}$ the mean square velocity saturates at a non-zero value
determined by $f$. The interparticle forces never manage to balance completely
the active foces and the systems relax on the time scale shorter than the
persistence time. Therefore, the infinite $\tau_{p}$ limits of our systems lie
in the un-jammed phase of the active yielding phase diagram studied (in two
dimensions) by Liao and Xu LiaoXu . Several properties of these systems become
$\tau_{p}$-independent and scale as non-trivial power laws with $f$. In the
following we describe the systems we studied and then present and discuss our
observations.

We study a three-dimensional 50:50 binary mixture of spherically symmetric
active particles interacting via the Weeks-Chandler-Andersen potential,
$V_{\alpha\beta}=4\epsilon\left[\left(\sigma_{\alpha\beta}/r\right)^{12}-\left(\sigma_{\alpha\beta}/r\right)^{6}\right]$
for $r<\varsigma_{\alpha\beta}=2^{1/6}\sigma_{\alpha\beta}$ and 0 otherwise.
Here, $\alpha$, $\beta$ denote the particles species $A$ or $B$ and $\epsilon$
is the unit of energy. The distance unit is set by $\sigma_{BB}=1.0$,
$\sigma_{AA}=1.4$, and $\sigma_{AB}=1.2$. We study the number density
$N/V=0.451$, which corresponds to the volume fraction $\phi=\pi
N[\varsigma_{AA}^{3}+\varsigma_{BB}^{3}]/(12V)=0.625$.

We use the athermal active Ornstein-Uhlenbeck particle model Szamel2014 ;
Maggi2015 ; Fodor2016 . The equation of motion for the position
$\mathbf{r}_{n}$ of particle $n$ is

$\xi_{0}\dot{\mathbf{r}}_{n}=\mathbf{F}_{n}+\mathbf{f}_{n},$ (1)

where $\mathbf{F}_{n}=-\sum_{m\neq n}\nabla_{n}V(r_{nm})$ and $\mathbf{f}_{n}$
is the active force. $\xi_{0}=1$ is the friction coefficient of an isolated
particle and $\xi_{0}\sigma_{BB}/\epsilon$ sets the unit of time. The equation
of motion for $\mathbf{f}_{n}$ reads

$\tau_{p}\dot{\mathbf{f}}_{n}=-\mathbf{f}_{n}+\bm{\zeta}_{n},$ (2)

where $\tau_{p}$ is the persistence time of the self-propulsion and
$\bm{\zeta}_{n}$ is a Gaussian white noise with zero mean and variance
$\left<\bm{\zeta}_{n}(t)\bm{\zeta}_{m}(t^{\prime})\right>_{\mathrm{noise}}=2\xi_{0}T_{a}\mathbf{I}\delta_{nm}\delta(t-t^{\prime})$,
where $\left<\ldots\right>_{\mathrm{noise}}$ denotes averaging over the noise
distribution, $T_{a}$ is a single particle effective temperature, $\mathbf{I}$
is the unit tensor and we set the Boltzmann constant $k_{B}=1$. The root-mean
square strength of active forces is $f=\sqrt{3T_{a}/\tau_{p}}$.

Figure 1: The persistence time dependence of the mean square velocity
$v^{2}=\left<\dot{\mathbf{r}}^{2}\right>$ for fixed active force strength $f$.
The velocity decreases with $\tau_{p}$ until $\tau_{p}(f)$ method  and then
saturates. The vertical dashed lines indicate $\tau_{p}(f)$.

We start be examining the persistence time dependence of the mean square
velocity,
$v^{2}\equiv\left<\sum_{n}\dot{\mathbf{r}}_{n}^{2}\right>=\left<\sum_{n}\mathbf{F}_{n}^{2}+\sum_{n}2\mathbf{F}_{n}\cdot\mathbf{f}_{n}\right>+Nf^{2}$,
which is shown in Fig. 1. We observe that with increasing persistence time
$v^{2}$ decreases and then saturates. For each active force strength $f$ we
define a characteristic persistence time $\tau_{p}(f)$ at which $v^{2}$ stops
changing method . The cancellation of the interparticle and active forces is
never complete, unlike in systems investigated in Ref. Mandal2021 . For the
range of $f$ that we studied, our systems are never at the bottom of an
effective potential consisting of the potential energy tilted by the terms
originating from the active forces Mandal2021 ; Keta2023  and do not become
arrested on the time scale of the persistence time. Therefore, in the infinite
$\tau_{p}$ limit our systems fall into the fluid phase of the three-
dimensional version of the phase diagram of Liao and Xu LiaoXu .

Figure 2: The mean-square-displacement $\left<\delta r^{2}(t)\right>$ at fixed
$f$ for several $\tau_{p}$. For shorter $\tau_{p}$ the mean squared
displacements display a glassy plateau followed by diffusive motion. With
increasing $\tau_{p}$ an extended superdiffusive region appears that is
analyzed in Fig. 4.

Next, we examine the persistence time dependence of the mean squared
displacement (MSD)

$\left<\delta
r^{2}(t)\right>=N^{-1}\left<\sum_{n}[\mathbf{r}_{n}(t)-\mathbf{r}_{n}(0)]^{2}\right>,$
(3)

shown in Fig. 2. At short times the motion is ballistic and it is determined
by the mean square velocity, $\left<\delta r^{2}(t)\right>=v^{2}t^{2}$
Szamel2015 . Thus, the short-times MSD decreases with increasing $\tau_{p}$
and becomes constant beyond $\tau_{p}(f)$.

Figure 3: The long time diffusion coefficient for fixed active force strength
$f$ as a function of persistence time $\tau_{p}$. The diffusion coefficient
grows with increasing $\tau_{p}$ and is proportional to $\tau_{p}$ in the
large $\tau_{p}$ limit.

At long times, the MSD exhibits diffusive behavior. The self-diffusion
coefficient, $D=\lim_{t\to\infty}\left<\delta r^{2}(t)\right>/(6t)$, is shown
in Fig. 3. For a given $f$ it monotonically increases with increasing
$\tau_{p}$. At large $\tau_{p}$ we find that $D\sim\tau_{p}$, indicated by the
dashed lines.

Figure 4:  The mean-square displacement $\left<\delta
r^{2}(t)\right>/(v^{2}t^{2})$ for $f=5.48$. For large $\tau_{p}$, at short
times the motion is ballistic (not shown), and then there is a super-diffusive
regime. For very large $\tau_{p}$ the super-diffusive motion speeds up and the
resulting behavior is approximately ballistic. For $t>\tau_{p}$ long-time
diffusive motion is observed. In the $\tau_{p}\rightarrow\infty$ limit the
system stays in the second ballistic regime, with velocity $v_{1}$.

We found a surprising time-dependence of the MSD between the initial ballistic
and the long-time diffusive regimes. In Fig. 4 we show the MSD divided by
$v^{2}t^{2}$ to show this time-dependence more clearly. The MSD exhibits a
superdiffusive behavior that seems to approach a large $\tau_{p}$ master
curve. The superdiffusive behavior does not follow a single power law.
Instead, a second, intermediate time ballistic regime appears, with velocity
$v_{1}$. This is in contrast to the finding of Keta et al. Keta2022  who
observed intermediate power law behavior $\left<\delta r^{2}(t)\right>\sim
t^{\beta}$ with $\beta\approx 1.6$. In the $\tau_{p}\to\infty$ limit the
systems stays in the second ballistic regime.

Results shown in Fig. 4 suggest that in the large $\tau_{p}$ limit the
diffusion can be thought of as a random walk consisting of of steps of length
$v_{1}\tau_{p}$ taken every $\tau_{p}$. This picture rationalizes the observed
scaling $D\sim\tau_{p}$.

Figure 5: The distribution of velocities for $f=0.0548$ for a broad range of
persistence times $\tau_{p}$. The distributions are non-Gaussian; the tails
become more prominent with increasing $\tau_{p}$ until $\tau_{p}(f)$ when the
distributions overlap. The dashed line represents a Maxwell distribution.

In Fig. 5 we show the velocity distributions. As found by Keta et al. Keta2022
, the distributions are strongly non-Gaussian. Their broad tails become more
prominent with increasing $\tau_{p}$ until $\tau_{p}(f)$. For
$\tau_{p}>\tau_{p}(f)$ the distributions overlap.

Figure 6: The self-intermediate scattering function $F_{s}(k;t)$ for
$f=0.0548$ and a range of persistence times $\tau_{p}$. The relaxation time
decreases with increasing persistence time until around $\tau_{p}(f)$ where
$F_{s}(k;t)$ becomes independent of persistence time. The inset shows the
$\tau_{p}$ dependence of parameter $\Gamma$ of fit to
$F_{s}(k;t)=ae^{-(t/\tau_{s})^{\Gamma}}$ where $a\leq 1$.

The evolution of the mean square displacement with the persistence time is
reflected in the $\tau_{p}$ dependence of the self-intermediate scattering
function

$F_{s}(k;t)=\frac{1}{N}\left<\sum_{n}e^{i\mathbf{k}\cdot(\mathbf{r}_{n}(t)-\mathbf{r}_{n}(0))}\right>.$
(4)

We chose $k=5.3$, which is approximately equal the first peak of the total
static structure factor. In Fig. 6 we show $F_{s}(k;t)$ for $f=0.0548$. With
increasing $\tau_{p}$ the intermediate time glassy plateau disappears and the
decay changes from stretched exponential, to exponential, then to compressed
exponential. Shown in the inset to Fig. 6 is the parameter $\Gamma$ obtained
from fits to $F_{s}(k;t)=ae^{-(t/\tau_{s})^{\Gamma}}$ where we restrict $a\leq
1$. $\Gamma$ increases with increasing $\tau_{p}$ and reaches a plateau above
$\tau_{p}\approx 94$.

Figure 7:  The large $\tau_{p}$ limit of $v^{2}$, $D/\tau_{p}$ and
$1/\tau_{s}$ as a function of the self-propulsion strength $f$. The lines are
power law fits where the exponent for $v^{2}$, $D/\tau_{p}$ and $1/\tau_{s}$
are $2.6\pm 0.1$, $2.7\pm 0.1$ and $1.3\pm 0.1$, respectively.

We find that the large persistence time limits of several properties discussed
above depend on the strength of the active forces as power laws. In Fig. 7 we
show the large $\tau_{p}$ limits of $v^{2}$ (squares), $D/\tau_{p}$ (circles)
and $\tau_{s}$ (triangles). We find that the former two quantities follow a
power law with $f$ with statistically the same exponent, $2.6\pm 0.1$ for
$v^{2}$ and $2.5\pm 0.1$ for $D/\tau_{p}$. The power law of the relaxation
time, $\tau_{s}\sim f^{-1.3}$ can be related to that of $v^{2}$; in the large
$\tau_{p}$ limit $F_{s}$ decays on the time scale on which a particle moves
over its diamater, which scales as $v$.

Figure 8: Normalized shear-stress correlation function for $f=0.0548$ and a
large range of persistence times. The shear stress relaxation time initially
increases with $\tau_{p}$, but becomes constant for $\tau_{p}>\tau_{p}(f)$.
Additionally, $\Sigma_{xy}(0)$ grows with $\tau_{p}$ until approximately
$\tau_{p}(f)$ and then it becomes approximately constant (inset).

The above discussed quantities describe the single particle motion in our
many-particle systems. To access collective properties of these systems we
investigated the $\tau_{p}$ dependence of the stress fluctuations and the
rheological response. First, we examined the shear-stress correlation function
$\Sigma_{\alpha\beta}(t)=V^{-1}\left<\sigma_{\alpha\beta}(t)\sigma_{\alpha\beta}(0)\right>$,
where

$\sigma_{\alpha\beta}(t)=-\frac{1}{2}\sum_{n}\sum_{m\neq
n}\frac{r_{nm}^{\alpha}r_{nm}^{\beta}}{r_{nm}}\frac{dV(r_{nm})}{dr_{nm}},$ (5)

and $r_{nm}^{\alpha}$ is the $\alpha$ component of the distance vector between
particle $n$ and particle $m$.

In Fig. 8 we report the normalized shear stress correlation function,
$\Sigma_{xy}(t)/\Sigma_{xy}(0)$, for $f=0.0548$ and a large range of
persistence times. For small $\tau_{p}$ there is a rapid decay to an emerging
plateau followed by a slow decay to zero. With increasing $\tau_{p}$, the
decay of $\Sigma_{xy}(t)$ becomes more exponential and it is exponential above
$\tau_{p}(f)$. In the inset we show the dependence of the initial value,
$\Sigma_{xy}(0)$, on $\tau_{p}$. We see that the initial value first grows
with $\tau_{p}$ and then plateaus.

Figure 9: Average shear stress $\left<\sigma_{xy}\right>$ divided by the shear
rate $\dot{\gamma}$ at constant $f=0.548$ for a wide range of persistence
times. The zero-shear-rate viscosity $\eta$ can be obtained from the small
shear rate plateau. There is a decrease in $\eta$ with increasing $\tau_{p}$
$\eta$ followed by a saturation.

To probe the rheological response of our active systems we simulated shear
flow by adding to Eq. (1) a bulk non-conservative force
$\mathbf{F}_{n}^{\dot{\gamma}}=\xi_{0}\dot{\gamma}y_{n}\mathbf{e}_{x}$ with
Lees-Edwards boundary conditions LeesE . In Fig. 9 we show the average shear
stress, $\left<\sigma_{xy}\right>/\dot{\gamma}$, for $f=0.548$ and a large
range of range of $\tau_{p}$. The flow curves for other strengths of the self-
propulsion have the same main features. The limiting zero-shear-rate viscosity
$\eta$ can be obtained from the small $\dot{\gamma}$ pleateus.

Figure 10: The viscosity $\eta$ at fixed average active forces strength versus
$\tau_{p}$. The viscosity initially decreases with increasing $\tau_{p}$ and
then becomes constant above $\tau_{p}(f)$, which is shown as vertical dashed
lines.

In Fig. 10 we show the $\tau_{p}$ dependence of the zero-shear-rate viscosity.
We find that $\eta$ initially decreases and reaches a $\tau_{p}$-independent
plateau above $\tau_{p}(f)$.

Figure 11:  The large $\tau_{p}$ limit of the relaxation time of
$\Sigma_{(}xy)(t)/\Sigma_{xy}(0)$ (defined as when this function equals
$e^{-1}$) and of viscosity $\eta$ as a function of the self-propulsion
strength $f$. The lines are power law fits where the exponents for
$\tau_{\eta}$ and $\eta$ $-1.2\pm 0.1$ and $-0.60\pm 0.1$, respectively.

Again, we find that large $\tau_{p}$ limits of collective properties depend on
$f$ a power laws. In Fig. 11 we show the dependence of the large $\tau_{p}$
limits of the relaxation time of the normalized stress tensor autocorrelation
function and of the viscosity on the strength of the self-propulsion.

When analyzing the dynamics in passive systems, one usually tries to make
connection between the average distribution of the particles and their
dynamics. To check how the average arrangement of the particles in our active
systems changes with increasing persistence time we evaluated the steady-state
structure factor

$S(k)=\frac{1}{N}\left<\sum_{n,m}e^{i\mathbf{k}\cdot(\mathbf{r}_{n}-\mathbf{r}_{m})}\right>.$
(6)

In Fig. 12 we show that the peak height of the structure factor initially
decreases with increasing persistence time, which nicely correlates with
relaxation getting faster and viscosity decreasing. The peak height then
saturates at persistence time around $\tau_{p}(f)$. However, the structure
factors for $\tau_{p}\geq\tau_{p}(f)$ still look liquid-like homogeneous . It
is not obvious at all from these structure factors that the MSD exhibits two
ballistic regimes and $F_{s}(k;t)$ is well fitted by a compressed exponential.
We conclude that to describe the dynamics of extremely persistent dense active
fluids one cannot rely upon static structure factors only.

Figure 12: The persistence time dependence of the steady-state structure
factor $S(k)$ for $f=0.548$. The peak height decreases with increasing
persistence time and then saturates aroung $\tau_{p}(f)$.

We presented here a new class of extremely persistent active matter systems.
Whereas earlier investigations Mandal2021 ; Keta2023  revealed systems that
relax on the time scale of the self-propulsion and exhibit intermittent
dynamics with system-size spanning elastic and plastic events, we uncovered
systems that relax on the time scale that, in the large persistence time
limit, depends only on the strength of the self-propulsion. Curiously, the
single-particle motion exhibits two ballistic regions separated by a
superdiffusive regime. Classic signatures of two-step relaxation are absent
both in the mean square displacement and in the intermediate scattering
function. Many properties that quantify the large persistence time limit of
the relaxation depend on the strength of the active forces as a power law.

We expect that for higher volume fractions there is a transition between the
regime in which the relaxation becomes independent of the persistence time of
the self-propulsion, which is the regime we analyzed, and the regime in which
the system flows only on the time scale of the self-propulsion, which is the
regime investigated earlier Mandal2021 ; Keta2023 . At a fixed volume fraction
the transition would be driven by the strength of the active forces while at a
fixed strength of the active forces it would be driven by the density. We hope
that future work will determine the corresponding phase diagram, which would
be the three-dimensional analog of the diagram uncovered by Liao and Xu LiaoXu
.

Finally, while for small and moderate persistence times there are approximate
theories that can be used to describe the relaxation in active fluids
SzamelMCT ; LiluashviliMCT ; FengMCT1 ; FengMCT2 ; DebetsMCT , these theories
are not expected to work in the large persistence time limit. Thus, the
discovery of a new different paradigm of extremely persistent active fluids
with non-trivial power laws call for additional theoretical work.

We thank L. Berthier and P. Sollich for discussions and comments on the
manuscript. Part of this work was done when GS was on sabbatical at Georg-
August Universität Göttingen. He thanks his colleagues there for their
hospitality. We gratefully acknowledge the support of NSF Grant No. CHE
2154241.

## References

  * (1) M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, ”Hydrodynamics of soft active matter”, J. Prost, Rev. Mod. Phys. 85, 1143 (2013).
  * (2) J. Elgeti, R.G. Winkler, and G. Gompper, ”Physics of microswimmers–single particle motion and collective behavior: a review”, Rep. Prog. Phys. 78, 056601 (2015).
  * (3) C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, ”Active particles in complex and crowded environments”, Rev. Mod. Phys. 88, 045006 (2016).
  * (4) R. Mandal, P. J. Bhuyan, P. Chaudhuri, C. Dasgupta, and M. Rao, ”Extreme active matter at high densities”, Nat. Commun. 11, 2581 (2020).
  * (5) R. Mandal and P. Sollich, ”How to study a persistent active glassy system”, J. Phys.: Condens. Matt. 33, 184001 (2021).
  * (6) Y.-E. Keta, R.L. Jack, and L. Berthier, ”Disordered Collective Motion in Dense Assemblies of Persistent Particles”, Phys. Rev. Lett. 129, 048002 (2022).
  * (7) Y.-E. Keta, R. Mandal, P. Sollich, R. L. Jack, and L. Berthier, ”Intermittent relaxation and avalanches in extremely persistent active matter”, Soft Matter 19, 3871 (2023).
  * (8) L. Berthier and J. Kurchan, ”Non-equilibrium glass transitions in driven and active matter”, Nat. Phys. 9, 310 (2013).
  * (9) L. Berthier, ”Nonequilibrium Glassy Dynamics of Self-Propelled Hard Disks”, Phys. Rev. Lett. 112, 220602 (2014).
  * (10) R. Ni, M. A. Cohen Stuart, and M. Dijkstra, ”Pushing the glass transition towards random close packing using self-propelled hard spheres”, Nat. Commun. 4, 1 (2013).
  * (11) G. Szamel, E. Flenner, and L. Berthier, ”Glassy dynamics of athermal self-propelled particles: Computer simulations and a nonequilibrium microscopic theory”, Phys. Rev. E 91, 062304 (2015).
  * (12) R. Mandal, P. J. Bhuyan, M. Rao, and C. Dasgupta, ”Active fluidization in dense glassy systems”, Soft Matter 12, 6268 (2016).
  * (13) L. Berthier, E. Flenner, and G. Szamel, ”Perspective: Glassy dynamics in dense systems of active particles ”, J. Chem. Phys. 150, 200901 (2019).
  * (14) N. Klongvessa, F. Ginot, C. Ybert, C. Cottin-Bizonne, and M. Leocmach, ”Active Glass: Ergodicity Breaking Dramatically Affects Response to Self-Propulsion”, Phys. Rev. Lett. 123, 248004 (2019).
  * (15) L. Janssen, ”Active glasses”, J. Phys.: Condens. Matter 31, 503002 (2019).
  * (16) E. Flenner, G. Szamel, and L. Berthier, ”The nonequilibrium glassy dynamics of self-propelled particles”, Soft Matter 12, 7136 (2016).
  * (17) L. Berthier, E. Flenner, and G. Szamel, ”How active forces influence nonequilibrium glass transitions”, New J. Phys. 19, 125006 (2017).
  * (18) R. Mandal and P. Sollich, ”Multiple Types of Aging in Active Glasses”, Phys. Rev. Lett. 125, 218001 (2020).
  * (19) Q. Liao and N. Xu, “Criticality of the zero-temperature jamming transition probed by self-propelled particles”, Soft Matter 14, 853 (2018).
  * (20) G. Szamel, ”Self-propelled particle in an external potential: Existence of an effective temperature”, Phys. Rev. E 90, 012111 (2014).
  * (21) U.M.B. Marconi, N. Gnan, M. Paoluzzi, C. Maggi, and R. Di Leonardo, ”Velocity distribution in active particles systems”, Sci. Rep. 6, 23297 (2016).
  * (22) E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, ”How Far from Equilibrium Is Active Matter?”, Phys. Rev. Lett. 117, 038103 (2016).
  * (23) R. Wiese, K. Kroy, and D. Levis, ”Fluid-Glass-Jamming Rheology of Soft Active Brownian Particles”, arXiv:2303.11245 (2023).
  * (24) To approximately quantify $\tau_{p}(f)$ we fit $v^{2}$ for small $\tau_{p}$ to a power law, which approximately describes the small $\tau_{p}$ behavior, and determine when this power law is equal to the average of the large $\tau_{p}$ value of $b$. We find that a reasonable value of $\tau_{p}(f)$ is approximately twice when the power law equals the average value.
  * (25) A. W. Lees and S. F. Edwards, ”The computer study of transport processes under extreme conditions”, J. Phys. C: Solid State Phys. 5, 1921 (1972).
  * (26) The absence of small wavevector peaks implies that the systems are homogenenous. We confirmed this observation by evaluating local density histograms at several simulated state points.
  * (27) G. Szamel, ”Theory for the dynamics of dense systems of athermal self-propelled particles”, Phys. Rev. E 93, 012603 (2016).
  * (28) A. Liluashvili, J. Ónody, and T. Voigtmann, ”Mode-coupling theory for active Brownian particles”, Phys. Rev. E 96, 062608 (2017).
  * (29) M. Feng and Z. Hou, ”Mode coupling theory for nonequilibrium glassy dynamics of thermal self-propelled particles”, Soft Matter 13, 4464 (2017).
  * (30) M. Feng and Z. Hou, ”Mode-coupling theory for the dynamics of dense underdamped active Brownian particle system”, J. Chem. Phys. 158, 024102 (2023).
  * (31) V.E. Debets and L.M.C. Janssen, “Mode-coupling theory for mixtures of athermal self-propelled particles”, arXiv:2304.08936.