# Unique extension of the maximum entropy principle to phase coexistence in
heat conduction

Naoko Nakagawa  Department of Physics, Ibaraki University, Mito 310-8512,
Japan  Shin-ichi Sasa  Department of Physics, Kyoto University, Kyoto
606-8502, Japan

###### Abstract

The maximum entropy principle determines the values of thermodynamic variables
in thermally isolated equilibrium systems. This paper extends the principle to
a variational principle that applies to liquid–gas coexistence in heat
conduction. We show the uniqueness of the extension under the assumption that
the variational principle and the fundamental thermodynamic relation are
simultaneously extended in the linear response regime with the total energy
fixed. Using the extended variational principle, we calculate the
thermodynamic quantities in this steady state and find that the temperature of
the liquid–gas interface deviates from the equilibrium transition temperature,
which should be verified in experiments.

##  I Introduction

The values of extensive unconstrained variables in thermally isolated
equilibrium systems maximize the total entropy. This variational principle is
called the maximum entropy principle and plays a fundamental role in
equilibrium thermodynamics Callen . The principle leads to the definition of
temperature and provides the equation of state. Additionally, the fluctuation
law of thermodynamic variables is naturally conjectured from the variational
principle and turns out to be consistent with the principle of equal weight in
statistical mechanics Einstein . The main question addressed in this paper is
whether the maximum entropy principle extends to out-of-equilibrium systems.

The first approach to answering the above question relies on nonequilibrium
statistical mechanics. In the last two decades, several universal properties
of systems out of equilibrium have been understood with the aid of the
fluctuation theorem and its several variants Evans-Cohen-Morriss ; Gallavotti
; JarzynskiPRL ; Kurchan ; LS ; Maes ; Crooks-WR ; Hatano-Sasa ; Sekimoto-book
; SeifertRPP . In particular, the difference in the stationary distribution
from nonequilibrium to equilibrium has been studied when the Shannon entropy
has been assumed to be the nonequilibrium extension of equilibrium entropy.
Specifically, for general systems with local detailed balance, the
representation of the stationary distribution has been known to include the
time integration of the entropy production in the relaxation to the
equilibrium state Crooks . In the linear response regime, the representation
takes the form proposed by Zubarev and Mclennan Zubarev ; Mclennan .
Additionally, a representation to higher order beyond the linear response
regime has been proposed KN ; KNST . Although these representations are valid,
they are not directly used to analyze thermodynamic properties owing to the
difficulty of treating the time integration. For some particular models,
stationary distributions without the time integration have been derived DLS ;
Jona-Lasinio . However, these forms are too specific to apply to general out-
of-equilibrium systems. There has been no successful extension of the maximum
entropy principle based on nonequilibrium statistical mechanics.

The second approach is to construct a phenomenological framework of the
extension of thermodynamics. There has been a long history of attempting to
construct this framework Keizer ; Eu ; Jou ; Oono-Paniconi ; Sasa-Tasaki ;
Bertin ; Seifert-contact ; Dickman ; Holyst . All attempts have adopted the
same basic idea of adding a nonequilibrium variable and its conjugate variable
to equilibrium thermodynamic variables. However, the method adopted for the
extension depends on the theory, and indeed there are many different versions.
For each specific setup, a nonequilibrium variable is chosen by noting the
intensive or extensive nature. The conjugate variable of the nonequilibrium
variable is defined from the thermodynamic relation using extended entropy.
However, to introduce extended entropy, the theory requires an additional
assumption, which is at the heart of the argument for each theory. It has not
yet been established which assumption is most reasonable, and the extension of
the maximum entropy principle, whether or not it is aimed at, is not a
requisite for the construction of the framework for each theory.

The central concept of the paper is that the maximum entropy principle and the
fundamental thermodynamic relation are simultaneously extended. That is, we
first assume an extension of the equilibrium variational function with an
undetermined function, and then impose the value of the variational function
in the steady state to satisfy a fundamental thermodynamic relation. In cases
of equilibrium, the maximum entropy principle is closely related to the law of
entropy increase for thermally isolated systems, where the total energy is
conserved without external operations. Thus, we should start with the design
of energy-conserving systems in heat conduction. An example of such a system
is shown in Fig. 1, where the in-flow heat current is controlled as it is
strictly the same as the out-flow current at any moment global-fluc . We
attempt to formulate an extended maximum entropy principle for determining the
thermodynamic quantities of such systems.

The important thing here is that the extended maximum entropy principle should
provide a novel prediction of quantitative phenomena that can be observed in
experiments. From this viewpoint, the liquid–gas phase coexistence in heat
conduction has turned out to be a good target even in the linear response
regime. Our previous papers predicted that the interface temperature deviates
from the equilibrium transition temperature global-PRL ; global-JSP .
Numerical simulations have quantitatively confirmed this prediction KNS . Such
phase coexistence may become studied by extending the maximum entropy
principle to the linear response regime in energy-conserving heat-conduction
systems.

Figure 1: Schematic illustration of thermally isolated systems in and out of
equilibrium. At constant pressure of $p$, the total conserved energy of the
system is enthalpy $H$. $J$ is the amount of heat current per unit area
imposed at the left and right boundary walls. See the main text for further
details.

Here, we briefly review the maximum entropy principle for the liquid–gas
coexistence at equilibrium. At constant pressure, the thermodynamic entropy
$S$ is a function of the energy and the number of particles of the whole
system, whereas the variational function $\mathscr{S}$ is a function of those
of the liquid with the total energy and the total number of particles of the
system fixed. The two entropies are functionally different and connected by
the maximum entropy principle; i.e., the value of $S$ is equal to the maximum
value of $\mathscr{S}$. The maximization determines the unique equilibrium
state. The equilibrium entropy possesses a remarkable property that the two
entropies, $\mathscr{S}$ and $S$, are similarly expressed as the sum of the
entropy of the liquid and the entropy of the gas.

We proceed to the heat conduction state in the linear response regime. The
respective entropies of the liquid and the gas are given by the integration of
the local entropy density in the respective regions. One may then guess that
the variational function $\mathscr{S}$ is obtained by summing these entropies
similarly to the equilibrium cases. However, there is no reason to assume the
additivity of entropy for out-of-equilibrium systems. Instead of assuming the
additivity, we impose that the value of the variational function in the steady
state satisfies the thermodynamic relation with the global temperature
introduced in Refs. global-PRL .

From this fundamental assumption and other assumptions of the
extensive/intensive nature of variables, we can uniquely determine the form of
the variational function. The remarkable result obtained in this paper is that
the variational function has a non-additive contribution even in the linear
response regime. This term is specific to phase coexistence because it is
proportional to the latent heat. Furthermore, we show that the steady state
determined within this framework is identical to that determined by the free
energy minimum principle with the global temperature fixed global-PRL . As a
consequence, the interface temperature deviates from the equilibrium
transition temperature.

The remainder of the paper is organized as follows. Section II reviews the
maximum entropy principle for determining the thermodynamic state of
coexisting liquid and gas phases with the total enthalpy, pressure and total
particle number fixed. Section III describes the setup for the heat conduction
states that we study. Section IV addresses the problem that we want to solve;
i.e., we seek a form of the variational function as an extension of the
maximum entropy principle. Section V presents the form of the variational
function, which gives the unique answer to the problem. Section VI argues the
thermodynamic equivalence of steady states in heat conduction. Section VII
provides a quantitative result for the steady state. The final section is
devoted to concluding remarks. Derivations of the results and formulas are
separately presented in the appendices. In this paper, we define the
temperature using the energy unit $k_{\mathrm{B}}=1$ for simplicity.

##  II Brief review on the role of entropy in equilibrium

Let us start with a familiar liquid–gas transition observed in a fluid at
constant pressure $p$ in contact with a heat bath of temperature $T$. The
density of the fluid exhibits a discontinuous jumps at $T=T_{\mathrm{c}}(p)$.
The fluid behaves as a liquid at $T<T_{\mathrm{c}}(p)$ and as gas at
$T>T_{\mathrm{c}}(p)$. Letting the equation of state for the liquid and gas be
$\rho=\rho^{\mathrm{L}}(T,p)$ and $\rho=\rho^{\mathrm{G}}(T,p)$, respectively,
we obtain the particle number density at the liquid–gas transition as
$\rho_{\mathrm{c}}^{\mathrm{L}}(p)=\lim_{T\rightarrow
T_{\mathrm{c}}(p)^{-}}\rho^{\mathrm{L}}(T,p)$ and
$\rho_{\mathrm{c}}^{\mathrm{G}}(p)=\lim_{T\rightarrow
T_{\mathrm{c}}(p)^{+}}\rho^{\mathrm{G}}(T,p)$. See Fig. 2(a). The latent heat
generated in a transition to gas from liquid is equivalent to the jump in
enthalpy $H$, where $H=U+pV$ with the internal energy $U$ and the volume $V$
of the system. As shown in Fig. 2(b), letting $\hat{h}^{\mathrm{L}}(T,p)$ and
$\hat{h}^{\mathrm{G}}(T,p)$ be the enthalpy per particle for liquid and gas,
the latent heat per particle is

$\displaystyle\hat{q}(p)=\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)-\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p),$
(II.1)

where $\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)=\lim_{T\rightarrow
T_{\mathrm{c}}(p)^{-}}\hat{h}^{\mathrm{L}}(T,p)$ and
$\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)=\lim_{T\rightarrow
T_{\mathrm{c}}(p)^{+}}\hat{h}^{\mathrm{G}}(T,p)$.

Figure 2:  (a) Particle number density $\rho$ and (b) enthalpy per particle
$\hat{h}$ as functions of $T$ for a given $p$.

For a thermally isolated fluid from the environment of constant pressure,
enthalpy $H$ is the conserved energy, which corresponds to the sum of the
energy of the system and the interaction potential between the system and the
environment. Figure 3 shows the equilibrium temperature of the system as a
function of $\hat{h}=H/N$ with $p$ fixed. This figure is equivalent to Fig.
2(b) except for the swapping of the vertical and horizontal axes. The system
exhibits stable liquid–gas coexistence for a certain range of enthalpy, say
$\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)<\hat{h}<\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)$,
while it is occupied by liquid for
$\hat{h}<\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)$ or by gas for
$\hat{h}>\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)$. Choosing $\hat{h}$ as a
certain value in the range of coexistence, the distribution of $H$ and $N$ to
the liquid and the gas are uniquely determined as $H^{\mathrm{L}}$ with
$H^{\mathrm{G}}=H-H^{\mathrm{L}}$ and $N^{\mathrm{L}}$ with
$N^{\mathrm{G}}=N-N^{\mathrm{L}}$.

Figure 3: Temperature as a function of $\hat{h}=H/N$ for equilibrium systems
with $p$ fixed. The plateau corresponds to the transition temperature
$T_{\mathrm{c}}(p)$. Swapping the vertical and horizontal axes of the graph in
Fig. 2(b) yields this figure.

A remarkable point in equilibrium thermodynamics is the correspondence of the
thermodynamic entropy $S$ with the entropy as the variational function
$\mathscr{S}$ in their functional forms. Suppose that ${\cal H}^{\mathrm{L}}$
and ${\cal N}^{\mathrm{L}}$ are the variational variables corresponding to the
distribution of the enthalpy and the number of particles to liquid, which
should be uniquely determined in equilibrium as $H^{\mathrm{L}}$ and
$N^{\mathrm{L}}$. The entropy function as the variational function is given by

$\displaystyle{\mathscr{S}}({\cal H}^{\mathrm{L}},{\cal
N}^{\mathrm{L}};H,p,N)=S({\cal H}^{\mathrm{L}},p,{\cal
N}^{\mathrm{L}})+S({\cal H}^{\mathrm{G}},p,{\cal N}^{\mathrm{G}})$ (II.2)

with ${\cal H}^{\mathrm{G}}=H-{\cal H}^{\mathrm{L}}$ and ${\cal
N}^{\mathrm{G}}=N-{\cal N}^{\mathrm{L}}$. Meanwhile, the thermodynamic entropy
$S$ of the total system satisfies

$\displaystyle
S(H,p,N)=S(H^{\mathrm{L}},p,N^{\mathrm{L}})+S(H^{\mathrm{G}},p,N^{\mathrm{G}}),$
(II.3)

where $H^{\mathrm{G}}=H-H^{\mathrm{L}}$ and $N^{\mathrm{G}}=N-N^{\mathrm{L}}$.
The functional form of the right-hand side of (II.3) is the same as that of
(II.2). However, we note the difference in variables between $\mathscr{S}$ and
$S$; i.e., $\mathscr{S}$ is a five-variable function whereas $S$ is a three-
variable function. The two functions appear in different contexts.

Figure 4: Schematic figure for the entropy $\mathscr{S}$ as the variational
function. The axis of ${\cal N}^{\mathrm{L}}$ is omitted for clarity.

In equilibrium thermodynamics, thermodynamic entropy $S$ is related to the
variational function $\mathscr{S}$ as follows. Equilibrium states maximize the
entropy (variational function) $\mathscr{S}$ according to the second law of
thermodynamics, and the equilibrium states are expressed as

$\displaystyle(H^{\mathrm{L}},N^{\mathrm{L}})=\operatorname*{arg\,max}_{{\cal
H}^{\mathrm{L}},~{}{\cal N}^{\mathrm{L}}}{\mathscr{S}}({\cal
H}^{\mathrm{L}},{\cal N}^{\mathrm{L}};H,p,N).$ (II.4)

Thus, $H^{\mathrm{L}}$ and $N^{\mathrm{L}}$ are the solutions to the
simultaneous equations for ${\cal H}^{\mathrm{L}}$ and ${\cal
N}^{\mathrm{L}}$,

$\displaystyle\left.\left(\frac{\partial\mathscr{S}}{\partial{\cal
H}^{\mathrm{L}}}\right)_{N^{\mathrm{L}},H,p,N}\right|_{{\cal
H}^{\mathrm{L}}=H^{\mathrm{L}}}=0,$ (II.5)
$\displaystyle\left.\left(\frac{\partial\mathscr{S}}{\partial{\cal
N}^{\mathrm{L}}}\right)_{H^{\mathrm{L}},H,p,N}\right|_{{\cal
N}^{\mathrm{L}}=N^{\mathrm{L}}}=0,$ (II.6)

from which we identify the equilibrium state as

$\displaystyle H^{\mathrm{L}}=H^{\mathrm{L}}(H,p,N),\quad
N^{\mathrm{L}}=N^{\mathrm{L}}(H,p,N).$ (II.7)

Substituting this solution into (II.2), we obtain

$\displaystyle{\mathscr{S}}(H^{\mathrm{L}}(H,p,N),N^{\mathrm{L}}(H,p,N);H,p,N)=S(H,p,N),$
(II.8)

which means that the maximum value of the variational function $\mathscr{S}$
is equal to the thermodynamic entropy $S$. We also have

$\displaystyle S(H,p,N)=$ $\displaystyle
S(H^{\mathrm{L}}(H,p,N),p,N^{\mathrm{L}}(H,p,N))$
$\displaystyle+S(H^{\mathrm{G}}(H,p,N),p,N^{\mathrm{G}}(H,p,N))$ (II.9)

as the exact expression of (II.3). The relation of $S$ with $\mathscr{S}$ is
presented in Fig. 4.

The temperatures of the liquid and gas are defined as

$\displaystyle\frac{1}{T^{\mathrm{L}}}\equiv\left(\frac{\partial S}{\partial
H^{\mathrm{L}}}\right)_{p,N^{\mathrm{L}}},\quad\frac{1}{T^{\mathrm{G}}}\equiv\left(\frac{\partial
S}{\partial H^{\mathrm{G}}}\right)_{p,N^{\mathrm{G}}}.$ (II.10)

Substituting (II.2) with ${\cal H}^{\mathrm{G}}=H-{\cal H}^{\mathrm{L}}$ into
the variational equation (II.5), we have

$\displaystyle T^{\mathrm{L}}=T^{\mathrm{G}},$ (II.11)

which is the balance of temperature between the liquid and gas in equilibrium.
Similarly, because the chemical potentials are given by

$\displaystyle\frac{\mu^{\mathrm{L}}}{T^{\mathrm{L}}}\equiv-\left(\frac{\partial
S}{\partial
N^{\mathrm{L}}}\right)_{p,H^{\mathrm{L}}},\quad\frac{\mu^{\mathrm{G}}}{T^{\mathrm{G}}}\equiv-\left(\frac{\partial
S}{\partial N^{\mathrm{G}}}\right)_{p,H^{\mathrm{G}}},$ (II.12)

the variational equation (II.6) together with $T^{\mathrm{L}}=T^{\mathrm{G}}$
and ${\cal N}^{\mathrm{G}}=N-{\cal N}^{\mathrm{L}}$ yields

$\displaystyle\mu^{\mathrm{L}}=\mu^{\mathrm{G}},$ (II.13)

which corresponds to the balance of the chemical potential between the liquid
and gas. With these two balances, we obtain the fundamental relation of
thermodynamics for the total system as

$\displaystyle dS=\frac{1}{T}dH-\frac{V}{T}dp-\frac{\mu}{T}dN,$ (II.14)

in which we write $T=T^{\mathrm{L}}=T^{\mathrm{G}}$ and
$\mu=\mu^{\mathrm{L}}=\mu^{\mathrm{G}}$. $V=V^{\mathrm{L}}+V^{\mathrm{G}}$ is
the total volume of the system.

As we have reviewed so far, the correspondence of the two entropy functions,
$\mathscr{S}$ and $S$, is concluded from the second law of thermodynamics, and
the correspondence is a strong property of the equilibrium entropy describing
macroscopic phenomena.

##  III Setup

We focus on enthalpy-conserving systems at constant pressure in heat
conduction. See Fig. 1. $N$ particles are enclosed in a container made of
thermally insulating material except for the left and right boundaries. The
top plate of the container at the position $L_{z}$ is set freely movable to
keep the pressure constant, whereas the other five walls are fixed with
constant lengths $L_{x}$ and $L_{y}$. The volume of the system
$V=L_{x}L_{y}L_{z}$ is not fixed whereas the total enthalpy $H$ of the system
is kept constant. We assume that the transition temperature
$T_{\mathrm{c}}(p)$ is far below the critical temperature. For simplicity,
gravity effects are ignored such that the pressure is uniform over the system.

To conserve enthalpy, we control the in- and out-flow heat currents from the
left and right boundaries to be strictly $JL_{y}L_{z}$ per unit time. For
$J<0$, the liquid localizes on the left side of the system, and there is an
interface that separates the liquid from the gas. Such violation of left–right
symmetry continues to $J\rightarrow 0^{-}$. The steady state for $J>0$ is
obtained by the space-inversion of the system with $-J$ as shown in Fig. 5.
When $J$ is continuously changed from negative to positive values, local
thermodynamic quantities may jump from those of liquid to gas or vise versa at
$J=0$. This indicates the singularity at $J=0$.

Figure 5: Violation of left–right symmetry in heat conduction. The steady
state at $J<0$ is symmetric to that at $J>0$. Both mirror-image states become
stable exactly at $J=0$.

For fixed values of $H$, $p$, $N$, and $J$, the enthalpy and particle
distributions are determined uniquely. We assume that $|J|$ is so small that
each local thermodynamic quantity is uniform in the section perpendicular to
$x$, and we therefore write the local quantity as a function of $x$. This
means that the heat conduction system remains extensive in $y$ and $z$
directions even for $J\neq 0$ because the value of $L_{y}$ or $L_{z}$ does not
affect any local state. Using local temperature $T(x)$, we write the
temperatures at the two ends as

$\displaystyle T_{1}=\lim_{x\rightarrow 0^{+}}T(x),\quad
T_{2}=\lim_{x\rightarrow L_{x}^{-}}T(x).$ (III.1)

We observe liquid–gas coexistence when $T_{1}<T_{\mathrm{c}}(p)<T_{2}$ for
$J<0$, or $T_{1}>T_{\mathrm{c}}(p)>T_{2}$ for $J>0$. As a dimensionless
parameter representing the degree of nonequilibrium, we set

$\displaystyle\varepsilon=\frac{T_{2}-T_{1}}{T_{\mathrm{c}}(p)}.$ (III.2)

Suppose that the liquid–gas interface is situated at $x=X$, where the
thickness of the interface is not macroscopic and is ignored in the present
description. We express the enthalpy and the particle number of the phase in
the left (right) region of $x<X$ ($x>X$) as

$\displaystyle H^{\mathrm{\ell}}=L_{y}L_{z}\int_{0}^{X}dx~{}h(x),\quad
H^{\mathrm{r}}=L_{y}L_{z}\int_{X}^{L_{x}}dx~{}h(x),$ (III.3) $\displaystyle
N^{\mathrm{\ell}}=L_{y}L_{z}\int_{0}^{X}dx~{}\rho(x),\quad
N^{\mathrm{r}}=L_{y}L_{z}\int_{X}^{L_{x}}dx~{}\rho(x),$ (III.4)

where $h(x)$ and $\rho(x)$ denote local enthalpy density and number density.
The total enthalpy and total number of particles in the system are

$\displaystyle H=H^{\mathrm{\ell}}+H^{\mathrm{r}},\quad
N=N^{\mathrm{\ell}}+N^{\mathrm{r}}.$ (III.5)

Conventionally, one may interpret $\mathrm{\ell}=\mathrm{L}$ and
$\mathrm{r}=\mathrm{G}$ for $J<0$ and vise versa for $J>0$. The variational
variables ${\cal H}^{\mathrm{L/G}}$ and ${\cal N}^{\mathrm{L/G}}$ in Sec. II
are replaced with ${\cal H}^{\mathrm{\ell/r}}$ and ${\cal
N}^{\mathrm{\ell/r}}$.

When we know the values of $T(x)$ and $p$ and the local phase as liquid or
gas, we can identify any local thermodynamic quantity from equilibrium
thermodynamics regardless of whether the local state is stable or metastable.
For instance, when the local phase is liquid, local entropy density is written
as $s(x)=\rho^{\mathrm{L}}(T(x),p)\hat{s}^{\mathrm{L}}(T(x),p)$ with
equilibrium entropy $\hat{s}(T,p)$ per particle. The entropies for the left
(right) region are expressed as the spatial integration of the local density;
i.e.,

$\displaystyle S^{\mathrm{\ell}}=L_{y}L_{z}\int_{0}^{X}dx~{}s(x),\quad
S^{\mathrm{r}}=L_{y}L_{z}\int_{X}^{L_{x}}dx~{}s(x).$ (III.6)

These entropies are defined in heat conduction; however they are written by
the equilibrium entropy functions,

$\displaystyle
S^{\mathrm{\ell}}=S(H^{\mathrm{\ell}},p,N^{\mathrm{\ell}}),\quad
S^{\mathrm{r}}=S(H^{\mathrm{r}},p,N^{\mathrm{r}})$ (III.7)

with an error of $O(\varepsilon^{2})$, as shown in Sec.3 of global-JSP . See
Appendix A.1.

In the last of this section, we mention the treatment of the singularity at
$J=0$ associated with the left–right symmetry demonstrated in Fig. 5. When $J$
is varied continuously from negative to positive values, thermodynamic
quantities for the left region become discontinuous at $J=0$. We assume that
this discontinuity can be expressed as the limit of smooth functions of
$\varepsilon$ by introducing some regularization parameters. We set the
regularization parameters to be zero in the last step of the calculation of
the thermodynamic quantities. Here, we choose the regularization parameters in
such a way that the global thermodynamics is formulated consistently.

##  IV Problem

Our aim in this paper is to extend the maximum entropy principle to heat
conduction states. We introduce a nonequilibrium intensive variable $\phi$ and
its conjugate extensive variable $\Psi$ with the condition

$\displaystyle\phi=O(\varepsilon),\quad\Psi=O(\varepsilon^{0}).$ (IV.1)

We then seek extended forms of the entropy $S$ and variational function
${\mathscr{S}}$ that satisfy three conditions.

The first condition is that $S$ and ${\mathscr{S}}$ take the same form

$\displaystyle
S(H,p,N,\phi)=S(H^{\mathrm{\ell}},p,N^{\mathrm{\ell}})+S(H^{\mathrm{r}},p,N^{\mathrm{r}})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\phi\Psi(H,p,N,\phi),$
(IV.2) $\displaystyle{\mathscr{S}}({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N,\phi)=S({\cal H}^{\mathrm{\ell}},p,{\cal
N}^{\mathrm{\ell}})$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+S({\cal
H}^{\mathrm{r}},p,{\cal N}^{\mathrm{r}})+\phi{\psi}({\cal
H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}};H,p,N),$ (IV.3)

where the function $\psi$ does not depend on $\phi$ explicitly.

The second condition for $S$ and ${\mathscr{S}}$ is that the thermodynamic
entropy $S$ corresponds to the maximum value of the variational function
$\mathscr{S}$; i.e.,

$\displaystyle S(H,p,N,\phi)$
$\displaystyle~{}={\mathscr{S}}(H^{\mathrm{\ell}}(H,p,N,\phi),N^{\mathrm{\ell}}(H,p,N,\phi);H,p,N,\phi),$
(IV.4)

where the function $\psi$ is connected to $\Psi$ as

$\displaystyle\Psi(H,p,N,\phi)=\psi(H^{\mathrm{\ell}},N^{\mathrm{\ell}};H,p,N)$
(IV.5)

with the steady state values

$\displaystyle H^{\mathrm{\ell}}=H^{\mathrm{\ell}}(H,p,N,\phi),\quad
N^{\mathrm{\ell}}=N^{\mathrm{\ell}}(H,p,N,\phi).$ (IV.6)

We assume that the steady state maximizes $\mathscr{S}$. Then,
$H^{\mathrm{\ell}}$ and $N^{\mathrm{\ell}}$ are determined as

$\displaystyle(H^{\mathrm{\ell}},N^{\mathrm{\ell}})=\operatorname*{arg\,max}_{{\cal
H}^{\mathrm{\ell}},~{}{\cal N}^{\mathrm{\ell}}}{\mathscr{S}}({\cal
H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}};H,p,N,\phi).$ (IV.7)

The third condition for $S$ and ${\mathscr{S}}$ is that $S(H,p,N,\phi)$
satisfies the fundamental relation of thermodynamics

$\displaystyle
dS=\frac{1}{{\tilde{T}}}dH-\frac{V}{{\tilde{T}}}dp-\frac{\tilde{\mu}}{{\tilde{T}}}dN+\Psi
d\phi,$ (IV.8)

where ${\tilde{T}}$ is the global temperature defined by

$\displaystyle{\tilde{T}}=\frac{\int_{0}^{L_{x}}dx~{}\rho(x)T(x)}{\int_{0}^{L_{x}}dx~{}\rho(x)},$
(IV.9)

which was introduced in global-PRL . We also define the global chemical
potential as

$\displaystyle\tilde{\mu}=\frac{\int_{0}^{L_{x}}dx~{}\rho(x)\mu(x)}{\int_{0}^{L_{x}}dx~{}\rho(x)},$
(IV.10)

where $\mu(x)$ is the local chemical potential. We emphasize that, once we
decide to adopt ${\tilde{T}}$ as a temperature to characterize the heat
conduction states, $\tilde{\mu}$ is concluded to be a unique extension of the
equilibrium chemical potential. See Sec. 7 of global-JSP .

Summarizing the above, we attempt to determine an entropy extended to
liquid–gas coexistence in heat conduction using simultaneous equations,
specifically two equations from the variational principle (IV.7),

$\displaystyle\left.\left(\frac{\partial\mathscr{S}}{\partial{\cal
H}^{\mathrm{\ell}}}\right)_{N^{\mathrm{\ell}},H,p,N,\phi}\right|_{{\cal
H}^{\mathrm{\ell}}=H^{\mathrm{\ell}}}=0,$ (IV.11)
$\displaystyle\left.\left(\frac{\partial\mathscr{S}}{\partial{\cal
N}^{\mathrm{\ell}}}\right)_{H^{\mathrm{\ell}},H,p,N,\phi}\right|_{{\cal
N}^{\mathrm{\ell}}=N^{\mathrm{\ell}}}=0,$ (IV.12)

and four equations from the fundamental relation (IV.8),

$\displaystyle\left(\frac{\partial S}{\partial
H}\right)_{p,N,\phi}=\frac{1}{{\tilde{T}}},$ (IV.13)
$\displaystyle\left(\frac{\partial S}{\partial
p}\right)_{H,N,\phi}=-\frac{V}{{\tilde{T}}},$ (IV.14)
$\displaystyle\left(\frac{\partial S}{\partial
N}\right)_{H,p,\phi}=-\frac{\tilde{\mu}}{{\tilde{T}}},$ (IV.15)
$\displaystyle\left(\frac{\partial S}{\partial\phi}\right)_{H,p,N}=\Psi.$
(IV.16)

##  V Results

Solving the equations (IV.11), (IV.12), (IV.13), (IV.14), (IV.15), and (IV.16)
under the conditions described in Sec. III, we determine the functional form
of $\phi$ and $\psi$ as

$\displaystyle\phi=\varepsilon$ (V.1)

and

$\displaystyle\psi({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N)=\frac{{\cal H}^{\mathrm{r}}{\cal
N}^{\mathrm{\ell}}-{\cal H}^{\mathrm{\ell}}{\cal
N}^{\mathrm{r}}}{2NT_{\mathrm{c}}(p)}$ (V.2)

up to a multiplicative constant. This multiplicative constant does not affect
any quantitative prediction and we thus set it as a specific value as
discussed in Appendix B.4

The steady-state value of $\psi$ is expressed as

$\displaystyle\Psi(H,p,N,\phi)=\frac{\phi}{|\phi|}\frac{\hat{q}(p)}{2T_{\mathrm{c}}(p)}\frac{N^{\mathrm{L}}N^{\mathrm{G}}}{N}$
(V.3)

using (IV.5) and (V.2), where $N^{\mathrm{L/G}}$ is the steady number of
particles in the liquid and gas and we have used the relation for the steady
state value of enthalpy as
$H^{\mathrm{L/G}}=\hat{h}_{\mathrm{c}}^{\mathrm{L/G}}(p)N^{\mathrm{L/G}}+O(\varepsilon)$.
The expression (V.3) clarifies that the new quantity $\Psi$ is associated with
the phase coexistence because it is connected to the latent heat $\hat{q}(p)$.
Meanwhile, $\phi$ is a nonequilibrium variable that is never associated with
equilibrium properties. Moreover, we emphasize that $\phi$ is uniquely
determined by (V.1) as the difference in temperature at the two boundaries and
not as the steady heat current $J$ although $J$ is an operational parameter
that controls the enthalpy conservation in heat conduction.

Using the variational function (IV.3) with (V.1) and (V.2), the variational
equations (IV.11) and (IV.12) are expressed as

$\displaystyle\frac{1}{{\tilde{T}}^{\mathrm{\ell}}}-\frac{1}{{\tilde{T}}^{\mathrm{r}}}=\frac{\phi}{2T_{\mathrm{c}}(p)},$
(V.4)
$\displaystyle\frac{\tilde{\mu}^{\mathrm{\ell}}}{{\tilde{T}}^{\mathrm{\ell}}}-\frac{\tilde{\mu}^{\mathrm{r}}}{{\tilde{T}}^{\mathrm{r}}}=\frac{\phi}{2T_{\mathrm{c}}(p)}\frac{H}{N}.$
(V.5)

Here, $(\mathrm{\ell},\mathrm{r})=(L,G)$ for $\varepsilon>0$ and
$(\mathrm{\ell},\mathrm{r})=(G,L)$ for $\varepsilon<0$, where
${\tilde{T}}^{\mathrm{L}}$ and ${\tilde{T}}^{\mathrm{G}}$ are global
temperatures for the liquid and gas, respectively, and
$\tilde{\mu}^{\mathrm{L}}$ and $\tilde{\mu}^{\mathrm{G}}$ are the respective
global chemical potentials. (V.4) and (V.5) at $\varepsilon=0$ correspond to
the equilibrium balances of temperature and chemical potential between the
liquid and gas. Thus, these relations at $\varepsilon\neq 0$ provide the
first-order correction of the balances. Indeed, (V.4) is transformed as

$\displaystyle{\tilde{T}}^{\mathrm{r}}-{\tilde{T}}^{\mathrm{\ell}}=\frac{\phi}{2}T_{\mathrm{c}}(p)+O(\varepsilon^{2}),$
(V.6)

which leads to (III.2) with $\phi=\varepsilon$ as derived from (A.18).

We emphasize that (V.4) and (V.5) are sufficient to identify steady states. As
we will demonstrate in Sec. VII, (V.4) and (V.5) lead to all thermodynamic
quantities for the liquid–gas coexistence in heat conduction. For instance,
(VII.2) and (VII.14) provide experimentally accessible relations for the
steady states, which may act as the check of the validity of the present
theory. We will also see that $\theta$ is not necessarily equal to
$T_{\mathrm{c}}(p)$ because metastable states, such as super-cooled gas or
super-heated liquid, may be stably observed in heat conduction. Moreover, the
steady states are consistent with those determined by the variational
principle for heat conduction systems attached to two heat baths of $T_{1}$
and $T_{2}$ global-JSP . See Sec. VI for the correspondence between the
enthalpy-conserving and nonconserving systems.

##  VI Thermodynamic equivalence of steady states in heat conduction

The thermodynamic relations of equilibrium systems are independent of whether
the system is adiabatic or isothermal and whether the system is at constant
pressure or constant volume. Remarkably, the equivalence between the
fundamental thermodynamic relations can be formulated using the Legendre
transformation. We then ask whether there exist similar relations among
different formulations for heat conduction systems. See Table 1. Specifically,
we study the three cases presented in Fig. 6. So far, we have studied enthalpy
conserving systems at constant pressure in heat conduction, which is shown in
the top panel. In the second panel, we attach a heat bath of $T_{1}$ at the
left wall and a heat bath of $T_{2}$ at the right wall, where $T_{1}\leq
T_{2}$ is assumed without loss of generality. Below, we assume that there are
no temperature gaps between the system and the heat baths (i.e.,
$T(0^{+})=T_{1}$ and $T(L_{x}^{-})=T_{2}$) in steady states. In the bottom
panel of Fig. 6, the system is kept at constant volume instead of at constant
pressure. While the fundamental thermodynamic relations and the variational
principles for the systems of constant pressure or constant volume were
investigated in global-JSP , we reformulate them from the results in the
previous section.

| Variational function | Thermodynamic relation
---|---|---
| constant enthalpy
---
$(H,p,N,\phi)$
${\mathscr{S}}=L_{y}L_{z}\int_{0}^{L_{x}}s(x)~{}dx+\phi\psi$ |  ${\displaystyle dS=\frac{dH}{{\tilde{T}}}-\frac{V}{{\tilde{T}}}dp-\frac{\tilde{\mu}}{{\tilde{T}}}dN+\Psi d\phi}$
| constant pressure
---
$({\tilde{T}},p,N,\phi)$
${\cal G}=L_{y}L_{z}\int_{0}^{L_{x}}\mu(x)\rho(x)~{}dx$ |  ${\displaystyle dG=-Sd{\tilde{T}}+{V}dp+{\tilde{\mu}}dN-{\tilde{T}}\Psi d\phi}$
| constant volume
---
$({\tilde{T}},V,N,\phi)$
${\cal F}=L_{y}L_{z}\int_{0}^{L_{x}}f(x)~{}dx$ |  ${\displaystyle dF=-Sd{\tilde{T}}-pdV+{\tilde{\mu}}dN-{\tilde{T}}\Psi d\phi}$
Table 1: Summary of variational functions and fundamental relations of
thermodynamics. $S$ is the maximum value of $\mathscr{S}$ whereas $G$ and $F$
are the minimum values of $\cal G$ and $\cal F$, respectively. The fundamental
relations are associated with one another via a Legendre transformation.
Figure 6: Thermodynamic equivalence of heat conduction systems.

###  VI.1 Thermodynamic relations

We define the Gibbs free energy in heat conduction as

$\displaystyle G=L_{y}L_{z}\int_{0}^{L_{x}}dx~{}\mu(x)\rho(x).$ (VI.1)

The entropy $S$ given in (IV.2) with $\phi=\varepsilon$ and $\Psi$ as (V.3)
satisfies

$\displaystyle S=\frac{H-G}{{\tilde{T}}}+O(\varepsilon^{2})$ (VI.2)

with the global temperature ${\tilde{T}}$ in (IV.9). See Appendix C for the
derivation of (VI.2). Substituting (VI.2) into (IV.8), we obtain

$\displaystyle dG=-Sd{\tilde{T}}+Vdp+\tilde{\mu}dN-{\tilde{T}}\Psi d\phi.$
(VI.3)

From this relation, we find that the functional form

$\displaystyle G=G({\tilde{T}},p,N,\phi)$ (VI.4)

is useful. Because the global Helmholtz free energy

$\displaystyle F=L_{y}L_{z}\int_{0}^{L_{x}}dx~{}f(x)$ (VI.5)

satisfies

$\displaystyle F=G-pV,$ (VI.6)

(VI.3) is transformed into the fundamental relation

$\displaystyle dF=-Sd{\tilde{T}}-pdV+\tilde{\mu}dN-{\tilde{T}}\Psi d\phi,$
(VI.7)

which leads to the expression

$\displaystyle F=F({\tilde{T}},V,N,\phi).$ (VI.8)

Thus, the fundamental relations of global thermodynamics are formed
consistently among $(H,p,N,\phi)$, $({\tilde{T}},p,N,\phi)$, and
$({\tilde{T}},V,N,\phi)$. The transformation into $(U,V,N,\phi)$ is also
straightforward. These transformations are nothing but the Legendre
transitions. See Appendix E for further arguments on the additivity of $G$ and
$F$.

We here present a remark. The fundamental relations of global thermodynamics
derived in global-JSP  were expressed in slightly different forms from (VI.3)
and (VI.7); i.e.,

$\displaystyle
dG=-(S^{\mathrm{L}}+S^{\mathrm{G}})d{\tilde{T}}+Vdp+\tilde{\mu}dN-\Psi d\Xi,$
(VI.9) $\displaystyle
dF=-(S^{\mathrm{L}}+S^{\mathrm{G}})d{\tilde{T}}-pdV+\tilde{\mu}dN-\Psi d\Xi,$
(VI.10)

where $\Xi=T_{2}-T_{1}$. (VI.9) and (VI.10) are obtained by substituting
$\phi=\Xi/{\tilde{T}}$ into (VI.3) and (VI.7). While the expression (VI.9) is
equivalent to (VI.3), the former implies the extension of the Gibbs free
energy in the form $G=G({\tilde{T}},p,N,\Xi)$, in which the nonequilibrium
variable is not $\phi$ but $\Xi$. As long as we discuss thermodynamic
properties for the systems attached to heat baths, both the relations work
consistently. However, the connection to the thermodynamic relation in
enthalpy-conserving systems is more clearly seen in (VI.3).

###  VI.2 Variational principles

We first explain the variational principle corresponding to the minimization
of the Gibbs free energy $\cal G$. In equilibrium, this minimization is
performed by fixing $(T,p,N)$. The extension of this variational principle to
heat conduction is not straightforward because the temperature profile cannot
be given as an external parameter of the setup. Nevertheless, referring to the
thermodynamic relation (VI.3), we formally adopt the fixed condition as
$({\tilde{T}},p,N,\phi)$. To fix ${\tilde{T}}$ and $\phi$ in the variation at
constant $p$, we shift the two temperatures by the same amount while fixing
$T_{2}-T_{1}(=\Xi)$ and find heuristically the shift required to keep the
right-hand side of (IV.9) as ${\tilde{T}}$. Thus, fixing
$({\tilde{T}},p,N,\phi)$ is the same as fixing $({\tilde{T}},p,N,\Xi)$, whose
variational principle and the steady states given by it have already been
investigated in global-JSP .

In brief, taking a variational function as a natural extension from
equilibrium,

$\displaystyle{\cal G}=L_{y}L_{z}\int_{0}^{L_{x}}dx~{}\mu(x)\rho(x),$ (VI.11)

we obtain

$\displaystyle{\cal G}({\cal
N}^{\mathrm{L}};{\tilde{T}},p,N,\phi)=G^{\mathrm{L}}({\cal
T}^{\mathrm{L}}({\cal N}^{\mathrm{L}};{\tilde{T}},N,\phi),p,{\cal
N}^{\mathrm{L}})$ $\displaystyle~{}~{}~{}~{}~{}+G^{\mathrm{G}}({\cal
T}^{\mathrm{G}}({\cal N}^{\mathrm{L}};{\tilde{T}},N,\phi),p,N-{\cal
N}^{\mathrm{L}}),$ (VI.12)

where $G^{\mathrm{L}}({\cal T}^{\mathrm{L}},p,{\cal
N}^{\mathrm{L}})=\mu^{\mathrm{L}}({\cal T}^{\mathrm{L}},p){\cal
N}^{\mathrm{L}}$ and $G^{\mathrm{G}}({\cal T}^{\mathrm{G}},p,{\cal
N}^{\mathrm{G}})=\mu^{\mathrm{G}}({\cal T}^{\mathrm{G}},p)(N-{\cal
N}^{\mathrm{G}})$. ${\cal T}^{\mathrm{L}}$ and ${\cal T}^{\mathrm{G}}$ are the
global temperatures for the liquid and gas, which are considered as
variational quantities because they depend on ${\cal N}^{\mathrm{L}}$
according to

$\displaystyle{\cal T}^{\mathrm{L}}={\tilde{T}}\left(1-\phi\frac{N-{\cal
N}^{\mathrm{L}}}{2N}\right),\quad{\cal
T}^{\mathrm{G}}={\tilde{T}}\left(1+\phi\frac{{\cal
N}^{\mathrm{L}}}{2N}\right).$ (VI.13)

(VI.13) results from the definition of ${\tilde{T}}$ and $\phi$ with
${\tilde{T}}=T_{\mathrm{c}}(p)+O(\varepsilon)$ assuming that local temperature
is continuous everywhere, even at the liquid–gas interface. See the discussion
around (A.19).

The steady state is specified by determining the distribution of particles to
liquid as

$\displaystyle N^{\mathrm{L}}=\operatorname*{arg\,min}_{{\cal
N}^{\mathrm{L}}}{\cal G}({\cal N}^{\mathrm{L}};{\tilde{T}},p,N,\phi),$ (VI.14)

which is rewritten as

$\displaystyle\left.\left(\frac{\partial\cal G}{\partial{\cal
N}^{\mathrm{L}}}\right)_{{\tilde{T}},p,N,\phi}\right|_{{\cal
N}^{\mathrm{L}}=N^{\mathrm{L}}}=0.$ (VI.15)

Using (VI.12) together with (VI.13), the variational equation (VI.15) leads to

$\displaystyle\mu^{\mathrm{L}}-\mu^{\mathrm{G}}=\phi{\tilde{T}}\frac{S^{\mathrm{L}}+S^{\mathrm{G}}}{2N},$
(VI.16)

where we applied the fundamental relation of global thermodynamics, which is
briefly reviewed in Appendix A.1. We show the equivalence between (V.5) and
(VI.16) in Appendix D.

We next deal with constant-volume systems in heat conduction. We assume that
the steady state for a given $({\tilde{T}},V,N,\phi)$ is obtained as the state
minimizing the total Helmholtz free energy of the system,

$\displaystyle{\cal F}=L_{y}L_{z}\int_{0}^{L_{x}}dx~{}f(x).$ (VI.17)

More precisely, we formulate the variational function as

$\displaystyle{\cal F}({\cal V}^{\mathrm{L}},{\cal
N}^{\mathrm{L}};{\tilde{T}},V,N,\phi)=F({\cal T}^{\mathrm{L}}({\cal
N}^{\mathrm{L}};{\tilde{T}},N,\phi),{\cal V}^{\mathrm{L}},{\cal
N}^{\mathrm{L}})$ $\displaystyle~{}~{}~{}~{}~{}+F({\cal T}^{\mathrm{G}}({\cal
N}^{\mathrm{L}};{\tilde{T}},N,\phi),V-{\cal V}^{\mathrm{L}},N-{\cal
N}^{\mathrm{L}}),$ (VI.18)

where $\cal T^{\mathrm{L}}$ and $\cal T^{\mathrm{G}}$ are functions given by
(VI.13). The steady state is obtained as

$\displaystyle(V^{\mathrm{L}},N^{\mathrm{L}})=\operatorname*{arg\,min}_{{\cal
V}^{\mathrm{L}},~{}{\cal N}^{\mathrm{L}}}{\cal F}({\cal V}^{\mathrm{L}},{\cal
N}^{\mathrm{L}};{\tilde{T}},V,N,\phi).$ (VI.19)

That is to say,

$\displaystyle\left.\left(\frac{\partial\cal F}{\partial{\cal
V}^{\mathrm{L}}}\right)_{{\tilde{T}},V,N,\phi,N^{\mathrm{L}}}\right|_{{\cal
V}^{\mathrm{L}}=V^{\mathrm{L}}}=0,$ (VI.20)
$\displaystyle\left.\left(\frac{\partial\cal F}{\partial{\cal
N}^{\mathrm{L}}}\right)_{{\tilde{T}},V,N,\phi,V^{\mathrm{L}}}\right|_{{\cal
N}^{\mathrm{L}}=N^{\mathrm{L}}}=0.$ (VI.21)

The first equation (VI.20) leads to

$\displaystyle p^{\mathrm{L}}=p^{\mathrm{G}}$ (VI.22)

corresponding to the balance of pressure in the liquid and gas. The second
equation (VI.21) yields (VI.16).

In summary, the minimization of free energy $\cal G$ with
$({\tilde{T}},p,N,\phi)$ fixed and the minimization of $\cal F$ with
$({\tilde{T}},V,N,\phi)$ fixed provide the same results as the maximization of
entropy $\mathscr{S}$ with $(H,p,N,\phi)$ fixed; i.e., the steady states are
equivalent among the three.

##  VII Steady states

Figure 7:  Schematic figures comparing a heat conduction state with an
equilibrium state. (a) Equilibrium state at $J\rightarrow 0^{-}$. Local
temperature is $T_{\mathrm{c}}(p)$ at any $x$. (b) Heat conduction state with
$J<0$ and the temperature profile $T(x)$. Super-cooled gas occupies a rather
wide region.

This section summarizes the thermodynamic quantities in the steady heat
conduction state, which result from the variational principle. The derivations
of the formulas are presented in Appedix F.

In Sec. VII.1, we refer to the expression of the temperature at the liquid–gas
interface, which was derived in global-JSP . We demonstrate various quantities
in the steady state of the enthalpy-conserving systems with $J\neq 0$ as
slight changes from corresponding equilibrium quantities in Sec. VII.2. Such
perturbative treatment is difficult to apply to systems in contact with two
heat baths at constant pressure because the equilibrium position of the
liquid–gas interface is not unique at $T_{\mathrm{c}}(p)$, whereas the liquid
and gas do not coexist at $T\neq T_{\mathrm{c}}(p)$. This indicates that we
cannot formulate a naive perturbative expansion in heat conduction near
equilibrium. The thermodynamic equivalence holds in steady states as discussed
in Sec. VI, and we may therefore transform the formulas for the steady state
in $(H,p,N,\phi)$ into $({\tilde{T}},p,N,\phi)$. See Sec. VII.3. Below, we
omit the dependence of $p$ for the equilibrium quantities at the transition
temperature.

When we set $J$ as a parameter, the nonequilibrium variable
$\phi=(T_{2}-T_{1})/T_{\mathrm{c}}$ is determined as

$\displaystyle\phi=-J\frac{L_{x}}{T_{\mathrm{c}}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}\kappa_{\mathrm{c}}^{\mathrm{G}}}\frac{\kappa_{\mathrm{c}}^{\mathrm{L}}\rho_{\mathrm{c}}^{\mathrm{L}}(\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}})+\kappa_{\mathrm{c}}^{\mathrm{G}}\rho_{\mathrm{c}}^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}{\rho_{\mathrm{c}}^{\mathrm{L}}(\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}})+\rho_{\mathrm{c}}^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}+O(\varepsilon^{2}),$
(VII.1)

where $\kappa_{\mathrm{c}}^{\mathrm{L/G}}$ is heat conductivity. Note that
(VII.1) holds regardless of the sign of $J$ and $\phi J\leq 0$. In the
estimates below, we use $\phi$ as it means (VII.1).

A simple sketch of the steady state is presented in Fig. 7 for $J<0$ in
comparison with equilibrium liquid–gas coexistence at $J\rightarrow 0^{-}$.
There is a region occupied by super-cooled gas on the adjacent to the
liquid–gas interface. When we take $J>0$, the steady state is obtained as the
mirror image of $J<0$ as shown in Fig. 5, where the physical properties are
equivalent to those for $J<0$. Thus, all steady-state properties should be
expressed by functions of $|J|$ or $|\phi|$.

###  VII.1 Temperature at the liquid–gas interface

From the variational principle (V.4) and (V.5), the temperature $\theta$ at
the interface is formulated as

$\displaystyle\theta={T_{\mathrm{c}}}-\frac{X(L_{x}-X)}{2L_{x}}\left[|J|\left(\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}-\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}\right)+\frac{|\phi|T_{\mathrm{c}}}{L_{x}}\frac{\rho_{\mathrm{c}}^{\mathrm{L}}-\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}\right],$
(VII.2)

in which $\bar{\rho}=N/V$ and $X$ is the width of the liquid region as
reported in global-JSP . Another form of (VII.2) is

$\displaystyle\theta=T_{\mathrm{c}}-\frac{X(L_{x}-X)}{2L_{x}}|J|\left(\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{\rho_{\mathrm{c}}^{\mathrm{L}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}\right)$
(VII.3)

as shown in Appendix F.7. (VII.2) indicates that $\theta$ becomes lower than
$T_{\mathrm{c}}$ in the linear response regime. Moreover, the absolute values
$|J|$ and $|\phi|$ in (VII.2) imply that the true equilibrium state with a
vanishing current $J=0$ is singular. It is noted that $\theta$ may deviate
from $T_{c}$ in experiments, as has been quantitatively discussed for a
specific material such as water global-JSP .

###  VII.2 Steady-state expressions as functions of $(H,p,N,\phi)$

From the variational equations (V.4) and (V.5), we identify the width of the
liquid $X=X(H,p,N,\phi)$, the volume $V=V(H,p,N,\phi)$, the number of
particles $N^{\mathrm{L}}=N^{\mathrm{L}}(H,p,N,\phi)$, and the enthalpy
$H^{\mathrm{L}}=H^{\mathrm{L}}(H,p,N,\phi)$. The following estimates have
precision of $O(\varepsilon)$ and error of $O(\varepsilon^{2})$. The detailed
derivation for each estimate is presented in Appendix F.

In the equilibrium limit $|J|\rightarrow 0$, the interface corresponds to the
Gibbs dividing surface. For a given $(H,p,N)$, we obtain the numbers of
particles as

$\displaystyle\frac{N_{\mathrm{eq}}^{\mathrm{L}}}{N}=\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}},\quad\frac{N_{\mathrm{eq}}^{\mathrm{G}}}{N}=\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}},$
(VII.4)

the respective volumes as

$\displaystyle
V_{\mathrm{eq}}^{\mathrm{L}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{L}}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}},\quad
V_{\mathrm{eq}}^{\mathrm{G}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{G}}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}},$
(VII.5)

and the width of the liquid as

$\displaystyle\frac{X_{\mathrm{eq}}}{L_{x}}=\frac{\rho_{\mathrm{c}}^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}{\rho_{\mathrm{c}}^{\mathrm{L}}(\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}})+\rho^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}.$
(VII.6)

Imposing the heat current $J$, the distributions of conserved quantities to
each bulk deviate as follows. Letting $\hat{c}_{p}^{\mathrm{L/G}}$ be the
specific heat at constant pressure, the numbers of particles are

$\displaystyle\frac{N^{\mathrm{L}}-N_{\mathrm{eq}}^{\mathrm{L}}}{N}=-|\phi|\frac{T_{\mathrm{c}}}{2\hat{q}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right],$
(VII.7)
$\displaystyle\frac{N^{\mathrm{G}}-N_{\mathrm{eq}}^{\mathrm{G}}}{N}=|\phi|\frac{T_{\mathrm{c}}}{2\hat{q}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right],$
(VII.8)

the enthalpies per particle are

$\displaystyle\hat{h}^{\mathrm{L}}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}=-|\phi|\frac{T_{\mathrm{c}}}{2}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\hat{c}_{p}^{\mathrm{L}},$
(VII.9)
$\displaystyle\hat{h}^{\mathrm{G}}-\hat{h}_{\mathrm{c}}^{\mathrm{G}}=|\phi|\frac{T_{\mathrm{c}}}{2}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\hat{c}_{p}^{\mathrm{G}},$
(VII.10)

and the volumes are

$\displaystyle
V^{\mathrm{L}}-V_{\mathrm{eq}}^{\mathrm{L}}=-|\phi|\frac{T_{\mathrm{c}}}{2}\frac{N}{\rho_{\mathrm{c}}^{\mathrm{L}}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\alpha_{\mathrm{c}}^{\mathrm{L}}\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+\left(\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\frac{\hat{c}_{p}}{\hat{q}}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\right)\right],$
(VII.11) $\displaystyle
V^{\mathrm{G}}-V_{\mathrm{eq}}^{\mathrm{G}}=|\phi|\frac{T_{\mathrm{c}}}{2}\frac{N}{\rho_{\mathrm{c}}^{\mathrm{G}}}\left[\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\alpha_{\mathrm{c}}^{\mathrm{G}}\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+\left(\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\right)\right],$
(VII.12)

where $\alpha_{\mathrm{c}}^{\mathrm{L}}$ and
$\alpha_{\mathrm{c}}^{\mathrm{G}}$ are thermal expansion coefficients at
constant pressure for saturated liquid and gas. The scaled width of the liquid
region, $x\equiv X/L_{x}$, changes as

$\displaystyle
x-x_{\mathrm{eq}}=-|\phi|\frac{T_{\mathrm{c}}}{2}x_{\mathrm{eq}}(1-x_{\mathrm{eq}})\left[\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\alpha_{\mathrm{c}}^{\mathrm{G}}+\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\alpha_{\mathrm{c}}^{\mathrm{L}}\right.$
$\displaystyle~{}~{}\left.+\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}-\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\right].$
(VII.13)

Using (VII.6), (VII.13) is reformulated as

$\displaystyle
x-x_{\mathrm{eq}}=-|\phi|\frac{T_{\mathrm{c}}}{2}x_{\mathrm{eq}}(1-x_{\mathrm{eq}})\frac{\alpha_{\mathrm{c}}^{\mathrm{G}}\rho_{\mathrm{c}}^{\mathrm{G}}(1-x_{\mathrm{eq}})+\alpha_{\mathrm{c}}^{\mathrm{L}}\rho_{\mathrm{c}}^{\mathrm{L}}x_{\mathrm{eq}}}{\rho_{\mathrm{c}}^{\mathrm{G}}(1-x_{\mathrm{eq}})+\rho_{\mathrm{c}}^{\mathrm{L}}x_{\mathrm{eq}}}$
$\displaystyle-|\phi|\frac{T_{\mathrm{c}}}{2}\left[\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}\frac{\rho_{\mathrm{c}}^{\mathrm{L}}}{\rho_{\mathrm{c}}^{\mathrm{G}}}x_{\mathrm{eq}}^{2}-\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\rho_{\mathrm{c}}^{\mathrm{L}}}(1-x_{\mathrm{eq}})^{2}\right].$
(VII.14)

When the liquid and gas satisfy
$\rho_{\mathrm{c}}^{\mathrm{L}}\gg\rho_{\mathrm{c}}^{\mathrm{G}}$ and
$\alpha_{\mathrm{c}}^{\mathrm{L}}\ll 1$, which are usually satisfied when
$T_{\mathrm{c}}$ is much lower than the critical temperature, formula (VII.14)
can be simplified as

$\displaystyle
x-x_{\mathrm{eq}}\simeq-|\phi|\frac{T_{\mathrm{c}}}{2}\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}\frac{\rho_{\mathrm{c}}^{\mathrm{L}}}{\rho_{\mathrm{c}}^{\mathrm{G}}}x_{\mathrm{eq}}^{2}.$
(VII.15)

Because $\rho_{\mathrm{c}}^{\mathrm{L}}\gg\rho_{\mathrm{c}}^{\mathrm{G}}$, the
interface may shift drastically with a violent increase in volume when
imposing a slight heat current while maintaining the amount of enthalpy.

###  VII.3 Steady-state expressions as functions of $({\tilde{T}},p,N,\phi)$

Owing to the thermodynamic equivalence, the above estimates can be translated
to functions of $({\tilde{T}},p,N,\phi)$ once specifying the global
temperature as ${\tilde{T}}(H,p,N,\phi)$ and its inverse function as
$H({\tilde{T}},p,N,\phi)$.

As derived in Appendix F.2, the global temperature for a given $(H,p,N,\phi)$
is written as

$\displaystyle{\tilde{T}}=T_{\mathrm{c}}+|\phi|\frac{T_{\mathrm{c}}}{\hat{q}}\left(\hat{h}-\frac{\hat{h}_{\mathrm{c}}^{\mathrm{L}}+\hat{h}_{\mathrm{c}}^{\mathrm{G}}}{2}\right),$
(VII.16)

which has two limiting cases of ${\tilde{T}}\rightarrow
T_{\mathrm{c}}(1-|\phi|/2)$ for
$\hat{h}\rightarrow\hat{h}_{\mathrm{c}}^{\mathrm{L}}$ and
${\tilde{T}}\rightarrow T_{\mathrm{c}}(1+|\phi|/2)$ for
$\hat{h}\rightarrow\hat{h}_{\mathrm{c}}^{\mathrm{G}}$. The former limit
indicates that $T_{\mathrm{c}}(1-|\phi|)<T(x)<T_{\mathrm{c}}$, where the
system is almost completely occupied by liquid with $x\rightarrow 1$. The
latter limit indicates that $T_{\mathrm{c}}<T(x)<T_{\mathrm{c}}(1+|\phi|)$
(i.e., $x\rightarrow 0$), where the system is occupied by gas.

Solving (VII.16), we obtain $\hat{h}$ as

$\displaystyle\hat{h}=\frac{\hat{h}_{\mathrm{c}}^{\mathrm{L}}+\hat{h}_{\mathrm{c}}^{\mathrm{G}}}{2}+\frac{\hat{q}}{|\phi|}\frac{{\tilde{T}}-T_{\mathrm{c}}}{T_{\mathrm{c}}}+O(\varepsilon),$
(VII.17)

which is a function of $({\tilde{T}},p,\phi)$. Note that the relation (VII.17)
does not provide the value of $\hat{h}$ out of equilibrium owing to the error
term of $O(\varepsilon)$. However, compared with the equilibrium liquid–gas
coexistence at $T_{\mathrm{c}}$ at which $\hat{h}$ can take an arbitrary value
in
$\hat{h}_{\mathrm{c}}^{\mathrm{L}}<\hat{h}<\hat{h}_{\mathrm{c}}^{\mathrm{G}}$,
(VII.17) specifies a value of $\hat{h}$ uniquely in the equilibrium limit
$|\phi|\rightarrow 0$ once we set ${\tilde{T}}$ as
${\tilde{T}}-T_{\mathrm{c}}=r|\phi|T_{\mathrm{c}}$ with a constant $r$; i.e.,
$\hat{h}$ is expressed as a function of $(p,r)$. For instance, for the system
with $\rho_{\mathrm{c}}^{\mathrm{L}}\gg\rho_{\mathrm{c}}^{\mathrm{G}}$,
(VII.6) can be rewritten as

$\displaystyle
x_{\mathrm{eq}}=\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\rho_{\mathrm{c}}^{\mathrm{L}}}~{}\lim_{\phi\rightarrow
0}\frac{|\phi|T_{\mathrm{c}}-2({\tilde{T}}-T_{\mathrm{c}})}{|\phi|T_{\mathrm{c}}+2({\tilde{T}}-T_{\mathrm{c}})}=\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\rho_{\mathrm{c}}^{\mathrm{L}}}\frac{1-2r}{1+2r}.$
(VII.18)

From (VII.15), we estimate the deviation of the interface as slight as
$x-x_{\mathrm{eq}}\sim|\phi|\rho_{\mathrm{c}}^{\mathrm{G}}/\rho_{\mathrm{c}}^{\mathrm{L}}$
except for ${\tilde{T}}\rightarrow T_{\mathrm{c}}(1-|\phi|/2)$ as
$\hat{h}\rightarrow\hat{h}_{\mathrm{c}}^{\mathrm{L}}$. There, the system is
almost occupied by liquid as $x_{\mathrm{eq}}\rightarrow 1$, and the
liquid–gas interface may jump with a sudden increase in the volume once a
temperature difference between the two heat baths is imposed.

##  VIII Concluding Remarks

Figure 8: Schematic explanation for the violation of additivity in heat
conduction. If there is a region occupied by super-cooled gas, the local
states are not kept in the separation of the system.

We derived a unique extension of the entropy function to heat conduction. The
extension works as a variational function whose maximum gives the steady state
and it maintains the fundamental thermodynamic relation by adopting the global
temperature as the temperature of the whole system. The different fundamental
thermodynamic relations are connected by the Legendre transformation. See
Table 1 for the overall view of the thermodynamic structure of the global
thermodynamics. The steady state is equivalent between the energy-conserving
heat conduction system imposing a constant heat current and the standard heat
conduction system in contact with two heat baths of different temperature,
which suggests the robustness of the steady state and the firm structure of
the global thermodynamics.

The extended entropy is not an additive quantity but comprises the spatial
integration of local entropy density and an extra term associated with phase
coexistence. Recalling that the additivity of extensive quantities is
intrinsic in equilibrium thermodynamics, such non-additivity is a
manifestation of nonequilibrium.

Physically, the non-additivity may be related to the existence of the
metastable state, which appears in a macroscopic region adjacent to the
interface. As a thought experiment, we consider the separation of the system
into three parts as shown in Fig. 8, where the same heat current as for the
original system is imposed on the three parts. Immediately after the
separation, the middle part is occupied by super-cooled gas, whose local
temperature is $\theta<T(x)<T_{\mathrm{c}}(p)$. Once the middle part is
separated from the other parts, this super-cooled gas is no longer in a steady
state, and some portion of the middle part may transition into liquid. In this
way, we expect that the heat conduction state with a liquid–gas interface is
not additive once super-cooled gas is stabilized by the heat current.

We are able to transform the non-additive expression (IV.2) into an additive
form by considering an isentropic surface, $S(H,p,N,\phi)=S(H_{\phi},p,N)$,
which connects a nonequilibrium state $(H,p,N,\phi)$ with an equilibrium state
$(H_{\phi},p,N)$. Here, $H_{\phi}=H+T_{\mathrm{c}}(p)\phi\Psi$ for $J\neq 0$.
The equilibrium entropy is the sum of that for liquid and that for gas. We
thus write

$\displaystyle
S(H,p,N,\phi)=S(H_{\phi}^{\mathrm{L}},p,N_{\phi}^{\mathrm{L}})+S(H_{\phi}^{\mathrm{G}},p,N_{\phi}^{\mathrm{G}})$
(VIII.1)

with $H_{\phi}^{\mathrm{L}}+H_{\phi}^{\mathrm{G}}=H_{\phi}$ and
$N_{\phi}^{\mathrm{L}}+N_{\phi}^{\mathrm{G}}=N$. $H_{\phi}$,
$H_{\phi}^{\mathrm{L/G}}$, and $N_{\phi}^{\mathrm{L/G}}$ are determined in
Appendix G. Although (VIII.1) may look additive, we note that
$H_{\phi}^{\mathrm{L}}+H_{\phi}^{\mathrm{G}}\neq H$ for $J\neq 0$, and
therefore, $S$ and $H$ cannot be simultaneously additive quantities no matter
what transformation is performed. Non-additivity is essential for the
coexistence of liquid and gas under heat conduction.

When the transition temperature $T_{\mathrm{c}}(p)$ is close to the critical
temperature, the metastable region may disappear. Equation (VII.2) suggests
that the interface temperature $\theta$ does not deviate from
$T_{\mathrm{c}}(p)$ because both thermal conductivity and density become equal
in liquid and gas. The extensive variable $\Psi$ vanishes because it is
proportional to the latent heat $\hat{q}$. Moreover, our initial assumption of
the negligible thickness of the interface may not be valid at the critical
point because fluctuations grow up and the width of the liquid–gas interface
may become macroscopic.

We proposed in this paper a scheme with which we construct the variational
function $\mathscr{S}$ using the correspondence with thermodynamic entropy
$S$. For equilibrium systems, the variational function of thermodynamics is
closely related to the probability density of unconstrained thermodynamic
variables in fluctuation theory Callen . This idea may apply to systems out of
equilibrium. We thus naturally conjecture that, in the setup we study, the
probability density of the unconstrained thermodynamic variables is also
expressed by the variational function $\mathscr{S}$. To confirm the validity
of the conjecture, we will study the potential for the fluctuation of
thermodynamic variables by analyzing mesoscopic stochastic systems.

Apart from the maximum entropy principle, there are several variational
principles aimed at explaining nonequilibrium phenomena. In the linear
response regime from equilibrium, the principle of minimum entropy production
has been formulated MEP . The principle states that the stationary
distribution minimizes the entropy production rate MEP-measure , which is well
founded from the dynamical fluctuation theory based on statistical mechanics
Maes-LD . Moreover, as an attempt to apply the variational principle to
various nonequilibrium phenomena, the principle of the maximum entropy
production rate has also been studied MMEP . However, these variational
principles are not yet connected to the maximum entropy principle in
thermodynamics. The extended maximum entropy principle proposed in this paper
describes phenomena in the linear response regime, and the connection with the
principle of minimum entropy production should thus be studied in the future.
Furthermore, the concept of using global quantities may be applied to a wide
class of phenomena far from equilibrium. It may then be interesting to find an
interface between our theory and variational principles far from equilibrium.

Last but not least, we remark that several important concepts of
nonequilibrium thermodynamics have not yet been studied. For example, excess
heat is the key concept for an extension of the Clausius formula Landauer ;
Oono-Paniconi ; Hatano-Sasa ; Ruelle ; KNST-SST ; KNST-SST-Twist ; Maes-SST ;
Jona-Lasinio-SST . The basic idea is that the heat in the time evolution is
decomposed into a house-keeping part and an excess part, where the former is
identified as the part necessary for maintaining the nonequilibrium steady
state. However, there are several versions of the decomposition, and it has
not been established which decomposition is most useful. We constructed an
extended entropy for the phase coexistence in heat conduction, and we may
therefore ask what is a measurable heat $Q$ defined by an extended Clausius
equality $Q=\tilde{T}\Delta S$ with the entropy and the global temperature
formulated in this paper. If such a quantity $Q$ is explored in the present
framework, the isentropic surface defined by $S(H,p,N,\phi)=S(H_{\phi},p,N)$
provides quasi-static adiabatic processes. Generalizing the argument, we may
further consider general adiabatic processes in which the entropy never
decreases. There, $-T_{\mathrm{c}}(p)\phi\Psi$, which corresponds to
$H-H_{\phi}$, may be regarded as the minimum work that induces steady heat
current. It is a challenging problem to operationally identify adiabatic
conditions for heat conduction systems.

Acknowledgment.— The present study was supported by JSPS KAKENHI (Nos.
17H01148, 19K03647, 19H05795, 20K20425).

##  Appendix A Preliminaries

###  A.1 Global thermodynamics for liquid and gas in heat conduction

In this section, we briefly review global thermodynamics that holds in each
region of liquid and gas. As derived in global-JSP , liquid or gas in the
linear response regime of heat conduction is thermodynamically equivalent to a
corresponding equilibrium system. This is because the profile of any
thermodynamic density is approximately linear and therefore all global
thermodynamic quantities defined averaged over the whole system are identified
by the trapezoidal rule applied to the integration.

Below, we assume $J<0$ without loss of generality. Similarly to (IV.9), global
temperatures of liquid and gas are defined by

$\displaystyle{\tilde{T}}^{\mathrm{L}}=\frac{\int_{0}^{X}dx~{}\rho(x)T(x)}{\int_{0}^{X}dx~{}\rho(x)},\quad{\tilde{T}}^{\mathrm{G}}=\frac{\int_{X}^{L_{x}}dx~{}\rho(x)T(x)}{\int_{X}^{L_{x}}dx~{}\rho(x)},$
(A.1)

and corresponding inverse temperatures by

$\displaystyle{\tilde{\beta}}^{\mathrm{L}}=\frac{\int_{0}^{X}dx~{}\rho(x)\beta(x)}{\int_{0}^{X}dx~{}\rho(x)},\quad{\tilde{\beta}}^{\mathrm{G}}=\frac{\int_{X}^{L_{x}}dx~{}\rho(x)\beta(x)}{\int_{X}^{L_{x}}dx~{}\rho(x)},$
(A.2)

where $\beta(x)=1/T(x)$. We derive

$\displaystyle{\tilde{\beta}}^{\mathrm{L}}=\frac{1}{{\tilde{T}}^{\mathrm{L}}}+O(\varepsilon^{2}),\quad{\tilde{\beta}}^{\mathrm{G}}=\frac{1}{{\tilde{T}}^{\mathrm{G}}}+O(\varepsilon^{2}),$
(A.3)

which are the same as the relations at equilibrium. Then, defining

$\displaystyle\alpha^{\mathrm{L}}=-{\tilde{\beta}}^{\mathrm{L}}\mu^{\mathrm{L}}({\tilde{T}}^{\mathrm{L}},p)+O(\varepsilon^{2}),$
(A.4)
$\displaystyle\alpha^{\mathrm{G}}=-{\tilde{\beta}}^{\mathrm{G}}\mu^{\mathrm{G}}({\tilde{T}}^{\mathrm{G}},p)+O(\varepsilon^{2})$
(A.5)

with equilibrium chemical potential $\mu^{\mathrm{L}}(T,p)$ and
$\mu^{\mathrm{G}}(T,p)$ of liquid and gas, we have the fundamental relation of
thermodynamics in each region as

$\displaystyle
dS^{\mathrm{L}}={{\tilde{\beta}}^{\mathrm{L}}}dH^{\mathrm{L}}-{\tilde{\beta}}^{\mathrm{L}}{V^{\mathrm{L}}}dp+\alpha^{\mathrm{L}}dN^{\mathrm{L}},$
(A.6) $\displaystyle
dS^{\mathrm{G}}={{\tilde{\beta}}^{\mathrm{G}}}dH^{\mathrm{G}}-{{\tilde{\beta}}^{\mathrm{G}}}{V^{\mathrm{G}}}dp+\alpha^{\mathrm{G}}dN^{\mathrm{G}},$
(A.7)

where $S^{\mathrm{L}}$ and $S^{\mathrm{G}}$ are the spatial integration of
entropy density in (III.6). These relations indicate

$\displaystyle S^{\mathrm{L}}=S(H^{\mathrm{L}},p,N^{\mathrm{L}}),\quad
S^{\mathrm{G}}=S(H^{\mathrm{G}},p,N^{\mathrm{G}}).$ (A.8)

That is, $S^{\mathrm{L}}$ and $S^{\mathrm{G}}$ are expressed by the
equilibrium entropy function $S(H,p,N)$. We emphasize that both liquid and gas
are out of equilibrium with conducting heat.

###  A.2 Functions in liquid–gas coexistence

Let $\cal H^{\mathrm{\ell}}$ and $\cal N^{\mathrm{\ell}}$ be the variational
variables describing the enthalpy and the particle number in the left region
of $0<x<X$. In the argument below, six variable function $A({\cal
H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}};H,p,N,\phi)$ and four variable
function $A_{*}(H,p,N,\phi)$ appear. More precisely, for any six variable
function $A({\cal H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}};H,p,N,\phi)$,
$A_{*}$ denotes the four variable function defined by

$\displaystyle A_{*}(H,p,N,\phi)=$
$\displaystyle~{}~{}~{}A(H^{\mathrm{\ell}}(H,p,N,\phi),N^{\mathrm{\ell}}(H,p,N,\phi);H,p,N,\phi),$
(A.9)

where $H^{\mathrm{\ell}}(H,p,N,\phi)$ and $N^{\mathrm{\ell}}(H,p,N,\phi)$ are
the steady-state values when $(H,p,N,\phi)$ are given. Here, we assume that
$\phi$ is also determined uniquely for a given $J$. The relation (IV.4) is
simply expressed by $S={\mathscr{S}}_{*}$. Obviously, $H^{\mathrm{L}}={\cal
H}^{\mathrm{L}}_{*}$ and $N^{\mathrm{L}}={\cal N}^{\mathrm{L}}_{*}$

Although we do not specify the function type once they are defined, the
function types are fixed throughout this section, which is not the standard
convention in thermodynamics. Thus, for example, a partial derivative is
simply expressed as

$\frac{\partial A}{\partial H}.$ (A.10)

Accordingly, the partial derivative at the steady state is defined. For
instance, the partial derivative with respect to $\cal H^{\mathrm{\ell}}$ is
written as

$\displaystyle\left(\frac{\partial A}{\partial\cal
H^{\mathrm{\ell}}}\right)_{*}=\left.\left(\frac{\partial A}{\partial\cal
H^{\mathrm{\ell}}}\right)_{{\cal N}^{\mathrm{\ell}},H,p,N,\phi}\right|_{({\cal
H^{\mathrm{\ell}}},{\cal
N^{\mathrm{\ell}}})=(H^{\mathrm{\ell}},N^{\mathrm{\ell}})}.$ (A.11)

###  A.3 Basic quantities for the liquid–gas coexistence

Thermodynamic relations in each liquid or gas hold for given ${\cal
H}^{\mathrm{\ell}}$ and ${\cal N}^{\mathrm{\ell}}$. Therefore, for the
thermodynamic states, $({\cal H}^{\mathrm{\ell}},p,{\cal N}^{\mathrm{\ell}})$
and $({\cal H}^{\mathrm{r}},p,{\cal N}^{\mathrm{r}})$ with ${\cal
H}^{\mathrm{r}}=H-{\cal H}^{\mathrm{\ell}}$ and ${\cal N}^{\mathrm{r}}=N-{\cal
N}^{\mathrm{\ell}}$, ${\tilde{\beta}}^{\mathrm{\ell}}$ and
${\tilde{\beta}}^{\mathrm{r}}$ can be defined from the fundamental
thermodynamic relations (A.6) and (A.7) as

$\displaystyle{\tilde{\beta}}^{\mathrm{\ell}}=\left(\frac{\partial
S^{\mathrm{\ell}}}{\partial\cal H^{\mathrm{\ell}}}\right)_{p,{\cal
N^{\mathrm{\ell}}}},\qquad{\tilde{\beta}}^{\mathrm{r}}=\left(\frac{\partial
S^{\mathrm{r}}}{\partial\cal H^{\mathrm{r}}}\right)_{p,{\cal
N^{\mathrm{r}}}}.$ (A.12)

We then introduce the global inverse temperature
${\tilde{\beta}}={\tilde{\beta}}({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N)$ in the liquid–gas coexistence as

$\displaystyle{\tilde{\beta}}=\frac{{\cal
N}^{\mathrm{\ell}}}{N}{\tilde{\beta}}^{\mathrm{\ell}}+\frac{{\cal
N}^{\mathrm{r}}}{N}{\tilde{\beta}}^{\mathrm{r}},$ (A.13)

which is consistent with

$\displaystyle{\tilde{\beta}}=\frac{\int_{0}^{L_{x}}dx~{}\rho(x)\beta(x)}{\int_{0}^{L_{x}}dx~{}\rho(x)}.$
(A.14)

Developing a similar argument for $\alpha=-\mu/T$, we have

$\displaystyle\alpha^{\mathrm{\ell}}=\left(\frac{\partial
S^{\mathrm{\ell}}}{\partial\cal N^{\mathrm{\ell}}}\right)_{{\cal
H^{\mathrm{\ell}}},p},\quad\alpha^{\mathrm{r}}=\left(\frac{\partial
S^{\mathrm{r}}}{\partial\cal N^{\mathrm{r}}}\right)_{{\cal
N^{\mathrm{r}}},p},$ (A.15)

from which we define the global quantity in the liquid–gas coexistence
${\tilde{\alpha}}={\tilde{\alpha}}({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N,\phi)$ as

$\displaystyle{\tilde{\alpha}}=\frac{{\cal
N}^{\mathrm{\ell}}}{N}\alpha^{\mathrm{\ell}}+\frac{{\cal
N}^{\mathrm{r}}}{N}\alpha^{\mathrm{r}}$ (A.16)

consistently with

$\displaystyle{\tilde{\alpha}}=\frac{\int_{0}^{L_{x}}dx~{}\rho(x)\alpha(x)}{\int_{0}^{L_{x}}dx~{}\rho(x)}.$
(A.17)

In the steady state, we expect that the temperature profile $T(x)$ is
continuous in $0<x<L_{x}$ including the liquid–gas interface. This is because
there is no particle current to produce latent heat at the interface. Since
the temperature profile satisfies $T(x)=T_{\mathrm{c}}(p)+O(\varepsilon)$, we
have ${\tilde{\beta}}_{*}=\beta_{\mathrm{c}}(p)+O(\varepsilon)$. Let $\theta$
be the temperature at the interface. The global temperatures of left and right
regions are estimated as

$\displaystyle{\tilde{T}}^{\mathrm{\ell}}_{*}=\frac{T_{1}+\theta}{2}+O(\varepsilon^{2}),\quad{\tilde{T}}^{\mathrm{r}}_{*}=\frac{T_{2}+\theta}{2}+O(\varepsilon^{2})$
(A.18)

by applying the trapezoidal rule to the integration (A.1). These expressions
result in

$\displaystyle\frac{{\tilde{T}}^{\mathrm{r}}_{*}-{\tilde{T}}^{\mathrm{\ell}}_{*}}{T_{\mathrm{c}}(p)}=\frac{\varepsilon}{2}+O(\varepsilon^{2}),$
(A.19)

which is equivalent to

$\displaystyle\frac{{\tilde{\beta}}^{\mathrm{\ell}}_{*}-{\tilde{\beta}}^{\mathrm{r}}_{*}}{\beta_{\mathrm{c}}(p)}=\frac{\varepsilon}{2}+O(\varepsilon^{2}).$
(A.20)

Together with (A.13), this relation leads to

$\displaystyle{\tilde{\beta}}^{\mathrm{\ell}}_{*}={\tilde{\beta}}_{*}\left(1+\frac{\varepsilon}{2}\frac{N^{\mathrm{r}}}{N}\right),\quad{\tilde{\beta}}^{\mathrm{r}}_{*}={\tilde{\beta}}_{*}\left(1-\frac{\varepsilon}{2}\frac{N^{\mathrm{\ell}}}{N}\right).$
(A.21)

Multiplying the formula (A.13) and ${\tilde{T}}=({\cal
N}^{\mathrm{\ell}}{\tilde{T}}^{\mathrm{\ell}}+{\cal
N}^{\mathrm{r}}{\tilde{T}}^{\mathrm{r}})/N$, and applying (A.3), we have

$\displaystyle{\tilde{\beta}}_{*}{\tilde{T}}_{*}=\left(\frac{N^{\mathrm{\ell}}}{N}\right)^{2}+\left(\frac{N^{\mathrm{r}}}{N}\right)^{2}+\frac{N^{\mathrm{\ell}}N^{\mathrm{r}}}{N^{2}}\left(\frac{{\tilde{\beta}}^{\mathrm{\ell}}_{*}}{{\tilde{\beta}}^{\mathrm{r}}_{*}}+\frac{{\tilde{\beta}}^{\mathrm{r}}_{*}}{{\tilde{\beta}}^{\mathrm{\ell}}_{*}}\right).$
(A.22)

Since (A.21) leads to
${\tilde{\beta}}^{\mathrm{\ell}}_{*}/{\tilde{\beta}}^{\mathrm{r}}_{*}=1+\varepsilon/2+O(\varepsilon^{2})$,
we transform (A.22) into

$\displaystyle{\tilde{\beta}}_{*}=\frac{1}{{\tilde{T}}_{*}}+O(\varepsilon^{2}),$
(A.23)

using $N^{\mathrm{\ell}}+N^{\mathrm{r}}=N$. By a similar argument, we can show

$\displaystyle{\tilde{\alpha}}_{*}=-\frac{\tilde{\mu}_{*}}{{\tilde{T}}_{*}}+O(\varepsilon^{2}).$
(A.24)

Thus, we can write the fundamental relation of thermodynamics (IV.8) as

$\displaystyle
dS={\tilde{\beta}}_{*}dH-{\tilde{\beta}}_{*}V_{*}dp+{\tilde{\alpha}}_{*}dN+\Psi
d\phi.$ (A.25)

##  Appendix B Derivation of (V.1) and (V.2)

We first argue the concept of intensive and extensive variables in Appendix
B.1. Next, in Appendix B.2, we determine $\Psi$ from the variational equations
and fundamental thermodynamic relations. In Appendix B.3, we formulate the
singularity at $\varepsilon=0$. Finally, in Appendix B.4, we determine $\psi$
and $\lambda$. Appendix B.5 is devoted to discussion on the singularity.

###  B.1 Intensive and extensive variables

Figure 9: Extensivity of the heat conduction state with a liquid–gas
interface. $J$ is the amount of heat current per unit area. For a system with
area $aL_{y}L_{z}$ in a cross section perpendicular to the $x$-axis, all
profiles of local thermodynamic quantities are equal to those of the system
with $L_{y}L_{z}$ if $J$ is the same.

Since heat conduction states are inhomogeneous along the horizontal direction
parallel to the heat current, the extensive property is not trivial for global
thermodynamic variables. On the other hand, thermodynamic states are
homogeneous along the section perpendicular to the heat current, and therefore
we can consider the extensive property for the size scaling procedure in this
direction. As schematically shown in Fig. 9, we fix $L_{x}$ and $J$ in the
argument below, where $J$ is defined as the current per unit area. The total
amount of heat current is proportional to $L_{y}L_{z}$.

We define an intensive variable $\phi$ indicating the degree of
nonequilibrium, which is proportional to $\varepsilon$ and invariant for the
scaling transformation of $L_{y}L_{z}\rightarrow aL_{y}L_{z}$ with
$H\rightarrow aH$ and $N\rightarrow aN$ in Fig. 9, where $a$ is an arbitrary
constant. This invariance allows $\phi$ to depend on $\hat{h}=H/N$ and $p$.
Thus, we express $\phi$ as

$\displaystyle\phi=\frac{\varepsilon}{\lambda(\hat{h},p)}$ (B.1)

without loss of generality, where $\lambda$ is an intensive function to be
determined in the steady state.

As the conjugate variable to $\phi$, we set $\Psi$ as it is an extensive
variable. More precisely, $\Psi(H,p,N,\phi)$ is equal to the steady state
value $\psi_{*}$ determined as (IV.5) from $\psi({\cal
H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}};H,p,N)$. For the scaling
transformation in Fig. 9, the extensivity is expressed as

$\displaystyle a\psi({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N)=\psi(a{\cal H}^{\mathrm{\ell}},a{\cal
N}^{\mathrm{\ell}};aH,p,aN).$ (B.2)

Differentiating both sides in $a$ and setting $a=1$, we have the Euler
relation,

$\displaystyle\psi=\frac{\partial\psi}{\partial{\cal H}^{\mathrm{\ell}}}{{\cal
H}^{\mathrm{\ell}}}+\frac{\partial\psi}{\partial{\cal
N}^{\mathrm{\ell}}}{{\cal N}^{\mathrm{\ell}}}+\frac{\partial\psi}{\partial
H}{H}+\frac{\partial\psi}{\partial N}{N}.$ (B.3)

###  B.2 Determination of $\psi$ and $\Psi$

Differentiating the variational function $\mathscr{S}$ in (IV.3), we have

$\displaystyle\frac{\partial\mathscr{S}}{\partial{\cal
H}^{\mathrm{\ell}}}=({\tilde{\beta}}^{\mathrm{\ell}}-{\tilde{\beta}}^{\mathrm{r}})+\phi\frac{\partial\psi}{\partial{\cal
H}^{\mathrm{\ell}}},$ (B.4)
$\displaystyle\frac{\partial\mathscr{S}}{\partial{\cal
N}^{\mathrm{\ell}}}=(\alpha^{\mathrm{\ell}}-\alpha^{\mathrm{r}})+\phi\frac{\partial\psi}{\partial{\cal
N}^{\mathrm{\ell}}},$ (B.5) $\displaystyle\frac{\partial\mathscr{S}}{\partial
H}={\tilde{\beta}}^{\mathrm{r}}+\phi\frac{\partial\psi}{\partial H},$ (B.6)
$\displaystyle\frac{\partial\mathscr{S}}{\partial
p}=-{\tilde{\beta}}^{\mathrm{\ell}}V^{\mathrm{\ell}}-{\tilde{\beta}}^{\mathrm{r}}V^{\mathrm{r}}+\phi\frac{\partial\psi}{\partial
p},$ (B.7) $\displaystyle\frac{\partial\mathscr{S}}{\partial
N}=\alpha^{\mathrm{r}}+\phi\frac{\partial\psi}{\partial N},$ (B.8)
$\displaystyle\frac{\partial\mathscr{S}}{\partial\phi}=\psi.$ (B.9)

Comparing these with (IV.11), (IV.12), (IV.13), (IV.14), (IV.15), and (IV.16),
we obtain five differential equations for $\psi$ at the steady state as

$\displaystyle\phi\left(\frac{\partial\psi}{\partial{\cal
H}^{\mathrm{\ell}}}\right)_{*}={\tilde{\beta}}^{\mathrm{r}}_{*}-{\tilde{\beta}}^{\mathrm{\ell}}_{*},$
(B.10) $\displaystyle\phi\left(\frac{\partial\psi}{\partial{\cal
N}^{\mathrm{\ell}}}\right)_{*}=\alpha^{\mathrm{r}}_{*}-\alpha^{\mathrm{\ell}}_{*},$
(B.11) $\displaystyle\phi\left(\frac{\partial\psi}{\partial
H}\right)_{*}={\tilde{\beta}}_{*}-{\tilde{\beta}}^{\mathrm{r}}_{*},$ (B.12)
$\displaystyle\phi\left(\frac{\partial\psi}{\partial
p}\right)_{*}=-{\tilde{\beta}}_{*}V_{*}+{\tilde{\beta}}^{\mathrm{\ell}}_{*}V^{\mathrm{\ell}}_{*}+{\tilde{\beta}}^{\mathrm{r}}_{*}V^{\mathrm{r}}_{*},$
(B.13) $\displaystyle\phi\left(\frac{\partial\psi}{\partial
N}\right)_{*}={\tilde{\alpha}}_{*}-\alpha^{\mathrm{r}}_{*}.$ (B.14)

Substituting these five relations into (B.3) at the steady state, we have

$\displaystyle\phi\Psi=-({\tilde{\beta}}^{\mathrm{\ell}}_{*}-{\tilde{\beta}}^{\mathrm{r}}_{*})\left(H^{\mathrm{\ell}}-\frac{N^{\mathrm{\ell}}}{N}H\right).$
(B.15)

Using (A.20) and (B.1), we further rewrite it as

$\displaystyle\Psi=-\frac{\lambda(\hat{h},p)}{2}\beta_{\mathrm{c}}(p)\left(H^{\mathrm{\ell}}-\frac{N^{\mathrm{\ell}}}{N}H\right).$
(B.16)

The functional form of $\lambda(\hat{h},p)$ still remains to be determined.

###  B.3 Relation emerged from singularity

Figure 10: (a) Transition of the steady state when $J$ is changed from $J>0$
($\varepsilon<0$) to $J<0$ ($\varepsilon>0)$. Correspondingly, the fraction of
particles in the left region, $c^{\mathrm{\ell}}=N^{\mathrm{\ell}}/N$, behaves
as a singular function of $\varepsilon$ shown in (b). Similarly,
$\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}}$ shows discontinuous change as
in (c).

Noting that the trivial relation $H=H^{\mathrm{\ell}}+H^{\mathrm{r}}$ with the
number fraction $c^{\mathrm{\ell}}\equiv N^{\mathrm{\ell}}/N$ and the enthalpy
per particle
$\hat{h}^{\mathrm{\ell/r}}\equiv{H^{\mathrm{\ell/r}}}/{N^{\mathrm{\ell/r}}}$
gives

$\displaystyle
H^{\mathrm{\ell}}-\frac{N^{\mathrm{\ell}}}{N}H=-N(\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}})(1-c^{\mathrm{\ell}})c^{\mathrm{\ell}},$
(B.17)

we rewrite (B.16) as

$\displaystyle{\Psi}={N}\frac{\lambda}{2}\beta_{\mathrm{c}}(\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}})(1-c^{\mathrm{\ell}})c^{\mathrm{\ell}}.$
(B.18)

From the left–right symmetry, $\Psi$ should change its sign for
$\varepsilon\to-\varepsilon$. Indeed, the expression in (B.18) explicitly
shows this property because $c^{\mathrm{\ell}}\leftrightarrow
c^{\mathrm{r}}=1-c^{\mathrm{\ell}}$ for
$\varepsilon\leftrightarrow-\varepsilon$ and
$\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}}$ changes its sign at
$\varepsilon=0$ as shown in Fig. 10(c), where
$\hat{q}=\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}$
with

$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\hat{h}^{\mathrm{\ell}}(\phi)=\hat{h}_{\mathrm{c}}^{\mathrm{L}},\quad\lim_{\varepsilon\rightarrow
0^{-}}\hat{h}^{\mathrm{\ell}}(\phi)=\hat{h}_{\mathrm{c}}^{\mathrm{G}},$ (B.19)
$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\hat{h}^{\mathrm{r}}(\phi)=\hat{h}_{\mathrm{c}}^{\mathrm{G}},\quad\lim_{\varepsilon\rightarrow
0^{-}}\hat{h}^{\mathrm{r}}(\phi)=\hat{h}_{\mathrm{c}}^{\mathrm{L}}.$ (B.20)

The left side region of the interface is occupied by liquid for
$\varepsilon>0$ whereas by gas for $\varepsilon<0$. The position of the
interface may jump at $\varepsilon=0$ and the number fraction
$c^{\mathrm{\ell}}$ suddenly changes its value; i.e.,

$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}c^{\mathrm{\ell}}(\phi)=c_{\mathrm{c}}^{\mathrm{L}},\quad\lim_{\varepsilon\rightarrow
0^{-}}c^{\mathrm{\ell}}(\phi)=1-c_{\mathrm{c}}^{\mathrm{L}}$ (B.21)

with the equilibrium fraction of liquid $c_{\mathrm{c}}^{\mathrm{L}}$. See
Fig. 10(b).

The singularity at $\varepsilon=0$ is critical for the extension of entropy.
To see it, we differentiate (IV.2) in $\phi$ with $H$, $p$, and $N$ fixed and
obtain

$\displaystyle\frac{\partial
S}{\partial\phi}=({\tilde{\beta}}_{*}^{\mathrm{\ell}}-{\tilde{\beta}}_{*}^{\mathrm{r}})\frac{\partial
H^{\mathrm{\ell}}}{\partial\phi}+(\alpha^{\mathrm{\ell}}_{*}-\alpha^{\mathrm{r}}_{*})\frac{\partial
N^{\mathrm{\ell}}}{\partial\phi}+\Psi+O(\varepsilon)$ (B.22)

by noting $H^{\mathrm{\ell}}=H^{\mathrm{\ell}}(H,p,N,\phi)$ and
$N^{\mathrm{\ell}}=N^{\mathrm{\ell}}(H,p,N,\phi)$. Comparing (B.22) with
(IV.16) and applying (A.20), we obtain

$\displaystyle\frac{\varepsilon}{2}\beta_{\mathrm{c}}\frac{\partial
H^{\mathrm{\ell}}}{\partial\phi}+(\alpha^{\mathrm{\ell}}_{*}-\alpha^{\mathrm{r}}_{*})\frac{\partial
N^{\mathrm{\ell}}}{\partial\phi}=O(\varepsilon).$ (B.23)

At first glance, (B.23) is trivial because its left-hand side appears to be
$O(\varepsilon)$. However, we should be careful to differentiate
$H^{\mathrm{\ell}}$ and $N^{\mathrm{\ell}}$ in $\phi$ due to the singular
behavior in Fig. 10. Using $\hat{h}^{\mathrm{\ell}}$ and $c^{\mathrm{\ell}}$,
(B.23) is transformed into

$\displaystyle\left(\alpha^{\mathrm{\ell}}_{*}-\alpha^{\mathrm{r}}_{*}+\frac{\varepsilon}{2}\beta_{\mathrm{c}}\hat{h}\right)\frac{\partial
c^{\mathrm{\ell}}}{\partial\phi}=\frac{\varepsilon}{2}\beta_{\mathrm{c}}\frac{\partial(\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}})c^{\mathrm{\ell}}(1-c^{\mathrm{\ell}})}{\partial\phi}.$
(B.24)

We thus need to define the singular functions $c^{\mathrm{\ell}}$ and
$\hat{h}^{\mathrm{\ell}}$. We introduce a regularization parameter
$\varepsilon_{1}$ to make $c^{\mathrm{\ell}}(\phi)$ a smooth function in
$\varepsilon$:

$\displaystyle
c^{\mathrm{\ell}}(\phi)=\frac{1}{2}+\left(c_{\mathrm{c}}^{\mathrm{L}}-\frac{1}{2}\right)\tanh\left(\frac{\varepsilon}{\varepsilon_{1}}\right).$
(B.25)

Since $\hat{h}^{\mathrm{r}}(\phi)-\hat{h}^{\mathrm{\ell}}(\phi)$ is
antisymmetric in $\varepsilon$, we write

$\displaystyle\hat{h}^{\mathrm{r}}(\phi)-\hat{h}^{\mathrm{\ell}}(\phi)=\hat{q}\tanh\left(\frac{\varepsilon}{\varepsilon_{2}}\right)$
(B.26)

with another regularization parameter $\varepsilon_{2}$. Note that
$\varepsilon_{1}$ and $\varepsilon_{2}$ should be set to zero after the
calculation.

Accepting these regularizations, we can calculate the derivatives with $\phi$
at $\varepsilon=0$ as

$\displaystyle\left.\frac{\partial
c^{\mathrm{\ell}}}{\partial\phi}\right|_{\varepsilon=0}=\lambda\frac{c_{\mathrm{c}}^{\mathrm{\ell}}-\frac{1}{2}}{\varepsilon_{1}},\quad\left.\frac{\partial(\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}})}{\partial\phi}\right|_{\varepsilon=0}=\lambda\frac{\hat{q}}{\varepsilon_{2}}.$
(B.27)

See Sec. B.5 for the interpretation of the regularization parameters.

Substituting the derivatives (B.27) into (B.24), we obtain

$\displaystyle\alpha^{\mathrm{\ell}}_{*}-\alpha^{\mathrm{r}}_{*}+\frac{\varepsilon}{2}\beta_{\mathrm{c}}\hat{h}=\varepsilon\frac{\varepsilon_{1}}{\varepsilon_{2}}\beta_{\mathrm{c}}\hat{q}\frac{c^{\mathrm{\ell}}(1-c^{\mathrm{\ell}})}{2c^{\mathrm{\ell}}-1},$
(B.28)

where we ignored the contribution of $O(\varepsilon^{2})$. Now, we consider
the limit $\varepsilon_{1}\to 0$ and $\varepsilon_{2}\to 0$ with
$\varepsilon_{1}/\varepsilon_{2}$ fixed. The right-hand side of (B.28)
diverges at $c^{\mathrm{\ell}}=1/2$ if
$\varepsilon_{1}/\varepsilon_{2}\not=0$, while the left-hand side remains
regular. To avoid such inconsistency, the only possible case is that
$\varepsilon_{1}/\varepsilon_{2}=0$, which leads to

$\displaystyle\alpha^{\mathrm{\ell}}_{*}-\alpha^{\mathrm{r}}_{*}+\frac{\varepsilon}{2}\beta_{\mathrm{c}}\hat{h}=0.$
(B.29)

We mention that (B.29) provides the balance of chemical potential in heat
conduction.

###  B.4 Determination of $\psi$ and $\lambda$

Since the functional form of $\Psi$ is expressed as (B.16), we set $\psi$ as

$\displaystyle\psi({\cal H}^{\mathrm{\ell}},{\cal
N}^{\mathrm{\ell}};H,p,N)=-\frac{\cal B}{2}{\beta_{\mathrm{c}}(p)}\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal N}^{\mathrm{\ell}}}{N}H\right),$ (B.30)

where $\cal B$ is an intensive function

$\displaystyle{\cal B}={\cal B}\left(\frac{{\cal
H}^{\mathrm{\ell}}}{N},\frac{{\cal
N}^{\mathrm{\ell}}}{N},\frac{H}{N},p\right).$ (B.31)

Note that $\cal B$ is independent of $\phi$ because $\psi$ does not depend on
$\phi$. To be consistent with (B.16), we have

$\displaystyle{\cal B}_{*}=\lambda(\hat{h},p).$ (B.32)

Differentiating (B.30), we have

$\displaystyle\frac{\partial\psi}{\partial{\cal
H}^{\mathrm{\ell}}}=-\frac{\cal
B}{2}\beta_{\mathrm{c}}(p)-\frac{\beta_{c}(p)}{2N}\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal
N}^{\mathrm{\ell}}}{N}H\right)\frac{\partial\cal B}{\partial({\cal
H}^{\mathrm{\ell}}/N)},$ (B.33)
$\displaystyle\frac{\partial\psi}{\partial{\cal N}^{\mathrm{\ell}}}=\frac{\cal
B}{2}\beta_{\mathrm{c}}(p)\frac{H}{N}-\frac{\beta_{c}(p)}{2N}\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal
N}^{\mathrm{\ell}}}{N}H\right)\frac{\partial\cal B}{\partial({\cal
N}^{\mathrm{\ell}}/N)},$ (B.34) $\displaystyle\frac{\partial\psi}{\partial
H}=\frac{\cal B}{2}\beta_{\mathrm{c}}(p)\frac{{\cal
N}^{\mathrm{\ell}}}{N}-\frac{\beta_{c}(p)}{2N}\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal
N}^{\mathrm{\ell}}}{N}H\right)\frac{\partial\cal B}{\partial(H/N)},$ (B.35)
$\displaystyle\frac{\partial\psi}{\partial p}=-\frac{\cal
B}{2}\frac{d\beta_{\mathrm{c}}}{dp}\left({\cal H}^{\mathrm{\ell}}-\frac{{\cal
N}^{\mathrm{\ell}}}{N}H\right)-\frac{\beta_{c}(p)}{2}\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal
N}^{\mathrm{\ell}}}{N}H\right)\frac{\partial\cal B}{\partial p}.$ (B.36)

Comparing (B.10), (B.11), and (B.12) with (B.33), (B.34), and (B.35),
respectively, we obtain

$\displaystyle\left(\frac{\partial\cal B}{\partial({\cal
H}^{\mathrm{\ell}}/N)}\right)_{*}=0,$ (B.37)
$\displaystyle\left(\frac{\partial\cal B}{\partial(H/N)}\right)_{*}=0,$ (B.38)
$\displaystyle\left(\frac{\partial\cal B}{\partial({\cal
N}^{\mathrm{\ell}}/N)}\right)_{*}=0,$ (B.39)

where we have used (A.13), (A.16), (A.20), (A.21), (B.29), and (B.32).
Meanwhile, (B.13) and (B.36) bring a relation

$\displaystyle\left(\frac{\partial\cal B}{\partial p}\right)_{*}=-{\cal
B_{*}}\left(\frac{1}{\beta_{\mathrm{c}}}\frac{d\beta_{\mathrm{c}}}{dp}+\frac{N^{\mathrm{r}}V^{\mathrm{\ell}}_{*}-N^{\mathrm{\ell}}V^{\mathrm{r}}_{*}}{N^{\mathrm{r}}H^{\mathrm{\ell}}-N^{\mathrm{\ell}}H^{\mathrm{r}}}\right).$
(B.40)

The Clausius–Clapeyron relation

$\displaystyle\frac{dT_{\mathrm{c}}}{dp}=T_{\mathrm{c}}\frac{\hat{v}^{\mathrm{\ell}}-\hat{v}^{\mathrm{r}}}{\hat{h}^{\mathrm{\ell}}-\hat{h}^{\mathrm{r}}}$
(B.41)

leads to the simplification of (B.40). Indeed, noting
$N^{\mathrm{r}}V_{*}^{\mathrm{\ell}}-N^{\mathrm{\ell}}V_{*}^{\mathrm{r}}=(\hat{v}^{\mathrm{\ell}}-\hat{v}^{\mathrm{r}})N^{\mathrm{\ell}}N^{\mathrm{r}}$
with specific volumes $\hat{v}^{\mathrm{\ell/r}}$ and
$N^{\mathrm{r}}H^{\mathrm{\ell}}-N^{\mathrm{\ell}}H^{\mathrm{r}}=(\hat{h}^{\mathrm{\ell}}-\hat{h}^{\mathrm{r}})N^{\mathrm{\ell}}N^{\mathrm{r}}$,
we obtain

$\displaystyle\left(\frac{\partial\cal B}{\partial p}\right)_{*}=0.$ (B.42)

Summarizing (B.37), (B.38), (B.39), and (B.42), we find that $\cal B$ takes
the form

$\displaystyle\lambda_{0}+\\!\\!\sum_{k+l\geq 2}\\!a_{kl}({\cal
H}^{\mathrm{\ell}}\\!-H^{\mathrm{\ell}}(H,p,N,\phi))^{k}({\cal
N}^{\mathrm{\ell}}\\!-N^{\mathrm{\ell}}(H,p,N,\phi))^{l}$ (B.43)

with a numerical constant $\lambda_{0}$, where $a_{kl}$ are functions of
$({\cal H}^{\mathrm{\ell}},{\cal N}^{\mathrm{\ell}},H,p,N)$. However, since
$\cal B$ does not explicitly depend on $\phi$, $H^{\mathrm{\ell}}(H,p,N,\phi)$
should not appear in $\cal B$. This means that $a_{kl}=0$, and we conclude

$\displaystyle{\cal B}=\lambda_{0}.$ (B.44)

Thus, (B.30) is written as

$\displaystyle\psi=-\frac{\lambda_{0}}{2}\beta_{\mathrm{c}}(p)\left({\cal
H}^{\mathrm{\ell}}-\frac{{\cal N}^{\mathrm{\ell}}}{N}H\right).$ (B.45)

Recalling (IV.2), (IV.3), and the fundamental relation of thermodynamics
(IV.8), the value of the numerical constant $\lambda_{0}$ does not affect
experimental observations at all. Thus, we set $\lambda_{0}=1$ without loss of
generality. We then reach (V.1) and (V.2).

###  B.5 Interpretation of the regularization parameters

Mathematically, the regularization parameters $\varepsilon_{1}$ and
$\varepsilon_{2}$ are introduced to set the discontinuous functions in Figs.
10(b) and (c) to be a limit of continuous functions (B.25) and (B.26). We here
provide their interpretation. As demonstrated in Fig. 5, two configurations,
$(\mathrm{\ell},\mathrm{r})=(\mathrm{L},\mathrm{G})$ and
$(\mathrm{\ell},\mathrm{r})=(\mathrm{G},\mathrm{L})$, are equivalent at
$\varepsilon=0$ due to the left–right symmetry of the system. The symmetry is
violated by imposing heat current: In $\varepsilon>0$, the probability of
observing the configuration with
$(\mathrm{\ell},\mathrm{r})=(\mathrm{L},\mathrm{G})$ becomes higher than that
with $(\mathrm{\ell},\mathrm{r})=(\mathrm{G},\mathrm{L})$ because the
temperature of the left region is lower than the right region. The probability
approaches unity beyond some magnitude of $\varepsilon$, which corresponds to
the regularization parameters. For $\varepsilon$ beyond the regularization
parameters, we observe the steady states as shown in Fig. 10(a).

Recalling that the enthalpy and particle number are both conserved quantities,
their spatial distributions are not necessarily functionally dependent. This
indicates that $\varepsilon_{1}$ can be intrinsically independent of
$\varepsilon_{2}$. The two regularization parameters should satisfy
$\varepsilon_{1}/\varepsilon_{2}\to 0$ in the limit $\varepsilon_{1}\to 0$ and
$\varepsilon_{2}\to 0$ for the present framework to be consistent as discussed
in Appendix B.3. Although we do not have a physical interpretation of this
result, we expect that such regularization parameters may naturally appear in
the analysis of statistical mechanics of the phase separation in heat
conduction. Performing molecular dynamics simulations of finite-size systems,
we observe the obvious separation of the liquid from gas when $|J|$ is not too
much small. The separation becomes subtle for sufficiently small but non-
vanishing values of $|J|$. For instance, liquid and gas regions fluctuate and
sometimes exchange, which leads to the smearing of the discontinuous change at
$J=0$. The crossover value of $|J|$ depends on the system size. Characterizing
such finite-size smearing effects may shed light on the properties of the
regularization.

##  Appendix C Derivation of (VI.2)

From local thermodynamic relation $h(x)-\mu(x)\rho(x)=T(x)s(x)$, we have

$\displaystyle H-G=L_{y}L_{z}\int_{0}^{L_{x}}dx~{}T(x)s(x).$ (C.1)

Decompose the integral of the right-hand side as

$\displaystyle\int_{0}^{L_{x}}dx~{}T(x)s(x)=$
$\displaystyle~{}~{}~{}~{}{\tilde{T}}\int_{0}^{L_{x}}dx~{}s(x)+\int_{0}^{L_{x}}dx~{}(T(x)-{\tilde{T}})\rho(x)\hat{s}(x),$
(C.2)

where $\hat{s}(x)=s(x)/\rho(x)$. $\hat{s}(x)$ is discontinuous at the
interface; i.e., $\hat{s}(x)=\hat{s}^{\mathrm{\ell}}(T(x),p)$ for $0<x<X$ and
$\hat{s}(x)=\hat{s}^{\mathrm{r}}(T(x),p)+O(\varepsilon)$ for $X<x<L_{x}$. We
then estimate
$\hat{s}^{\mathrm{\ell/r}}(T(x),p)=\hat{s}^{\mathrm{\ell/r}}_{\mathrm{c}}+O(\varepsilon)$,
where
$\hat{s}^{\mathrm{\ell/r}}_{\mathrm{c}}=\hat{s}^{\mathrm{\ell/r}}(T_{\mathrm{c}}(p),p)$.
Ignoring the contribution of $O(\varepsilon^{2})$, we transform the last term
of (C.2) as

$\displaystyle\hat{s}^{\mathrm{\ell}}_{\mathrm{c}}\int_{0}^{X}dx~{}(T(x)-{\tilde{T}})\rho(x)+\hat{s}^{\mathrm{r}}_{\mathrm{c}}\int_{X}^{L_{x}}dx~{}(T(x)-{\tilde{T}})\rho(x)$
$\displaystyle=\hat{s}^{\mathrm{\ell}}_{\mathrm{c}}({\tilde{T}}^{\mathrm{\ell}}-{\tilde{T}})\frac{N^{\mathrm{\ell}}}{L_{y}L_{z}}+\hat{s}^{\mathrm{r}}_{\mathrm{c}}({\tilde{T}}^{\mathrm{r}}-{\tilde{T}})\frac{N^{\mathrm{r}}}{L_{y}L_{z}},$
(C.3)

where the second line is obtained from the definition of the global
temperature for each region. From
$N{\tilde{T}}={N^{\mathrm{\ell}}}{\tilde{T}}^{\mathrm{\ell}}+{N^{\mathrm{r}}}{\tilde{T}}^{\mathrm{r}}$,
we also have

$\displaystyle{\tilde{T}}^{\mathrm{\ell}}-{\tilde{T}}=-\frac{N^{\mathrm{r}}}{N}({\tilde{T}}^{\mathrm{r}}-{\tilde{T}}^{\mathrm{\ell}}),\quad{\tilde{T}}^{\mathrm{r}}-{\tilde{T}}=\frac{N^{\mathrm{\ell}}}{N}({\tilde{T}}^{\mathrm{r}}-{\tilde{T}}^{\mathrm{\ell}}).$
(C.4)

Substituting these two relations into (C.3), we express the right-hand side as

$\displaystyle\phi
T_{\mathrm{c}}(p)(\hat{s}^{\mathrm{r}}_{\mathrm{c}}-\hat{s}^{\mathrm{\ell}}_{\mathrm{c}})\frac{N^{\mathrm{\ell}}N^{\mathrm{r}}}{2N}\frac{1}{L_{y}L_{z}},$
(C.5)

where we have used (D.9) and (D.10). Note that
$\hat{s}^{\mathrm{r}}_{\mathrm{c}}-\hat{s}^{\mathrm{\ell}}_{\mathrm{c}}=\hat{q}T_{\mathrm{c}}$
when $(\mathrm{\ell},\mathrm{r})=(\mathrm{L},\mathrm{G})$ and
$-\hat{q}T_{\mathrm{c}}$ when
$(\mathrm{\ell},\mathrm{r})=(\mathrm{G},\mathrm{L})$. As
$(\mathrm{\ell},\mathrm{r})=(\mathrm{L},\mathrm{G})$ for $\phi>0$ and
$(\mathrm{G},\mathrm{L})$ for $\phi<0$, we further rewrite (C.5) as

$\displaystyle|\phi|{\hat{q}(p)}\frac{N^{\mathrm{L}}N^{\mathrm{G}}}{2N}\frac{1}{L_{y}L_{z}}.$
(C.6)

Comparing (C.6) with $\Psi$ in (V.3), we conclude

$\displaystyle\int_{0}^{L_{x}}dx~{}(T(x)-{\tilde{T}})\rho(x)\hat{s}(x)=\frac{T_{\mathrm{c}}(p)\phi\Psi}{L_{y}L_{z}}.$
(C.7)

Then, noting ${\tilde{T}}=T_{\mathrm{c}}(p)+O(\varepsilon)$, we write (C.1) as

$\displaystyle
H-G={\tilde{T}}\left(L_{y}L_{x}\int_{0}^{L_{x}}dx~{}s(x)+\phi\Psi\right),$
(C.8)

which is (VI.2).

##  Appendix D Equivalence of (V.5) to (VI.16)

According to (A.4) and (A.5), we write

$\displaystyle\alpha^{\mathrm{\ell}}=\alpha^{\mathrm{\ell}}({\tilde{\beta}}^{\mathrm{\ell}},p),\quad\alpha^{\mathrm{r}}=\alpha^{\mathrm{r}}({\tilde{\beta}}^{\mathrm{r}},p).$
(D.1)

We expand these functions around $\beta_{\mathrm{c}}(p)$ as

$\displaystyle\alpha^{\mathrm{\ell}}=\alpha^{\mathrm{\ell}}(\beta_{\mathrm{c}}(p),p)-({\tilde{\beta}}^{\mathrm{\ell}}-\beta_{\mathrm{c}}(p))\hat{h}^{\mathrm{\ell}}+O(\varepsilon^{2}),$
(D.2)
$\displaystyle\alpha^{\mathrm{r}}=\alpha^{\mathrm{r}}(\beta_{\mathrm{c}}(p),p)-({\tilde{\beta}}^{\mathrm{r}}-\beta_{\mathrm{c}}(p))\hat{h}^{\mathrm{r}}+O(\varepsilon^{2}),$
(D.3)

where we have used
$\left({\partial\alpha}/{\partial\beta}\right)_{p}=-\hat{h}$ resulted from the
Gibbs-Duhem relation. Let us substitute these equations into a re-expressed
form of (V.5); i.e., into

$\displaystyle\alpha^{\mathrm{\ell}}-\alpha^{\mathrm{r}}+\phi\beta_{\mathrm{c}}(p)\frac{H}{2N}=0.$
(D.4)

Then, using
$\alpha^{\mathrm{\ell}}(\beta_{\mathrm{c}}(p),p)=\alpha^{\mathrm{r}}(\beta_{\mathrm{c}}(p),p)$
and
$H=\hat{h}^{\mathrm{\ell}}N^{\mathrm{\ell}}+\hat{h}^{\mathrm{r}}N^{\mathrm{r}}$,
we obtain

$\displaystyle-({\tilde{\beta}}^{\mathrm{\ell}}-\beta_{\mathrm{c}}(p))\hat{h}^{\mathrm{\ell}}$
$\displaystyle+({\tilde{\beta}}^{\mathrm{r}}-\beta_{\mathrm{c}}(p))\hat{h}^{\mathrm{r}}$
$\displaystyle+$
$\displaystyle\frac{\phi{\tilde{\beta}}}{2}\left(\frac{N^{\mathrm{\ell}}}{N}\hat{h}^{\mathrm{\ell}}+\frac{N^{\mathrm{r}}}{N}\hat{h}^{\mathrm{r}}\right)=0,$
(D.5)

where $\phi\beta_{\mathrm{c}}(p)$ is replaced by $\phi{\tilde{\beta}}$ by
ignoring the difference of $O(\varepsilon^{2})$. Substituting (A.21) into
${\tilde{\beta}}^{\mathrm{\ell}}$ and ${\tilde{\beta}}^{\mathrm{r}}$, we
further transform (D.5) into

$\displaystyle(\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}})\left[{\tilde{\beta}}\left(1+\frac{\phi}{2}\frac{N^{\mathrm{r}}-N^{\mathrm{\ell}}}{N}\right)-\beta_{\mathrm{c}}(p)\right]=0.$
(D.6)

Since $|\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}}|$ at $\varepsilon=0$
corresponds to the latent heat $\hat{q}$,
$\hat{h}^{\mathrm{r}}-\hat{h}^{\mathrm{\ell}}$ is non-zero. Thus, we conclude
that the variational principle (V.5) is written as

$\displaystyle{\tilde{\beta}}=\beta_{\mathrm{c}}(p)\left(1-\frac{\phi}{2}\frac{N^{\mathrm{r}}-N^{\mathrm{\ell}}}{N}\right),$
(D.7)

which is equal to

$\displaystyle{\tilde{T}}=T_{\mathrm{c}}(p)\left(1+\frac{\phi}{2}\frac{N^{\mathrm{r}}-N^{\mathrm{\ell}}}{N}\right).$
(D.8)

Substituting (D.7) into (A.21), we have

$\displaystyle{\tilde{T}}^{\mathrm{\ell}}=T_{\mathrm{c}}(p)\left(1-\frac{\phi}{2}\frac{N^{\mathrm{\ell}}}{N}\right),$
(D.9)
$\displaystyle{\tilde{T}}^{\mathrm{r}}=T_{\mathrm{c}}(p)\left(1+\frac{\phi}{2}\frac{N^{\mathrm{r}}}{N}\right).$
(D.10)

We can repeat a similar procedure starting with (VI.16). We expand the two
chemical potentials
$\mu^{\mathrm{\ell}}=\mu^{\mathrm{\ell}}({\tilde{T}}^{\mathrm{\ell}},p)$ and
$\mu^{\mathrm{r}}=\mu^{\mathrm{r}}({\tilde{T}}^{\mathrm{r}},p)$ as

$\displaystyle\mu^{\mathrm{\ell}}=\mu^{\mathrm{\ell}}(T_{\mathrm{c}}(p),p)-({\tilde{T}}^{\mathrm{\ell}}-T_{\mathrm{c}}(p))\hat{s}^{\mathrm{\ell}}+O(\varepsilon^{2}),$
(D.11)
$\displaystyle\mu^{\mathrm{r}}=\mu^{\mathrm{r}}(T_{\mathrm{c}}(p),p)-({\tilde{T}}^{\mathrm{r}}-T_{\mathrm{c}}(p))\hat{s}^{\mathrm{r}}+O(\varepsilon^{2}),$
(D.12)

where we have used $(\partial\mu/\partial T)_{p}=-\hat{s}$. Here,
${\tilde{T}}^{\mathrm{\ell}}$ and ${\tilde{T}}^{\mathrm{r}}$ are estimated by
(A.21) with ${\tilde{T}}^{\mathrm{\ell}}=1/{\tilde{\beta}}^{\mathrm{\ell}}$
and ${\tilde{T}}^{\mathrm{r}}=1/{\tilde{\beta}}^{\mathrm{r}}$. Substituting
these into (VI.16) and noting that
$\mu^{\mathrm{\ell}}(T_{\mathrm{c}}(p),p)=\mu^{\mathrm{r}}(T_{\mathrm{c}}(p),p)$,
we have

$\displaystyle(\hat{s}^{\mathrm{r}}-\hat{s}^{\mathrm{\ell}})\left({\tilde{T}}-T_{\mathrm{c}}(p)-\frac{\phi{\tilde{T}}}{2}\frac{N^{\mathrm{r}}-N^{\mathrm{\ell}}}{N}\right)=0.$
(D.13)

Since $\hat{s}^{\mathrm{r}}\neq\hat{s}^{\mathrm{\ell}}$, we obtain the same
relation as (D.8). That is, the steady state is equivalent.

We add a comment. Summing up the last two formulas and using (A.18) and (D.8),
we have the temperature relation

$\displaystyle{\tilde{T}}-\frac{T_{1}+T_{2}}{2}=\theta-T_{\mathrm{c}}(p),$
(D.14)

which was derived in global-JSP .

##  Appendix E Additivity of $G$ and $F$

According to (VI.1) and (VI.5), the free energies are expressed by additive
forms,

$\displaystyle G=G^{\mathrm{L}}+G^{\mathrm{G}},$ (E.1) $\displaystyle
F=F^{\mathrm{L}}+F^{\mathrm{G}},$ (E.2)

where $G^{\mathrm{L}}=L_{y}L_{z}\int_{0}^{X}dx~{}\mu^{\mathrm{L}}(x)\rho(x)$,
$G^{\mathrm{G}}=L_{y}L_{z}\int_{X}^{L_{x}}dx~{}\mu^{\mathrm{G}}(x)\rho(x)$,
$F^{\mathrm{L}}=L_{y}L_{z}\int_{0}^{X}dx~{}f(x)$, and
$F^{\mathrm{G}}=L_{y}L_{z}\int_{X}^{L_{x}}dx~{}f(x)$. As has been derived in
global-JSP  and shortly reviewed in Appendix A.1, respective global
thermodynamics hold in liquid and gas, and therefore

$\displaystyle
G^{\mathrm{L}}=\mu^{\mathrm{L}}({\tilde{T}}^{\mathrm{L}},p)N^{\mathrm{L}},\quad
G^{\mathrm{G}}=\mu^{\mathrm{G}}({\tilde{T}}^{\mathrm{G}},p)N^{\mathrm{G}},$
(E.3) $\displaystyle
F^{\mathrm{L}}=F({\tilde{T}}^{\mathrm{L}},V^{\mathrm{L}},N^{\mathrm{L}}),\quad
F^{\mathrm{G}}=F({\tilde{T}}^{\mathrm{G}},V^{\mathrm{G}},N^{\mathrm{G}})$
(E.4)

with respective global temperatures

$\displaystyle{\tilde{T}}^{\mathrm{L}}={\tilde{T}}\left(1-\phi\frac{N^{\mathrm{G}}}{2N}\right),\quad{\tilde{T}}^{\mathrm{G}}={\tilde{T}}\left(1+\phi\frac{N^{\mathrm{L}}}{2N}\right),$
(E.5)

which are consistent with (VI.13).

We notice
${\tilde{T}}^{\mathrm{L}}\neq{\tilde{T}}^{\mathrm{G}}\neq{\tilde{T}}$ for
$\varepsilon\neq 0$. Respecting the global temperature ${\tilde{T}}$, we
rewrite (E.1) and (E.2) as parallel expressions to the non-additive entropy
(IV.2);

$\displaystyle
G=G^{\mathrm{L}}({\tilde{T}},p,N^{\mathrm{L}})+G^{\mathrm{G}}({\tilde{T}},p,N^{\mathrm{G}})-\phi{\tilde{T}}{\Psi},$
(E.6) $\displaystyle
F=F({\tilde{T}},V^{\mathrm{L}},N^{\mathrm{L}})+F({\tilde{T}},V^{\mathrm{G}},N^{\mathrm{G}})-\phi{\tilde{T}}{\Psi}$
(E.7)

by expanding $G^{\mathrm{L/G}}$ and $F^{\mathrm{L/G}}$ around ${\tilde{T}}$
and ignoring the contribution of $O(\varepsilon^{2})$. Here,
$G^{\mathrm{L}}({\tilde{T}},p,N^{\mathrm{L}})=\mu^{\mathrm{L}}({\tilde{T}},p)N^{\mathrm{L}}$
and
$G^{\mathrm{G}}({\tilde{T}},p,N^{\mathrm{G}})=\mu^{\mathrm{G}}({\tilde{T}},p)N^{\mathrm{G}}$.
Despite the apparent consistency of (E.6) and (E.7) with (IV.2) for the non-
additive entropy, we do not adopt ${\tilde{T}}$ as the temperature to
characterize liquid and gas because the fundamental thermodynamic relations of
liquid and gas are (A.6) and (A.7) with ${\tilde{T}}^{\mathrm{L/G}}$ as each
global temperature. Indeed, these fundamental relations provide

$\displaystyle
dS^{\mathrm{L/G}}\neq\frac{dH^{\mathrm{L/G}}}{{\tilde{T}}}-\frac{V^{\mathrm{L/G}}}{{\tilde{T}}}dp-\frac{\mu^{\mathrm{L/G}}({\tilde{T}},p)}{{\tilde{T}}}dN^{\mathrm{L/G}},$
(E.8)

which implies that ${\tilde{T}}$ does not work as global temperature for
liquid or gas. Nevertheless, we can show

$\displaystyle
S^{\mathrm{L}}=\frac{H^{\mathrm{L}}-G^{\mathrm{L}}({\tilde{T}}^{\mathrm{L}},p,N^{\mathrm{L}})}{{\tilde{T}}^{\mathrm{L}}}=\frac{H^{\mathrm{L}}-G^{\mathrm{L}}({\tilde{T}},p,N^{\mathrm{L}})}{{\tilde{T}}},$
(E.9) $\displaystyle
S^{\mathrm{G}}=\frac{H^{\mathrm{G}}-G^{\mathrm{G}}({\tilde{T}}^{\mathrm{G}},p,N^{\mathrm{G}})}{{\tilde{T}}^{\mathrm{G}}}=\frac{H^{\mathrm{G}}-G^{\mathrm{G}}({\tilde{T}},p,N^{\mathrm{L}})}{{\tilde{T}}}.$
(E.10)

This suggests that $\tilde{T}$ plays some role of the temperature in each
region.

##  Appendix F Derivations of relations in Sec. VII

We derive various relations for the steady state in heat conduction reported
in Sec. VII with assuming $J\leq 0$. We can change $\phi$ and $J$ into
$|\phi|$ and $|J|$ for the obtained relations with taking care of $\phi J\leq
0$, because these formulas are common in $J\geq 0$. We then arrive at the
formulas in Sec. VII.

###  F.1 Preliminaries

We introduce a ratio representing the particle distribution as

$\displaystyle r\equiv\frac{1}{2}-\frac{N^{\mathrm{L}}}{N},$ (F.1)

by which we have

$\displaystyle\frac{N^{\mathrm{L}}}{N}=-r+\frac{1}{2},\quad\frac{N^{\mathrm{G}}}{N}=r+\frac{1}{2}.$
(F.2)

We first consider equilibrium liquid–gas coexistence with $(H,p,N)$ fixed. Let
$H^{\mathrm{L/G}}_{\mathrm{eq}}$ and $N^{\mathrm{L/G}}_{\mathrm{eq}}$ be the
enthalpy and the particle number of liquid and gas at equilibrium. In the
coexistence state at given $p$, there are constraints among
$H_{\mathrm{eq}}^{\mathrm{L/G}}$ and $N_{\mathrm{eq}}^{\mathrm{L/G}}$ such
that

$\displaystyle
H_{\mathrm{eq}}^{\mathrm{L}}=\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)N_{\mathrm{eq}}^{\mathrm{L}},\quad
H_{\mathrm{eq}}^{\mathrm{G}}=\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)N_{\mathrm{eq}}^{\mathrm{G}}.$
(F.3)

Combining with $H=H_{\mathrm{eq}}^{\mathrm{L}}+H_{\mathrm{eq}}^{\mathrm{G}}$
and $N=N_{\mathrm{eq}}^{\mathrm{L}}+N_{\mathrm{eq}}^{\mathrm{G}}$, we have

$\displaystyle
r_{\mathrm{eq}}=\frac{\hat{h}-\hat{h}_{\mathrm{m}}(p)}{\hat{q}(p)},$ (F.4)

where

$\displaystyle\hat{h}_{\mathrm{m}}(p)=\frac{\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)+\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)}{2}.$
(F.5)

###  F.2 Derivation of (VII.16)

The formulas (D.8), (D.9), and (D.10) for the global temperatures
${\tilde{T}}$, ${\tilde{T}}^{\mathrm{L}}$, and ${\tilde{T}}^{\mathrm{G}}$ are
rewritten with $r$ as

$\displaystyle{\tilde{T}}=T_{\mathrm{c}}(1+\phi r)+O(\varepsilon^{2}),$ (F.6)
$\displaystyle{\tilde{T}}^{\mathrm{L}}=T_{\mathrm{c}}\left(1+\frac{\phi}{2}\left(r-\frac{1}{2}\right)\right),$
(F.7)
$\displaystyle{\tilde{T}}^{\mathrm{G}}=T_{\mathrm{c}}\left(1+\frac{\phi}{2}\left(r+\frac{1}{2}\right)\right).$
(F.8)

By substituting (F.4) with (F.5) into these equations, we obtain

$\displaystyle{\tilde{T}}=T_{\mathrm{c}}\left(1+\phi\frac{\hat{h}-\hat{h}_{\mathrm{m}}}{\hat{q}}\right),$
(F.9)
$\displaystyle{\tilde{T}}^{\mathrm{L}}=T_{\mathrm{c}}\left(1-\frac{\phi}{2}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right),$
(F.10)
$\displaystyle{\tilde{T}}^{\mathrm{G}}=T_{\mathrm{c}}\left(1+\frac{\phi}{2}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right).$
(F.11)

Here, (F.9) corresponds to (VII.16).

###  F.3 Derivation of (VII.9), (VII.10) and (VII.7), (VII.8)

As demonstrated in Appendix A.1, heat conduction states for liquid and gas in
the linear response regime are mapped to equilibrium states. We then express

$\displaystyle
H^{\mathrm{L}}=\hat{h}^{\mathrm{L}}({\tilde{T}}^{\mathrm{L}},p)N^{\mathrm{L}}.$
(F.12)

Substituting (F.10) into it and then expanding $\hat{h}^{\mathrm{L}}$ in
$\varepsilon$, we obtain

$\displaystyle
H^{\mathrm{L}}=\left(\hat{h}_{\mathrm{c}}^{\mathrm{L}}-\phi\frac{T_{\mathrm{c}}}{2}\hat{c}_{p}^{\mathrm{L}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)N^{\mathrm{L}},$
(F.13)

which is (VII.9). Similarly, we have

$\displaystyle
H^{\mathrm{G}}=\left(\hat{h}_{\mathrm{c}}^{\mathrm{G}}+\phi\frac{T_{\mathrm{c}}}{2}\hat{c}_{p}^{\mathrm{G}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)N^{\mathrm{G}},$
(F.14)

which is (VII.10). Recalling that $H=H^{\mathrm{L}}+H^{\mathrm{G}}$ and
applying (F.2) and (F.4), we have

$\displaystyle\hat{h}=\left(\frac{1}{2}-r\right)\hat{h}_{\mathrm{c}}^{\mathrm{L}}+\left(\frac{1}{2}+r\right)\hat{h}_{\mathrm{c}}^{\mathrm{G}}$
$\displaystyle-\phi\frac{T_{\mathrm{c}}}{2}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right].$
(F.15)

Solving (F.15) in $r$, we obtain

$\displaystyle
r=\frac{\hat{h}-\hat{h}_{\mathrm{m}}}{\hat{q}}+\phi\frac{T_{\mathrm{c}}}{2\hat{q}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right].$
(F.16)

Thus, the value of $r$ is uniquely determined for given $\hat{h}$ and $p$.
Explicitly, the number of particles is written as

$\displaystyle\frac{N^{\mathrm{L}}}{N}=\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}-\phi\frac{T_{\mathrm{c}}}{2\hat{q}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right],$
(F.17)
$\displaystyle\frac{N^{\mathrm{G}}}{N}=\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}+\phi\frac{T_{\mathrm{c}}}{2\hat{q}}\left[\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right],$
(F.18)

which correspond to (VII.7) and (VII.8).

###  F.4 Derivation of (VII.11) and (VII.12)

The volumes of liquid and gas are connected to the densities as

$\displaystyle
V^{\mathrm{L/G}}=\frac{N^{\mathrm{L/G}}}{\rho^{\mathrm{L/G}}}=\frac{N^{\mathrm{L/G}}}{N}\frac{N}{\rho^{\mathrm{L/G}}}.$
(F.19)

By substituting (F.7) and (F.8) into
$\rho^{\mathrm{L}}=\rho^{\mathrm{L}}({\tilde{T}}^{\mathrm{L}},p)$ and
$\rho^{\mathrm{G}}=\rho^{\mathrm{G}}({\tilde{T}}^{\mathrm{G}},p)$ and
expanding them in $\varepsilon$, we have

$\displaystyle\rho^{\mathrm{L}}=\rho_{\mathrm{c}}^{\mathrm{L}}\left[1-\phi\frac{T_{\mathrm{c}}}{2}\left(r-\frac{1}{2}\right)\alpha_{\mathrm{c}}^{\mathrm{L}}\right],$
(F.20)
$\displaystyle\rho^{\mathrm{G}}=\rho_{\mathrm{c}}^{\mathrm{G}}\left[1-\phi\frac{T_{\mathrm{c}}}{2}\left(r+\frac{1}{2}\right)\alpha_{\mathrm{c}}^{\mathrm{G}}\right],$
(F.21)

where $\alpha^{\mathrm{L}}$ and $\alpha^{\mathrm{G}}$ are thermal expansion
coefficients of liquid and gas at constant pressure, defined by

$\displaystyle\alpha^{\mathrm{L}}=-\frac{1}{\rho^{\mathrm{L}}}\left(\frac{\partial\rho^{\mathrm{L}}}{\partial
T}\right)_{p},\quad\alpha^{\mathrm{G}}=-\frac{1}{\rho^{\mathrm{G}}}\left(\frac{\partial\rho^{\mathrm{G}}}{\partial
T}\right)_{p}.$ (F.22)

$\alpha_{\mathrm{c}}^{\mathrm{L}}$ and $\alpha_{\mathrm{c}}^{\mathrm{G}}$ are
the values at the saturation. Thus, (F.19) leads to

$\displaystyle
V^{\mathrm{L}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{L}}}\frac{N^{\mathrm{L}}}{N}\left[1+\phi\frac{T_{\mathrm{c}}}{2}\left(r-\frac{1}{2}\right)\alpha_{\mathrm{c}}^{\mathrm{L}}\right],$
(F.23) $\displaystyle
V^{\mathrm{G}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{G}}}\frac{N^{\mathrm{G}}}{N}\left[1+\phi\frac{T_{\mathrm{c}}}{2}\left(r+\frac{1}{2}\right)\alpha_{\mathrm{c}}^{\mathrm{G}}\right].$
(F.24)

Substituting (F.16), (F.17), and (F.18) into these, $V^{\mathrm{L}}$ and
$V^{\mathrm{G}}$ for given $(\hat{h},p,\phi)$ are written as

$\displaystyle
V^{\mathrm{L}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{L}}}\left[\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}-\phi\frac{T_{\mathrm{c}}}{2}\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\alpha_{\mathrm{c}}^{\mathrm{L}}\right.$
$\displaystyle~{}~{}~{}\left.-\phi\frac{T_{\mathrm{c}}}{2\hat{q}}\left(\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right)\right],$
(F.25) $\displaystyle
V^{\mathrm{G}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{G}}}\left[\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}+\phi\frac{T_{\mathrm{c}}}{2}\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\alpha_{\mathrm{c}}^{\mathrm{L}}\right.$
$\displaystyle~{}~{}~{}\left.+\phi\frac{T_{\mathrm{c}}}{2\hat{q}}\left(\left(\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{L}}-\left(\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}\right)^{2}\hat{c}_{p}^{\mathrm{G}}\right)\right],$
(F.26)

which are (VII.11) and (VII.12).

###  F.5 Derivation of (VII.13)

We define the normalized position of the liquid–gas interface as

$\displaystyle
x\equiv\frac{X}{L_{x}}=\frac{V^{\mathrm{L}}}{V^{\mathrm{L}}+V^{\mathrm{G}}}.$
(F.27)

From (F.25) and (F.26), the volumes in equilibrium are written as

$\displaystyle
V_{\mathrm{eq}}^{\mathrm{L}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{L}}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}},\quad
V_{\mathrm{eq}}^{\mathrm{G}}=\frac{N}{\rho_{\mathrm{c}}^{\mathrm{G}}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}},$
(F.28)

and the ratio of the volume in heat conduction deviates from the equilibrium
ratio as

$\displaystyle\frac{V^{\mathrm{G}}}{V^{\mathrm{L}}}=\frac{V_{\mathrm{eq}}^{\mathrm{G}}}{V_{\mathrm{eq}}^{\mathrm{L}}}\left(1+\phi\frac{T_{\mathrm{c}}}{2}A\right),$
(F.29)

where

$\displaystyle A$
$\displaystyle=\alpha_{\mathrm{c}}^{\mathrm{L}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}+\alpha_{\mathrm{c}}^{\mathrm{G}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}+\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}-\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}.$
(F.30)

From (F.27) and (F.29), it is found that

$\displaystyle x-x_{\mathrm{eq}}$
$\displaystyle=-\phi\frac{T_{\mathrm{c}}}{2}\left(1+\frac{V_{\mathrm{eq}}^{\mathrm{G}}}{V_{\mathrm{eq}}^{\mathrm{L}}}\right)^{-2}\frac{V_{\mathrm{eq}}^{\mathrm{G}}}{V_{\mathrm{eq}}^{\mathrm{L}}}A.$
(F.31)

Here, the equilibrium position $x_{\mathrm{eq}}$ is calculated from (F.27) and
(F.28) as

$\displaystyle
x_{\mathrm{eq}}=\frac{\rho_{\mathrm{c}}^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}{\rho_{\mathrm{c}}^{\mathrm{L}}(\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}})+\rho^{\mathrm{G}}(\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h})}.$
(F.32)

Since $V_{\mathrm{eq}}^{\mathrm{G}}/V_{\mathrm{eq}}^{\mathrm{L}}$ is connected
to $x_{\mathrm{eq}}$, we have

$\displaystyle
x-x_{\mathrm{eq}}=-\phi\frac{T_{\mathrm{c}}}{2}x(1-x_{\mathrm{eq}})A$
$\displaystyle=-\phi\frac{T_{\mathrm{c}}}{2}x_{\mathrm{eq}}(1-x_{\mathrm{eq}})\times$
$\displaystyle\left[\alpha_{\mathrm{c}}^{\mathrm{L}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{q}}+\alpha_{\mathrm{c}}^{\mathrm{G}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{q}}+\frac{\hat{c}_{p}^{\mathrm{L}}}{\hat{q}}\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}-\frac{\hat{c}_{p}^{\mathrm{G}}}{\hat{q}}\frac{\hat{h}-\hat{h}_{\mathrm{c}}^{\mathrm{L}}}{\hat{h}_{\mathrm{c}}^{\mathrm{G}}-\hat{h}}\right],$
(F.33)

which corresponds to (VII.13).

###  F.6 Derivation of (VII.1)

Since the steady heat current $J$ is homogeneous in the container, we have

$\displaystyle J=-\kappa_{\mathrm{c}}^{\mathrm{L}}\frac{\theta-
T_{1}}{X}=-\kappa_{\mathrm{c}}^{\mathrm{G}}\frac{T_{2}-\theta}{L_{x}-X},$
(F.34)

which leads to

$\displaystyle\theta=\frac{\kappa_{\mathrm{c}}^{\mathrm{L}}(L_{x}-X)T_{1}+\kappa_{\mathrm{c}}^{\mathrm{G}}XT_{2}}{\kappa_{\mathrm{c}}^{\mathrm{L}}(L_{x}-X)+\kappa_{\mathrm{c}}^{\mathrm{G}}X}.$
(F.35)

Using this expression, (F.34) is written as

$\displaystyle
J=-\frac{\kappa_{\mathrm{c}}^{\mathrm{L}}\kappa_{\mathrm{c}}^{\mathrm{G}}}{\kappa_{\mathrm{c}}^{\mathrm{L}}(1-x_{\mathrm{eq}})+\kappa_{\mathrm{c}}^{\mathrm{G}}x_{\mathrm{eq}}}\frac{T_{2}-T_{1}}{L_{x}}.$
(F.36)

Substituting (F.32) into (F.36), we arrive at (VII.1).

###  F.7 Derivation of (VII.2)

The uniformity of the heat current in $x$,

$\displaystyle J=-\kappa_{\mathrm{c}}^{\mathrm{L}}\frac{\theta-
T_{1}}{X}=-\kappa_{\mathrm{c}}^{\mathrm{G}}\frac{T_{2}-\theta}{L_{x}-X},$
(F.37)

leads to

$\displaystyle{\tilde{T}}^{\mathrm{L}}=\theta+J\frac{X}{2\kappa_{\mathrm{c}}^{\mathrm{L}}},$
(F.38)
$\displaystyle\phi=-\frac{J}{T_{\mathrm{c}}(p)}\left(\frac{X}{\kappa_{\mathrm{c}}^{\mathrm{L}}}+\frac{L_{x}-X}{\kappa_{\mathrm{c}}^{\mathrm{G}}}\right),$
(F.39)

where ${\tilde{T}}^{\mathrm{L}}=(T_{1}+\theta)/2$ and
$\phi=(T_{2}-T_{1})/T_{\mathrm{c}}(p)$. Substituting (F.38) and (F.39) into
(D.9), we obtain

$\displaystyle\theta=T_{\mathrm{c}}(p)-\frac{JL_{x}}{2}\left(\frac{X}{L_{x}}\frac{N^{\mathrm{G}}}{N}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{L_{x}-X}{L_{x}}\frac{N^{\mathrm{L}}}{N}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}\right).$
(F.40)

Since (F.19) yields

$\displaystyle\frac{N^{\mathrm{L}}}{N}=\frac{X}{L_{x}}\frac{\rho^{\mathrm{L}}}{\bar{\rho}},\quad\frac{N^{\mathrm{G}}}{N}=\frac{L_{x}-X}{L_{x}}\frac{\rho^{\mathrm{G}}}{\bar{\rho}},$
(F.41)

where $\bar{\rho}=N/V$, we write (F.40) as

$\displaystyle\theta=T_{\mathrm{c}}(p)-J\left(\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{\rho_{\mathrm{c}}^{\mathrm{L}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}\right)\frac{X(L_{x}-X)}{2L_{x}}.$
(F.42)

Using

$\displaystyle\bar{\rho}=\frac{X}{L_{x}}\rho^{\mathrm{L}}+\frac{L_{x}-X}{L_{x}}\rho^{\mathrm{G}},$
(F.43)

we have

$\displaystyle\frac{\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{\rho_{\mathrm{c}}^{\mathrm{L}}}{\bar{\rho}}\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}$
$\displaystyle=\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}+\left(\frac{X}{\kappa_{\mathrm{c}}^{\mathrm{L}}L_{x}}+\frac{L_{x}-X}{\kappa_{\mathrm{c}}^{\mathrm{G}}L_{x}}\right)\frac{\rho_{\mathrm{c}}^{\mathrm{L}}-\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}$
(F.44)
$\displaystyle=\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}-\frac{\phi
T_{\mathrm{c}}}{JL_{x}}\frac{\rho_{\mathrm{c}}^{\mathrm{L}}-\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}},$
(F.45)

where we have applied (F.36) to transform into the second line. Substituting
this into (F.42), we obtain

$\displaystyle\theta=T_{\mathrm{c}}(p)-\left[J\left(\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{L}}}-\frac{1}{\kappa_{\mathrm{c}}^{\mathrm{G}}}\right)+\frac{\phi
T_{\mathrm{c}}}{L_{x}}\frac{\rho_{\mathrm{c}}^{\mathrm{L}}-\rho_{\mathrm{c}}^{\mathrm{G}}}{\bar{\rho}}\right]\frac{X(L_{x}-X)}{2L_{x}}.$
(F.46)

which corresponds to (VII.2) for $J<0$. We can repeat the same argument for
$J>0$ and conclude (VII.2).

##  Appendix G Derivation of (VIII.1)

We start with an expansion in $\phi$ as

$\displaystyle S(H,p,N,\phi)$ $\displaystyle=S(H,p,N)+\phi\left(\frac{\partial
S}{\partial\phi}\right)_{H,p,N}\\!+O(\varepsilon^{2})$
$\displaystyle=S(H,p,N)+\phi\Psi+O(\varepsilon^{2}),$ (G.1)

where we used (IV.16) to obtain the second line. Let $H_{\phi}$ be a slightly
shifted enthalpy from $H$, whose form will be determined. Assuming
$H-H_{\phi}=O(\varepsilon)$, we have

$\displaystyle S(H,p,N)=S(H_{\phi},p,N)+\left(\frac{\partial S}{\partial
H}\right)_{p,N}\\!\\!\\!(H-H_{\phi})+O(\varepsilon^{2}).$ (G.2)

Comparing this form with (G.1), we find that the term $\phi\Psi$ in (G.1)
cancels out by setting $H_{\phi}$ to

$\displaystyle H_{\phi}=H+T_{\mathrm{c}}(p)\phi\Psi.$ (G.3)

Thus, we can write

$\displaystyle S(H,p,N,\phi)=S(H_{\phi},p,N)$ (G.4)

in the linear response regime. We interpret that (G.4) defines an isentropic
surface to connect a heat conduction state $(H,p,N,\phi)$ with an equilibrium
state $(H_{\phi},p,N)$.

Since the right-hand side of (G.4) is the equilibrium entropy, the
nonequilibrium entropy can be expressed by the sum of the entropy of liquid
and gas in (VIII.1); i.e.,

$\displaystyle
S(H,p,N,\phi)=S(H_{\phi}^{\mathrm{L}},p,N_{\phi}^{\mathrm{L}})+S(H_{\phi}^{\mathrm{G}},p,N_{\phi}^{\mathrm{G}})$
(G.5)

with $N=N_{\phi}^{\mathrm{L}}+N_{\phi}^{\mathrm{G}}$ and
$H_{\phi}=H_{\phi}^{\mathrm{L}}+H_{\phi}^{\mathrm{G}}$. We emphasize that
$H_{\phi}\neq H$ indicates $H\neq
H_{\phi}^{\mathrm{L}}+H_{\phi}^{\mathrm{G}}$. This may be another aspect of
the violation of the additivity of the liquid–gas coexistence in heat
conduction.

We can identify $H_{\phi}^{\mathrm{L/G}}$ and $N_{\phi}^{\mathrm{L/G}}$ as
functions of $(H,p,N,\phi)$. Substituting
$H_{\phi}^{\mathrm{L}}=\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)N_{\phi}^{\mathrm{L}}$,
$H_{\phi}^{\mathrm{G}}=\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)N_{\phi}^{\mathrm{G}}$,
and (G.3) into $N=N_{\phi}^{\mathrm{L}}+N_{\phi}^{\mathrm{G}}$ and
$H_{\phi}=H_{\phi}^{\mathrm{L}}+H_{\phi}^{\mathrm{G}}$, we obtain

$\displaystyle
N_{\phi}^{\mathrm{L}}=\frac{\hat{h}_{\mathrm{c}}^{\mathrm{G}}(p)N-H-
T_{\mathrm{c}}(p)\phi\Psi}{\hat{q}(p)},$ (G.6) $\displaystyle
N_{\phi}^{\mathrm{G}}=-\frac{\hat{h}_{\mathrm{c}}^{\mathrm{L}}(p)N-H-
T_{\mathrm{c}}(p)\phi\Psi}{\hat{q}(p)}.$ (G.7)

Comparing (G.6) with (F.17), we confirm $N_{\phi}^{\mathrm{L}}\neq N^{L}$ for
$\phi\neq 0$.

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