# Configurational temperature in active-matter models

Shibu Saw  Glass and Time, IMFUFA, Department of Science and Environment,
Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark  Lorenzo
Costigliola  Glass and Time, IMFUFA, Department of Science and Environment,
Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark  Jeppe C. Dyre
<EMAIL_ADDRESS>Glass and Time, IMFUFA, Department of Science and Environment,
Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark

###### Abstract

Abstract: We show that the liquid-state-theory configurational temperature
${T}_{\rm conf}$ defines an energy scale, which for active-matter models of
point particles based on a potential-energy function can be used to determine
how to adjust model parameters to achieve approximately invariant structure
and dynamics when the density is changed. The required parameter changes are
calculated from the variation of a single configuration’s ${T}_{\rm conf}$
upon a uniform scaling of all coordinates. The formalism developed applies for
models involving a potential-energy function with hidden scale invariance. The
procedure is illustrated by computer simulations of the Kob-Andersen binary
Lennard-Jones model with active Ornstein-Uhlenbeck dynamics in three
dimensions and the two-dimensional single-component Yukawa model with active
Brownian particle dynamics. For the latter model we also show how ${T}_{\rm
conf}$ may be applied for estimating the MIPS phase boundary, in effect
reducing by one the dimension of the parameter space. We finally propose that
the ratio between the equilibrium-system temperature corresponding to actual
potential energy and ${T}_{\rm conf}$ provides a useful measure of how far an
active-matter system is from thermal equilibrium.

##  I Introduction

Any system in thermal equilibrium has a well-defined temperature, and
temperature is fundamental in thermodynamics and statistical mechanics. In
view of this it is obvious to try to generalize the concept of temperature to
non-equilibrium systems. Excellent reviews of different temperatures that have
been proposed throughout the years are given in Refs. Casas-Vazquez and Jou,
2003; Powles  _et al._ , 2005; Leuzzi, 2009; Puglisi  _et al._ , 2017; Zhang
_et al._ , 2019. Examples are the effective temperature quantifying deviations
from the fluctuation-dissipation theorem Cugliandolo and Kurchan (1994);
Cugliandolo (2011); Petrelli  _et al._ (2020) and the fictive temperature
characterizing a glass’ structure in terms of the temperature at which the
liquid solidified Tool (1946); Scherer (1986). Non-equilibrium temperatures
are generally motivated by the prospect of connecting the non-equilibrium
system in question to thermal-equilibrium properties of the same system. That
is not the motivation, however, of the below proposed application of liquid-
state theory’s configurational temperature Landau and Lifshitz (1958); Rugh
(1997); Powles  _et al._ (2005); Himpel and Melzer (2019) to active-matter
models, which instead focuses on scaling properties.

Active matter is an umbrella term used to describe any physical system whose
building blocks can autonomously perform mechanical work. This includes fluids
consisting of self-propelled particles ()e.g., Quincke rollers, suspensions of
swimming bacteria, animal groups), mutually-propelled particles (e.g.,
cytoskeletal filaments and motor proteins), immobile dividing cells (e.g.,
sessile bacteria), immobile contractile cells (e.g., epithelia), etc. A common
theme of active matter is the self propulsion as found, e.g., in connection
with cell migration, bacterial motion, and bird or insect flock dynamics
Angelini  _et al._ (2011); Marchetti  _et al._ (2013); Bechinger  _et al._
(2016); Ramaswamy (2017); Saintillan (2018); Das  _et al._ (2020); Shaebani
_et al._ (2021); Bowick  _et al._ (2022). Active matter is not time reversible
(although in some instances like that of active turbulence there may be
statistically time-reversible states). This means that a multitude of
different dynamics may come into play O’Byrne  _et al._ (2022), presenting a
much richer field of study than that of ordinary time-reversible dynamics
Mandal  _et al._ (2020). A noted example of the intriguing features of active
matter is motility-induced phase separation (MIPS), which is the finding that
even a purely repulsive system may phase separate into high- and low-density
phases Vicsek  _et al._ (1995); Das  _et al._ (2014); Cates and Tailleur
(2015); Ramaswamy (2017); Geyer  _et al._ (2019); Das  _et al._ (2020);
Merrigan  _et al._ (2020).

Active matter does not have states of ordinary thermal equilibrium, but there
have been suggestions for how to map active-matter states to equilibrium,
implying the existence of a non-equilibrium active-matter temperature. For
instance, Szamel proposed an effective temperature for a single self-propelled
particle Szamel (2014), and Fodor et al. showed Fodor  _et al._ (2016) that
for active Ornstein-Uhlenbeck particles (AOUP) at small persistence time one
can identify an effective temperature from the analog of the fluctuation-
dissipation theorem (see also Refs. Loi  _et al._ , 2008; Wang and Wolynes,
2011; Flenner and Szamel, 2020). In a parallel development, Takatori and Brady
formulated a thermodynamic-type temperature for active matter based on the
swim-pressure concept Takatori and Brady (2015).

For an ordinary time-reversal-invariant system in thermal equilibrium, the
temperature $T$ is equal to ${T}_{\rm conf}$ that is defined Rugh (1997);
Powles  _et al._ (2005) as follows. For a system of $N$ particles with
collective coordinate vector
$\mathbb{R}\equiv(\mathbb{r}_{1},...,\mathbb{r}_{N})$ and potential-energy
function $U(\mathbb{R})$, $k_{B}{T}_{\rm conf}\equiv\langle(\nabla
U)^{2}\rangle/\langle\nabla^{2}U\rangle$ in which $k_{B}$ is the Boltzmann
constant, $\nabla$ is the gradient operator in the $3N$-dimensional
configuration space, and the sharp brackets denote canonical-ensemble
averages. The proof that ${T}_{\rm conf}=T$ in equilibrium is so simple that
it deserves to be repeated here Landau and Lifshitz (1958): If $Z$ is the
configuration-space partition integral, a partial integration of
$\langle\nabla^{2}U\rangle=\int\nabla^{2}U(\mathbb{R})\exp(-U(\mathbb{R})/k_{B}T)d\mathbb{R}/Z$
leads to $\langle\nabla^{2}U\rangle=-\int\nabla
U(\mathbb{R})\cdot\nabla\exp(-U(\mathbb{R})/k_{B}T)d\mathbb{R}/Z=\langle(\nabla
U)^{2}\rangle/k_{B}T$ from which ${T}_{\rm conf}=T$ follows.

Approaching the thermodynamic limit, the relative fluctuations of both the
numerator and the denominator of ${T}_{\rm conf}$ vanish. This means that if
one defines an $\mathbb{R}$-dependent configurational temperature by

$k_{B}{T}_{\rm conf}(\mathbb{R})\,\equiv\,\frac{(\nabla
U(\mathbb{R}))^{2}}{\nabla^{2}U(\mathbb{R})}\,,$ (1)

the identity ${T}_{\rm conf}(\mathbb{R})\cong T$ applies in the sense that
deviations vanish as $N\to\infty$. We have this limit in mind throughout and
shall (mostly) ignore that ${T}_{\rm conf}(\mathbb{R})$ fluctuates slightly
for any finite system. Note that, in contrast to the “standard” kinetic-energy
based temperature definition, the configurational temperature is not defined
for a system of free particles. Note also that configurations with
$\nabla^{2}U(\mathbb{R})=0$ become less likely as $N\to\infty$ so the fact
that Eq. (1) is not defined for such configurations is irrelevant; by the same
argument one can ignore the existence of configurations with
$\nabla^{2}U(\mathbb{R})<0$. We return below briefly to a discussion of
${T}_{\rm conf}$ fluctuations in simulations (Fig. 3). Finally, we note that
the focus of the present paper is on the subset of active-matter models
without orientational interactions.

Since the derivation of the configurational temperature ${T}_{\rm conf}$ is
based on the fact that the probability in the canonical ensemble of
configuration $\mathbb{R}$ is proportional to $\exp(-U(\mathbb{R})/k_{B}T)$,
it would appear that ${T}_{\rm conf}$ cannot be relevant for systems that are
far from thermal equilibrium. We show below that ${T}_{\rm conf}(\mathbb{R})$
may nevertheless be used for tracing out lines of invariant structure and
dynamics in the phase diagram of certain active-matter models with hidden
scale invariance. The latter is the symmetry that the ordering of
configurations according to their potential energy at a given density is
maintained if these are scaled uniformly to a different density (compare Eq.
(6) below), a property that applies to a good approximation for a number of
well-known pair potentials, including the Lennard-Jones and Yukawa interaction
Gnan  _et al._ (2009); Schrøder and Dyre (2014); Ingebrigtsen and Tanaka
(2015); Dyre (2018).

##  II Approximately invariant physics at different densities of the Kob-
Andersen AOUP model

This section studies the active Ornstein-Uhlenbeck particle (AOUP) dynamics of
the Kob-Andersen (KA) binary Lennard-Jones (LJ) model Kob and Andersen (1995),
which is popular in numerical studies of viscous liquids and the glass
transition because it is easily supercooled into a metastable highly viscous
liquid state Ingebrigtsen  _et al._ (2019). We simulated a KA system of
$10000$ particles consisting of two spheres, A (80%) and B (20%). Writing the
LJ pair potential between particles of type $\alpha$ and $\beta$ as
$v_{\alpha\beta}(r)=4\varepsilon_{\alpha\beta}((r/\sigma_{\alpha\beta})^{-12}-(r/\sigma_{\alpha\beta})^{-6})$
with $\alpha,\beta=A,B$, the KA parameters are Kob and Andersen (1995)
$\sigma_{AA}=1.0$, $\sigma_{AB}=\sigma_{BA}=0.8$, $\sigma_{BB}=0.88$,
$\varepsilon_{AA}=1.0$, $\varepsilon_{AB}=\varepsilon_{BA}=1.5$,
$\varepsilon_{BB}=0.5$. A shifted-force cutoff of $v_{\alpha\beta}(r)$ at
$r_{\textrm{cut}}=2.5\sigma_{\alpha\beta}$ was used Toxvaerd and Dyre (2011).
The simulations employed the time step $\Delta
t=\Delta\tilde{t}/(D~{}\rho^{2/3})$ in which $\Delta\tilde{t}=0.4$ and $\rho$
is the (particle) density. At the reference density $\rho=1.2$ the value of
$\Delta t=0.0001$ was used. The simulations were carried out on GPU cards,
with the active-matter simulations using a home-made code while the MD
simulations used RUMD Bailey  _et al._ (2017).

The AOUP model is a model which has no momentum conservation and for which
hydrodynamics is not taken into account; all information about particle
interactions is contained in the potential-energy function Marchetti  _et al._
(2013); Fily  _et al._ (2017); Shaebani  _et al._ (2021). In the
$3N$-dimensional configuration space the AOUP equation of motion Farage  _et
al._ (2015); Maggi  _et al._ (2015); Szamel  _et al._ (2015); Fodor  _et al._
(2016) is

$\dot{\mathbb{R}}\,=\,\mu\mathbb{F}(\mathbb{R})\,+\,\bm{\eta}(t)\,.$ (2)

Here $\mu$ is the mobility (velocity over force) and the force vector is given
by $\mathbb{F}(\mathbb{R})=-\nabla U(\mathbb{R})$. The noise vector
$\bm{\eta}(t)$ is colored according to an Ornstein-Uhlenbeck process, i.e., is
a Gaussian stochastic process characterized by

$\langle\eta_{i}^{\alpha}(t)\eta_{j}^{\beta}(t^{\prime})\rangle\,=\,\delta_{ij}\delta_{\alpha\beta}\frac{D}{\tau}\,e^{-|t-t^{\prime}|/\tau}\,$
(3)

in which $i$ and $j$ are particle indices, $\alpha$ and $\beta$ represent
spatial $xyz$ indices, and $D$ and $\tau$ are constants. The diffusion
coefficient $D$ is that of a single AOUP, which may be used to introduce a
single-particle effective temperature. We are interested in how the physics is
affected when the density is changed. The question addressed below is whether
approximately invariant physics can be obtained by adjusting $D$ and $\tau$
(regarding $\mu$ as a material constant throughout) in the sense that a movie
of the particle motion will show the same except for a scaling of space and
time.

Figure 1:  Radial distribution functions (RDF) of the Kob-Andersen system with
active Ornstein-Uhlenbeck (AOUP) dynamics at densities between 1.2 and 3.0 for
the model parameters $D$ and $\tau$ of Table 1 determined by means of Eq. (II)
and Eq. (1). The first column shows the three partial RDFs along the proposed
line of invariance generated from the reference state point
$(\rho_{0},D_{0},\tau_{0})=(1.2,3000,10)$, plotted as functions of the pair
distance $r$. The second column shows the same data plotted as functions of
the reduced pair distance $\tilde{r}\equiv\rho^{1/3}r$, revealing a good
collapse except at the first peak of $g_{AB}$. For comparison, the third
column shows data for the same values of $D$ and $\tau$ as the two previous
columns at the fixed density $\rho=1.2$; the fourth column shows AOUP data at
the reference state point (brown) and standard MD thermal equilibrium data
(indigo) at the density $\rho=1.2$ and the MD temperature that results in the
same average potential energy as that of the AOUP simulation ($T_{MD}=1.57$).

The dimension of $\mu$ is length squared over energy times time. Thus, if
$l_{0}$ is a length unit, $t_{0}$ a time unit, and $e_{0}$ an energy unit, the
quantity $\mu(t_{0}e_{0}/l_{0}^{2})$ is dimensionless. Likewise,
$D(t_{0}/l_{0}^{2})$ and $\tau/t_{0}$ are dimensionless because $D$ has the
dimension of a diffusion coefficient and $\tau$ of a time. It is reasonable to
expect that when the density is changed, invariant physics can come about only
if these three dimensionless quantities do not change. As length unit we take
the average interparticle spacing, $l_{0}=\rho^{-1/3}$. The colored-noise
correlation time $\tau$ of Eq. (3) is a natural choice for the time unit,
$t_{0}=\tau$. The idea is now to investigate the consequences of using for the
energy unit the configurational temperature, i.e., choosing
$e_{0}=k_{B}{T}_{\rm conf}$ (Sec. III justifies this choice). If the above two
dimensionless quantities are to be invariant when density varies, the
following must apply: $\mu\propto l_{0}^{2}/(t_{0}e_{0})=\rho^{-2/3}/(\tau
k_{B}{T}_{\rm conf})$ and to $D\propto l_{0}^{2}/t_{0}=\rho^{-2/3}/\tau$.
Since $\mu$ is assumed constant, this leads to
$\tau\propto\rho^{-2/3}/k_{B}{T}_{\rm conf}$ and $D\propto k_{B}{T}_{\rm
conf}$, i.e., the following equations determine $D$ and $\tau$ at the density
$\rho$ from their values $D_{0}$ and $\tau_{0}$ at a reference state point of
density $\rho_{0}$,

Figure 2:  Mean-square displacement (MSD) at the same state points as in Fig.
1. The first column shows the MSD of the A and B particles along the predicted
line of invariance, plotted as a function of the time $t$. The second column
shows the same data in reduced units revealing a good collapse. The third
column shows reduced data for the same values of $D$ and $\tau$ as the
previous figures at the fixed density $\rho=1.2$. The fourth column shows a
comparison of the reduced MSD AOUP data at the reference state point (brown)
to standard reduced MD thermal-equilibrium data (indigo) at $\rho=1.2$, where
the temperature was determined to result in the same average potential energy
as the AOUP simulation ($T_{MD}=1.57$); the MD reduced units are defined below
in Eq. (7).

$\displaystyle D$ $\displaystyle\,=\,$ $\displaystyle D_{0}\,\,\frac{{T}_{\rm
conf}(\rho)}{{T}_{\rm conf}(\rho_{0})}\,,$ $\displaystyle\tau$
$\displaystyle\,=\,$
$\displaystyle\tau_{0}\left(\frac{\rho_{0}}{\rho}\right)^{2/3}\frac{{T}_{\rm
conf}(\rho_{0})}{{T}_{\rm conf}(\rho)}\,.$ (4)

We note that $D$ scales in proportion to the configurational temperature
${T}_{\rm conf}$, which is analogous to the equilibrium result $D\propto T$
where $T$ is the temperature.

As mentioned, fluctuations are small for a large system, so ${T}_{\rm
conf}(\rho_{0})$ may be evaluated from a single configuration of a steady-
state simulation, $\mathbb{R}_{0}$ by means of ${T}_{\rm
conf}(\rho_{0})\cong{T}_{\rm conf}(\mathbb{R}_{0})$. In order to find
${T}_{\rm conf}(\rho)$ one scales $\mathbb{R}_{0}$ uniformly to the density
$\rho$ using $\mathbb{R}=(\rho_{0}/\rho)^{1/3}\mathbb{R}_{0}$; the
configurational temperature is then identified from ${T}_{\rm
conf}(\rho)\cong{T}_{\rm conf}(\mathbb{R})$. When substituted into Eq. (II)
this leads to the following recipe for calculating $D$ and $\tau$ at density
$\rho$

$\displaystyle D$ $\displaystyle\,=\,$ $\displaystyle D_{0}\,\,\frac{{T}_{\rm
conf}\left[(\rho_{0}/\rho)^{1/3}\mathbb{R}_{0}\right]}{{T}_{\rm
conf}(\mathbb{R}_{0})}\,,$ $\displaystyle\tau$ $\displaystyle\,=\,$
$\displaystyle\tau_{0}\left(\frac{\rho_{0}}{\rho}\right)^{2/3}\frac{{T}_{\rm
conf}(\mathbb{R}_{0})}{{T}_{\rm
conf}\left[(\rho_{0}/\rho)^{1/3}\mathbb{R}_{0}\right]}\,.$ (5)

Table 1 shows the resulting values of $D$ and $\tau$ for densities ranging
from 1.2 to 3.0, starting from the reference state point
$(\rho,D,\tau)=(1.2,3000,10)$.

Figure 3:  Configurational temperature variation. (a) shows the configurational temperature ${T}_{\rm conf}$ as a function of the density for the selected scaling configuration $\mathbb{R}_{0}$. At the highest temperatures, ${T}_{\rm conf}$ scales with density in proportion to $\rho^{4}$ (red dashed line); this is where the repulsive $r^{-12}$ term of the LJ pair potential dominates the potential energy. At the lowest temperatures, the scaling is approximately proportional to $\rho^{5.3}$ (blue dashed line), showing that the scaling is nontrivial. (b) shows the distribution of ${T}_{\rm conf}$ for $N=10000$ particles at the reference state point. The spread is much larger than expected from a simple statistical $1/\sqrt{N}$ argument. The blue star marks ${T}_{\rm conf}$ of the reference scaling configuration $\mathbb{R}_{0}$. (c) shows how $D$ varies according to Eq. (II) for three different configurations: one is the $\mathbb{R}_{0}$ used in Fig. 1 and Fig. 2, which was selected from the center of the distribution (red), the two others are from the lowest and highest ends of the distribution (black and blue). (d) shows how $\tau$ varies according to Eq. (II) for the same three configurations. $\rho$ | $D$ | $\tau$ | ${T}_{\rm conf}$
---|---|---|---
$1.2$ | $3000$ | $10.000$ | 0.2742
$1.5$ | $9859$ | $2.622$ | 0.9014
$2.0$ | $39160$ | $0.5450$ | 3.580
$2.5$ | $105600$ | $0.1741$ | 9.657
$3.0$ | $230800$ | $0.0706$ | 21.10
Table 1:  Density $\rho$ and model parameters $D$ and $\tau$ along the
predicted line of invariance calculated from Eq. (II) in which ${T}_{\rm
conf}(\rho)$ is determined from a single configuration $\mathbb{R}_{0}$ by
means of Eq. (1) after first scaling it uniformly to density $\rho$.

The two left columns of Fig. 1 show the three partial radial distribution
functions (RDFs) along the predicted line of invariance plotted, respectively,
as a function of the radial distance $r$ and of the reduced radial distance
$\tilde{r}\equiv\rho^{1/3}r$. The latter shows good invariance, except that
the height of the first peak is not invariant, in particular for the AB RDF.
The third column of Fig. 1 shows the results for the same values of $D$ and
$\tau$ as previously (see Table 1) but with the fixed density $\rho=1.2$, in
which case no invariance is observed. Finally, the fourth column compares the
reference density RDFs with those of a thermal-equilibrium molecular-dynamics
(MD) simulation at the reference density and the temperature at which the
average potential energy is equal to that of the reference-state-point AOUP
simulation ($T_{MD}=1.57$), showing little resemblance. This temperature is
quite different from the configurational temperature (${T}_{\rm conf}=0.27$,
which corresponds to a such a deeply supercooled state for the Newtonian
system that the metastable liquid presently cannot be equilibrated using MD).

Figure 2 shows the mean-square displacement (MSD) of the A and B particles as
functions of the time. The four columns are similar to those of Fig. 1 with
the time $t$ as the x-coordinate in the first column and the reduced time
$\tilde{t}\equiv(D\rho^{2/3})t\propto t/\tau$ in the second, where the MSD is
also given in reduced units, i.e., multiplied by $\rho^{2/3}$. The latter
shows approximate invariance of the dynamics. It is instructive to consider
the limits of short and long times. For $t\to 0$, in the “ballistic” regime,
Eq. (2) and Eq. (3) imply that the MSD is proportional to $(D/\tau)t^{2}$,
while for $t\to\infty$ the MSD is proportional to $Dt$. Thus the reduced-unit
short- and long-time limit MSDs are proportional to
$\rho^{2/3}D\tau\tilde{t}^{2}$ and $\rho^{2/3}D\tau\tilde{t}$, respectively.
Since Eq. (II) implies that $\rho^{2/3}D\tau$ is a constant, this means that
in these limit the MSDs are simply proportional to $\tilde{t}^{2}$ and
$\tilde{t}$, respectively, which is confirmed by Fig. 2(b). The third column
of Fig. 2 gives the reduced MSD using the predicted $D$ and $\tau$ at the
reference density. The fourth column compares the reference state point MSDs
to those of the $T_{MD}=1.57$ MD simulation. We conclude from Fig. 1 and Fig.
2 that there is a nontrivial approximate invariance of structure and dynamics.

Figure 3 investigates the robustness of the procedure used for generating
lines of approximately invariant structure and dynamics in the general AOUP
phase diagram. Figure 3(a) shows ${T}_{\rm conf}$ as a function of $\rho$ in a
log-log plot for selected scaling configuration $\mathbb{R}_{0}$ used in Eq.
(II). At high density one finds almost ${T}_{\rm conf}(\rho)\propto\rho^{4}$,
which reflects the dominance here of the $r^{-12}$ repulsive term of the LJ
pair potential – it is straightforward to show from Eq. (1) that ${T}_{\rm
conf}(\rho)\propto\rho^{n/3}$ for a system of inverse power-law pair
potentials $\propto r^{-n}$. At lower densities this does not apply, however,
showing that the invariance of structure and dynamics is not a trivial
consequence of the scale-invariant repulsive $r^{-12}$ term of the LJ pair
potential. Figure 3(b) shows the distribution of configurational temperatures
for the system simulated at the reference state point. We find a fairly broad
distribution. This motivates an investigation into how much the prediction of
the invariance line depends on the choice of the $\mathbb{R}_{0}$. Figures
3(c) and (d) show the predictions for $D$ and $\tau$ using three different
configurations in Eq. (1). The red curve is for the scaling configuration
$\mathbb{R}_{0}$ used above that was selected from the center of the
distribution in (b), while the black and blue curves are for two
configurations taken from the lower and higher ends of the distribution. For
both $D$ and $\tau$ there is little visible difference; in the two latter
cases we find that the RDFs and MSDs are virtually indistinguishable from
those of Fig. 1 and Fig. 2 (data not shown). Only a ratio of configurational
temperatures is present in Eq. (II), and it appears that a significant
cancellation occurs. This means that, despite the relatively large spread of
${T}_{\rm conf}$, $N=10000$ particles are enough for Eq. (II) to be used for
predicting model parameters resulting in approximately invariant structure and
dynamics.

##  III Justification of the procedure

How can the canonical-ensemble-based characteristic energy $k_{B}{T}_{\rm
conf}$ provide the relevant energy scale for identifying invariant active-
matter physics upon a density change? While the energy $k_{B}{T}_{\rm conf}$
per se may not have physical relevance in active-matter models, we argue below
that the ratio ${T}_{\rm conf}(\rho)/{T}_{\rm conf}(\rho_{0})$ determines the
ratio of the relevant energy scales at the two densities. To arrive at this
conclusion, it is necessary to first summarize some recent findings.

The starting point is that the KA model to a good approximation obeys the
hidden-scale-invariance symmetry defined Schrøder and Dyre (2014); Dyre (2018)
by the following logical implication for the potential-energy function
$U(\mathbb{R})$,

$U({\bf R}_{\rm a})<U({\bf R}_{\rm b})\,\,\Rightarrow\,\,U(\lambda{\bf R}_{\rm
a})<U(\lambda{\bf R}_{\rm b})\,.$ (6)

Here ${\bf R}_{\rm a}$ and ${\bf R}_{\rm b}$ are configurations of the same
density and $\lambda$ is a uniform-scaling parameter. Physically, Eq. (6)
expresses that the ordering of configurations at one density according to
their potential energy is maintained when configurations are scaled uniformly
to a different density. At a given thermodynamic equilibrium state point, the
excess entropy ${S}_{\rm ex}$ is defined as the entropy minus the ideal-gas
entropy at the same density and temperature Hansen and McDonald (2013). In the
case of ordinary Newtonian mechanics, Eq. (6) implies that structure and
dynamics in reduced units (see below) are invariant along the curves of
constant excess entropy Gnan  _et al._ (2009); Schrøder and Dyre (2014); Dyre
(2018). These curves are termed isomorphs, and systems with isomorphs are
termed R-simple.

Isomorph invariance is exact whenever Eq. (6) applies without exception, but
this is never the case for pair potentials with both attractions and
repulsions. Isomorph invariance is still a good approximation, however, if Eq.
(6) applies for most of the physically relevant configurations at the state
points in question. This is believed to be the case for most metals and van
der Waals bonded systems, whereas systems with strong directional interactions
like hydrogen-bonded and covalently bonded systems generally do not conform to
Eq. (6) and violate isomorph-theory predictions Dyre (2014) (ionic and dipolar
systems constitute a class in-between). Realistic pair-potential models with
approximate hidden scale invariance include the standard Lennard-Jones model
in single-component, binary, and polydisperse versions and with exponents
other than 6 and 12 (the so-called Mie potentials), as well as the Yukawa
(screened Coulomb) pair potential Schrøder and Dyre (2014); Veldhorst  _et
al._ (2015). For systems with inverse-power-law interactions, isomorph theory
is exact.

For R-simple systems with Newtonian dynamics, the structure and dynamics of
the condensed liquid and solid phases are isomorph invariant to a good
approximation when made dimensionless using as “reduced” units the length
$l_{0}$, energy $e_{0}$, and time $t_{0}$ given by (where $m$ is the particle
mass)

$l_{0}=\rho^{-1/3}\,,\,\,\,e_{0}=k_{B}T\,,\,\,\,t_{0}=\rho^{-1/3}\sqrt{m/k_{B}T}\,.$
(7)

The microscopic excess entropy is defined Schrøder and Dyre (2014) by
${S}_{\rm ex}(\mathbb{R})\equiv{S}_{\rm ex}(\rho,U(\mathbb{R}))$ in which
${S}_{\rm ex}(\rho,U)$ is the thermodynamic excess entropy of the equilibrium
state point with density $\rho$ and average potential energy $U$. Note that
the function ${S}_{\rm ex}(\mathbb{R})$ is defined for any configuration of
any system, whether R-simple or not. It can be shown that if Eq. (6) applies,
however, ${S}_{\rm ex}(\mathbb{R})$ depends only on the configuration’s
reduced coordinates, ${\tilde{\mathbb{R}}}\equiv\rho^{1/3}\mathbb{R}$ Schrøder
and Dyre (2014). Inverting the relation ${S}_{\rm ex}(\mathbb{R})={S}_{\rm
ex}(\rho,U(\mathbb{R}))$ for an R-simple system thus leads to

$U(\mathbb{R})\,=\,U(\rho,{S}_{\rm ex}({\tilde{\mathbb{R}}}))\,$ (8)

in which $U(\rho,{S}_{\rm ex})$ is the average potential energy of the
thermodynamic equilibrium state point with density $\rho$ and excess entropy
${S}_{\rm ex}$.

Consequences of Eq. (8) have so far been worked out only for systems with
standard time-reversible Newtonian dynamics Schrøder and Dyre (2014); Dyre
(2018), but since Eq. (8) follows from Eq. (6) that has no reference to
thermal equilibrium, Eq. (8) may also be applied to active-matter models with
a hidden-scale-invariant potential-energy function. Note that the function
${S}_{\rm ex}(\mathbb{R})$ still refers to the standard microcanonical
ensemble according to which ${S}_{\rm ex}(\mathbb{R})$ is basically the
logarithm of the number of configurations with the same density and potential
energy as $\mathbb{R}$ Schrøder and Dyre (2014).

We proceed to rewrite the AOUP equation of motion in terms of dimensionless
variables. Writing $\mathbb{R}=l_{0}{\tilde{\mathbb{R}}}$ and
$t=t_{0}\tilde{t}$ in which $l_{0}=\rho^{-1/3}$ and $t_{0}=\tau$ as in Sec.
II, Eq. (2) becomes (in which $\tilde{\nabla}=\rho^{-1/3}\nabla$)

$\frac{l_{0}}{\tau}\,\dot{{\tilde{\mathbb{R}}}}\,=\,-\mu\,\frac{1}{l_{0}}\,\tilde{\nabla}U(\mathbb{R})\,+\,\bm{\eta}(t)\,.$
(9)

The reduced noise is given by $\tilde{\bm{\eta}}=(\tau/l_{0})\bm{\eta}$ in
terms of which Eq. (3) becomes

$\langle\tilde{\eta}_{i}^{\alpha}(\tilde{t})\tilde{\eta}_{j}^{\beta}(\tilde{t}^{\prime})\rangle\,=\,\delta_{ij}\,\delta_{\alpha\beta}\,\frac{\tau\,D}{l_{0}^{2}}\,e^{-|\tilde{t}-\tilde{t}^{\prime}|}\,.$
(10)

Equation (9) leads to

$\dot{{\tilde{\mathbb{R}}}}\,=\,-\mu\,\frac{\tau}{l_{0}^{2}}\,\tilde{\nabla}U(\mathbb{R})\,+\,\tilde{\bm{\eta}}(\tilde{t})\,.$
(11)

For any configuration $\mathbb{R}$ the systemic temperature ${T}_{\rm
s}(\mathbb{R})$ is defined Dyre (2020) by

${T}_{\rm s}(\mathbb{R})\,\equiv\,\left.\left(\frac{\partial
U}{\partial{S}_{\rm ex}}\right)_{\rho}\right|_{{S}_{\rm ex}={S}_{\rm
ex}(\mathbb{R})}\,.$ (12)

In a steady-state situation the fluctuations of the systemic temperature go to
zero in the thermodynamic limit, just as those of ${T}_{\rm
conf}(\mathbb{R})$. We henceforth often leave out $\mathbb{R}$ when convenient
and write simply ${T}_{\rm s}$. In practice, to determine ${T}_{\rm
s}(\mathbb{R})$ one utilizes the fact that ${T}_{\rm s}(\mathbb{R})$ is the
equilibrium temperature ${T}_{\rm eq}$ of the thermodynamic state point with
the density of $\mathbb{R}$ and excess entropy ${S}_{\rm ex}(\mathbb{R})$,
implying that Dyre (2020)

${T}_{\rm s}(\mathbb{R})\,=\,{T}_{\rm eq}(\rho,{S}_{\rm
ex}({\tilde{\mathbb{R}}}))={T}_{\rm eq}(\rho,U(\mathbb{R}))\,.$ (13)

Equation (8) leads to $\tilde{\nabla}U(\mathbb{R})={T}_{\rm
s}\tilde{\nabla}{S}_{\rm ex}({\tilde{\mathbb{R}}})$. When substituted into Eq.
(11) this results in

$\dot{{\tilde{\mathbb{R}}}}\,=\,-\mu\,\frac{\tau{T}_{\rm
s}}{l_{0}^{2}}\,\tilde{\nabla}{S}_{\rm
ex}({\tilde{\mathbb{R}}})\,+\,\tilde{\bm{\eta}}(\tilde{t})\,.$ (14)

We conclude from Eq. (10) and Eq. (14) that the reduced AOUP equation of
motion is invariant upon a density change if $\tau D/l_{0}^{2}$ and
$\tau{T}_{\rm s}/l_{0}^{2}$ do not vary. This implies $D(\rho)\propto{T}_{\rm
s}(\rho)$ and $\tau(\rho)\propto\rho^{-2/3}/{T}_{\rm s}(\rho)$ in which
${T}_{\rm s}(\rho)$ is short-hand notation for ${T}_{\rm eq}(\rho,{S}_{\rm
ex}({\tilde{\mathbb{R}}}))$, compare Eq. (13). Working from the reference
state point $(\rho_{0},D_{0},\tau_{0})$, this means that the function
${T}_{\rm s}(\rho)$ determines how to scale $D$ and $\tau$ to ensure invariant
AOUP dynamics,

$\displaystyle D(\rho)\,$ $\displaystyle=$
$\displaystyle\,D(\rho_{0})\,\frac{{T}_{\rm s}(\rho)}{{T}_{\rm s}(\rho_{0})}$
$\displaystyle\tau(\rho)\,$ $\displaystyle=$
$\displaystyle\,\tau(\rho_{0})\,\left(\frac{\rho_{0}}{\rho}\right)^{2/3}\,\frac{{T}_{\rm
s}(\rho_{0})}{{T}_{\rm s}(\rho)}\,.$ (15)

We next link to the configurational temperature. There is no reason to expect
${T}_{\rm conf}={T}_{\rm s}$, and these quantities indeed differ by up to a
factor of six in our simulations. However, Eq. (III) still applies with
${T}_{\rm conf}$ instead of ${T}_{\rm s}$ if the two temperatures are
proportional in their density variation. To show this we note that Eq. (8)
implies $\tilde{\nabla}U(\mathbb{R})={T}_{\rm s}\tilde{\nabla}{S}_{\rm
ex}({\tilde{\mathbb{R}}})$ and $\tilde{\nabla}^{2}U(\mathbb{R})={T}_{\rm
s}\tilde{\nabla}^{2}{S}_{\rm ex}({\tilde{\mathbb{R}}})$, so ${T}_{\rm
conf}(\mathbb{R})=(\nabla
U(\mathbb{R}))^{2}/\nabla^{2}U(\mathbb{R})=(\tilde{\nabla}U(\mathbb{R}))^{2}/\tilde{\nabla}^{2}U(\mathbb{R})={T}_{\rm
s}(\tilde{\nabla}{S}_{\rm
ex}({\tilde{\mathbb{R}}}))^{2}/\tilde{\nabla}^{2}{S}_{\rm
ex}({\tilde{\mathbb{R}}})$ Dyre (2020). Here we ignored the dependence of
${T}_{\rm s}(\mathbb{R})$ on the configuration $\mathbb{R}$ at a given state
point which, as argued above, vanishes in the thermodynamic limit. In terms of
$\phi({\tilde{\mathbb{R}}})\equiv(\tilde{\nabla}{S}_{\rm
ex}({\tilde{\mathbb{R}}}))^{2}/\tilde{\nabla}^{2}{S}_{\rm
ex}({\tilde{\mathbb{R}}})$ we thus have

$\frac{{T}_{\rm conf}(\mathbb{R})}{{T}_{\rm
conf}(\mathbb{R}_{0})}=\frac{{T}_{\rm
s}(\rho)\,\phi({\tilde{\mathbb{R}}})}{{T}_{\rm
s}(\rho_{0})\,\phi({\tilde{\mathbb{R}}}_{0})}\,.$ (16)

Since ${\tilde{\mathbb{R}}}={\tilde{\mathbb{R}}}_{0}$ this implies ${T}_{\rm
conf}(\mathbb{R})/{T}_{\rm conf}(\mathbb{R}_{0})={T}_{\rm s}(\rho)/{T}_{\rm
s}(\rho_{0})$. In this way Eq. (III) leads to Eq. (II). Note that by using
${T}_{\rm conf}$ rather than ${T}_{\rm s}$, one does not have to identify the
equilibrium state point with the same potential energy as the active-matter
state point in question.

Figure 4:  Master isomorph (dashed curve, Eq. (17)) expressing the virial $W$
as a function of the potential energy $U$ for LJ systems along any isomorph;
$W_{0}$ is the virial at the state point of zero potential energy on the
isomorph. This is Fig. 8 of Ref. Schrøder  _et al._ , 2011 to which we have
added points for the AOUP KA model (open circles). The abbreviations KA, WBLJ,
SCLJ stand for the Kob-Andersen system, the Wahnstrom binary LJ mixture
Wahnström (1991), and the standard single-component LJ system.

As mentioned, lines of invariance are termed isomorphs in the case of standard
Newtonian equilibrium dynamics. This concept was recently generalized to out-
of-equilibrium Newtonian systems like that of a shear flow or an aging glass,
leading to the introduction of the systemic temperature (Eq. (12)); the name
“systemic isomorph” was introduced for the lines of approximately invariant
structure and dynamics of the relevant phase diagram Dyre (2020). The results
of the present paper demonstrate the existence of systemic isomorphs for non-
time-reversible systems like active matter.

We end this section by checking a consequence of the above. It was shown in
Ref. Schrøder  _et al._ , 2011 for LJ systems that if $W_{0}$ is the virial at
zero potential energy of a given isomorph, the following relation between the
virial $W$ and the potential energy $U$ applies along the isomorph in question

$2\,\frac{W}{W_{0}}\,=\,1\,+\,8\,\frac{U}{W_{0}}\,+\,\sqrt{1\,+\,8\,\frac{U}{W_{0}}}\,.$
(17)

This identity is solely a consequence of the reduced-unit RDF isomorph
invariance that applies in the R-simple region (liquid or solid, not gas) of
any single- or multicomponent LJ system Schrøder  _et al._ (2011). Since the
reduced RDF is also invariant to a good approximation along the above studied
KA active-matter systemic isomorph (Fig. 1), $W$ should also in this case be
related to $U$ according to Eq. (17). This is validated in Fig. 4 that
reproduces the Newtonian-dynamics equilibrium data of Ref. Schrøder  _et al._
, 2011. It is clear from Fig. 4 that the states considered in the present
paper are qualitatively different from the equilibrium states represented in
the figure; basically the active systems are dense and corresponds to high-
temperature equilibrium fluids.

##  IV The Yukawa Active Brownian-particle model in two dimensions

In order to illustrate the generality of the procedure, this section
illustrates the procedure for generating lines of approximately invariant
structure and dynamics – systemic isomorphs – for a different active-matter
model. We study the active Brownian-particle (ABP) model in two dimensions of
the single-component Yukawa pair potential Yukawa (1935); Meacock  _et al._
(2021)

$v(r)\,=\,\frac{Q^{2}\,\sigma}{r}\,e^{-r/(\lambda\sigma)}\,.$ (18)

The Yukawa pair potential has hidden scale invariance Veldhorst  _et al._
(2015), so the above developed procedure should apply here as well.

If $\mathbb{r}_{i}$ is the position vector of particle $i$, the ABP equations
of motion in two dimensions is

$\dot{\mathbb{r}}_{i}\,=\,\mu\mathbb{F}_{i}+\bm{\xi}_{i}(t)+v_{0}\,\mathbf{o}_{i}(t)\,.$
(19)

Here, $\mu$ is again the mobility,
$\mathbb{F}_{i}(\mathbb{R})=-\nabla_{i}U(\mathbb{R})$ is the force on particle
$i$, $\bm{\xi}_{i}(t)$ is a Gaussian random white-noise vector, $v_{0}$ is a
constant velocity, and
$\mathbf{o}_{i}(t)=(\cos(\theta_{i}(t)),\sin(\theta_{i}(t)))$ is a stochastic
unit vector. The direction vector angle $\theta_{i}(t)$ is determined by a
white Gaussian noise of magnitude $D_{r}$,

$\langle\dot{\theta}_{i}(t)\dot{\theta}_{j}(t^{\prime})\rangle\,=\,2D_{r}\delta_{ij}\,\delta(t-t^{\prime})\,,$
(20)

and the white-noise vector has magnitude $D_{t}$,

$\langle\bm{\xi}_{i}^{\alpha}(t)\bm{\xi}_{j}^{\beta}(t^{\prime})\rangle\,=\,2D_{t}\delta_{ij}\delta_{\alpha\beta}\delta(t-t^{\prime})\,.$
(21)

The ABP model has four parameter. Regarding again $\mu$ as a material
constant, in order to have invariant physics the dimensionless versions of the
three other parameters are required to be constant when the density is
changed. Following the procedure of Sec. III we take as units
$l_{0}=\rho^{-1/2}$ (the exponent is $-1/2$ because the model is two-
dimensional) and $t_{0}=1/D_{r}$, and write the equation of motion in terms of
the corresponding reduced variables. Substituting
$\mathbb{r}_{i}=\rho^{-1/2}\tilde{\mathbb{r}}_{i}$ and $t=(1/D_{r})\tilde{t}$
into Eq. (19) and making use of Eq. (8) as well as the definition of the
systemic temperature ${T}_{\rm s}$ (Eq. (13)), we get

$\dot{\tilde{\mathbb{r}}}_{i}\,=\,-\mu\rho({T}_{\rm
s}/D_{r})\tilde{\nabla}_{i}{S}_{\rm
ex}({\tilde{\mathbb{R}}})+\tilde{\bm{\xi}}_{i}(t)+\tilde{v}_{0}\,\mathbf{o}_{i}(t)\,.$
(22)

Here $\tilde{v}_{0}=(\rho^{1/2}/D_{r})v_{0}$,
$\tilde{\bm{\xi}}_{i}=(\rho^{1/2}/D_{r})\bm{\xi}_{i}$,

$\langle\tilde{\bm{\xi}}_{i}^{\alpha}(t)\tilde{\bm{\xi}}_{j}^{\beta}(t^{\prime})\rangle\,=\,2\rho(D_{t}/D_{r})\delta_{ij}\delta_{\alpha\beta}\delta(\tilde{t}-\tilde{t}^{\prime})\,,$
(23)

and (with a dot denoting the derivative with respect to $\tilde{t}$)

$\langle\dot{\theta}_{i}(t)\dot{\theta}_{j}(t^{\prime})\rangle\,=\,2\delta_{ij}\,\delta(\tilde{t}-\tilde{t}^{\prime})\,.$
(24)

These equations are invariant under a change of density if $\mu\rho{T}_{\rm
s}/D_{r}$, $\rho D_{t}/D_{r}$, and $\tilde{v}_{0}$ are all constant. Since
$\mu$ is a material constant, this implies (where the subscript zero refers to
a reference state of density $\rho_{0}$ and ${T}_{\rm s}(\rho)\equiv{T}_{\rm
eq}(\rho,{S}_{\rm ex}({\tilde{\mathbb{R}}}))$

$\displaystyle D_{r}$ $\displaystyle\,=\,$ $\displaystyle
D_{r,0}\,\frac{\rho}{\rho_{0}}\,\frac{{T}_{\rm s}(\rho)}{{T}_{\rm
s}(\rho_{0})}$ $\displaystyle D_{t}$ $\displaystyle\,=\,$ $\displaystyle
D_{t,0}\,\frac{{T}_{\rm s}(\rho)}{{T}_{\rm s}(\rho_{0})}$ (25) $\displaystyle
v_{0}$ $\displaystyle\,=\,$ $\displaystyle
v_{0,0}\,\left(\frac{\rho}{\rho_{0}}\right)^{1/2}\frac{{T}_{\rm
s}(\rho)}{{T}_{\rm s}(\rho_{0})}\,.$

By the same argument as in Sec. III one can here replace ${T}_{\rm s}(\rho)$
by ${T}_{\rm conf}(\mathbb{R})$, leading to

$\displaystyle D_{r}$ $\displaystyle\,=\,$ $\displaystyle
D_{r,0}\,\frac{\rho}{\rho_{0}}\,\frac{{T}_{\rm
conf}\left[(\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right]}{{T}_{\rm
conf}(\mathbb{R}_{0})}$ $\displaystyle D_{t}$ $\displaystyle\,=\,$
$\displaystyle D_{t,0}\,\frac{{T}_{\rm
conf}\left[(\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right]}{{T}_{\rm
conf}(\mathbb{R}_{0})}$ (26) $\displaystyle v_{0}$ $\displaystyle\,=\,$
$\displaystyle v_{0,0}\,\left(\frac{\rho}{\rho_{0}}\right)^{1/2}\frac{{T}_{\rm
conf}\left[(\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right]}{{T}_{\rm
conf}(\mathbb{R}_{0})}\,.$

We note in passing that while the Peclet number $v_{0}/\sqrt{2D_{r}D_{t}}$
Bechinger  _et al._ (2016); Hecht  _et al._ (2021) is invariant along the
systemic isomorph, requiring this is not enough to determine how to scale the
model parameters; thus Peclet-number invariance is a necessary but not
sufficient condition for identifying a systemic isomorph. For certain systems
of ABPs the ratio of $D_{r}$ and $D_{t}$ is fixed (e.g., by the size of the
colloidal sphere). This may bring into question the relevance of the above
transformation, but even for such systems Eq. (IV) may used for comparing
state points of different models.

$\rho$ | $D_{r}$ | $D_{t}$ | $v_{0}$ | ${T}_{\rm conf}$
---|---|---|---|---
$1.0$ | $3.000$ | $1.000$ | $25.00$ | $1.489$
$1.5$ | $12.37$ | $2.750$ | $84.20$ | $4.093$
$2.0$ | $30.43$ | $5.072$ | $179.3$ | $7.550$
$2.5$ | $58.13$ | $7.751$ | $306.4$ | $11.54$
$3.0$ | $95.82$ | $10.65$ | $461.0$ | $15.85$
Table 2: Values of $\rho$, $D_{r}$, $D_{t}$, $v_{0}$, and ${T}_{\rm conf}$
along the systemic isomorph. As in Table 1, ${T}_{\rm conf}(\rho)$ is
determined from a single configuration $\mathbb{R}_{0}$ by means of Eq. (1).

To validate the above prediction we simulated $N=10000$ particles of the 2d
Yukawa system with $Q=50$, $\lambda=0.16$, $\sigma=1$ defining the length
unit, and a cutoff at $4.5\sigma$. The time step used is given by $\Delta
t=\Delta\tilde{t}(D_{t}/{v_{0}}^{2})$, where $\Delta\tilde{t}=0.0625$ so that
$\Delta t=0.0001$ at the reference state point defined by
$(\rho,D_{r},D_{t},v_{0})=(1.0,3.0,1.0,25.0)$. All simulations were carried
out on GPU cards using a home-made code. A systemic isomorph was traced out
for densities varying a factor of three using Eq. (IV) for a configuration
$\mathbb{R}_{0}$ selected from a simulation at the reference state point.
Table 2 gives the parameters obtained from Eq. (IV).

Figure 5:  Structure and dynamics of the Yukawa ABP model in two dimensions.
(a) The left panel shows the RDF as a function of the pair distance $r$ along
the systemic isomorph, the middle panel shows the same data in reduced units,
and the right panel shows the RDF for the same parameters (Table 2) at the
reference density $\rho=1.0$. (b) The left panel shows the MSD as a function
of time $t$ along the systemic isomorph, the middle panel shows the same data
in reduced units where the dashed lines mark slope unity (ordinary diffusion),
and the right panel shows the MSD for the same parameters at the reference
density.

Figure 5(a) shows the radial distribution function (RDF). The left two panels
show the RDF along the systemic isomorph as a function of $r$ and $\tilde{r}$,
respectively, while the right panel for comparison shows the results for the
same parameters at the reference state-point density $\rho=1.0$. We find a
good invariance of the reduced-unit RDF along the systemic isomorph. The same
is the case for the reduced-unit MSD shown in the middle panel of (b); the
left panel gives the MSD as a function of time while the right panel for
comparison is the reduced-unit MSD along the $\rho=1.0$ isochore.

Figure 6:  Phase diagrams showing MIPS state points as red stars and
homogeneous state points as black squares; green circles are gas-like states.
The MIPS phase consists of coexisting phases that differ by the density, the
denser of which corresponds to the “solid” homogeneous phase that consists of
hexagonal solid crystals. The reference state point
$(\rho,D_{r},D_{t},v_{0})=(1.01,3,1,367)$ is located in the homogeneous
(solid) phase close to the phase boundary. From it a systemic isomorph has
been traced out using Eq. (IV) (black line). The figure gives data in the
$(\rho,D_{t})$ phase diagram with $D_{r}$ and $v_{0}$ given by Eq. (IV) at
density $\rho$. The blue dashed line marks $\pm 5$% variations in density.

##  V The MIPS boundary of the 2d ABP Yukawa model

The 2d ABP Yukawa model is a model for which motility-induced phase separation
(MIPS) appears for certain parameter choices. Identifying which constants lead
to MIPS and which do not may be tedious because the phase diagram is four
dimensional. We show in this section that the configurational-temperature
provides a method for reducing the number of simulations needed to map out the
phase diagram. The idea is that, because the physics is approximately
invariant along a systemic isomorph, this cannot cross the MIPS boundary. Thus
if one has identified a state point in the homogeneous solid phase that is
close to the MIPS boundary and use this as reference state point, all state
points identified by Eq. (IV) should also be close to the MIPS boundary. A
similar line of reasoning has been validated for the melting line of the
standard Lennard-Jones system Costigliola  _et al._ (2016).

Figure 7:  Structure and dynamics probed along the MIPS phase boundary of the
2d ABP Yukawa system, slightly into the homogeneous phase (compare Fig. 6).
The left figures (a) and (c) show the radial distribution function and the
mean-square displacement, respectively, and the right figures (b) and (d) show
the same data in reduced units. The reduced-unit physics is invariant to a
good approximation along the boundary, confirming that it is a close to being
a systemic isomorph.

To investigate this use of the configurational temperature, we studied the
Yukawa model with parameters $Q=1000$ and $\lambda=0.12$ with a cutoff at
$4.2\sigma$. We then fixed the model parameters to the values
$(D_{r},D_{t},v_{0})=(3,1,367)$ and systematically decreased the density from
a high value where one is well within the homogeneous solid phase. A system of
40000 particles was simulated for 40 million time steps and the occurrence of
MIPS was detected by visual inspection (subsequent simulations involved just
10000 particles). The lowest density before observing MIPS was $\rho=1.01$. We
then used $(\rho,D_{r},D_{t},v_{0})=(1.01,3,1,367)$ as reference state point
for generating a systemic isomorph according to Eq. (IV). This is the black
full line in Fig. 7, which shows the results of investigating the existence of
MIPS in a $(\rho,D_{t})$ phase diagram where the parameters $D_{r}(\rho)$,
$D_{t}(\rho)$, and $v_{0}(\rho)$ are given by Eq. (IV). The black squares
denote state points of the homogeneous solid phase, the red stars denote state
points where MIPS appears (the low-density green circles denote gas-phase
state points). The blue dashed lines mark the systemic isomorph $\pm$5% in
density. We see that the phase transition line is predicted well though not
entirely accurately; this is consistent with the approximate nature of the
argument. Nevertheless, the simulations demonstrate that Eq. (IV) can be used
to help in identifying the MIPS phase by significantly reducing the required
number of simulations.

In order to validate that the black line of Fig. 7 is indeed a line of
approximately invariant physics we show in Fig. 7 how structure and dynamics
vary along it. The left figures show the RDF and MSD in standard units while
the right figures show the same data in reduced units.

##  VI How far is a given active-matter system from equilibrium?

We have shown that, despite the fact that its justification is based on the
canonical ensemble of equilibrium statistical mechanics, the configurational
temperature ${T}_{\rm conf}$ may be used for predicting lines of approximately
invariant physics of certain active-matter models of particles without
orientational interactions. In contrast to other temperatures discussed in
connection with active matter, the use of ${T}_{\rm conf}$ does not embody a
mapping to equilibrium.

We end the paper by suggesting a further application of ${T}_{\rm conf}$,
namely that it may be used for quantifying how far an active matter system is
from ordinary thermal equilibrium. In the context of ordinary non-active
matter researchers have introduced thermostats based on the idea of
configurational temperature and used them in the context of non-equilibrium
simulations Powles  _et al._ (2005), but the application proposed below is
simpler and does not involve any thermostat. Quantifying the degree of non-
equilibrium is often done by calculating one or the other form of entropy
production, which is zero in thermal equilibrium and thus provides a measure
of how far a system is from equilibrium Fodor  _et al._ (2016); Flenner and
Szamel (2020); O’Byrne  _et al._ (2022). However, using a quantity that goes
to zero in some limit to quantify the degree of deviation from that limit does
not in any obvious way make it possible to identify when deviations from
equilibrium are to be regarded as “large”. If deviations from equilibrium are
instead quantified by means of a quantity that goes to unity in the
equilibrium limit, deviations from equilibrium can be regarded as “small”
whenever that quantity does not deviate substantially from unity.

To illustrate this, we have performed simulations of the 2d ABP Yukawa model
at different values of the constant velocity $v_{0}$. Figure 8(a) shows the
systemic temperature ${T}_{\rm s}$ and the configurational temperature
${T}_{\rm conf}$ as functions of $v_{0}$ for fixed values of the remaining
three model parameters; here ${T}_{\rm s}$ was determined numerically by
identifying the equilibrium temperature at which the system under a MD
simulation has the same average potential energy as the active-matter system
in question. Both ${T}_{\rm s}$ and ${T}_{\rm conf}$ converge to 2 as
$v_{0}\to 0$, which is the equilibrium Brownian-dynamics temperature
($D_{t}=2$, $\mu=1$). Figure 8(b) shows the dissipated “active” power, i.e.,
the average of the dot product of the particle velocity with the $v_{0}$ term
of Eq. (19). The proposal is now to quantify the deviation from equilibrium in
terms of the ratio ${T}_{\rm s}/{T}_{\rm conf}$, and Fig. 8(c) plots the
dissipated power against this ratio. We here also included data where the
power is given in isomorph-reduced units (red points), which interestingly
show an almost linear dependence on ${T}_{\rm s}/{T}_{\rm conf}$. Figure 8(d)
shows log(Power) versus ${T}_{\rm s}/{T}_{\rm conf}$. From these simulations
one would conclude that the active matter system starts to deviate
substantially from an equilibrium system when $v_{0}$ is larger than about 3
because here ${T}_{\rm s}/{T}_{\rm conf}$ is roughly 1.1.

Figure 8:  Using the configurational temperature to quantify how far the
system is from thermal equilibrium, corresponding to $v_{0}=0$ in Eq. (19);
the parameters used here are $\rho=1$, $D_{r}=3$, and $D_{t}=2$. (a) and (b)
show how the systemic temperature ${T}_{\rm s}$ and the configurational
temperature ${T}_{\rm conf}$ vary with $v_{0}$ (left) and how the dissipation
(Power) in MD units varies with $v_{0}$ (right). From (a) we see that when
$v_{0}\to 0$, the two temperatures become identical (equal to $2$ because
$D_{t}=2$ and $D_{t}$ corresponds to the equilibrium temperature); at the same
time the dissipation goes to zero (b). (c) and (d) show the Power as a
function of ${T}_{\rm s}/{T}_{\rm conf}$. The quantity ${T}_{\rm s}/{T}_{\rm
conf}$ goes to unity as thermal equilibrium is approached, which presents an
advantage compared to using the dissipated Power for quantifying deviations
from equilibrium.

##  VII Outlook

Active-matter models involve more parameters than conventional equilibrium
models. We have shown that the configurational temperature provides a tool for
reducing significantly the amount of simulations needed to get an overview of
the phase diagram. If the pair potential is known for an experimental system,
the same method may be useful for obtaining information about regions of the
phase diagram that are not easily accessible. Unfortunately, the method is
limited to models based on a potential-energy function with hidden scale
invariance, but fortunately this constitutes a fairly large class that
includes, e.g., Mie-type pair potentials and the Yukawa pair potential. Mixing
two and three components, even considering broad polydisperse mixtures, does
not ruin the hidden-scale-invariance property Ingebrigtsen and Tanaka (2015).
It is our hope that ${T}_{\rm conf}$ besides its ability to identify systemic
isomorphs may also turn out to be useful for clarifying some of the puzzles of
active matter, for instance the recently discovered non-trivial dependence of
the apparent glass transition on the persistence time Berthier  _et al._
(2017). More generally, since the above reasoning is not dependent on specific
properties of active matter, the configurational temperature may also turn out
to be useful in other out-of-equilibrium situations, for instance that of a
nonlinear flow or of physical aging.

###### Acknowledgements.

This work was supported by the VILLUM Foundation’s Matter grant (16515).

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# Results of a Single Blind Literary Taste Test
with Short Anonymized Novel Fragments

Andreas van Cranenburgh
CLCG, University of Groningen
<EMAIL_ADDRESS>
Corina Koolen
Independent Scholar
<EMAIL_ADDRESS>
###### Abstract

It is an open question to what extent perceptions of literary quality are
derived from text-intrinsic versus social factors. While supervised models can
predict literary quality ratings from textual factors quite successfully, as
shown in the Riddle of Literary Quality project (Koolen et al., 2020), this
does not prove that social factors are not important, nor can we assume that
readers make judgments on literary quality in the same way and based on the
same information as machine learning models. We report the results of a pilot
study to gauge the effect of textual features on literary ratings of Dutch-
language novels by participants in a controlled experiment with 48
participants. In an exploratory analysis, we compare the ratings to those from
the large reader survey of the Riddle in which social factors were not
excluded, and to machine learning predictions of those literary ratings. We
find moderate to strong correlations of questionnaire ratings with the survey
ratings, but the predictions are closer to the survey ratings. Code and data:
https://github.com/andreasvc/litquest

##  1 Introduction

It remains an open question why some novels are considered literary, while
other novels such as thrillers are considered less prestigious. One position
in this debate is that formal features of the text play an important role,
i.e., formalism. The notion that foregrounding language is involved with the
aesthetic appreciation of literary texts is supported by experimental studies
(e.g., Hakemulder, 2004). Recent work investigates the role of the text in
literary quality empirically on a larger scale by collecting literary ratings
of novels in a large reader survey (henceforth the Riddle survey, Koolen et
al., 2020). This survey provides a way to investigate the association of
literary judgments and textual features directly. Can human judgments of
literary quality be predicted by machine learning models from purely textual
features? The answer turns out to be yes, to a substantial extent. The model
by van Cranenburgh and Bod (2017) showed that textual features can explain 61%
of the variation in the literary ratings.

However, machine learning may pick up on various subtle frequency differences
that humans do not notice; compare the case of authorship attribution, which
is arguably more difficult to do by humans by hand, than it is for a computer
with a simple table of function word frequencies and cosine distances. In
order to place the success of the machine learning model that can predict
literariness in context, we would have to ask humans to assign ratings purely
based on a text fragment, without being influenced by the prestige associated
with an author’s name or the title of the novel. The setup of such a
‘challenge’ was proposed in van Cranenburgh and Koolen (2019); however, no
results were presented. Wine experts perform at chance level in blind taste
tests (Hodgson, 2008); will readers be able to recognize literature?

In this paper we report on the results of a pilot version of such an
experiment and analyze the results. This allows us to see the extent to which
our participants agree with ratings in the Riddle survey who rate based on
author and title, as well as how they compare to the machine learning models
that perform a similar task. This also allows us to estimate the importance of
textual features for human readers when making judgments. By looking into the
motivations that the participants have given for their ratings, we also hope
to identify interesting linguistic features of literary quality, which can be
investigated further in future computational and literary studies work.

##  2 Questionnaire setup

We selected 8 originally Dutch novels for the pilot; 4 of which were rated as
highly literary (5.5–7.0) in the Riddle survey, and 4 rated as neither non-
literary or very literary (3-5–5.0) in that survey. We deliberately did not
select non-literary novels with low ratings of less than 3.5, because previous
research showed that the genre of such novels is very easy to recognize (van
Cranenburgh and Bod, 2017), and we are interested in the more subtle
differences between literary novels and ‘not quite literary’ novels. In
addition, author gender is balanced, both for the literary and the not quite
literary novels.

From these novels we selected fragments of 250 words, from the beginning of a
chapter, though not the first chapter. We rule out the influence of author
prestige by not showing any metadata of the fragments that are presented to
the participants. In addition to not presenting metadata, we also anonymize
the fragments; i.e., names of the main characters in the text are abbreviated
to initials. Fragments are presented in an arbitrary order (although we did
not shuffle the fragments across participants).

We use the same 7-point Likert scale as in the Riddle survey, which ranges
from not at all literary to very literary. As in that survey, we did not give
a definition of what ‘literary’ is, because we did not want to influence the
participants. For a given fragment, participants were asked the following
questions:

“How literary do you think this fragment is?” (7-point Likert scale)

“Briefly explain why you chose this rating” (text field)

“If a specific phrase contributed to this judgment, cite it” (text field)

We recruited participants from our social network. They range from casual to
enthusiastic readers, of various genres, with ages ranging from 30–60. We
collected responses from 48 participants. Compared to the 14k participants of
the Riddle survey, this is a small pilot. However, for the Riddle survey, we
look only at the ratings of books the participants had rated, which means that
the number of ratings varies per book. It was shown that 50–100 ratings are
sufficient to compute a reliable mean (van Cranenburgh and Bod, 2017). In
contrast, all fragments were rated by all 48 participants in our
questionnaire.

|  |  | Author | Rating | Riddle
---|---|---|---|---|---
Rank | Novel | Genre | gender | M | SD | rank
1 | Mortier, Godenslaap | literary | male | 5.8 | 1.2 | 2
2 | Durlacher, De Held | literary | female | 4.9 | 1.4 | 56
3 | Dijkzeul, Gouden Bergen | suspense | female | 4.7 | 1.3 | 156
4 | Den Tex, Wachtwoord | suspense | male | 4.6 | 1.4 | 147
5 | Smit, Vloed | literary | female | 4.3 | 1.6 | 103
6 | Van der Heijden, Tonio | literary | male | 4.3 | 1.3 | 7
7 | Dorrestein, De stiefmoeder | literary | female | 4.0 | 1.7 | 64
8 | Appel, Van twee kanten | suspense | male | 3.4 | 1.5 | 184
Table 1: Results of the questionnaire with a comparison to the ranking in the
Riddle survey (401 novels).

##  3 Literary ratings

In this first analysis, we consider the mean literary rating for each novel
fragment in the questionnaire. We compare these means to the ratings in the
Riddle survey and to predicted ratings from supervised models.

###  3.1 Comparison with the large reader survey

Table 1 presents the results as a ranking, ordered by the rating in the
questionnaire. We note 4 differences in ranking. Compared to the Riddle
survey, the thrillers at rank 3 and 4 are rated as more literary, while the
literary novels at rank 6 and 7 are rated substantially less literary by the
participants. The gap between male and female authors is less pronounced than
in the Riddle survey, where literary female authors were rated 0.5 points
lower than male authors, on average (Koolen, 2018). Similarly, the genre
effect of suspense novels being rated lower than literary novels disappears
for the novels by Dijkzeul and Den Tex. That their writing stands out is not
surprising; Dijkzeul has been nominated several times for the _Gouden Strop_ ,
the most well-known award for Dutch suspense novels, while Den Tex is a three
time laureate of the award.

It is possible that these differences are due to author prestige not playing a
role in the questionnaire, but it may also be due to the difference in rating
a small fragment on stylistic aspects versus a complete novel where plot may
also play a role. Rating a small fragment in isolation is very different from
rating a a novel one has read from memory, based only on the author and title.

When a reader rates a novel from memory, social factors will most likely come
into play. If an author is known to have been acknowledged by literary
institutions, for example through literary prizes and literary studies, this
will add to their prestige (Verboord, 2003). And in turn, this prestige could
influence a reader to award a higher rating on a scale of literary quality.
Alternatively, a lack of prestige might result in a lower judgment, as is the
case with suspense novels. Respondents in the National Reader Survey would
explain a low rating for a suspense novel with comments such as “It’s a
suspense novel,” signifying a tendency to associate lower literary quality
with a specific genre. On the other hand, novels by female authors are seen as
less literary overall, according to our larger reader survey, even when the
novel judged won a literary prize (Koolen et al., 2020). In sum, prestige—of
an author and/or novel—constitutes a complex web of factors, which likely
plays a part in literariness judgments, as the results of the questionnaire
show. Female authors and suspense authors are no longer separated in literary
judgment from the male and literary authors, respectively.

There are problems with this assumption, however. For example, the literary
novel by female author Durlacher has a suspenseful plot which might have
resulted in lower literary ratings in the Riddle survey. At the same time the
writing style was praised as layered and literary, which may explain the high
rating in the questionnaire—since the latter mainly invites judgments based on
writing style. Dijkzeul’s suspense novel is another example. One reviewer
calls the plot of this novel
predictable;111https://www.hebban.nl/recensie/a-vd-heide-over-gouden-bergen
this could explain why it is rated lower in the Riddle survey than in our
questionnaire, where plot plays no role. Further research is necessary to
confirm these explanations.

However, it could also be that the scales are not comparable. Overall it seems
that the participants in our questionnaire are more conservative, because only
one novel got a rating above 5. This may be due to difficulty of rating such
short fragments, which might be compared to judging a wine by tasting a drop
of it; in the absence of sufficient information, we observe a regression to
the mean.

###  3.2 Comparison with machine learning models

The plot on the left of Figure 1 shows a comparison of the ratings in our
questionnaire, the Riddle survey, and two predictive models from previous
work. The predictive models take the mean ratings from the Riddle survey as
ground truth by training and evaluatinng on them using crossvalidation.

The first model, “BoW + tree fragments,” uses a large number of textual
features including word bigrams and syntactic tree fragments and is based on
long novel fragments of 1000 sentences (van Cranenburgh and Bod, 2017); this
model reached an $R^{2}$ of 61.1. This score was obtained using
crossvalidation of a regularized linear SVM regression model. An $R^{2}$ score
is a measure of the variation explained and ranges from 0 to 100, with 0 being
baseline performance and 100 being a perfect score. Tree fragments where
extracted based on their frequency and correlation with literary ratings in
the training fold.

The second model, “LDA + paragraph vectors,” addresses the harder task of
predicting literary ratings from short fragments of 1000 words and only uses
LDA topic weights (Blei et al., 2003) and DBOW paragraph vectors (Le and
Mikolov, 2014) based on the 401 novels as document representations (van
Cranenburgh et al., 2019); this model reached an $R^{2}$ of 52.2. This score
was also obtained using crossvalidation, with a regularized linear model
(Ridge regression). The topic weights are obtained using Mallet, while the
paragraph vectors were obtained with Gensim.

For the latter model, we have identified the matching 1000 word fragment
containing the 250 word fragment presented in the questionnaire, and show the
corresponding prediction in Figure 1. While the comparison would be even more
faithful if we evaluated the prediction on the exact same 250 word fragments,
we opt for using the existing tuned and benchmarked model. If we take the
Riddle survey as ground truth, the predictions of the models are closer than
the ratings in our questionnaire, with two exceptions, Durlacher and Dijkzeul.
This suggests that the predictive models may pick up on cues in the text that
are not apparent to an untrained observer. However, it could also be that the
participants in the questionnaire are more accurate because they were not
influenced by social factors.

Figure 1: Left: Comparison of ratings from the questionnaire, Riddle survey
and predictions from machine learning models. For the Riddle survey the
average number of ratings is indicated. Error bars on ratings show standard
error (note that the error bars of the Riddle survey are so small they are
barely visible). Right: A histogram of Spearman correlations, showing how
strongly the responses of the participants in the questionnaire are correlated
with the Riddle survey.

We can also compare our questionnaire and the Riddle survey in terms of the
participants. How similar are their ratings? Because we are not interested in
the precise differences in ratings, we want to focus on their rankings. We
therefore compute a Spearman correlation coefficient of the ratings of each
participant in the questionnaire with the mean ratings in the Riddle survey
for the 8 novels. However, due this limited number of 8 data points per
participant, these correlations are only presented as an exploratory analysis
to give an indication of effect sizes, and we do not perform a formal
hypothesis test with p-values. The plot on the right of Figure 1 shows a
histogram of these correlations, showing that most participants of the
questionnaire displayed a moderate to large agreement ($\rho>0.4$) with the
Riddle survey. These correlations support the construct validity of
literariness as a measurable variable, since the human ratings from very
different experiments agree to a large extent. If the participants would have
‘failed the challenge,’ we would expect the distribution of correlations to be
centered around 0, but the majority shows a positive correlation. In the
Riddle survey, participants had read the complete novel they rated, but may
also have been influenced by author prestige. In our questionnaire,
participants were only presented with a short fragment, but were not
influenced by other factors (except in the case that the participant
recognized an author or novel from the fragment, which is not highly likely
considering the briefness of the fragments and the anonymization of the names
of the main characters).

##  4 Analysis of motivations

Overall, the common thread in the motivations is that elaborate and
sophisticated language is seen as more literary. Most motivations refer to
specific stylistic aspects. Participants also remarked that the task was hard
due to the short text fragments. The following sentence was cited by multiple
participants who agreed on its high literariness (our translation): “She
brings out the world’s blissful mongoloid smile, the grinning, wet shining zen
of dumb objects, from which she blows the names off like chaff.”

However, the choice of fragment likely plays a large role. The fragments were
selected arbitrarily, and are not necessarily representative of the whole
novel. Especially for Dorrestein, this likely played a role. In the fragment,
a woman observes a group of children, followed by the following free indirect
discourse (our translation): “She notices that she moves her jaws. I am Mrs
Pacman. Bite, swallow, gone. Bite swallow gone.” This phrase struck many
participants, but it is striking that they disagreed on whether it made the
fragment more or less literary. One participant cited this phrase as
motivation for a 6 out of 7 (‘no fixed pattern, surprising, would like to read
more’), while another cited it to give a rating of 1 out of 7 due to the
sudden shift in style (‘a hodgepodge of old fashioned words and modern
imagery’). Incidentally, such examples may be an issue for the notion of
foregrounding as an explanation of literariness: while we find phrases that
stand out for many of the participants, they do not agree on their effect on
the literary status of the fragment. This requires further study; for example,
would the participants’ judgment change if more context is given?

While the fragment by Den Tex scored high in the questionnaire, several of the
participants cited the mention of a soccer player, or the fact that soccer was
discussed in the fragment, as a reason for judging the fragment as non-
literary. This suggests that there is a perception that references to low
brow, popular culture are associated with lower prestige, while literature
should be about more timeless and serious matters. We can hypothesize that
this is part of a heuristic: by default, references to low-brow popular
culture trigger a low rating; on the other hand, such references can also
appear in high literature, but in this case more context (i.e., a longer
fragment) may be needed to see how such references fit in the story.

##  5 Discussion and Conclusion

We found a reasonable consensus for the strongest style differences in the
fragments. The correlations of the Riddle survey ratings with our
questionnaire ratings range from moderate to strong, which supports the
construct validity of literariness as a variable. The difference between the
ranking of the most and least literary fragment was preserved, and some of the
changes in ranking could be explained as genre and gender effects that
disappear when rating an anonymized fragment.

Participants found the task of rating short fragments hard, and the
predictions by the machine learning models are closer to the survey ratings.
However, the machine learning models are trained, while the participants are
not. The ability of predictive models to reproduce the quality ratings should
be interpreted carefully since the models may pick up on more than just
literariness from the textual features, such as the aforementioned genre and
gender effects.

The participants agree on criteria (e.g., word usage), agree on salience of
phrases, but sometimes disagree on how literary they are. This suggests that
foregrounded language is not necessarily seen as literary language, and raises
the question of how to identify potential linguistic markers of literary
language for further study.

Our work is related to work on experimental aesthetics. In one branch of
experimental aesthetics based on the ideas of Fechner (1876), the subjects’
ideas on aesthetics are researched. Respondents are asked to recall as many
adjectives related to aesthetic judgment as possible, for instance of literary
novels. In Knoop et al. (2016), this approach leads to the conclusion that
‘beautiful’ and ‘suspenseful’ are central to descriptions of aesthetic
judgment of fictional literature. This type of research is related to the
influence of the text, but Knoop et al. (2016) did not ask readers to reflect
on specific works. Other work on experimental aesthetics considers
foregrounding effects and does present participants with novel fragments
(e.g., Hakemulder, 2004) or lines of poetry (e.g., Blohm et al., 2018).
However, more generally, there is more to literary quality than aesthetics.
Even though there has been a focus on the aesthetic experience of literary
language (cf. Van Peer, 2008), ethical, moral, affective and other motivations
can play a role in literary judgments as well.

In future work, we want to determine the influence of author prestige by
conducting a controlled experiment in which one group sees fragments with
author names, while the other only sees the fragments. After this, a larger
experiment should be conducted, with more novels, more fragments per novel,
and more participants. It would be interesting to contrast general readers
with readers with particular backgrounds (e.g., literature professors). Once
promising linguistic markers are identified, we want to manipulate these
markers in the fragments, to confirm their effect on literariness, similar to
the work of Hakemulder (2004) on novel fragments and Blohm et al. (2018) on
poetry.

## Acknowledgments

We are grateful to Ana Guerberof, Antonio Toral, and the anonymous reviewers
for helpful comments.

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