# A classification of nonequilibrium steady states based on temperature
correlations

Sergio Davis1,2 1Research Center on the Intersection in Plasma Physics, Matter
and Complexity, P2mc, Comisión Chilena de Energía Nuclear, Casilla 188-D,
Santiago, Chile  2Departamento de Física, Facultad de Ciencias Exactas,
Universidad Andres Bello. Sazié 2212, piso 7, 8370136, Santiago, Chile.
<EMAIL_ADDRESS>
###### Abstract

Although generalized ensembles have now been in use in statistical mechanics
for decades, including frameworks such as Tsallis’ nonextensive statistics and
superstatistics, a classification of these generalized ensembles outlining the
boundaries of validity of different families of models, is still lacking. In
this work, such a classification is proposed in terms of _supercanonical_ and
_subcanonical_ ensembles, according to a newly defined parameter, the inverse
temperature covariance parameter $\mathcal{U}$. This parameter is non-negative
in superstatistics (and is equal to the variance of the inverse temperature)
but can be negative for other families of statistical ensembles, adquiring
then a broader meaning. It is shown that $\mathcal{U}$ is equal for every
region of a composite system in a steady state, and examples are given of
supercanonical and subcanonical states.

##  1 Introduction

Ensembles beyond the canonical ensemble of equilibrium thermodynamics have
already been used for decades. On the one hand, they have been successful in
describing nonequilibrium systems in steady states such as plasmas [1, 2, 3,
4], self-gravitating systems [5, 6, 7] and other complex systems [8, 9, 10].
On the other hand, they have extended existing computer simulation methods
[11]. Among these generalized ensembles, the $q$-canonical ensemble, commonly
associated with Tsallis’ nonextensive statistics [12], is widely used in the
description of a variety of complex systems that follow power laws [13].
Instead of the traditional canonical ensemble

$P(\bm{x}|\beta)=\frac{\exp(-\beta\mathcal{H}(\bm{x}))}{Z(\beta)},$ (1)

that describes thermal equilibrium for a system with Hamiltonian $\mathcal{H}$
at inverse temperature $\beta=1/(k_{B}T)$, the $q$-canonical ensemble
considers a microstate distribution of the form

$P(\bm{x}|q,\beta)=\frac{\exp(-\beta\mathcal{H}(\bm{x});q)}{Z_{q}(\beta)},$
(2)

having an additional parameter $q$, where

$\exp(x;q)\mathrel{\mathop{:}}=\big{[}1+(1-q)x\big{]}_{+}^{\frac{1}{1-q}}$ (3)

is the $q$-exponential function. Because $\exp(x;1)=\exp(x)$, the limit
$q\rightarrow 1$ of (2) recovers (1), so the canonical ensemble is contained
as a particular case.

Despite the success of these generalized ensembles, there are still
fundamental questions about their range of validity. For instance, it has been
clearly established [14] that states with $q>1$ are qualitatively different
than those with $q<1$, and have been classified as subadditive and
superadditive, respectively [15]. Moreover, only states with $q\geq 1$ can be
described using superstatistics [16, 17], an elegant framework which assumes
that temperature is a random variable with distribution $P(\beta|S)$, such
that the joint distribution of $\beta$ and the microstate $\bm{x}$ is given by

$P(\bm{x},\beta|S)=P(\bm{x}|\beta,S)P(\beta|S)=\left[\frac{\exp(-\beta\mathcal{H}(\bm{x}))}{Z(\beta)}\right]P(\beta|S).$
(4)

Beyond the studies regarding the $q$-canonical ensemble, however, no attempt
has been made to classify the nonequilibrium steady states in general, or to
search for any general explanation of the differences between the two separate
regimes of $q$-canonical states. In this work we propose such a classification
of nonequilibrium steady states into two large categories, namely
supercanonical and subcanonical states, using a statistical measure of the
covariance between temperature fluctuations.

This paper is organized as follows. In Section 2 we first review two
definitions of temperature for generalized ensembles, then in Sections 3 and 4
we introduce the inverse temperature covariance $\mathcal{U}$, in terms of
which we will express our main results, and present the proposed
classification of nonequilibrium steady states. Section 5 presents some
examples of nonequilibrium steady states in each category, and we provide some
concluding remarks in Section 6.

##  2 Steady states, temperatures and superstatistics

We will define a generalized ensemble $S$ as any non-equilibrium steady state
with microstate probabilities of the form

$P(\bm{x}|S)=\rho(\mathcal{H}(\bm{x});S),$ (5)

where the function $\rho$ is the ensemble function associated to $S$. If an
arbitrary value of $\mathcal{H}$ is denoted by $E$, the energy distribution in
this ensemble is given by

$P(\mathcal{H}=E|S)=\rho(E;S)\Omega(E)$ (6)

with

$\Omega(E)\mathrel{\mathop{:}}=\int d\bm{x}\delta(E-\mathcal{H}(\bm{x}))$ (7)

the density of states. A particular class among these steady states is
described by superstatistics, as defined by (4). By integrating over $\beta$
we obtain the microstate distribution

$P(\bm{x}|S)=\int_{0}^{\infty}d\beta P(\bm{x},\beta|S)=\int_{0}^{\infty}d\beta
P(\beta|S)\left[\frac{\exp(-\beta\mathcal{H}(\bm{x}))}{Z(\beta)}\right],$ (8)

and therefore the superstatistical ensemble function is

$\rho(E;S)=\int_{0}^{\infty}d\beta P(\beta|S)\left[\frac{\exp(-\beta
E)}{Z(\beta)}\right].$ (9)

From (8) it is clear that the canonical ensemble at $\beta_{0}$ is recovered
from superstatistics when $\beta_{0}$ is the only possible value of
temperature, that is, when

$P(\beta|\beta_{0})=\delta(\beta-\beta_{0}).$ (10)

Every steady state defined by (5) has a _fundamental inverse temperature_
function $\beta_{F}(E;S)$, given by

$\beta_{F}(E;S)\mathrel{\mathop{:}}=-\frac{\partial}{\partial E}\ln\rho(E;S)$
(11)

and such that the canonical ensemble is the only ensemble where $\beta_{F}$ is
a constant function, $\beta_{F}(E;\beta_{0})=\beta_{0}$. Another definition of
temperature, this time being intrinsic to the Hamiltonian, i.e. independent of
the ensemble function, is provided by the _microcanonical inverse temperature_
,

$\beta_{\Omega}(E)\mathrel{\mathop{:}}=\frac{\partial}{\partial
E}\ln\Omega(E),$ (12)

also known in the community of computer simulation as the _statistical
temperature_ [18]. This temperature corresponds to the thermodynamic
definition

$\frac{1}{T}\mathrel{\mathop{:}}=\frac{\partial S}{\partial E},$ (13)

with $S(E)\mathrel{\mathop{:}}=k_{B}\ln\Omega(E)$ the Boltzmann entropy. Note
that the most probable energy, denoted by $E^{*}$, is a solution of the
extremum equation

$0=\frac{\partial}{\partial E}\ln
P(\mathcal{H}=E|S)\Big{|}_{E=E^{*}}=\beta_{F}(E^{*})-\beta_{\Omega}(E^{*}),$
(14)

therefore it is such that it equalizes the fundamental and microcanonical
inverse temperatures. The specific heat $C_{E}$ at constant energy is
connected to this microcanonical temperature through the relation

$C_{E}\mathrel{\mathop{:}}=\left(\frac{\partial T(E)}{\partial
E}\right)^{-1}=-\frac{\beta_{\Omega}(E)^{2}}{{\beta_{\Omega}}^{\prime}(E)},$
(15)

and is in general a function of $E$. From this definition is clear that the
sign of $C_{E}$ is always the opposite of the sign of the derivative
${\beta_{\Omega}}^{\prime}(E)$.

By assuming that the energy distribution in (6) has zero probability at its
boundaries, and invoking the conjugate variables theorem (CVT) [19, 20], we
obtain the identity

$\left<\frac{\partial\omega}{\partial
E}\right>_{S}=-\left<\omega\frac{\partial}{\partial E}\ln
P(E|S)\right>_{S}=\Big{<}\omega\big{(}\beta_{F}-\beta_{\Omega})\Big{>}_{S},$
(16)

where $\omega=\omega(E)$ is an arbitrary, differentiable function of the
energy. Taking $\omega(E)=1$ we readily see that

$\big{<}\beta_{F}\big{>}_{S}=\big{<}\beta_{\Omega}\big{>}_{S},$ (17)

and we will refer to this mean inverse temperature as the inverse temperature
$\beta_{S}$ of the ensemble,

$\beta_{S}\mathrel{\mathop{:}}=\big{<}\beta_{\Omega}\big{>}_{S}=\big{<}\beta_{F}\big{>}_{S}.$
(18)

In the case of superstatistics, it follows directly from the definition of
$\beta_{F}$ in (11) that

$\beta_{F}(E;S)=-\frac{\partial}{\partial
E}\ln\rho(E;S)=\int_{0}^{\infty}d\beta\left[\frac{P(\beta|S)\exp(-\beta
E)}{\rho(E;S)Z(\beta)}\right]\beta=\big{<}\beta\big{>}_{E,S},$ (19)

where we have recognized the conditional distribution

$P(\beta|E,S)=\frac{P(E,\beta|S)}{P(E|S)}=\frac{P(\beta|S)\exp(-\beta
E)}{\rho(E;S)Z(\beta)}.$ (20)

By taking expectation of (19) in the form

$\big{<}\beta\big{>}_{E,S}=\beta_{F}(E),$ (21)

using $P(\mathcal{H}=E|S)$, we can also assert that
$\big{<}\beta\big{>}_{S}=\beta_{S}$, where the expectation of the left-hand
side is the mean superstatistical inverse temperature.

##  3 The inverse temperature covariance $\mathcal{U}$

In previous works [21] it was shown that the variance of the superstatistical
inverse temperature $\beta$ can, in principle, be measured through the
equality

$\text{Var}\left(\beta;S\right)=\text{Var}\left(\beta_{\Omega};S\right)+\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}.$
(22)

where $\text{Var}\left(A;S\right)$ denotes the variance of the quantity $A$ in
the steady state $S$, defined as

$\text{Var}\left(A;S\right)\mathrel{\mathop{:}}=\big{<}(\delta
A)^{2}\big{>}_{S}=\big{<}A^{2}\big{>}_{S}-\big{<}A\big{>}_{S}^{2}.$ (23)

The quantity in the right-hand side of (22),

$\mathcal{U}\mathrel{\mathop{:}}=\text{Var}\left(\beta_{\Omega};S\right)+\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S},$
(24)

only involves the statistics of the microcanonical inverse temperature and,
according to (22), is non-negative in the case of superstatistics.

Because the variance of $\beta$ is equal to zero only in the canonical
ensemble, that case corresponds to $\mathcal{U}=0$. However, the condition
$\mathcal{U}\geq 0$ does not always hold for more general steady states, as
$\mathcal{U}$ has been shown to take negative values in certain states outside
superstatistics [22].

The fact that $\mathcal{U}$ can be either positive or negative, and
furthermore, that $\mathcal{U}=0$ for the canonical ensemble, suggests a
classification of steady states into _supercanonical_ states, where
$\mathcal{U}>0$, and _subcanonical_ states, with $\mathcal{U}<0$. Although in
the subcanonical case $\mathcal{U}$ can no longer be associated with a
variance, it is always equal to the covariance between $\beta_{F}$ and
$\beta_{\Omega}$,

$\mathcal{U}=\big{<}\delta\beta_{F}\delta\beta_{\Omega}\big{>}_{S}$ (25)

which can be negative. This follows directly by choosing
$\omega(E)=\beta_{\Omega}(E)$ in (16). By instead choosing
$\omega(E)=\beta_{F}(E)$ we obtain an alternative expression, this time in
terms of the fundamental inverse temperature, namely

$\mathcal{U}=\text{Var}\left(\beta_{F};S\right)-\big{<}{\beta_{F}}^{\prime}\big{>}_{S},$
(26)

by which we can immediately see that a constant $\beta_{F}$ function, having
zero variance and zero derivative, implies $\mathcal{U}=0$. By using
$\omega(E)=\beta_{F}(E)-\beta_{\Omega}(E)$ in (16), we obtain

$\big{<}{\beta_{F}}^{\prime}\big{>}_{S}-\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}=\big{<}(\beta_{F}-\beta_{\Omega})^{2}\big{>}_{S}\geq
0$ (27)

hence we have the inequality

$\big{<}{\beta_{F}}^{\prime}\big{>}_{S}\geq\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}.$
(28)

From this inequality and using (24) and (26) it follows that

$\text{Var}\left(\beta_{F};S\right)>\text{Var}\left(\beta_{\Omega};S\right)\qquad\text{for}\;\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}>0,$
(29)

while on the other hand,

$\text{Var}\left(\beta_{\Omega};S\right)>\text{Var}\left(\beta_{F};S\right)\qquad\text{for}\;\big{<}{\beta_{F}}^{\prime}\big{>}_{S}<0.$
(30)

In both these cases where the ordering of the variances is well established we
see, by replacing in (24) or (26) respectively, that $\mathcal{U}>0$, which
means there are two clearly disjoint regimes within the supercanonical states,
that we will refer to as supercanonical-A when the variance of $\beta_{F}$ is
higher, and supercanonical-B otherwise. No such clear separation ocurrs for
the subcanonical ($\mathcal{U}<0$) states: they only require that
$\big{<}{\beta_{F}}^{\prime}\big{>}_{S}>0$ and
$\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}<0$, and are therefore always
consistent with (28).

Table 1 shows the proposed classification of steady states in this work. All
of superstatistics, including the canonical ensemble, is contained in the
supercanonical B class, so we see that there are three entire categories
outside superstatistics.

| Sign of $\mathcal{U}$ | Signature | Ordering of variances | Sign of $C_{E}$
---|---|---|---|---
Supercanonical A | $\mathcal{U}>0$ | $(+,+)$ | $\text{Var}\left(\beta_{F};S\right)>\text{Var}\left(\beta_{\Omega};S\right)$ | $C_{E}<0$
Supercanonical B | $\mathcal{U}\geq 0$ | $(-,-)$ | $\text{Var}\left(\beta_{\Omega};S\right)>\text{Var}\left(\beta_{F};S\right)$ | $C_{E}>0$
Subcanonical A | $\mathcal{U}\leq 0$ | $(-,+)$ | $\text{Var}\left(\beta_{F};S\right)>\text{Var}\left(\beta_{\Omega};S\right)$ | $C_{E}>0$
Subcanonical B | $\mathcal{U}\leq 0$ | $(-,+)$ | $\text{Var}\left(\beta_{\Omega};S\right)>\text{Var}\left(\beta_{F};S\right)$ | $C_{E}>0$
Table 1: Defining features of the classification of steady states proposed in
this work, consisting of supercanonical and subcanonical systems.
Superstatistical states, including the canonical ensemble, belong to the
supercanonical-B class. The _signature_ of a state is the sign of
$\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}$ followed by the sign of
$\big{<}{\beta_{F}}^{\prime}\big{>}_{S}$, where the signature $(+,-)$ is
forbidden by the inequality in (28).

###  3.1 Subsystem and environment

Let us consider a composite system in a nonequilibrium steady state. Its
Hamiltonian $\mathcal{H}$ is given by the sum of two terms, the energy
$H(\bm{x})$ of the subsystem $\bm{x}$ and the energy $G(\bm{y})$ of the
environment $\bm{y}$, that is,

$\mathcal{H}(\bm{x},\bm{y})=H(\bm{x})+G(\bm{y}),$ (31)

so that the distribution of microstates is

$P(\bm{x},\bm{y}|S)=\rho\big{(}H(\bm{x})+G(\bm{y});S\big{)}.$ (32)

Here $\bm{x}$ and $\bm{y}$ denote the entire set of degrees of freedom, both
kinetic and configurational. As the notation becomes intricate for all the
quantities in subsystem and environment, a summary is presented in Table 2.

The joint distribution of the energy $h$ of the subsystem and the energy $g$
of the environment is given by

$\begin{split}P(h,g|S)&=\Big{<}\delta(h-H(\bm{x}))\delta(g-G(\bm{y}))\Big{>}_{S}\\\
&=\int
d\bm{x}d\bm{y}\>\rho(H(\bm{x})+G(\bm{y});S)\>\delta(h-H(\bm{x}))\delta(g-G(\bm{y}))\\\
&=\rho(h+g;S)\Omega_{H}(h)\Omega_{G}(h),\end{split}$ (33)

where $\Omega_{H}$ and $\Omega_{G}$ are the densities of states of subsystem
and environment, respectively. We can see that these energies are in general
correlated unless $\rho(h+g)$ is separable, which is the case for the
canonical ensemble. The general form of the fluctuation-dissipation theorem
for the derivative of expectations [20] gives, for an arbitrary function
$\omega(h,g)$, the identity

$\frac{\partial}{\partial
h}\big{<}\omega\big{>}_{h,S}=\left<\frac{\partial\omega}{\partial
h}\right>_{h,S}+\left<\omega\frac{\partial}{\partial h}\ln
P(g|h,S)\right>_{h,S},$ (34)

which upon replacing

$P(g|h,S)=\frac{P(h,g|S)}{P(h|S)}=\frac{\rho(h+g;S)\Omega_{G}(g)}{\rho_{h}(h;S)}$
(35)

gives, for the logarithmic derivative,

$\frac{\partial}{\partial h}\ln P(g|h,S)=b_{F}(h)-\beta_{F}(h+g;S),$ (36)

and reduces (34) to

$\frac{\partial}{\partial
h}\big{<}\omega\big{>}_{h,S}=\left<\frac{\partial\omega}{\partial
h}\right>_{h,S}+\big{<}\omega\big{(}b_{F}-\beta_{F}\big{)}\big{>}_{h,S}.$ (37)

In a similar way, applying the CVT to $P(g|h,S)$ and using the logarithmic
derivative

$\frac{\partial}{\partial g}\ln
P(g|h,S)=\mathcal{B}_{\Omega}(g)-\beta_{F}(h+g;S),$ (38)

we obtain

$\left<\frac{\partial\omega}{\partial
g}\right>_{h,S}=\big{<}\omega\big{(}\beta_{F}-\mathcal{B}_{\Omega}\big{)}\big{>}_{h,S}.$
(39)

Setting $\omega(h,g)=1$ in (37) we readily obtain

$\big{<}b_{F}-\beta_{F}\big{>}_{h,S}=0,$ (40)

and because $b_{F}$ is only a function of $h$, we can write

$\big{<}\beta_{F}\big{>}_{h,S}=b_{F}(h).$ (41)

This rule expresses the fundamental inverse temperature of the subsystem as a
conditional mean of the fundamental inverse temperature of the entire system.
Taking expectation over $h$ in the state $S$ we see that

$\big{<}b_{F}\big{>}_{S}=\big{<}\beta_{F}\big{>}_{S}=\beta_{S}.$ (42)

In other words, the inverse temperature of the composite system must be equal
to the inverse temperature of the subsystem. Repeating the argument but using
$g$ instead of $h$, we see that

$\big{<}b_{F}\big{>}_{S}=\big{<}\mathcal{B}_{F}\big{>}_{S}=\beta_{S},$ (43)

and this is an extension of the defining property of temperature as the
quantity that equalizes in equilibrium. Additionally, using (39) with
$\omega(h,g)=1$ gives the important result

$\big{<}\mathcal{B}_{\Omega}\big{>}_{h,S}=\big{<}\beta_{F}\big{>}_{h,S}=b_{F}(h),$
(44)

where the last equality is due to (41). This new relation in (44) is a
generalization of

$b_{F}(h)=\mathcal{B}_{\Omega}(E-h)$ (45)

for the case where $h+g=E$ is fixed, as mentioned in Ref. [22]. It implies
that the fundamental (inverse) temperature of a subsystem is measurable if we
have access to the conditional energy distribution $P(g|h,S)$.

Now we have all the elements to evaluate the inverse temperature covariance
$\mathcal{U}_{h}$ for the subsystem. Using the choice
$\omega(h,g)=\beta_{F}(h+g;S)$ in (34) gives

$\big{<}{\beta_{F}}^{\prime}\big{>}_{h,S}=\big{<}{\beta_{F}}^{2}\big{>}_{h,S}-b_{F}(h)^{2}+{b_{F}}^{\prime}(h),$
(46)

and taking expectation under $S$ it follows that

$\big{<}{\beta_{F}}^{\prime}\big{>}_{S}=\big{<}{\beta_{F}}^{2}\big{>}_{S}-\big{<}(b_{F})^{2}\big{>}_{S}+\big{<}{b_{F}}^{\prime}\big{>}_{S}.$
(47)

Substracting $\beta_{S}$ from both sides and using (42), we finally obtain

$\text{Var}\left(\beta_{F};S\right)-\big{<}{\beta_{F}}^{\prime}\big{>}_{S}=\text{Var}\left(b_{F};S\right)-\big{<}{b_{F}}^{\prime}\big{>}_{S},$
(48)

that is, $\mathcal{U}=\mathcal{U}_{h}$. Repeating the same analysis for $g$
instead of $h$, it must also hold that $\mathcal{U}=\mathcal{U}_{g}$, so we
have

$\mathcal{U}=\mathcal{U}_{h}=\mathcal{U}_{g}.$ (49)

Property | Domain | Notation
---|---|---
Hamiltonian | Entire system | $\mathcal{H}(\bm{x},\bm{y})$
Hamiltonian | Subsystem | $H(\bm{x})$
Hamiltonian | Environment | $G(\bm{y})$
Value of energy | Entire system | $E$
Value of energy | Subsystem | $h$
Value of energy | Environment | $g$
Fundamental $\beta$ | Entire system | $\beta_{F}(E)$
Fundamental $\beta$ | Subsystem | $b_{F}(h)$
Fundamental $\beta$ | Environment | $\mathcal{B}_{F}(g)$
Microcanonical $\beta$ | Entire system | $\beta_{\Omega}(E)$
Microcanonical $\beta$ | Subsystem | $b_{\Omega}(h)$
Microcanonical $\beta$ | Environment | $\mathcal{B}_{\Omega}(g)$
Covariance of $\beta$ | Entire system | $\mathcal{U}$
Covariance of $\beta$ | Subsystem | $\mathcal{U}_{h}$
Covariance of $\beta$ | Environment | $\mathcal{U}_{g}$
Table 2: Notation used in this work when describing a composite system and its
parts. The quantity $\mathcal{U}$ is the inverse temperature covariance
defined by (25).

This is a new and important result for nonequilibrium steady states, which
tells us that in a steady state, the inverse temperature covariance
$\mathcal{U}$ is always the same for the composite system, the subsystem and
the environment. This result may help explain the findings by Nauenberg [23],
that subsystems with different entropic index $q$ cannot reach equilibrium, or
more generally, reach a steady state, provided that different values of $q$
produce different values of $\mathcal{U}$, as in the example presented in
subsection 5.1. From (49) we can readily obtain the classical result by Ray
[24] that small systems in the microcanonical ensemble have non-Maxwellian
momentum distributions: if $\mathcal{U}\neq 0$ for the configurational degrees
of freedom, the kinetic degrees of freedom cannot follow a canonical
distribution with $\mathcal{U}=0$.

In the case of superstatistics it is clear why (49) is true: for the composite
system we have

$\rho(h+g;S)=\int_{0}^{\infty}d\beta
P(\beta|S)\frac{\exp(-\beta(h+g))}{Z(\beta)},$ (50)

so the ensemble function of the subsystem is obtained by marginalization as

$\begin{split}\rho_{h}(h;S)&=\int dg\>\Omega_{G}(g)\rho(h+g;S)\\\
&=\int_{0}^{\infty}d\beta P(\beta|S)\frac{\exp(-\beta h)}{Z(\beta)}\int
dg\Omega_{G}(g)\exp(-\beta g)\\\ &=\int_{0}^{\infty}d\beta
P(\beta|S)\frac{\exp(-\beta h)}{Z_{h}(\beta)},\end{split}$ (51)

where we have used $Z(\beta)=Z_{h}(\beta)Z_{g}(\beta)$ and

$Z_{g}(\beta)=\int dg\Omega_{G}(g)\exp(-\beta g).$

In other words, the subsystem must be described by the same $P(\beta|S)$ as
the composite system and the environment. Because for superstatistics
$\mathcal{U}$ coincides with the variance of $\beta$, and is therefore
determined by $P(\beta|S)$, it follows that (49) must hold.

Furthermore, one can also prove that $\mathcal{U}$ has yet another definition,
this time connecting subsystem and environment, namely

$\mathcal{U}=\big{<}\delta b_{\Omega}\delta\mathcal{B}_{\Omega}\big{>}_{S},$
(52)

that is, $\mathcal{U}$ is the covariance between the microcanonical inverse
temperatures of subsystem and environment. In order to see why this is true,
we write the covariance in (52) in terms of the joint distribution $P(h,g|S)$,

$\begin{split}\big{<}\delta
b_{\Omega}\delta\mathcal{B}_{\Omega}\big{>}_{S}&=\int
dhdgP(h,g|S)b_{\Omega}(h)\mathcal{B}_{\Omega}(g)-(\beta_{S})^{2}\\\ &=\int
dhP(h|S)b_{\Omega}(h)\left[\int
dgP(g|h,S)\mathcal{B}_{\Omega}(g)\right]-(\beta_{S})^{2}\\\ &=\int
dhP(h|S)b_{\Omega}(h)b_{F}(h)-(\beta_{S})^{2}\\\ &=\big{<}\delta
b_{\Omega}\delta b_{F}\big{>}_{S}=\mathcal{U}_{h}=\mathcal{U},\end{split}$
(53)

where we have used (44) combined with (41) to replace the integral in square
brackets. Also, because of the law of total variance

$\text{Var}\left(X;I\right)=\big{<}\text{Var}\left(X;\bm{Y},I\right)\big{>}_{I}+\text{Var}\left(\big{<}X\big{>}_{\bm{Y},I};I\right),$
(54)

valid for any random scalar $X$ and random vector $\bm{Y}$, we have that, for
any set of parameters $\bm{\theta}$,

$\begin{split}\mathcal{U}-\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}&=\text{Var}\left(\beta_{\Omega};S\right)\\\
&=\big{<}\text{Var}\left(\beta_{\Omega};\bm{\theta},S\right)\big{>}_{S}+\text{Var}\left(\beta(\bm{\theta});S\right)\\\
&=\Big{<}\mathcal{U}_{\bm{\theta}}-\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{\bm{\theta},S}\Big{>}_{S}+\text{Var}\left(\beta(\bm{\theta});S\right)\\\
&=\big{<}\mathcal{U}_{\bm{\theta}}\big{>}_{S}-\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S}+\text{Var}\left(\beta(\bm{\theta});S\right)\end{split}$
(55)

where we have defined
$\beta(\bm{\theta})\mathrel{\mathop{:}}=\big{<}\beta_{\Omega}\big{>}_{\bm{\theta},S}$
and used the property

$\Big{<}\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{\bm{\theta},S}\Big{>}_{S}=\big{<}{\beta_{\Omega}}^{\prime}\big{>}_{S},$
(56)

hence we have the inequality

$\mathcal{U}\geq\big{<}\mathcal{U}_{\bm{\theta}}\big{>}_{S}.$ (57)

Not only this indicates that $\mathcal{U}$ cannot decrease when lifting
constraints, it gives a procedure to compute a lower bound for $\mathcal{U}$
in an arbitrary steady state if we can compute $\mathcal{U}_{\bm{\theta}}$ for
some fixed set of parameters $\bm{\theta}$, for instance, in microcanonical
steady states. An important result for the value of $\mathcal{U}$ in
microcanonical, short-range systems is given in the next section.

##  4 Regions of a large, microcanonical short-range system

Consider an homogeneous isolated system where
$\mathcal{H}(\bm{x},\bm{y})=H(\bm{x})+G(\bm{y})=E$ with $E$ its fixed total
energy. Here $\bm{x}$ and $\bm{y}$ form a partition of the system into a
subsystem and an environment, respectively. The joint distribution of $\bm{x}$
and $\bm{y}$ is microcanonical, so we have

$P(\bm{x},\bm{y}|E)=\frac{1}{\Omega(E)}\delta(H(\bm{x})+G(\bm{y})-E).$ (58)

In order to prove a simple but important result, we will make some
assumptions. First, the microcanonical inverse temperatures associated to each
region $\bm{x}$ and $\bm{y}$ will be given by the functions

$\displaystyle b_{\Omega}(h)$ $\displaystyle=f_{1}(h),$ (59a)
$\displaystyle\mathcal{B}_{\Omega}(g)$ $\displaystyle=f_{2}(g),$ (59b)

which are both strictly monotonic with $E$ and follow the same trend, i.e.,

$\operatorname*{sgn}(f_{1}^{\prime})=\operatorname*{sgn}(f_{2}^{\prime}).$
(60)

Second, the energy fluctuations of the subsystem are sufficiently small that
we can consider $h=h^{*}+\delta h$ with small $\delta h$. Under these
conditions we will show that the microcanonical system must be subcanonical,
that is,

$\mathcal{U}=\big{<}\delta b_{\Omega}\delta\mathcal{B}_{\Omega}\big{>}_{E}\leq
0,$ (61)

therefore $\mathcal{U}_{h}\leq 0$ and $\mathcal{U}_{g}\leq 0$, according to
(49). This follows directly from (52), which we can write in the form

$\mathcal{U}=\Big{<}\big{[}b_{\Omega}-\beta_{S}\big{]}\big{[}\mathcal{B}_{\Omega}-\beta_{S}\big{]}\Big{>}_{E}$
(62)

where
$\beta_{S}=\big{<}b_{\Omega}\big{>}_{E}=\big{<}\mathcal{B}_{\Omega}\big{>}_{E}$.
Considering $\delta h$ to be small, this expectation can be approximated as

$\beta_{S}=\big{<}f_{1}(h)\big{>}_{E}\approx\big{<}f_{1}(h^{*})+\delta
hf_{1}^{\prime}(h^{*})\big{>}_{E}=\beta_{0}+\big{<}\delta
h\big{>}_{E}f_{1}^{\prime}(h^{*})=\beta_{0}$ (63)

with $\beta_{0}\mathrel{\mathop{:}}=f_{1}(h^{*})$, but in the same way

$\beta_{S}=\big{<}f_{2}(g)\big{>}_{E}\approx\big{<}f_{2}(E-h^{*})+\delta
gf_{2}^{\prime}(E-h^{*})\big{>}_{E}=f_{2}(E-h^{*})$ (64)

thus $\beta_{0}=f_{2}(E-h^{*})$. The fluctuations of temperature in subsystem
and environment are given by

$\displaystyle\delta b_{\Omega}=f_{1}(h)-\beta_{0}$
$\displaystyle\approx\delta hf_{1}^{\prime}(h^{*}),$ (65a)
$\displaystyle\delta\mathcal{B}_{\Omega}=f_{2}(E-h)-\beta_{0}$
$\displaystyle\approx-\delta hf_{2}^{\prime}(E-h^{*}).$ (65b)

where we have used $\delta h+\delta g=\delta E=0$. It is clear that these
fluctuations always have opposite signs, therefore

$\delta b_{\Omega}\delta\mathcal{B}_{\Omega}=-(\delta
h)^{2}f_{1}^{\prime}(h^{*})f_{2}^{\prime}(E-h^{*})\leq 0$ (66)

for every value of $h$, and (61) follows by taking expectation. From Table 1
it also follows that $C_{E}>0$ for the composite system and its parts, so not
only are $f_{1}$ and $f_{2}$ strictly monotonic with $E$, it must be true that
$f_{1}^{\prime}<0$ and $f_{2}^{\prime}<0$.

##  5 Examples

###  5.1 The $q$-canonical ensemble

While we already know that $q\geq 1$ is described by the
$\chi^{2}$-superstatistics [17], the behavior of the $q$-canonical ensemble
for $q\leq 1$ remains to be explored. We can show that states with $q\leq 1$
can be subcanonical, that is, can have $\mathcal{U}<0$.

We start with the fundamental inverse temperature corresponding to (2), given
by [25]

$\beta_{F}(E;q,\beta_{0})=\frac{\beta_{0}}{1+(q-1)\beta_{0}E},$ (67)

and compute its derivative,

${\beta_{F}}^{\prime}(E;q,\beta_{0})=(1-q)\beta_{F}(E;q,\beta_{0})^{2}.$ (68)

We clearly see that this derivative is negative for $q>1$, consistent with
superstatistics, but positive for $q<1$. The inverse temperature covariance is

$\mathcal{U}=\big{<}(\delta\beta_{F})^{2}\big{>}_{q,\beta_{0}}-\big{<}{\beta_{F}}^{\prime}\big{>}_{q,\beta_{0}}=q\big{<}{\beta_{F}}^{2}\big{>}_{q,\beta_{0}}-{\beta_{S}}^{2},$
(69)

with $\mathcal{U}\geq 0$ for $q\geq 1$ as expected. In order to show that
$\mathcal{U}<0$ can be achieved in this ensemble for $q<1$, let us consider
the density of states $\Omega(E)=\Omega_{0}E^{\alpha}$ with $\alpha\geq-1/2$,
case which is solved in Ref. [26]. In that case we have

$\beta_{S}=\beta_{0}\big{(}1+(\alpha+1)(1-q)\big{)},$ (70)

and

$q\big{<}{\beta_{F}}^{2}\big{>}_{q,\beta_{0},\alpha}=\left[\frac{(1-q)\alpha+1}{(1-q)(\alpha+1)+1}\right]{\beta_{S}}^{2},$
(71)

so

$\mathcal{U}={\beta_{S}}^{2}\left[\frac{q-1}{1+(1-q)(\alpha+1)}\right]$ (72)

which is negative for $q<1$. For the case $q>1$ the condition $\mathcal{U}\geq
0$ imposes

$q\leq 1+\frac{1}{\alpha+1},$ (73)

precisely is the bound reported by Lutsko and Boon [14]. Because $q\geq 1$
implies $\big{<}{b_{F}}^{\prime}\big{>}_{S}\leq 0$ and
${\beta_{\Omega}}^{\prime}\leq 0$, it follows also that in this case
$\text{Var}\left(\beta_{\Omega};S\right)\geq\text{Var}\left(\beta_{F};S\right)$
so this confirms that $q\geq 1$ are supercanonical-B states. On the contrary,
states with $q\leq 1$ have

$\text{Var}\left(\beta_{F};S\right)\geq\text{Var}\left(\beta_{\Omega};S\right),$
(74)

so they are subcanonical-A states. The inequality in (74) follows from the
results in Ref. [26],

$\frac{\alpha+1}{\alpha}\big{<}\beta_{\Omega}^{2}\big{>}_{q,\beta_{0}}=q\big{<}\beta_{F}^{2}\big{>}_{q,\beta_{0}}$
(75)

using the fact that

$q\left(\frac{\alpha}{\alpha+1}\right)<q\leq 1.$

###  5.2 The Gaussian ensemble with a convex entropy region

Let us consider a Gaussian ensemble written as

$\rho(E;\lambda)=\frac{1}{\eta(\lambda)}\exp\left(-\frac{\lambda
E^{2}}{2}\right),$ (76)

where $\lambda>0$ and with the same density of states
$\Omega(E)=\Omega_{0}E^{\alpha}$. Its fundamental inverse temperature is given
by

$\beta_{F}(E;\lambda)=-\frac{\partial}{\partial E}\ln\rho(E;\lambda)=\lambda
E,$ (77)

and its derivative

${\beta_{F}}^{\prime}(E;\lambda)=\lambda$ (78)

is non-negative, therefore it cannot be described by superstatistics.
Nevertheless, we will show that it encompasses all the classes described in
Table 1. The $n$-th moment of the energy distribution is

$\big{<}E^{n}\big{>}_{S}=\sqrt{\Big{(}\frac{2}{\lambda}\Big{)}^{n}}\frac{\Gamma\big{(}\frac{1}{2}(\alpha+n+1)\big{)}}{\Gamma\big{(}\frac{1}{2}(\alpha+1)\big{)}}$
(79)

so we can construct

$\text{Var}\left(\beta_{F};S\right)=\big{<}(\lambda
E)^{2}\big{>}_{S}-\lambda^{2}\big{<}E\big{>}_{S}^{2}=\frac{\alpha+1}{\lambda}-\frac{2}{\lambda}\left[\frac{\Gamma(\alpha/2+1)}{\Gamma(\alpha/2+1/2)}\right]^{2}$
(80)

and finally obtain

$\mathcal{U}=\lambda\left[\alpha-\left(\frac{\Gamma(\alpha/2+1)}{\Gamma(\alpha/2+1/2)}\right)^{2}\right].$
(81)

This is a monotonically increasing function of $\alpha$ going from
$-2\lambda/\pi$ to $-\lambda/2$, thus it is always negative and all states
belong to the subcanonical class. The variance of $\beta_{\Omega}$ is

$\text{Var}\left(\beta_{\Omega};S\right)=\alpha^{2}2^{\frac{\alpha-3}{2}}\lambda^{\frac{1-\alpha}{2}}\Gamma\Big{(}\frac{\alpha-1}{2}\Big{)}-\left(\frac{2}{\lambda}\right)^{\alpha}\Gamma(1+\alpha/2)^{2}$
(82)

In the particular case $\lambda=2$ we can verify that there is a transition
from subcanonical-A for $\alpha<1$ to subcanonical-B behavior for $\alpha>1$.
However, as $\alpha\rightarrow\infty$ in the thermodynamic limit we have
subcanonical-B behavior for all $\lambda$.

Now let us generalize the density of states to the following expression,

$\Omega(E;\alpha,\mu,b)=\Omega_{0}\big{(}1+b^{2}(E-\mu)^{2}\big{)}E^{\alpha},$
(83)

which is mostly concave as in the previous case but has a convex region for
$b>0$, as seen in Fig. 1, what has been called a “convex intruder” [27, 28,
29, 30]. Its microcanonical inverse temperature is given by

$\beta_{\Omega}(E)=\frac{2b^{2}(E-\mu)}{1+b^{2}(E-\mu)^{2}}+\frac{\alpha}{E},$
(84)

with derivative

${\beta_{\Omega}}^{\prime}(E)=\frac{2b^{2}(b^{2}(E-\mu)^{2}-1)}{(1+b^{2}(E-\mu)^{2})^{2}}-\frac{\alpha}{E^{2}}.$
(85)

Here we see that ${\beta_{\Omega}}^{\prime}(E)$ can change sign for certain
values of $b$, $\mu$ and $\alpha$, and thus it can describe systems with
negative and positive $C_{E}$. This is then a candidate for a model that can
present supercanonical-A states, of which we have not shown an example yet. In
fact, such an example with a bimodal distribution of energy is seen in Fig. 2
(left panel). Figs. 3, 4 and 5 show that subcanonical states and even
transitions from one class to another are also possible within this model.

##  6 Concluding remarks

In this work, a classification of nonequilibrium steady states is proposed,
based on the sign of the inverse temperature covariance $\mathcal{U}$ between
inverse temperatures. This classification divides the space of nonequilibrium
models into supercanonical ($\mathcal{U}>0$) and subcanonical
($\mathcal{U}<0$) models, regarding all the superstatistical models as part of
the larger supercanonical-B class, disconnected from the supercanonical-A and
subcanonical classes.

The main theorem proved in this work, namely the equality in (49), indicates
that all regions of a system in a nonequilibrium steady state must have the
same value of $\mathcal{U}$. This is remarkably similar to the defining
property of temperature in thermodynamics, the fact that $\beta$ reaches the
same value for every region of a system in thermal equilibrium, but instead
connects the uncertainties $\text{Var}\left(\beta_{F};S\right)$ and
$\text{Var}\left(\beta_{\Omega};S\right)$ in different regions of a system.

Several results of general interest were also proven, such as the statement
that all regions of a large enough homogeneous, microcanonical system with
short-range interactions under strictly monotonic $\beta_{\Omega}$ must have
$\mathcal{U}\leq 0$ in a steady state, and therefore fall outside the domain
of superstatistics. Moreover, the results in this work show that
supercanonical behavior is not limited to superstatistics, but can ocurr
outside of it for systems with negative heat capacity, for instance metastable
states in first-order phase transitions.

## Acknowledgments

SD acknowledges financial support from ANID FONDECYT 1220651 grant.

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Figure 1: Logarithm of the density of states in (83) for $\mu$=2.60384,
$\alpha$=1.20401 and different values of $b$.

Figure 2: Left panel, example of a supercanonical A state, with $b$=1.94541,
$\lambda$=1.09747, $\mu$=2.53849 and $\alpha$=7.15724. In this case we have
$\mathcal{U}$=0.050368. Right panel, example of a supercanonical B state, with
$b$=2.17105, $\lambda$=0.58616, $\mu$=3.28635 and $\alpha$=5.62514. In this
case we have $\mathcal{U}$=0.13170.

Figure 3: Left panel, example of a subcanonical A state, with $b$=1.404639,
$\lambda$=2.27263, $\mu$=1.50866 and $\alpha$=8.29722. In this case we have
$\mathcal{U}$=-0.82574. Right panel, example of a subcanonical B state, with
$b$=0.594146, $\lambda$=1.73940, $\mu$=2.73567 and $\alpha$=2.48991. In this
case we have $\mathcal{U}$=-1.00845. Figure 4: Transition from subcanonical to
supercanonical behavior.

Figure 5: Left panel, transition from subcanonical B to subcanonical A
behavior. Right panel, transition from supercanonical B to supercanonical A
behavior.


based interferometers like LISA [90] and BBO [91] are most sensitive. These GW
signals generally lie below the sensitivity of the proposed detectors, but
there are suggestions that they may be detectable at BBO with
$T_{n}\sim\mathcal{O}$(10 GeV). We do note, finally, that these results are
derived from effective field theories that attempt to capture complex non-
perturbative behaviours. On the advent of more sophisticated, first-principles
lattice calculations, it may be that these signals are stronger than expected
and have greater prospects of detection.

##  VIII Conclusions

In this work, we explored a new dark QCD framework in which there is a
dynamical mechanism to relate the confinement scales of visible and dark QCD.
This provides a dark matter candidate, the dark baryon, whose mass is
naturally on a similar order to the proton mass. When combined with an
asymmetric dark matter mechanism that relates the number densities of visible
and dark matter, this provides an explanation of the coincidental similarity
of the cosmological mass densities of visible and dark matter. We followed on
from previous work in which new field content is introduced such that the
gauge couplings of visible and dark QCD evolve towards a fixed point with
small, related values for the two couplings – referred to as the finite-
coupling fixed point approach – and instead considered a framework in which
the couplings at the infrared fixed point are both zero. In this approach, the
initial coupling strengths in the UV both evolve to be very small, before the
new field content decouples at a high mass scale. These couplings then evolve
independently until the energy scale at which they become non-perturbative and
confinement occurs. The visible and dark confinement scales can then be
similar for a generic selection of initial values for the couplings in the UV,
with the caveat that this depends strongly on the number of light dark quarks.

We analysed models in this framework in order to find viable models that could
explain the similarity of the visible and dark confinement scales, looking at
dark gauge groups with varying number of colours $N_{d}$, and adding in field
content in various representations of $SU(3)_{\mathrm{QCD}}\times
SU(N_{d})_{\mathrm{dQCD}}$. We first looked at models where the required mass
scale of new physics is $\mathcal{O}$(TeV) to avoid naturalness constraints
from the running of the Higgs mass parameter. In that case we did find viable
models, but they were rare, and required a close cancellation between the
matter and gauge contributions to the $\beta$-function coefficients; this
cancellation is similar to that required for the viable models in the FCFP
approach. However, in the FCFP approach, the successful models required very
large multiplicities for the new field content; here, that was no longer the
case.

Secondly, we looked at models in which no such cancellation occurs between the
various contributions to the $\beta$-function coefficients. In these models,
the mass scale of new physics is generically very high (only a few orders of
magnitude below the UV scale) and so to avoid the naturalness issue for the
Higgs mass parameter required us to consider supersymmetric versions of these
models. In this case, for different values of $N_{d}$ we considered generic
sets of models with low multiplicities for the new field content we introduce.
For each model we chose the number of light dark quarks to maximise the
viability fraction $\epsilon_{v}$, our heuristic that characterises the
ability of a given model to explain the similarity of the confinement scales.

The results here were quite promising. In particular, when $N_{d}=4$, we found
that 65% of models have $\epsilon_{v}>0.3$, with most of these models
requiring 11 light dark quarks. For $N_{d}=3$ we found that 40% of models have
$\epsilon_{v}>0.25$, with most models requiring 6 dark quarks. For $N_{d}=2$,
however, all but a few models were not viable.

These sets of models are intended to be amenable to further development into
full models of asymmetric dark matter. In this direction, we showed examples
of a leptogenesis-like asymmetry generation mechanism for viable models in
both the non-SUSY and SUSY cases. We also highlighted some of the
phenomenology of these models. In particular we considered the gravitational
wave signals arising from the cosmological chiral phase transition in the dark
QCD sector. In this framework this transition is generically first-order and
thus features a stochastic GW signal. While these signals usually lie at mHz
peak frequencies sensitive to space-based interferometers like LISA and BBO,
they are thought to be too weak to be detectable, due to the short duration of
the phase transition. However, depending on calculational uncertainties in the
GW dynamics, there may be prospects for detection at BBO, or even the proposed
$\mu$Ares detector in the $\mu$Hz region.

The proposed ADM mechanisms in this paper are not the only possible way to
build upon models in this framework. Further studies of the phenomenology and
cosmology of complete ADM models incorporating other ADM mechanisms would be
welcome.

## Acknowledgements

ACR was supported by an Australian Government Research Training Program
Scholarship. This research was supported by the Australian Government through
the Australian Research Council Centre of Excellence for Dark Matter Particle
Physics (CDM, CE200100008).

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##  Appendix A Models with sextet representations and decent values of
$\epsilon_{v}^{10}$

In Section V.1 we looked at non-SUSY models with $N_{d}=3$, including new
field content in at most sextet representations of visible and dark QCD.
Allowing for at most three copies of each new field, and only specifying the
field content charged under visible QCD, we find 242 models that have
$b_{v}^{(0)}/2\pi=1/12$. The hope is that these models may have a small enough
one-loop $\beta$-function coefficient that they can have a decent proportion
of viable parameter space with $M<10$ TeV, $\epsilon_{v}^{10}$. When adding in
dark field content to these models – that is, both light and heavy
$(\mathbf{1},\mathbf{3})$ fermions (dark quarks) and heavy
$(\mathbf{1},\mathbf{6})$ fermions – we find that only 16 of the models obtain
a value of $\epsilon_{v}^{10}>0.25$. In Table 9 we show the field content of
these 16 models, along with their values for $\epsilon_{v}$ and
$\epsilon_{v}^{10}$.

fermions | $n_{dq}$ | scalars | $\epsilon_{v}$ | $\epsilon_{v}^{10}$
---|---|---|---|---
$\begin{pmatrix}&2&3\\\ 6&0&0\\\ 2&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix}$ | 0.42 | 0.28
$\begin{pmatrix}&2&3\\\ 6&0&0\\\ 2&0&0\end{pmatrix}$ | 2 | $\begin{pmatrix}&0&0\\\ 3&0&0\\\ 0&0&0\end{pmatrix}$ | 0.41 | 0.34
$\begin{pmatrix}&2&2\\\ 6&1&0\\\ 1&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 2&3&0\\\ 0&0&0\end{pmatrix}$ | 0.37 | 0.28
$\begin{pmatrix}&2&1\\\ 6&3&0\\\ 0&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 1&2&0\\\ 0&0&0\end{pmatrix}$ | 0.37 | 0.29
$\begin{pmatrix}&3&1\\\ 6&3&0\\\ 0&0&0\end{pmatrix}$ | 3 | $\begin{pmatrix}&0&0\\\ 2&0&0\\\ 1&0&0\end{pmatrix}$ | 0.38 | 0.29
$\begin{pmatrix}&3&2\\\ 7&1&0\\\ 1&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 1&2&0\\\ 0&0&0\end{pmatrix}$ | 0.40 | 0.30
$\begin{pmatrix}&4&2\\\ 7&1&0\\\ 1&0&0\end{pmatrix}$ | 3 | $\begin{pmatrix}&0&0\\\ 2&0&0\\\ 1&0&0\end{pmatrix}$ | 0.40 | 0.29
$\begin{pmatrix}&3&1\\\ 7&3&0\\\ 0&0&0\end{pmatrix}$ | 2 | $\begin{pmatrix}&0&0\\\ 3&0&0\\\ 0&0&0\end{pmatrix}$ | 0.38 | 0.34

fermions | $n_{dq}$ | scalars | $\epsilon_{v}$ | $\epsilon_{v}^{10}$
---|---|---|---|---
$\begin{pmatrix}&3&2\\\ 8&1&0\\\ 1&0&0\end{pmatrix}$ | 3 | $\begin{pmatrix}&0&0\\\ 0&1&0\\\ 0&0&0\end{pmatrix}$ | 0.39 | 0.33
$\begin{pmatrix}&4&2\\\ 8&1&0\\\ 1&0&0\end{pmatrix}$ | 2 | $\begin{pmatrix}&0&0\\\ 3&0&0\\\ 0&0&0\end{pmatrix}$ | 0.40 | 0.36
$\begin{pmatrix}&5&1\\\ 8&2&0\\\ 0&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 0&2&0\\\ 1&0&0\end{pmatrix}$ | 0.37 | 0.27
$\begin{pmatrix}&4&1\\\ 8&2&0\\\ 0&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 2&3&0\\\ 0&0&0\end{pmatrix}$ | 0.37 | 0.31
$\begin{pmatrix}&1&2\\\ 8&2&0\\\ 0&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 3&1&0\\\ 1&0&0\end{pmatrix}$ | 0.39 | 0.27
$\begin{pmatrix}&2&3\\\ 9&0&0\\\ 1&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 3&1&0\\\ 1&0&0\end{pmatrix}$ | 0.42 | 0.27
$\begin{pmatrix}&2&1\\\ 9&2&0\\\ 0&0&0\end{pmatrix}$ | 1 | $\begin{pmatrix}&0&0\\\ 1&0&1\\\ 0&0&0\end{pmatrix}$ | 0.37 | 0.27
$\begin{pmatrix}&4&1\\\ 9&2&0\\\ 0&0&0\end{pmatrix}$ | 4 | $\begin{pmatrix}&0&0\\\ 1&2&0\\\ 0&0&0\end{pmatrix}$ | 0.36 | 0.31

Table 9: These sixteen models have $b_{v}^{(0)}/2\pi=1/12\pi$ and at most
three of each new field, where we only include fields in at most sextet
representations of visible and dark QCD. For each model we have added dark
field content to achieve a decent value for $\epsilon_{v}^{10}$, the viable
proportion of UV coupling parameter space for which $M<10$ TeV. For each model
we give the viability fraction $\epsilon_{v}$ as well as $\epsilon_{v}^{10}$.
The field content of each model is specified in the same way as in Table 5,
but here the rows correspond to
$\mathbf{R}_{v}=\mathbf{1},\mathbf{3},\mathbf{6}$ from top to bottom, and the
columns correspond to $\mathbf{R}_{d}=\mathbf{1},\mathbf{3},\mathbf{6}$ from
left to right.