# Configurational temperature in active matter. II. Quantifying the deviation
from thermal equilibrium

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

This paper suggests using the configurational temperature ${T}_{\rm conf}$ for
quantifying how far an active-matter system is from thermal equilibrium. We
measure this “distance” by the ratio of the systemic temperature ${T}_{\rm s}$
to ${T}_{\rm conf}$, where ${T}_{\rm s}$ is the canonical-ensemble temperature
for which the average potential energy is equal to that of the active-matter
system. ${T}_{\rm conf}$ is “local” in the sense that it is the average of a
function, which only depends on how the potential energy varies in the
vicinity of a given configuration; in contrast ${T}_{\rm s}$ is a global
quantity. The quantity ${T}_{\rm s}/{T}_{\rm conf}$ is straightforward to
evaluate in a computer simulation; equilibrium simulations in conjunction with
a single steady-state active-matter configuration are enough to determine
${T}_{\rm s}/{T}_{\rm conf}$. We validate the suggestion that ${T}_{\rm
s}/{T}_{\rm conf}$ quantifies the deviation from thermal equilibrium by data
for the radial distribution function of 3d Kob-Andersen and 2d Yukawa active-
matter models with active Ornstein-Uhlenbeck and active Brownian Particle
dynamics. Moreover, we show that ${T}_{\rm s}/{T}_{\rm conf}$, structure, and
dynamics of the homogeneous phase are all approximately invariant along the
motility-induced phase separation (MIPS) boundary in the phase diagram of the
2d Yukawa model. The measure ${T}_{\rm s}/{T}_{\rm conf}$ is not limited to
active matter; it can be used for quantifying how far any system involving a
potential-energy function, e.g., a driven Hamiltonian system, is from thermal
equilibrium.

##  I Introduction

Temperature is fundamental in thermodynamics and statistical mechanics.
Generalizations of the temperature concept to deal with out-of-equilibrium
systems have been discussed in several publications, useful reviews of which
are given in Refs. Casas-Vazquez and Jou, 2003; Powles  _et al._ , 2005;
Leuzzi, 2009; Puglisi  _et al._ , 2017; Zhang  _et al._ , 2019. Non-
equilibrium temperatures generally attempt to relate the non-equilibrium
system to the its thermal equilibrium properties. This paper and its companion
Saw  _et al._ (2023), henceforth referred to as Paper I, propose two
applications of the so-called configurational temperature ${T}_{\rm conf}$
Landau and Lifshitz (1958); Rugh (1997); Powles  _et al._ (2005); Himpel and
Melzer (2019) to active-matter models, both of which are based on a different
philosophy. Paper I showed that ${T}_{\rm conf}$ defines an energy scale,
which can be used for tracing out lines of approximately invariant physics of
the 3d Kob-Andersen binary Lennard-Jones model with active Ornstein-Uhlenbeck
dynamics. The present paper shows that a similar procedure applies for the 2d
Yukawa model with active Brownian dynamics (ABP), after which we proceed to
the main focus: using ${T}_{\rm conf}$ for measuring how far an active-matter
system is from thermal equilibrium.

For an ordinary Hamiltonian system in thermal equilibrium, the temperature $T$
is identical to the configurational temperature ${T}_{\rm conf}$ that is
defined Rugh (1997); Powles  _et al._ (2005) as follows. If the system
consists of $N$ particles with collective coordinate vector
$\mathbb{R}\equiv(\mathbb{r}_{1},...,\mathbb{r}_{N})$ and potential-energy
function $U(\mathbb{R})$, one defines $k_{B}{T}_{\rm conf}\equiv\langle(\nabla
U)^{2}\rangle/\langle\nabla^{2}U\rangle$. Here $k_{B}$ is the Boltzmann
constant, $\nabla$ is the gradient operator, and the sharp brackets denote
canonical-ensemble averages. It is straightforward to prove that $T={T}_{\rm
conf}$ in equilibrium Landau and Lifshitz (1958), see, e.g., Paper I.
Approaching the thermodynamic limit, the relative fluctuations of both the
numerator and the denominator of ${T}_{\rm conf}$ goes to zero. Thus 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 thermal
equilibrium in the sense that deviations vanish as $N\to\infty$. Because
configurations with $\nabla^{2}U(\mathbb{R})\leq 0$ become less likely as
$N\to\infty$, the fact that Eq. (1) is not defined for such configurations
does not present a serious problem.

The derivation and justification of the configurational temperature ${T}_{\rm
conf}$ is based on the fact that the probability of configuration $\mathbb{R}$
in the canonical ensemble is proportional to $\exp(-U(\mathbb{R})/k_{B}T)$
Landau and Lifshitz (1958); Powles  _et al._ (2005); Saw  _et al._ (2023).
This is irrelevant, however, for the property demonstrated in Paper I that
${T}_{\rm conf}(\mathbb{R})$ may be used for tracing out lines of invariant
structure and dynamics in the phase diagram of active-matter models that
involve a potential-energy function obeying hidden scale invariance Dyre
(2014). This 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. Hidden scale invariance applies to a good
approximation for many well-known potentials, e.g., systems defined by the
Lennard-Jones and Yukawa interactions, density-functional derived atomic
interactions, and simple molecular models Gnan  _et al._ (2009); ing12b;
Schrøder and Dyre (2014); Hummel  _et al._ (2015); Dyre (2018).

This paper proposes an application of ${T}_{\rm conf}$ to active-matter
models, which addresses the problem of quantifying how far a system is from
ordinary canonical-ensemble thermal equilibrium. This question is important
because only if the system in question is close to thermal equilibrium, does
it make good sense to refer to the temperature of the corresponding canonical-
ensemble equilibrium system as a characteristic of the active-matter system.
As discussed in the next section, the ratio between the global “systemic”
temperature ${T}_{\rm s}$ and the “local” temperature ${T}_{\rm conf}$
provides such a measure. Section III sets the stage by detailing one example,
the 2d Yukawa model with active Brownian particle dynamics. Section IV
presents data for the radial distribution function of Kob-Andersen and 2d
Yukawa active-matter models, confirming that when ${T}_{\rm s}/{T}_{\rm conf}$
is close to unity, the structure is close to that of thermal equilibrium.
Also, Sec. IV evaluates a standard entropy-production-based measure of
deviations from thermal equilibrium and compared to the proposed new measure.
Section V shows that the new measure is roughly constant along the motility-
induced phase-separation line, which is consistent with the reasonable
assumption that all state points close to this line in the non-MIPS phase are
equally far from equilibrium. Finally, Sec. VI summarizes Papers I and II.

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

The investigations of Papers I and II are limited to active-matter point-
particle models characterized by a potential-energy function. Quantifying the
degree of non-equilibrium is usually done by calculating some form of
dissipation (entropy production). The idea is that since the entropy
production is zero in thermal equilibrium, this quantity measures how far a
given system is from thermal equilibrium Fodor  _et al._ (2016); Flenner and
Szamel (2020); O’Byrne  _et al._ (2022). Such measures can be applied to both
active-matter models and driven Hamiltonian systems. A fundamental issue with
these measures is the following. Using a quantity that goes to zero in some
limit to quantify the degree of deviation from that limit does not in an
obvious way make possible the identification of when deviations from
equilibrium are to be regarded as “large”. If deviations from thermal
equilibrium are instead quantified by means of a quantity that goes to unity
in the equilibrium limit, deviations from equilibrium are “small” whenever
that quantity does not deviate substantially from unity and “large” otherwise.

The configurational temperature is local in the sense that when regarded as a
function of $\mathbb{R}$, it only depends on how the potential energy
$U(\mathbb{R})$ varies in the immediate surroundings. Note that “local” here
refers to the 2N or 3N dimensional configuration space, not to the two- or
three-dimensional space in which the active particles move. This locality
means that by evaluating ${T}_{\rm conf}$ for a passive system’s configuration
at a given time, one cannot determine whether the system is in thermal
equilibrium corresponding to the temperature $T={T}_{\rm conf}(\mathbb{R})$.
For instance, for an aging glass annealed at temperature $T$, already after a
time on the phonon scale does ${T}_{\rm conf}(\mathbb{R})\cong T$ apply, i.e.,
long before equilibrium has been reached Powles  _et al._ (2005). A completely
different, global temperature concept is the systemic temperature ${T}_{\rm
s}$. This quantity was introduced for generalizing isomorph theory to out-of-
equilibrium conditions Dyre (2020), but ${T}_{\rm s}$ may be introduced for
any system as the equilibrium canonical-ensemble temperature of the
Hamiltonian system at the same density and average potential energy as that of
the out-of-equilibrium system. In thermal equilibrium one has ${T}_{\rm
conf}={T}_{\rm s}=T$.

The idea is to use the ratio of global to local temperature, ${T}_{\rm
s}/{T}_{\rm conf}$, for quantifying how far an active-matter system is from
thermal equilibrium. We showed in Paper I that the ratio ${T}_{\rm s}/{T}_{\rm
conf}$ is predicted to be constant along active-matter isomorphs. Since
structure and dynamics are also invariant along both active-matter isomorphs
and the corresponding Hamiltonian-system isomorphs, it is consistent to assume
that ${T}_{\rm s}/{T}_{\rm conf}$ measures how far the system is from thermal
equilibrium.

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

This section details the ABP model in two dimensions based on the single-
component Yukawa pair potential Yukawa (1935); Meacock  _et al._ (2021),

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

This potential obey hidden scale invariance Dyre (2014); Veldhorst  _et al._
(2015); Tolias and Castello (2019), so a procedure for identifying active-
matter isomorphs analogous to that introduced in Paper I for the active
Ornstein-Uhlenbeck particle (AOUP) model should apply here, as well. The idea
is that ${T}_{\rm s}/{T}_{\rm conf}$, as mentioned, is predicted to be
invariant along active-matter isomorphs where the deviations from thermal
equilibrium are also expected to be invariant.

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

$\dot{\mathbb{r}}_{i}\,=\,\mu\mathbb{F}_{i}+\bm{\xi}_{i}(t)+v_{0}\,\mathbf{o}_{i}(t)\,.$
(3)

Here, $\mu$ is 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 controlled by a
white Gaussian noise of magnitude $D_{r}$,

$\langle\dot{\theta}_{i}(t)\dot{\theta}_{j}(t^{\prime})\rangle\,=\,2D_{r}\delta_{ij}\,\delta(t-t^{\prime})\,,$
(4)

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

$\langle\bm{\xi}_{i}^{\alpha}(t)\bm{\xi}_{j}^{\beta}(t^{\prime})\rangle\,=\,2D_{t}\delta_{ij}\delta_{\alpha\beta}\delta(t-t^{\prime})\,.$
(5)

The ABP model has four parameters. Regarding $\mu$ as a system-specific
constant, the dimensionless versions of the three other parameters must be
constant in order to have invariant physics when the density is changed.
Following the procedure of Sec. III of Paper I, we take as length unit
$l_{0}=\rho^{-1/2}$ (the exponent is $-1/2$ and not $-1/3$ as in Paper I
because the model here is two-dimensional) and as time unit $t_{0}=1/D_{r}$,
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
$t=\tilde{t}/D_{r}$ into Eq. (3) and making use of Eq. (8) of Paper I and the
definition of the systemic temperature ${T}_{\rm s}$ Dyre (2020) in which
${S}_{\rm ex}(\mathbb{R})$ is the microscopic excess-entropy function Schrøder
and Dyre (2014); Dyre (2020),

${T}_{\rm s}(\mathbb{R})\,\equiv\,{T}_{\rm eq}(\rho,{S}_{\rm
ex}(\mathbb{R}))={T}_{\rm eq}(\rho,U(\mathbb{R}))\,,$ (6)

we get

$\dot{\tilde{\mathbb{r}}}_{i}\,=\,-\mu\rho({T}_{\rm
s}/D_{r})\tilde{\nabla}_{i}{S}_{\rm
ex}({\tilde{\mathbb{R}}})+\tilde{\bm{\xi}}_{i}(t)+\tilde{v}_{0}\,\mathbf{o}_{i}(t)\,.$
(7)

Here $\tilde{v}_{0}=(\rho^{1/2}/D_{r})v_{0}$,
$\tilde{\bm{\xi}}_{i}=(\rho^{1/2}/D_{r})\bm{\xi}_{i}$, ${T}_{\rm s}$ is brief
for ${T}_{\rm s}(\mathbb{R})$,

$\langle\tilde{\bm{\xi}}_{i}^{\alpha}(t)\tilde{\bm{\xi}}_{j}^{\beta}(t^{\prime})\rangle\,=\,2\rho(D_{t}/D_{r})\delta_{ij}\delta_{\alpha\beta}\delta(\tilde{t}-\tilde{t}^{\prime})\,,$
(8)

and dots now mark the derivative with respect to $\tilde{t}$,

$\langle\dot{\theta}_{i}(t)\dot{\theta}_{j}(t^{\prime})\rangle\,=\,2\delta_{ij}\,\delta(\tilde{t}-\tilde{t}^{\prime})\,.$
(9)

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 kept constant. Since
$\mu$ is a (system-specific) constant, this implies (where the subscript zero
refers to a reference state of density $\rho_{0}$ and ${T}_{\rm
s}(\rho)\equiv{T}_{\rm eq}(\rho,{S}_{\rm ex}({\tilde{\mathbb{R}}}))$ can be
used instead of ${T}_{\rm s}(\mathbb{R})$ because fluctuations go to zero in
the thermodynamic limit)

$\displaystyle D_{r}$ $\displaystyle\,=\,$ $\displaystyle
D_{r,0}\,\frac{\rho}{\rho_{0}}\,\frac{{T}_{\rm s}(\rho)}{{T}_{\rm
s}(\rho_{0})}$ $\displaystyle D_{t}$ $\displaystyle\,=\,$ $\displaystyle
D_{t,0}\,\frac{{T}_{\rm s}(\rho)}{{T}_{\rm s}(\rho_{0})}$ (10) $\displaystyle
v_{0}$ $\displaystyle\,=\,$ $\displaystyle
v_{0,0}\,\left(\frac{\rho}{\rho_{0}}\right)^{1/2}\frac{{T}_{\rm
s}(\rho)}{{T}_{\rm s}(\rho_{0})}\,.$

By the same argument as in Sec. III of Paper I one can here replace the
${T}_{\rm s}$ ratios by ${T}_{\rm conf}$ ratios, leading to

$\displaystyle D_{r}$ $\displaystyle\,=\,$ $\displaystyle
D_{r,0}\,\frac{\rho}{\rho_{0}}\,\frac{{T}_{\rm
conf}\left((\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right)}{{T}_{\rm
conf}(\mathbb{R}_{0})}$ $\displaystyle D_{t}$ $\displaystyle\,=\,$
$\displaystyle D_{t,0}\,\frac{{T}_{\rm
conf}\left((\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right)}{{T}_{\rm
conf}(\mathbb{R}_{0})}$ (11) $\displaystyle v_{0}$ $\displaystyle\,=\,$
$\displaystyle v_{0,0}\,\left(\frac{\rho}{\rho_{0}}\right)^{1/2}\frac{{T}_{\rm
conf}\left((\rho_{0}/\rho)^{1/2}\mathbb{R}_{0}\right)}{{T}_{\rm
conf}(\mathbb{R}_{0})}\,.$

In passing we note that while the Peclet number $v_{0}/\sqrt{2D_{r}D_{t}}$
Bechinger  _et al._ (2016); Hecht  _et al._ (2021) is invariant along the
active-matter isomorph, this requirement is not enough to determine how to
scale the model parameters – thus Peclet-number invariance is a necessary, but
not sufficient condition for identifying an active-matter isomorph.

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

To validate the existence of active-matter isomorphs according to 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
$(\rho,D_{r},D_{t},v_{0})=(1.0,3.0,1.0,25.0)$. The simulations were carried
out on GPU cards using a home-made code. An active-matter isomorph was traced
out for densities varying a factor of three using Eq. (III) for a
configuration $\mathbb{R}_{0}$ selected from a steady-state simulation at the
reference state point. Table 1 gives the parameters obtained from Eq. (III).

Figure 1:  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 active-matter isomorph, the middle panel shows the same data in reduced
units, and the right panel shows the reduced RDF for the same parameters
(Table 1) at the reference density $\rho=1.0$. (b) The left panel shows the
MSD as a function of time $t$ along the active-matter isomorph, the middle
panel shows the same data in reduced units where the dashed line marks slope
unity, i.e., ordinary diffusion; the right panel shows the reduced MSD for the
same parameters at the reference state-point density $\rho=1.0$.

Figure 1(a) shows the radial distribution function (RDF). The left two panels
show the RDF along the active-matter isomorph as a function of $r$ and
$\tilde{r}$, respectively. For comparison, the right panel shows the results
for the same parameters at the reference state-point density $\rho=1.0$. We
find a good invariance of the reduced RDF along the active-matter isomorph.
The same applies for the reduced mean-square displacement (MSD) shown in (b).

##  IV Deviations from thermal equilibrium quantified by ${T}_{\rm s}/{T}_{\rm
conf}$

Figure 2:  Determination of the ratio of systemic to configurational
temperature, ${T}_{\rm s}/{T}_{\rm conf}$, quantifying how far an active-
matter system is from thermal equilibrium. (a) shows data for ${T}_{\rm s}$
and ${T}_{\rm conf}$ for the 3d Kob-Andersen AOUP model (Paper I, Saw  _et
al._ (2023)) as functions of $\tau$ with the remaining model parameters kept
fixed. (b) shows ${T}_{\rm s}/{T}_{\rm conf}$ for the same data. For $\tau$
values around $10^{-4}$ the system begins to move away from equilibrium and
for $\tau>10^{-3}$ significant deviations from equilibrium are predicted. (c)
shows data for ${T}_{\rm s}$ and ${T}_{\rm conf}$ for the 2d Yukawa AOUP model
as functions of $\tau$ with the remaining model parameters kept fixed. (d)
shows ${T}_{\rm s}/{T}_{\rm conf}$ for the same data. For $\tau$ values above
$10^{-4}$ the system starts to deviate from equilibrium. (e) shows data for
${T}_{\rm s}$ and ${T}_{\rm conf}$ for the 2d Yukawa ABP model as functions of
$v_{0}$ with the remaining model parameters kept fixed. (f) shows ${T}_{\rm
s}/{T}_{\rm conf}$ for the same data. For $v_{0}$ values around $10$ the
system begins to move away from equilibrium.

Figure 2 gives data for the systemic and configurational temperatures of
different active-matter models, starting with the Kob-Andersen model studied
in Paper I. Figure 2(a) shows the systemic temperature ${T}_{\rm s}$ (black
symbols) and the configurational temperature ${T}_{\rm conf}$ (red symbols)
for the Kob-Andersen AOUP active-matter model as functions of the colored-
noise correlation time $\tau$ for fixed values of the other model parameters.
As mentioned, ${T}_{\rm s}$ is determined by identifying the equilibrium
temperature at which the system for a standard MD simulation has the same
average potential energy as the AOUP system. The system approaches an
equilibrium system for $\tau\to 0$, corresponding to the canonical-ensemble
temperature $T=1.6$. Figure 2(b) plots the ratio ${T}_{\rm s}/{T}_{\rm conf}$.
We see that for values of $\tau$ above $10^{-4}$, the system starts to move
away from thermal equilibrium. Figure 2(c) shows ${T}_{\rm s}$ and ${T}_{\rm
conf}$ as functions of $\tau$ for the 2d Yukawa AOUP model for fixed values of
the other model parameters. Both ${T}_{\rm s}$ and ${T}_{\rm conf}$ converge
to $5$ as $\tau\to 0$, confirming the fact that $T=5$ is the equilibrium
Brownian-dynamics temperature corresponding to the parameters $D_{t}=5$,
$\mu=1$. Figure 2(d) shows ${T}_{\rm s}/{T}_{\rm conf}$ and we see that for
$\tau$ above $10^{-4}$, the system begins to deviate from thermal equilibrium.
Figure 2(e) and (f) show ${T}_{\rm s}$ and ${T}_{\rm conf}$ and their ratio
for the 2d Yukawa ABP model as functions of $v_{0}$ for fixed values of the
other model parameters; here $v_{0}>10$ is the approximate criterion for
deviations from equilibrium.

Figure 3:  RDFs of active-matter states predicted to be close to (left column)
and not close to (right column) thermal equilibrium. The red curves are the
active-matter data and the black dashed lines are the RDFs of the
corresponding equilibrium system for $T={T}_{\rm s}$. (a)-(d) show results for
the AA and BB RDFs of the Kob-Andersen AOUP model for $\tau=10^{-4}$ and
$\tau=4\cdot 10^{-2}$ (red curves) corresponding to ${T}_{\rm s}/{T}_{\rm
conf}=1.13$ and ${T}_{\rm s}/{T}_{\rm conf}=6.59$. (e) and (f) show results
for the 2d Yukawa AOUP model at states with $\tau=10^{-4}$ and $\tau=8\cdot
10^{-3}$ corresponding to ${T}_{\rm s}/{T}_{\rm conf}=1.09$ and ${T}_{\rm
s}/{T}_{\rm conf}=2.59$. (g) and (h) show results for the 2d Yukawa ABP model
at states with $v_{0}=10$ and $v_{0}=50$ corresponding to ${T}_{\rm
s}/{T}_{\rm conf}=1.13$ and ${T}_{\rm s}/{T}_{\rm conf}=2.18$.

By reference to the data in Fig. 2, Fig. 3 compares the RDF of states
predicted to be close to and not close to thermal equilibrium. Each subfigure
reports ${T}_{\rm s}/{T}_{\rm conf}$; results for the cases where ${T}_{\rm
s}/{T}_{\rm conf}$ is close to unity are found in the left column. The RDFs
are compared to the equilibrium RDF for $T={T}_{\rm s}$, i.e., the temperature
corresponding to the potential energy of the active-matter configurations. The
black dashed lines give the equilibrium RDF, the red curves are the active-
matter RDFs. Figure 3(a)-(d) show data for $\textrm{RDF}_{\textrm{AA}}$ and
$\textrm{RDF}_{\textrm{BB}}$ of the Kob-Andersen AOUP model studied in Paper
I; $\textrm{RDF}_{\textrm{AB}}$ is similar to the AA (data not shown). Figure
3(e) and (f) give data for the 2d Yukawa AOUP model, while (g) and (h) give
data for the 2d Yukawa ABP model (Sec. III). Figure 3 confirms that when the
ratio ${T}_{\rm s}/{T}_{\rm conf}$ is close to unity, the configurations of
the active-matter model are close to thermal equilibrium configurations.

Figure 4:  Using the ratio of systemic to configurational temperature to
quantify how far the 2d Yukawa ABP system is from thermal equilibrium
(corresponding to $v_{0}=0$ in Eq. (3)); the parameters kept fixed here are
$\rho=1$, $D_{r}=3$, and $D_{t}=2$. (a) shows how the dissipation (“Power”)
varies with $v_{0}$ (MD units). From Fig. 2(e) we see that when $v_{0}\to 0$,
the two temperatures become identical (equal to $2$ because $D_{t}=2$
corresponds to that thermal equilibrium temperature); at the same time the
dissipation goes to zero. (b) and (c) 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 thermal
equilibrium.

Next we compare to a previously proposed measure of deviations from thermal
equilibrium, focusing on the 2d Yukawa ABP model. Figure 4(a) shows the
dissipated “active” power, i.e., the average of the scalar product of the
particle velocity with the ${v}_{0}\,\mathbf{o}_{i}(t)$ term of Eq. (3),
plotted as a function of $v_{0}$, keeping the three other model parameters
constant. From data like these one cannot easily determine when the system is
expected to be close to thermal equilibrium. Figure 4(b) shows the dissipated
power plotted against ${T}_{\rm s}/{T}_{\rm conf}$, demonstrating a one-to-one
correspondence between the two measures of deviations from thermal
equilibrium. Figure 4(b) also includes data for the reduced-unit power (red
points), which shows an interesting almost linear proportionality to ${T}_{\rm
s}/{T}_{\rm conf}-1$ for which we have no good explanation. Finally, Figure
4(c) plots the same data in a log-linear scale, which further illustrates that
measuring deviations from thermal equilibrium in terms of a quantity that is
zero in equilibrium is not useful for distinguishing between weak and stronger
deviations from equilibrium.

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

Figure 5:  $(\rho,D_{t})$ phase diagrams showing MIPS state points as red
stars and homogeneous state points as black squares (green circles are gas-
like states of minor relevance here). The MIPS phase consists of coexisting
phases that differ in density, the denser phase is a “solid” phase of
hexagonal crystal structure. 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 this an active-matter isomorph
was traced out using Eq. (III) (black line). The figure gives data in the
$(\rho,D_{t})$ phase diagram with $D_{r}$ and $v_{0}$ given by Eq. (III) at
density $\rho$. The blue dashed lines mark $\pm 5$% variations in density. We
see that the phase-transition line is an approximate active-matter isomorph,
which is consistent with the degree of deviation from thermal equilibrium
being constant along this line.

For certain parameters of the 2d ABP Yukawa model, motility-induced phase
separation (MIPS) is observed. This is the striking active-matter phenomenon
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). It is reasonable to assume that, when the
phase transition is approached from the homogeneous phase, the deviations from
thermal equilibrium are the same for all parameter values. Thus if ${T}_{\rm
s}/{T}_{\rm conf}$ indeed provides a measure of the deviation from
equilibrium, this quantity should be roughly constant approaching the MIPS
phase transition. Since the 2d Yukawa ABP model obeys hidden scale invariance,
this means that the phase transition should approximately follow an isomorph
(because the physics is approximately invariant along an active-matter
isomorph, such a curve cannot cross the MIPS boundary, compare Refs. Gnan  _et
al._ , 2009 and Costigliola  _et al._ , 2016; Pedersen  _et al._ , 2016). Thus
if one has identified a state point in the homogeneous solid phase close to
the MIPS boundary and uses this as reference state point for generating an
active-matter isomorph, all state points identified by Eq. (III) should be
close to the MIPS boundary. A similar line of reasoning was validated for the
melting line of the ordinary Lennard-Jones system Costigliola  _et al._
(2016); Pedersen  _et al._ (2016).

Figure 6:  Structure and dynamics probed along the active-matter isomorph
approximately delimiting the MIPS phase boundary of the 2d ABP Yukawa system,
slightly into the homogeneous phase (Fig. 5). (a) and (b) show log-log plots
of the RDF and MSD, respectively, (c) and (d) show the same data in reduced
units.

We studied the 2d Yukawa model with parameters $Q=1000$ and $\lambda=0.12$
with a cutoff at $4.2\sigma$ and $(D_{r},D_{t},v_{0})=(3,1,367)$, by
systematically decreasing the density from a high value well within the
homogeneous solid phase. Initially, a system of 40000 particles was simulated
for 40 million time steps, and the occurrence of MIPS was detected by visual
inspection. 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 an active-matter isomorph according to Eq. (III). This is the black
full line in Fig. 5, which shows the results of investigating the existence of
MIPS in a $(\rho,D_{t})$ phase diagram (along the isomorph the remaining
parameters $D_{r}(\rho)$ and $v_{0}(\rho)$ are given by Eq. (III)). The black
squares denote state points of the homogeneous solid phase, the red stars
denote state points where MIPS appears, and the green circles denote gas-phase
state points. The blue dashed lines mark the active-matter isomorph $\pm$5% in
density. We see that the phase transition line is predicted reasonably well
though not accurately; this is consistent with the approximate nature of the
argument. Nevertheless, the simulations demonstrate that Eq. (III) can be used
for roughly identifying the MIPS phase boundary. This confirms the physical
expectation that the deviation from thermal equilibrium is virtually constant
along the phase-transition line because it is an approximate active-matter
isomorph characterized by constant ${T}_{\rm s}/{T}_{\rm conf}$.

In order to confirm that the black line of Fig. 5 is a line of approximately
invariant physics, i.e., an active-matter isomorph, we show in Fig. 6 how
structure and dynamics vary along it. The upper figures show the RDF and MSD
in standard units, the lower figures show the same data in reduced units.

##  VI Summary of Papers I & II and Outlook

The configurational-temperature concept has traditionally been used in
connection with liquid models based on Newton’s laws of motion with forces
derived from a potential-energy function $U(\mathbb{R})$ Powles  _et al._
(2005). Indeed, the derivation of ${T}_{\rm conf}$ refers to the canonical
ensemble, and for this reason it is not obvious that ${T}_{\rm conf}$ has
relevance also for non-Hamiltonian and non-time-reversible systems like those
of active matter. We have suggested that the configurational temperature may
be useful also in that context and have presented two applications of
${T}_{\rm conf}$. Paper I demonstrated how ${T}_{\rm conf}$ may be used for
tracing out lines of approximately invariant structure and dynamics in the
phase diagram of models described by AOUP dynamics if the potential-energy
function obeys hidden scale invariance; such lines are referred to as active-
matter isomorphs. Specifically, Paper I gave the equations for how to change
the model parameters with density in order to have invariant physics, and
Paper II derived a similar procedure for ABP models. In both cases, by
effectively reducing the number of model parameters by one, this approach
provides a tool for simplifying the exploration of phase diagrams of active-
matter models with hidden scale invariance of the potential-energy function.

For the AOUP and the ABP models the ratio of systemic to configurational
temperature is predicted to be constant along an active-matter isomorph. Since
both the active-matter physics and the corresponding passive-matter physics
are invariant along their common systemic isomorph (defined as the thermal
equilibrium isomorph mapped into the density systemic-temperature phase
diagram Dyre (2020)), this is consistent with the present paper’s proposal
that ${T}_{\rm s}/{T}_{\rm conf}$ quantifies how far a given active-matter
system is from thermal equilibrium.

The ratio ${T}_{\rm s}/{T}_{\rm conf}$ is defined for any active-matter system
based on a potential-energy function, whether or not hidden scale invariance
applies. We suggest that an active-matter system may be regarded as “close to
thermal equilibrium” whenever ${T}_{\rm s}/{T}_{\rm conf}$ is close to unity
and “far from thermal equilibrium” whenever this is not the case. We
illustrated the use of ${T}_{\rm s}/{T}_{\rm conf}$ for quantifying deviations
from thermal equilibrium by showing that when this quantity is close to unity,
the RDF of the active-matter system is close to that of the corresponding
thermal-equilibrium system with $T={T}_{\rm s}$. Moreover, ${T}_{\rm
s}/{T}_{\rm conf}$ is roughly constant along the motility-induced phase
separation (MIPS) boundary along which the deviation from equilibrium are
expected not to vary, compare Fig. 6.

The advantages of using the quantity ${T}_{\rm s}/{T}_{\rm conf}$ for
quantifying how far an active-matter system is from thermal equilibrium are
threefold:

  * •

A measure that converges to unity when the system in question approaches
thermal equilibrium allows for answering the question: how to quantify the
deviation from thermal equilibrium? This is not the case for a measure that
converges to zero when equilibrium is approached.

  * •

${T}_{\rm s}/{T}_{\rm conf}$ is easy to evaluate because it can be determined
from a single configuration $\mathbb{R}$ of a steady-state simulation of the
active-matter system in conjunction with equilibrium simulations of the
corresponding Hamiltonian system.

  * •

${T}_{\rm s}/{T}_{\rm conf}$ is general measure because this quantity is
defined for any system characterized by a potential-energy function, whether
or not in the context of an active-matter model. For instance, in the case of
a non-linear steady-state shear flow of an ordinary Hamiltonian system, it is
also possible to quantify the deviation from thermal equilibrium by means of
${T}_{\rm s}/{T}_{\rm conf}$.

An interesting question that remains to be explored is the following: What is
the difference between the cases ${T}_{\rm s}/{T}_{\rm conf}>1$ and ${T}_{\rm
s}/{T}_{\rm conf}<1$?

###### Acknowledgements.

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

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