Nexus 4 | 760 | 911 | 1024 | 644 | 695 | 543 | 4577
Galaxy S3 | 664 | 889 | 944 | 560 | 635 | 474 | 4166
Galaxy S3 Mini | 409 | 501 | 524 | 297 | 396 | 280 | 2407
LG watch | 368 | 358 | 382 | 424 | 315 | 307 | 2154
Gear watch | 21 | 23 | 78 | 42 | 120 | 86 | 370
Total | 2222 | 2682 | 2952 | 1967 | 2161 | 1690 | 13674
Table 33: Domain proportions of labels in the HHAR Dataset

##### Architecture choice

As this data is similar to EEG recordings, we use the same deep convolution
network model as in the CAP and SEDFx datasets. The architecture is defined in
work from Schirrmeister et al. [94]. We use the implementation of the
BrainDecode Schirrmeister et al. [94] Toolbox. Temporal Convolutional Networks
(TCN) are powerful tools for processing time series data [8]. The architecture
we use combines temporal and spatial convolution, which fits this data well.
We found that it performed well on this task and obtained solid and stable
performance, which is great for our purpose of benchmarking.

####  A.7.2 Detailed results

##### ID evaluation

We show the ID results of ERM for the HHAR dataset in Table 34. We obtain
these results by doing a hyperparameter search with the methodology detailed
in Appendix D, but with no held-out test domain. In other words, all domains
are ID. We then perform train-domain validation to choose the best performing
model and report its accuracy.

Algorithm Nexus 4 Galazy S3 Galaxy S3 Mini LG watch Sam. Gear Average ID ERM
$98.91\pm 0.24$ $98.44\pm 0.15$ $98.68\pm 0.15$ $90.08\pm 0.28$ $80.63\pm
1.33$ 93.35

Table 34: ID results for the HHAR dataset

##### Baseline evaluation

We show the results of our set of baseline algorithms on the HHAR dataset in
Table 35. We obtain these results taking every domain as a held-out test set
and performing a hyperparameter search with the methodology detailed in
Appendix D. Hence, each column of Table 35 corresponds to the domain
performance when taken as a test set.

Train-domain validation Objective Nexus 4 Galazy S3 Galaxy S3 Mini LG watch
Sam. Gear Average ID ERM $98.91\pm 0.24$ $98.44\pm 0.15$ $98.68\pm 0.15$
$90.08\pm 0.28$ $80.63\pm 1.33$ 93.35 ERM $97.64\pm 0.15$ $97.64\pm 0.09$
$92.51\pm 0.46$ $71.69\pm 0.14$ $61.94\pm 1.04$ 84.28 IRM $96.02\pm 0.17$
$95.75\pm 0.22$ $89.46\pm 0.50$ $66.49\pm 0.94$ $57.66\pm 0.37$ 81.08 VREx
$95.81\pm 0.50$ $95.92\pm 0.23$ $90.72\pm 0.10$ $69.04\pm 0.23$ $56.42\pm
1.57$ 81.58 IB-ERM $97.56\pm 0.06$ $97.93\pm 0.21$ $91.76\pm 0.57$ $71.38\pm
1.02$ $59.01\pm 1.86$ 83.53 SD $98.14\pm 0.01$ $98.32\pm 0.19$ $92.71\pm 0.09$
$75.12\pm 0.18$ $63.85\pm 0.28$ 85.63 Oracle train-domain validation Objective
Nexus 4 Galazy S3 Galaxy S3 Mini LG watch Sam. Gear Average ID ERM $98.91\pm
0.24$ $98.44\pm 0.15$ $98.68\pm 0.15$ $90.08\pm 0.28$ $80.63\pm 1.33$ 93.35
ERM $97.98\pm 0.02$ $97.92\pm 0.05$ $93.09\pm 0.15$ $71.96\pm 0.04$ $64.08\pm
0.66$ 85.01 IRM $96.02\pm 0.17$ $95.75\pm 0.22$ $89.91\pm 0.25$ $68.00\pm
0.34$ $57.77\pm 0.42$ 81.49 VREx $96.65\pm 0.18$ $96.30\pm 0.05$ $90.98\pm
0.16$ $69.39\pm 0.27$ $59.12\pm 0.80$ 82.49 IB-ERM $98.16\pm 0.09$ $98.22\pm
0.09$ $93.18\pm 0.16$ $73.40\pm 0.68$ $64.64\pm 0.09$ 85.52 SD $98.48\pm 0.01$
$98.67\pm 0.11$ $94.36\pm 0.24$ $75.12\pm 0.18$ $64.86\pm 0.28$ 86.30

Table 35: Baseline results for the HHAR dataset

####  A.7.3 Credits and license

This dataset was adapted from the work of Stisen et al. [102] as made
available on the online UCI Machine Learning Repository [23]. This dataset is
licensed under the Open Data Commons Attribution license v1.0.

##  Appendix B Further details on OOD generalization gaps

This section discusses the multiple ways of defining the ID performance and
which is the best for measuring attainable generalization gaps. As mentioned
in Section 6, the generalization gap is the difference between the
performances of a model on data drawn ID and OOD of its training distribution.
The OOD performance is obtained by training a model on the training domains
$D^{\mathsf{train}}$ with ERM and measure the performance of this model on the
test domain $D^{\mathsf{test}}$. The ID performance, however, can be defined
in different ways. Koh et al. [51] provides multiple definitions for it:

  * •

Train-to-train Performance of a model on $D^{\mathsf{train}}$ when trained on
$D^{\mathsf{train}}$

  * •

Mixed-to-test Performance of a model on $D^{\mathsf{test}}$ when trained on a
mixture of $D^{\mathsf{train}}$ and $D^{\mathsf{train}}$.

  * •

Test-to-test Performance of a model on $D^{\mathsf{test}}$ when trained on
$D^{\mathsf{test}}$

In this list, we think that the measure that gives the best measure for
achievable generalization gaps is the mixed-to-test because Test-to-test and
Train-to-train can lead to erroneous measures.

To illustrate this problem, we explain how the train-to-train measures in the
Spurious-Fourier dataset lead to an inflated generalization gap that are
unattainable in reality. In the Spurious-Fourier dataset, performing ERM on
the training domains $D^{d}|_{d\in\\{80\%,\,90\%\\}}$ will lead to models
relying on the spurious features to make predictions, as they are a stronger
predictor of the label ($85\%$) than the invariant features ($75\%$). Thus,
the model will achieve $85\%$ accuracy on data sampled ID to domains
$d\in\\{80\%,\,90\%\\}$, but only achieve $10\%$ accuracy on the test domain
$D^{10\%}$. Comparing the ID and OOD performance would lead to a
generalization gap of $75\%$. However, this gap is misleading as a model could
never achieve $85\%$ accuracy on the test domain because the strongest
invariant predictor can only achieve $75\%$. A similar case can be made for
the test-to-test measure of ID performance, where the generalization gap can
lead to an unattainable performance on the test domain.

Instead, let us go through the same example with the mixed-to-test ID
performance. Performing ERM on the training domains
$D^{d}|_{d\in\\{80\%,\,90\%\\}}$ will lead to a predictor that relies on the
invariant features, as they are a stronger predictor of the label ($75\%$)
than the spurious features ($60\%$). Therefore, the ID performance will be
$75\%$, and the OOD performance will be $10\%$, leading to a generalization
gap of $65\%$. This gap is a much more significant measure of the upper bound
of the performance than the original definition, as it is an attainable gap.

##  Appendix C Evaluating OOD performance

In theory, the OOD performance of an algorithm is its performance on the test
domains $d\in\mathcal{E}_{\mathsf{all}}$ when trained on a set of training
domains $d\in\mathcal{E}_{\mathsf{train}}$ (see Problem 3.1). In practice,
however, a dataset only has a finite number of domains. Thus, a perfect
measure of performance on $\mathcal{E}_{\mathsf{all}}$ is impossible. Instead,
we estimate the OOD performance of an algorithm by either performing a

  * •

Single measure, where we train an algorithm on a set of training domains, then
evaluate on an held-out test domain, or a

  * •

Cross-validation measure, where take the average of the single measure for
every domain as the test domain.

Both of these approaches are used in the literature: Koh et al. [51] uses a
single measure in WILDS, while Gulrajani and Lopez-Paz [29] uses the cross-
validation measure in DomainBed. In this work we use the cross-validation
measure for our dataset as we think it is a more representative estimation of
the performance of an algorithm on test domains
$d\in\mathcal{E}_{\mathsf{all}}$.

##  Appendix D Evaluation framework workflow

In this section of the appendix, we detail the methodology employed to
evaluate the performance of OOD algorithms on our datasets. For the most part,
we follow the workflow used by Gulrajani and Lopez-Paz [29] in their DomainBed
testbed.

##### Domain cross validation

To get a complete evaluation of the performance of an algorithm on a given
dataset, we perform a domain cross-validation measure. We motivate this choice
in Appendix C. In summary, for every domain in a dataset, we hold this domain
out as the test domain and perform an hyperparameter search by training on the
remaining domain. After this hyperparameter search, we perform the model
selection associated with the dataset (see Section 8.1) in order to choose a
model given a specific validation set. We then report the performance of the
chosen model on the test set. All datasets in WOODS consist of at least 3
domains, therefore there are always at least 2 domains on which to apply an
invariance penalty when doing the cross validation.

##### Hyperparameter search

For every held-out domain in our domain cross validation, we perform a random
search [12] over the hyperparameter distribution of both training and
algorithm hyperparameter distribution (see Appendix E) search following the
random search. We train 20 models using randomly sampled hyperparameters. We
then perform the dataset appropriate model selection methods to select the
best models. For each domain, we split the data into 80% and 20% disjoint
subsets of the full domain data. We use the 80% split to train the models, and
we use the 20% split to validate ID on the training data and perform model
selection.

##### Statistically relevant

We repeat each hyperparameter search three times to obtain statistically
relevant results. This reduces the probability that some algorithm samples a
lucky configuration of hyper parameters. All the results reported in this work
are averaged over those three hyperparameter trials, we provide the estimated
standard deviation of those mean values.

##### Reducing bias

The search range is an important topic when discussing the fairness of this
evaluation strategy. Having reasonable hyperparameter distributions for
sampling in the random search is essential to remaining fair between the
algorithms and reducing the induced bias in the results Defining a narrow
hyperparameter distribution for which one knows the algorithm performs very
well on a dataset or test domain leads to a bias of the evaluation due to
queries of the testing domain under human intervention. This bias could lead
to algorithms getting better results by increasing the chance of the random
search finding a good value. When defining the hyperparameter range, one
should define a range wide enough as to cover the relevant search space for
this hyperparameter.

##  Appendix E Hyperparameter searches

We split the hyperparameter values into 2 categories.

##### Training hyperparameters

First, there are training hyperparameters that are functions of the dataset we
are training on. They are the base hyperparameters that ERM would need to
train. We show the Random distribution used to perform the random search in
Table 36.

Dataset | Hyperparameter | Random distribution
---|---|---
Spurious-Fourier | learning rate | $10^{\text{Uniform}(-4.5,-2.5)}$
batch size | $2^{\text{Uniform}(3,9)}$
class balance | True
weight decay | 0
TCMNIST-Source | learning rate | $10^{\text{Uniform}(-4.5,-2.5)}$
batch size | $2^{\text{Uniform}(3,9)}$
class balance | True
weight decay | 0
TCMNIST-Time | learning rate | $10^{\text{Uniform}(-4.5,-2.5)}$
batch size | $2^{\text{Uniform}(3,9)}$
class balance | True
weight decay | 0
CAP | learning rate | $10^{\text{Uniform}(-5,-3)}$
batch size | $2^{\text{Uniform}(3,4)}$
class balance | True
weight decay | 0
SEDFx | learning rate | $10^{\text{Uniform}(-5,-3)}$
batch size | $2^{\text{Uniform}(3,4)}$
class balance | True
weight decay | 0
PCL | learning rate | $10^{\text{Uniform}(-5,-3)}$
batch size | $2^{\text{Uniform}(3,5)}$
class balance | True
weight decay | 0
LSA64 | learning rate | $10^{\text{Uniform}(-5,-3)}$
batch size | $2^{\text{Uniform}(3,4)}$
class balance | True
weight decay | 0
HHAR | learning rate | $10^{\text{Uniform}(-4,-2)}$
batch size | $2^{\text{Uniform}(3,4)}$
class balance | True
weight decay | 0
Table 36: Distribution range of training hyperparameters for random search

##### Algorithm hyperparameters

Second, there are algorithm hyperparameters that are part of the algorithm
definition.

Dataset | Hyperparameter | Random distribution
---|---|---
Invariant Risk Minimization | penalty weight | $10^{\text{Uniform}(-1,5)}$
annealing iterations | $\text{Uniform}(0,2000)$
Variational REx | penalty weight | $10^{\text{Uniform}(-1,5)}$
annealing iterations | $\text{Uniform}(0,2000)$
Spectral Decoupling | penalty weight | $10^{\text{Uniform}(-5,-1)}$
IB-ERM | IB penalty weight | $10^{\text{Uniform}(-3,0)}$
Table 37: Distribution range of algorithm hyperparameters for random search


# Fluid-Glass-Jamming Rheology of Soft Active Brownian Particles

Roland Wiese<EMAIL_ADDRESS>Klaus Kroy  Institute for Theoretical
Physics, Leipzig University, 04103 Leipzig, Germany  Demian Levis
<EMAIL_ADDRESS>Departement de Física de la Materia Condensada, Facultat de
Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
University of Barcelona Institute of Complex Systems (UBICS), Facultat de
Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain

###### Abstract

We study the shear rheology of a binary mixture of soft Active Brownian
Particles, from the fluid to the disordered solid regime. At low shear rates,
we find a Newtonian regime, where a Green-Kubo relation with an effective
temperature provides the linear viscosity. It is followed by a shear-thinning
regime at larger shear rates. A solid regime at high densities is signalled by
the emergence of a finite yield stress. We construct a “fluid-glass-jamming”
phase diagram, in which the temperature axis is replaced by activity. Although
activity, like temperature, increases the amount of fluctuations in the
system, it plays a different role, as it changes the exponent characterizing
the decay of the diffusivity close to the glass transition and the shape of
the yield stress surface. The dense disordered active solid appears to be
mostly dominated by athermal jamming rather than glass rheology.

Ensembles of repulsive particles commonly undergo a phase transition from a
fluid to a solid state. If the tendency to crystallize is frustrated (e.g. by
size polydispersity), the system exhibits a glass transition to a disordered
solid [1, 2]. Solidity can also emerge upon compression in athermal systems of
non-Brownian particles, such as foams or grains: The so-called jamming
transition [3]. Both transitions share the existence of a critical density
beyond which solidity emerges, characterized by a dramatic slowing down of the
dynamics and the emergence of a yield stress [4, 5]: Thus a unified picture in
terms of a “jamming” phase diagram was proposed [6, 7]. The yield stress
surface has since been quantified with the help of idealized particle models
and rheological experiments [8, 9, 10, 11], helping to decipher the mechanisms
responsible for the emergence of rigidity in diverse soft materials [5].

A resurgence of interest in understanding the emergence of solidity is
recently observed in an _a priori_ completely new context, namely dense
disordered active matter (DDAM). Indeed, cell assemblies display a fluid to
solid transition, key to understand biological processes such as
morphogenesis. Again, this phenomenology has been rationalized in terms of a
jamming phase diagram [12, 13, 14, 15], where temperature is replaced by
activity, usually in the form of motility; see Fig. 1(a), where $\mathrm{Pe}$
quantifies activity. Dense assemblies of cells [16, 17] and synthetic active
colloids [18] feature collective dynamics reminiscent of supercooled liquids
approaching a glass transition. However, as thermal fluctuations can usually
be neglected in active systems, and fluctuations are non-thermal, it is
conceptually unclear whether the emergence of a disordered solid has to be
attributed to a jamming rather than a glass transition. The question of how
activity, in the form of self-propulsion, affects the glass transition, has
been addressed in numerous works using model systems [19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31]. However, the rheology of active matter [32, 33]
and its relation with jamming, remains poorly understood.

Figure 1: (a)“Glass-Jamming” phase diagram of harmonic ABPs at reduced
temperature $T=10^{-4}$ in terms of the yield stress surface. The black dotted
line corresponds to the athermal ($T=0$) passive jamming limit. The black
symbols in the Péclet number versus volume fraction plane locate the active
glass transition. (b) Cross section of the $xy$-plane of the three dimensional
simulation box, showing the imposed velocity profile.

In this Letter we investigate the shear rheology of self-propelled soft
particles by means of computer simulations. In previous studies with micro-
swimmer suspensions, both hydrodynamic and particle-wall interactions are
likely to be crucial [34, 35, 36, 37, 38]. Here, in order to decipher the role
played by self-propulsion alone, we consider a simplified model where none of
these two ingredients are at play [39]. More precisely, we consider a dry
model system of harmonic Active Brownian Particles (ABP) in three dimensions
[40, 41] with periodic boundary conditions (see Fig. 1(b)). This simplified
starting point allows us to cover the dilute and dense regimes in our
numerical simulations, and to explore both linear and non-linear response
(across eight orders of magnitude in the shear rate). The advantages of
choosing harmonic spheres are threefold:  i) Harmonic spheres have been
largely studied as models of foams [42], and its active version provides a
useful model of cell tissues [43, 44];  ii) the rheology of the equilibrium
limit of the model has been studied in detail [10, 11], allowing for a smooth
connection with previous results and thus helping for the identification of
the new features brought about by activity;  iii) computational speed-up
compared to hard spheres.

The rheology of model active particles has received recent interest, both for
individual ABPs [45], and for interacting many-body systems [46, 47, 48, 49].
An analogy has been drawn between shear (a global drive) and activity (a local
drive) [50], and indeed, in infinite spatial dimensions in the infinite-
persistence limit, their formal equivalence has been established [51, 52]. In
two dimensions, the mechanisms that govern yielding in the respective systems
were found to be different, though [53]. Active particles under shear
orientationally order in the presence of hard walls [48] and trigger shear
thickening as a result of clustering [49]. However, the rheology of dense
disordered ABPs with a finite persistence has not yet been explored.

We study sheared assemblies of $N$ soft repulsive ABPs, located in positions
$\\{\bm{r}_{i}\\}_{i=1}^{N}$ in a $V=L^{3}$ cubic box with PBC. Their dynamics
is overdamped and follows

$\displaystyle\begin{split}\dot{\bm{r}}_{i}&=\mu\sum_{j\neq
i}\bm{F}_{ij}+v_{0}\bm{n}_{i}+\dot{\gamma}y_{i}\bm{e}_{x}+\sqrt{2D_{t}}\bm{\xi}_{i}\,,\\\
\dot{\bm{n}}_{i}&=\sqrt{2D_{r}}\bm{n}_{i}\times\bm{\nu}_{i}\,.\\\ \end{split}$
(1)

Particles are self-propelled along their orientations $\bm{n}_{i}$ (with
$|\bm{n}_{i}|=1$), with a speed $v_{0}$. Interaction forces derive from a
harmonic repulsive pair potential
$V(r)~{}=~{}\epsilon~{}(1-r/a)^{2}~{}\Theta(a-r)$, $\Theta(r)$ being the
Heaviside step function. We consider a 50:50 bidisperse mixture of $N=10^{3}$
particles with diameter ratio $a_{B}/a_{A}=\sqrt{2}$ to suppress
crystallization [10, 54]. Both $\bm{\xi}_{i}$ and $\bm{\nu}_{i}$ are Gaussian
white noises of zero mean and unit variance, $D_{t}=\mu k_{B}T$ is the (bare)
translational diffusion coefficient and $D_{r}$ the rotational diffusivity
fixed at $D_{r}=3D_{t}/a_{A}^{2}$. With this choice, $T\to 0$ corresponds to
infinite persistence. To further explore the athermal case, $T=0$, we fix
$D_{t}=0$ and vary $D_{r}$ independently. The term
$\dot{\gamma}y_{i}\bm{e}_{x}$, together with Lees-Edwards boundary conditions
imposes a linear velocity profile to the particles with slope $\dot{\gamma}$,
the shear rate (see Fig. 1(b)) [55, 56]. The translational part of Eq.(1) is
integrated by an Euler-Mayurama scheme and the rotational part using the
algorithm described in [57] (see [58] for details).

Lengths are measured in units of the small particle diameter $a_{A}$, time in
units of $\hat{t}=a_{A}^{2}/(\mu\epsilon)$ and temperature in units of
$\hat{T}=\epsilon/k_{B}$. In the following, all observables will be given in
these units. From Eq.(1), one can identify a set of non-dimensional control
parameters: the volume fraction $\phi$, the Péclet number
$\mathrm{Pe}=v_{0}/a_{A}D_{r}$, quantifying activity, and the dimensionaless
shear rate $\dot{\gamma}$. We study the system at $T=10^{-6}...10^{3}$ (most
of the results presented are for $T=10^{-4}$), $\mathrm{Pe}=0,1,3,10,30$, and
a wide range of shear rates $\dot{\gamma}=10^{-7}...\,10$ including both the
linear and nonlinear response regimes. Keeping the activity at the
aforementioned values and $D_{r}$ constant, ensures that the repulsive force
is always several orders of magnitude larger than the self-propulsion,
avoiding any possibility of a reentrant gas phase, as seen in [59]. It is
known that monodisperse hard ABPs above a critical $\mathrm{Pe}\approx 30$
exhibit Motility-Induced Phase Separation (MIPS) [40, 41, 60]. Here we explore
a parameter regime where our system remains homogeneous.

In equilibrium ($\mathrm{Pe}=0$), as $\phi$ is increased, the system exhibits
a dramatic slowing down of the dynamics that one identifies with a glass
transition at $\phi_{G}$, characterized by the divergence of the viscosity and
the emergence of a yield stress for $\phi>\phi_{G}(0)$. As we show below,
similar behavior is observed in the presence of activity, although in this
case, the location of the glass transition is shifted to higher densities
$\phi_{G}(\mathrm{Pe})$ (as previously reported for different models [61, 20,
62, 21, 22, 23, 30]).

Figure 2:  (a) Mean-square displacement at fixed $T=10^{-4}$, $\mathrm{Pe}=0$
and $10$, for $\phi=0.66,\,0.72$, with and without shear, showing the melting
of the glass by activity (green curves) and by shear (red curves). The dashed
line indicates the initial diffusive regime.(b) Diffusion coefficients $D_{s}$
normalized by their ideal gas value $D_{a}$ (open green circles correspond to
$T=10^{-3}$ and $\mathrm{Pe}=0$), as a function of the distance to the
critical density $\phi_{G}-\phi$. Dashed lines are power law fits
$D_{s}\propto(\phi_{G}-\phi)^{\alpha}$.

We start our analysis by investigating the dynamics in the absence of shear by
means of the Mean-Square Displacement (MSD), defined as
$\Delta^{2}(t)=N^{-1}\sum_{i=1}^{N}\langle(\bm{r}_{i}(t)-\bm{r}_{i}(0))^{2}\rangle$,
where the average is taken over different noise realisations. As shown in Fig.
2(a), at $\phi=0.66$ the passive system exhibits caged dynamics, evidenced by
the sub-diffusive (plateau) regime in the MSD. For $\mathrm{Pe}=10$, particles
diffuse in the same time window, showing that activity is able to fluidize the
glass. From the long time MSD we measure the diffusion coefficient
$D_{s}\equiv\lim_{t\to\infty}\Delta^{2}(t)/6t$ in the range of parameters for
which a diffusive regime, $\Delta^{2}(t)\propto t$, is observed. At high
densities, $D_{s}(\phi)$ can be fitted by a power law
$D_{s}\propto(\phi_{G}-\phi)^{\alpha}$ that we use to locate the glass
transition density reported in Fig. 1(a) [21]. Figure 2(b) shows
$D_{s}(\phi,\mathrm{Pe})$ as a function of $(\phi_{G}-\phi)$ normalized by the
(active) ideal gas diffusion coefficient $D_{a}=D_{t}+\mathrm{Pe}^{2}D_{r}/6$,
for different $\mathrm{Pe}$. As the activity primarily enhances diffusion, it
moves the glass transition from $\phi_{G}(0)=0.62$ at $\mathrm{Pe}=0$ to ever
higher densites, up to $\phi_{G}(30)=0.72$ at $\mathrm{Pe}=30$ (see Fig.
1(a)). It might be tempting to simply interpret such a shift as resulting from
an increase of the single particle effective temperature, defined by
$T_{\text{eff}}=\mu D_{a}$, since the equilibrium glass transition density
would also be shifted upon increasing $T$ [63]. However, not only $\phi_{G}$
is affected by activity, but also the exponent $\alpha$ increases from $1.79$
at $\mathrm{Pe}=0$, up to $4.42$ at $\mathrm{Pe}=30$. While in equilibrium, a
slightly lower value ($\alpha=1.67$) is measured for higher temperature
$T=10^{-3}$, $\alpha$ significantly changes with $\mathrm{Pe}$, showing that
activity cannot be simply reduced to an effective temperature in this regime.

Applying shear provides another route to fluidize the disordered solid state.
As shown in Fig. 2(a), particles exhibiting caged dynamics in an active system
at $\phi=0.72>\phi_{G}(10)$, become mobile when shear is turned on. The MSD
displays super-diffusive, then diffusive behavior at long times. The obtained
long time diffusion is sensitive to finite-size effects: Lees-Edwards boundary
conditions introduce a discontinuity in the shearing profile which becomes
apparent in the MSD after a sufficiently long time (see [58] for details).

To characterize the flow properties we measure the $xy$-component of the
stress tensor, using the Irving-Kirkwood expression [64]
$\sigma_{xy}(t)=-(2V)^{-1}\sum_{j\neq k}x_{ij}(t)F^{y}_{ij}(t)\,,$ from which
we get the shear viscosity $\eta=\langle\sigma_{xy}\rangle/\dot{\gamma}$.
Activity contributes to the stress tensor with a self-term
$\sigma^{s}_{\alpha\beta}=-V^{-1}\sum_{i}r^{\alpha}_{i}v_{0}n^{\beta}_{i}(t)$,
but as the orientations ${\bm{n}}_{i}$ are decoupled from the shear flow,
$\sigma^{s}_{xy}$ fluctuates around zero [48]. The flow curves characterizing
the rheology of the system at $T=10^{-4}$ and $\mathrm{Pe}=0,10$ are depicted
in Fig. 3 (see [58] for more parameter values). In the passive case,
$\mathrm{Pe}=0$, we reproduce the flow curves reported for the same system in
[10]. Then, we explore the rheology in the presence of activity. In dense two-
dimensional active assemblies, shearing was observed to lead to orientational
order at large persistence time [48]. However, we did not find orientational
correlations in our three-dimensional model for the parameter range explored
(see [58]).

Figure 3:  Flow curves showing the dependence of the shear stress
$\langle\sigma_{xy}\rangle$ (top row) and viscosity $\eta$ (bottom row) on
$\dot{\gamma}$ (points, color code encoding $\phi$). The thick red lines
correspond to $\phi_{G}$ (as estimated via the diffusivity). Thinner lines
represent fits of the form
$\sigma_{xy}(\dot{\gamma})=\sigma_{Y}+(k\dot{\gamma})^{n}$, used to determine
the yield stress $\sigma_{Y}$.

At densities below $\phi_{G}(\mathrm{Pe})$, we find
$\sigma_{xy}\propto\dot{\gamma}$ for small enough applied shear. This
corresponds to the Newtonian fluid regime, defining a linear viscosity
$\eta_{0}=\lim_{\dot{\gamma}\to 0}\langle\sigma_{xy}\rangle/\dot{\gamma}$. For
higher values, $\dot{\gamma}\gtrsim 10^{-2}$, in the non-linear regime, we
find shear-thinning in all cases, meaning that the shear flow reduces the
system’s viscosity. In this regime, we find a decay of the viscosity
compatible with the $\eta\sim 1/|\dot{\gamma}|$ scaling expected from Mode-
Coupling Theory for Brownian suspensions [65]. At $\phi>\phi_{G}$, a finite
yield stress $\sigma_{Y}=\lim_{\dot{\gamma}\to 0}\sigma_{xy}(\dot{\gamma})$
appears, identified by a plateau in the stress flow curves. This results in a
divergent viscosity, signalling the emergence of solidity. Since activity
melts the solid, the system yields at higher densities at $\mathrm{Pe}=10$
compared to $\mathrm{Pe}=0$.

Figure 4:  Linear shear viscosities $\eta_{0}$, extracted from the flow curves
(symbols) and $\eta_{GK}/(\mathrm{Pe}+1)$, from Green-Kubo, (lines) as a
function of $\phi$. Dashed vertical lines indicate $\phi_{G}(\mathrm{Pe})$.
Inset: $\eta_{GK}\sim\phi^{2}$ at low densities .

In the fluid regime, one can access the linear viscosity by applying small
enough shear and, in equilibrium, it should correspond to the one given by the
Green-Kubo (GK) relation
$\eta_{GK}=\frac{V}{k_{B}T}\int_{0}^{\infty}\mathrm{d}t\,\langle\sigma_{\alpha\beta}(t)\sigma_{\alpha\beta}(0)\rangle_{0}\,,$
for $\alpha\neq\beta$, where $\langle*\rangle_{0}$ denotes an average over the
unperturbed ($\dot{\gamma}=0$) equilibrium distribution. As shown in Fig. 4,
the shear viscosity $\eta_{0}$ extracted from the low $\dot{\gamma}$ plateau
in the flow curves in Fig. 3(c) matches $\eta_{GK}$, measured from the GK
relation by direct integration of the equilibrium stress correlation function.
In the presence of activity, GK relations do not need to hold anymore,
although extensions of linear response theory to active systems have recently
been proposed [66, 67, 68, 69], providing GK relations involving steady-state
correlation functions [66, 68]. Here we apply the same procedure in the
presence of activity, thus replacing equilibrium by steady-state stress
correlations, and find a good agreement between $\eta_{0}$ and $\eta_{GK}$ if
we replace $T$ in the GK expression by an effective temperature
$T_{\mathrm{eff}}=T(\mathrm{Pe}+1)$, see Fig. 4. In all cases, $\eta_{0}$
increases with $\phi$ and eventually diverges at $\phi_{G}$, providing yet
another estimate for the onset of solidity. In dilute conditions, we find that
the GK viscosity grows like $\sim\phi^{2}$ (inset of Fig. 4), as predicted in
[70] for dilute Brownian suspensions: $\eta/\eta_{0}=1+2.5\phi+7.6\phi^{2}$.
We only observe the $\phi^{2}$ contributions, as there is no solvent in our
system.

Above $\phi_{G}$ the viscosity diverges and the system acquires a yield stress
$\sigma_{Y}$ that we measure by fitting a Herschel-Bulkley law,
$\sigma_{xy}(\dot{\gamma})=\sigma_{Y}+(k\dot{\gamma})^{n}$ [71], to the flow
curves, see Fig. 3. We report the obtained yield stress $\sigma_{Y}(\phi)$ as
a function of volume fraction for different values of $\mathrm{Pe}$ and $T$ in
Fig. 5. At $\mathrm{Pe}=0$ and finite $T$, the emergence of solidity is
controlled by the glass transition, at a density $\phi_{G}(T)$ that increases
with $T$. At $T=0$, it is instead controlled by the jamming transition, at
$\phi_{J}\approx 0.648$. Both glass and jamming physics affect the behavior of
$\sigma_{Y}(\phi,T)$. At $T\leqslant 10^{-4}$, for which $\phi_{G}<\phi_{J}$,
$\sigma_{Y}$ increases gently with $\phi$ and $T$ up to $\phi\approx\phi_{J}$
(given by the dotted line in Fig. 5). Above this value, the behavior of
$\sigma_{Y}$ changes qualitatively: it grows faster with $\phi$ close to
$\phi_{J}$, with little $T$-dependence, following
$\sigma_{Y}\sim(\phi-\phi_{J})^{\alpha^{\prime}}$ [10], see Fig. 5(a). Such
crossover allows us to differentiate glass- and jamming-dominated regimes.

Figure 5:  Yield stress $\sigma_{Y}$ as a function of $\phi$ for different
$\mathrm{Pe}$ and $T$. The dashed line given by
$(\phi-\phi_{J})^{\alpha^{\prime}}$ with $\phi_{J}=0.648$ and
$\alpha^{\prime}=1.04$, marks the athermal jamming limit.

At $\mathrm{Pe}=1$ and finite $T$, $\sigma_{Y}$ displays a $T$-sensitive
glass-like branch for $\phi<\phi_{J}$ followed by the $T$-insensitive jamming
branch, see Fig. 5(a). The yield stress curves do not follow the trend one
would expect if activity could be subsumed into an extra source of noise and
encoded by an increased effective temperature. The yield stress is smaller for
$\mathrm{Pe}=1$ than for $\mathrm{Pe}=0$, at a given $\phi$ and $T$, up to
$\phi_{J}$. A higher $T$ would result in a larger $\sigma_{Y}$. Moreover, at
$T=0$, we recover again the athermal jamming behavior, despite the presence of
random (active) forces. As we further increase the activity at fixed
$T=10^{-4}$, $\sigma_{Y}$ quickly collapses onto the jamming branch. For
$\mathrm{Pe}>3$, $\phi_{G}(\mathrm{Pe})>\phi_{J}$, and a finite yield stress
can only emerge in the regime controlled by jamming [54], where
$\sigma_{Y}\sim(\phi-\phi_{J})^{\alpha^{\prime}}$ universally applies,
independently of $\mathrm{Pe}$. The crossover between glass and jamming
rheology can thus be tuned by activity and is eventually lost, as it pushes
$\sigma_{Y}$ towards the $T=0$ behavior. The separation between $\phi_{G}$ and
$\phi_{J}$ progressively vanishes as activity increases. An overview over the
impact of activity on the yield stress at $T=10^{-4}$ is represented in the
fluid-glass-jamming phase diagram in Fig. 1, in terms of the shear stress,
density and activity.

We have studied ABPs under shear, from its fluid to disordered solid regime.
In the fluid, taking the zero shear limit, the Green-Kubo viscosities are
compatible with the ones extracted from the flow curves in the Newtonian
regime, once $T$ is rescaled by $\mathrm{Pe}$. Such effective temperatures
$T_{\text{eff}}$ have been introduced earlier to quantify the violations of
the fluctuation-dissipation theorem in active systems. In the dilute limit
$T_{\text{eff}}\sim\mathrm{Pe}^{2}$. Its $\mathrm{Pe}$ dependence at finite
densities is weaker and more complex and generically dependent on the
observables used to define it [22, 66, 72]. As the packing fraction is
increased towards the fluid-solid transition, the diffusivity decays
$D_{s}\sim(\phi_{G}-\phi)^{\alpha}$, with $\alpha$ increasing from
$\alpha\approx 1.8$ for $\mathrm{Pe}=0$, to $\alpha\approx 4.4$ for
$\mathrm{Pe}=30$, a behavior hardly interpretable on the grounds of an
effective temperature. In the solid regime, the glass-jamming phase diagram
(Fig. 1) reveals that ABPs rheology is mainly controlled by jamming. Although
both $T$ and $\mathrm{Pe}$ push $\phi_{G}$ to higher values, activity, as
opposed to temperature, eases the yielding. Our quantitative results for the
relationship between the fluid-solid transition in dense disordered active
matter and the glass and jamming transitions should serve as a helpful
reference for future studies. A possible application could be in interpreting
observations in dense assemblies of cells. We also hope that they stimulate
theoretical work to better understand the fundamental role played by non-
equilibrium fluctuations introduced by activity, in dense disordered systems.

## Acknowledgments

We warmly thank T. Voigtmann, A. Ikeda and L. Berthier for useful exchanges.
D.L. acknowledges MCIU/ AEI/FEDER for financial support under Grant Agreement
No. RTI2018-099032-J-I00.

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