text:
\section{Introduction}
\label{sec:1}

The mathematical theory of phase transitions in continuum particle
systems has much fewer results as compared to its counterpart
dealing with discrete underlying sets like lattices, graphs, etc. It
is then quite natural that the first steps in such theories are
being made by employing various mean field models. In \cite{LebP},
the mean field approach was mathematically realized by using a
Kac-like infinite range attraction combined with a two-body
repulsion. By means of rigorous upper and lower bounds for the
canonical partition function obtained in that paper the authors
derived the equation of state indicating the possibility of a
first-order phase transition. Later on, this result was employed in
\cite{Lebow,presutti} to go beyond the mean field, see  also
\cite{pulvirenti,PT} for recent results. Another way of realizing
the mean-field approach is to use Curie-Weiss interactions and then
appropriate methods of calculating asymptotics of integrals, cf.
\cite{EN}. Quite recently, this way was formulated as a coherent
mathematical theory based on the large deviation techniques, in the
framework of which the Gibbs states (thermodynamic phases) of the
system are constructed as probability measures on an appropriate
phase space, see \cite[Section II]{Opoku}.

In this work, we introduce a simple Curie-Weiss type model of a
single-sort continuum particle system in which the space
$\mathds{R}^d$ is divided into congruent (cubic) cells. For a
bounded region $V\subset\mathds{R}^d$ consisting of $N$ such cells,
the attraction between each two particles in $V$ is set to be
$J_1/N$, regardless their positions. If such two particles lie in
the same cell, they repel each other with intensity $J_2>J_1$.
Unlike to \cite{LebP}, we deal with the grand canonical ensemble.
Therefore, our initial thermodynamic variables are the inverse
temperature $\beta = 1/k_B T$ and the physical chemical potential.
However, for the sake of convenience we employ the variables
$p=\beta J_1$ and $\mu = \beta\times$\textit{(physical chemical
potential)} and define single-phase domains of the half-plane
$\{(p,\mu): p>0, \mu \in \mathds{R}\}$ (see Definition \ref{1df}) by
a condition that guaranties the existence of a unique $\bar{y}\in
\mathds{R}$, which determines a probability measure
$\mathbf{Q}_{p,\mu}$, given in (\ref{19a}) and (\ref{19}). This
measure is set to be the thermodynamic phase of the system. The
points $(p,\mu)$ where the mentioned single-phase condition fails to
hold due to the existence of multiple $\bar{y}$ correspond to the
coexistence of multiple thermodynamic phases. In Theorem \ref{4pn},
we show that there exists $p_0>0$ such that $\mathcal{R}(p_0):=\{(p,
\mu): p\in (0,p_0), \mu \in \mathds{R}\}$ is a single-phase domain,
that is, there is no phase-coexistence point in the strip
$\mathcal{R}(p_0)$ whenever the attraction is sufficiently small.
Note that, for some models on graphs, see, e.g., the example given
in \cite{KK}, there exist multiple phases for each positive
attraction. Thereafter, in Theorem \ref{1tm} we show that, for each
value of $J_2/J_1:=a>1$, there exist points in which two phases do
coexist. Namely,  we show that there exists $p_1>0$ such that, for
each $p\geq p_1$, there exists $\mu_c (p)\in \mathds{R}$ and small
enough $\epsilon >0$ such that the sets $\{(p,\mu): \mu\in ( \mu_c
(p)- \epsilon, \mu_c (p))\}$ and $\{(p,\mu): \mu\in (\mu_c(p), \mu_c
(p) + \epsilon)\}$ lie in different single-phase domains, and that
there exist at least two $\bar{y}$ whenever $\mu = \mu_c(p)$. In
Section 3, we present a number of results of the corresponding
numerical calculations which illustrate the facts proved in Theorems
\ref{4pn} and \ref{1tm}.

\section{The Model}
\label{sec:2}

By $\mathds{N}$, $\mathds{R}$ and $\mathds{C}$ we denote the sets of
natural, real and complex numbers, respectively. We also put
$\mathds{N}_0 = \mathds{N}\cup \{0\}$. For $d \in \mathds{N}$, by
$\mathds{R}^d$ we denote the Euclidean space of vectors $x=(x^1,
\dots, x^d)$, $x^i\in \mathds{R}$. In the sequel, its dimension $d$
will be fixed. By $ d x$ we mean the Lebesgue measure on
$\mathds{R}^d$.
 In this section,
we use some tools of the analysis on configuration spaces the main
aspects of which can be found in \cite{Albev}.

\subsection{The integration}
\label{ssec:2.1} For $n\in \mathds{N}$, let $\Gamma^{(n)}$ be the
set of all $n$-point subsets of $\mathds{R}^d$. Each such a subset,
$\gamma\in \Gamma^{(n)}$ (called \emph{configuration}), is  an
$n$-element set of distinct points $x\in \mathds{R}^d$. Let also
$\Gamma^{(0)}$ be the one--element set consisting of the empty
configuration. Each $\Gamma^{(n)}$ is equipped with the topology
related to the Euclidean topology of $\mathds{R}^d$. Then we define
\begin{equation*}

\Gamma_0 = \bigsqcup_{n \in \mathds{N}_0} \Gamma^{(n)},
\end{equation*}
that is, $\Gamma_0$ is the topological sum of the spaces
$\Gamma^{(n)}$. We equip $\Gamma_0$ with the corresponding Borel
$\sigma$-field $\mathcal{B}(\Gamma_0)$ that makes
$(\Gamma_0,\mathcal{B}(\Gamma_0))$ a standard Borel space. A
function $G: \Gamma_0 \to \mathds{R}$ is
$\mathcal{B}(\Gamma_0)$-measurable if and only if, for each $n\in
\mathds{N}$, there exists a symmetric Borel function
$G^{(n)}:(\mathds{R}^d)^n \to \mathds{R}$ such that $G(\gamma) =
G^{(n)}(x_1 , \dots x_n)$ for $\gamma = \{x_ 1, \dots , x_n\}$. For
such a function $G$, we also set $G^{(0)}= G(\varnothing)$. The
Lebesgue-Poisson measure $\lambda$ on
$(\Gamma_0,\mathcal{B}(\Gamma_0))$ is defined by the relation
\begin{equation}
  \label{2}
\int_{\Gamma_0} G (\gamma) \lambda ( d \gamma) = G^{(0)} +
\sum_{n=1}^\infty \frac{1}{n!} \int_{(\mathds{R}^d)^n} G^{(n)} ( x_1
, \dots , x_n) d x_1 \cdots d x_n,
\end{equation}
which has to hold for all measurable $G:\Gamma_0 \to \mathds{R}_+
:=[0, +\infty)$ for which the right-hand side of (\ref{2}) is
finite.

For some $c>0$, we let $\Delta= (-c/2, c/2]^d\subset \mathds{R}^d$
be a cubic cell of volume $\upsilon= c^d$ centered at the origin.
Let also $V\subset \mathds{R}^d$ be the union of $N\in \mathds{N}$
disjoint translates $\Delta_\ell$ of $\Delta$, i.e.,
\begin{equation*}
V = \bigcup_{\ell =1}^N \Delta_\ell.

\end{equation*}
As is usual for Curie-Weiss theories, cf. \cite{EN,Opoku}, the form
of the interaction energy of the system of particles placed in $V$
depends on $V$. In our model, the energy of a configuration $\gamma
\subset V$ is
\begin{eqnarray}
\label{4} W_N (\gamma) & = & \frac{1}{2} \sum_{x, y \in \gamma}
\Phi_N
(x, y), \nonumber \\[.2cm]
\Phi_N (x, y) & = & - J_1/N + J_2 \sum_{\ell =1}^N
\mathds{I}_{\Delta_\ell}(x) \mathds{I}_{\Delta_\ell}(y),
\end{eqnarray}
where $\mathds{I}_{\Delta_\ell}$ is the indicator of $\Delta_\ell$,
that is, $\mathds{I}_{\Delta_\ell}(x)=1$ if $x\in \Delta_\ell$ and
$\mathds{I}_{\Delta_\ell}(x) =0$ otherwise. For convenience, in
$W_N$ above we have included the self-interaction term
$\Phi_N(x,x)$, which does not affect the physics of the model. We
also write $W_N$ and $\Phi_N$ instead of $W_V$ and $\Phi_V$ since
these quantities depend only on the number of cells in $V$. The
first term in $\Phi_N$ with $J_1>0$ describes attraction. By virtue
of the Curie-Weiss approach, it is taken equal for all particles.
The second term with $J_2>0$ describes repulsion between two
particles contained in one and the same cell. That is, in our model
every two particles in $V$ attract each other independently of their
location, and repel if they are in the same cell. The intensities
$J_1$ and $J_2$ are assumed to satisfy the following condition
\begin{equation}
  \label{5}
  J_2 > J_1.
\end{equation}
The latter is to secure the stability of the interaction
\cite{Ruelle}, that is, to satisfy
\[
\int_V \Phi_N (x, y) d y >0, \qquad {\rm for} \ {\rm all} \ x \in V.
\]
Let $\beta=1/k_{\rm B}T$ be the inverse temperature. To optimize the
choice of the thermodynamic variables we introduce the following
ones
\begin{equation}
  \label{7a}
p = \beta J_1, \qquad a = J_2/J_1,
\end{equation}
and the dimensionless chemical potential $\mu= \beta
\times$(physical chemical potential). Then  $(p, \mu)\in
\mathds{R}_{+}\times \mathds{R}$ is considered as the basic set of
thermodynamic variables, whereas $a$ and $\upsilon$ are model
parameters.

The grand canonical partition function in region $V$ is
\begin{eqnarray}
  \label{6}
 \Xi_N (p, \mu)& = & \int_{\Gamma_V} \exp\bigg{(} \mu |\gamma| - \beta W_N (\gamma)
 \bigg{)} \lambda ( d \gamma), \\[.2cm]
 & = & \int_{\Gamma_V} \exp\bigg{(} \mu |\gamma| +
 \frac{p}{2N}|\gamma|^2 - \frac{ap}{2}\sum_{x,y\in \gamma}
 \sum_{\ell=1}^N \mathds{I}_{\Delta_\ell}(x)\mathds{I}_{\Delta_\ell}(y
 ) \bigg{)} \lambda ( d \gamma), \nonumber
\end{eqnarray}
where $|\gamma|$ stands for the number of points in configuration
$\gamma$ and $\Gamma_V$ is the subset of $\Gamma_0$ consisting of
all $\gamma$ contained in $V$. We write $\Xi_N$ instead of $\Xi_V$
for the  reasons mentioned above.

\subsection{Transforming the partition function}

Now we use the concrete form of the energy as in (\ref{4}) to bring
(\ref{6}) to a more convenient form. For a given $\ell =1, \dots ,
N$ and a configuration $\gamma\in \Gamma_V$, we set $\gamma_\ell =
\gamma \cap \Delta_\ell$, that is, $\gamma_\ell$ is the part of the
configuration contained in $\Delta_\ell$. Then $|\gamma_\ell|$ will
stand for the number of points of $\gamma$ contained in
$\Delta_\ell$. Note that
\begin{equation}
  \label{6a}
|\gamma_\ell| = \sum_{x\in \gamma_\ell} 1 = \sum_{x\in \gamma}
\mathds{I}_{\Delta_\ell}(x).
\end{equation}
Then, cf. (\ref{4}) and (\ref{6}),
\begin{equation*}
 \sum_{x,y\in \gamma} \Phi_N (x, y) = \sum_{\ell, \ell'
=1}^N \sum_{x\in \Delta_\ell} \sum_{y \in \Delta_{\ell'}} \Phi_N (x,
y)  = - \frac{J_1}{N} \left( \sum_{\ell =1}^N |\gamma_\ell|\right)^2
+ J_2 \sum_{\ell =1}^N \left\vert \gamma_\ell \right\vert^2.
\end{equation*}
To rewrite the integrand in (\ref{6}) in a more convenient form we
set
\begin{equation}
  \label{7b}
F_N(\varrho, p, \mu) = \exp\bigg{(}\frac{p}{2N}
\left(\sum_{\ell=1}^N \varrho_\ell \right)^2 +  \mu \sum_{\ell=1}^N
\varrho_\ell - \frac{a p }{2}\sum_{\ell=1}^N \varrho_\ell^2
\bigg{)},
\end{equation}
where $\varrho \in \mathds{N}_0^N$ is a vector with nonnegative
integer components $\varrho_\ell$, $\ell = 1, 2, \dots , N$. Then
(\ref{6}) takes the form
\begin{equation}
  \label{8}
 \Xi_N( p, \mu) = \int_{\Gamma_V} F_N (\nu(\gamma), p, \mu) \lambda ( d
 \gamma),
\end{equation}
where $F_N$ is as in (\ref{7b}) and $\nu(\gamma)\in \mathds{N}_0^N$
is the vector with component $|\gamma_\ell|$, $\ell = 1, \dots , N$.
For $n, m \in \mathds{N}_0$, the Kronecker $\delta$-symbol can be
written
\begin{equation*}

 \delta_{nm} = \int_0^1 \exp \bigg{(} 2 \pi {\rm i}t (n-m)\bigg{)} d t,
 \qquad {\rm i} = \sqrt{-1}.
\end{equation*}
Applying this in (\ref{8}) we get
\begin{eqnarray}
  \label{10}
 \Xi_N(p, \mu) &  = &   \sum_{\varrho \in \mathds{N}_0^N}
F_N(\varrho, p, \mu) \\[.2cm] \nonumber &\times & \int_{\Gamma_V} \int_{[0,1]^N} \exp\bigg{(}
2 \pi {\rm i} \sum_{\ell=1}^N \left( \varrho_\ell - |\gamma_\ell|
\right)t_\ell
\bigg{)} \lambda ( d \gamma) d t_1 \cdots d t_N \nonumber \\[.2cm]
\nonumber
 & = & \sum_{\varrho \in \mathds{N}_0^N} F_N(\varrho, p, \mu)\\[.2cm]
\nonumber & \times & \int_{[0,1]^N} \exp\bigg{(} 2 \pi {\rm i}
\sum_{\ell=1}^N \varrho_\ell t_\ell \bigg{)} R_N (t_1 , \dots , t_N)
d t_1 \cdots d t_N. \qquad
\end{eqnarray}
Here
\begin{eqnarray}
\label{10a} R_N (t_1 , \dots , t_N) & = & \int_{\Gamma_V}
\exp\bigg{(} - 2 \pi {\rm i} \sum_{\ell=1}^N |\gamma_\ell| t_\ell
\bigg{)} \lambda ( d \gamma)
\nonumber \\[.2cm] & = &\int_{\Gamma_V}  \exp\bigg{(} - 2 \pi {\rm
i} \sum_{\ell=1}^N \sum_{x\in \gamma} \mathds{I}_{\Delta_\ell} (x)
t_\ell \bigg{)}
 \lambda ( d \gamma) \\[.2cm]
& = &\sum_{n=0}^\infty \frac{1}{n!} \int_{V^n} \exp\bigg{(} - 2 \pi
{\rm i} \sum_{\ell=1}^N \sum_{j=1}^n \mathds{I}_{\Delta_\ell}(x_j)
t_\ell \bigg{)} d x_1 \cdots d x_n. \nonumber
\end{eqnarray}
In getting the second line of (\ref{10a}), we use (\ref{6a}), and
then the integral with $\lambda$ is written according to (\ref{2}).
Note that the expression under the integral in the last line of
(\ref{10a}) factors in $j$, which allows for writing it in the form
\begin{eqnarray*}
R_N (t_1 , \dots , t_N)& = & \sum_{n=0}^\infty \frac{1}{n!}\left(
\int_{V} \exp\bigg{(} - 2 \pi {\rm i} \sum_{\ell=1}^N
\mathds{I}_{\Delta_\ell}(x) t_\ell \bigg{)}
d x\right)^n \\[.2cm]
& = & \sum_{n=0}^\infty \frac{1}{n!} \left( \sum_{\ell=1}^N
\int_{\Delta_\ell} \exp\bigg{(} - 2 \pi {\rm i} t_\ell \bigg{)} d x
\right)^n \\[.2cm] & = & \exp\bigg{(} \upsilon \sum_{\ell =1}^N \exp\bigg{(} -
2 \pi {\rm i} t_\ell \bigg{)} \bigg{)} .
\end{eqnarray*}
Now we apply this in (\ref{10}) and obtain
\begin{eqnarray}
  \label{11}
 \Xi_N (p, \mu) & = & \sum_{\varrho\in \mathds{N}_0^N} F_N(\varrho, p, \mu)
 \prod_{\ell=1}^N \left(
 \frac{\upsilon^{\varrho_\ell}}{\varrho_\ell!}\right) \nonumber \\[.2cm] &
 = &
\sum_{\varrho\in \mathds{N}_0^N} \exp\bigg{(} \frac{p}{2N} \left(
\sum_{\ell=1}^N \varrho_\ell\right)^2\bigg{)} \prod_{\ell=1}^N \pi
(\varrho_\ell, \mu),
\end{eqnarray}
where $p$ is as in (\ref{7a}) and
\begin{equation}
  \label{12}
\pi (n, \mu) = \frac{\upsilon^n}{n!} \exp\bigg{(} \mu n- \frac{1}{2}
a p n^2\bigg{)}, \qquad n \in \mathds{N}_0.
\end{equation}
Note that, for $p=0$, $\pi$ turns into the (non-normalized) Poisson
distribution with parameter $\upsilon e^\mu$. Hence, alternating the
cell size amounts to shifting $\mu $.

\subsection{Single-phase domains}

By a standard identity
\[
  \exp\bigg{(} \frac{p}{2N} \left(
\sum_{\ell=1}^N \varrho_\ell\right)^2\bigg{)} = \sqrt{\frac{N}{2\pi
p}} \int_{\mathds{R}}\exp\bigg{(} - N \frac{y^2}{2p} + y
\sum_{\ell=1}^N \varrho_\ell\bigg{)} d y.
\]
we transform (\ref{11}) into the following expression
\begin{equation}
  \label{13}
 \Xi_N (p , \mu) = \sqrt{\frac{N}{2\pi
p}} \int_{\mathds{R}}\exp\bigg{(} N E (y, p , \mu) \bigg{)} dy,
\end{equation}
where
\begin{equation}
  \label{14}
E (y, p , \mu) = - \frac{y^2}{2p} + \ln K (y, p , \mu),
\end{equation}
and, cf. (\ref{7a}) and (\ref{12}),
\begin{equation}
  \label{15}
K (y, p , \mu) = \sum_{n=0}^\infty \frac{\upsilon^n}{n!}
\exp\bigg{(} (y+\mu)n - \frac{a p}{2} n^2 \bigg{)}.
\end{equation}
Note that $E$ is an infinitely differentiable function of all its
arguments. Set
\begin{equation}
  \label{15a}
  P_N (p , \mu) = \frac{1}{\upsilon N} \ln \Xi_N (p , \mu).
\end{equation}
By the following evident inequality
\[
(y +  \mu) n - \frac{ap}{2}n^2 \leq \frac{(y +  \mu)^2}{2 ap} ,
\qquad n\in \mathds{N}_0,
\]
we obtain from (\ref{15}) and  (\ref{14}) that
\begin{equation}
  \label{16}
E (y, p, \mu) \leq - \frac{a-1}{2ap} y^2 + \frac{ \mu}{2ap} (2y +
\mu) + \upsilon.
\end{equation}
By virtue of Laplace's method \cite{Fed}, to calculate the large $N$
limit in (\ref{15a})  we have to find global maxima of $E (y, p,
\mu)$ as a function of $y\in \mathds{R}$.
\begin{remark}
  \label{1rk}
From the estimate in (\ref{16}) it follows that: (a) the integral in
(\ref{13}) is convergent for all $p
>0$ and $\mu \in \mathds{R}$ since $a>1$, see (\ref{5}) and
(\ref{7a});  (b) for fixed $p$ and $\mu$,  as a bounded from above
function $E (y, p , \mu)$ has global maxima, each of which is also
its local maximum.
\end{remark}
To get (b) we observe that (\ref{16}) implies $\lim_{|y|\to
+\infty}E(y, p, \mu) = -\infty$; hence, each point $\bar{y}$ of
global maximum belongs to a certain interval $(\bar{y}-\varepsilon,
\bar{y}+\varepsilon)$, where it is also a maximum point. Since $E$
is everywhere differentiable in $y$, then $\bar{y}$ is the point of
global maximum only if it solves the following equation
\begin{equation}
  \label{16b}
E_1 (y, p, \mu) := \frac{\partial}{\partial y} E(y, p, \mu) =0.
\end{equation}
By (\ref{14}) and (\ref{15}) this equation can be rewritten in the
form
\begin{eqnarray}
  \label{16c}
&& -\frac{y}{p} + \frac{K_1 (y, p, \mu)}{K (y, p, \mu)} = 0, \\
&& K_1 (y, p, \mu):= \sum_{n=1}^\infty \frac{n\upsilon^n}{n!}
\exp\bigg{(}(y+\mu)n - \frac{ap}{2} n^2 \bigg{)}. \nonumber
\end{eqnarray}
\begin{remark}
  \label{RSrkk}
As we will see from the proof of Theorem \ref{4pn} below, the
equation in (\ref{16c}) has at least one solution for all $p>0$ and
$\mu \in \mathds{R}$. Since both $K_1$ and $K$ take only strictly
positive values, these solutions are also strictly positive.
\end{remark}
\begin{definition}
  \label{1df}
We say that  $(p, \mu)$ belongs to a single-phase domain if $E (y, p
, \mu)$ has a unique global maximum $\bar{y}\in\mathds{R}$ such that
\begin{equation}
  \label{16a}
E_2 (\bar{y}, p , \mu):=\frac{\partial^2}{\partial y^2} E (y, p ,
\mu)|_{y=\bar{y}} < 0.
\end{equation}
\end{definition}
Note that $\bar{y}$ can be a point of maximum if $E_1 (\bar{y}, p ,
\mu)= E_2 (\bar{y}, p , \mu)=0$. That is, not every point of global
maximum corresponds to a point in a single-phase domain.

The condition in (\ref{16c})  determines the unique probability
measure $Q_{p,\mu}$ on $\mathds{N}_0$ such that
\begin{equation}
  \label{19}
Q_{p,\mu} (n) = \frac{1}{K(\bar{y}, p , \mu)n!} \upsilon^n
\exp\bigg{(} (\bar{y}+ \mu)n -\frac{ap}{2}n^2 \bigg{)}, \qquad n \in
\mathds{N}_0,
\end{equation}
which yields the probability law of the occupation number of a
single cell. Then the unique thermodynamic phase of the model
corresponding to $(p, \mu)\in \mathcal{R}$ is the product
\begin{equation}
  \label{19a}
\mathbf{Q}_{p, \mu} = \bigotimes_{\ell=1}^\infty Q^{(\ell)}_{p, \mu}
\end{equation}
of the copies of the measure defined in (\ref{19}). It is a
probability measure on the space of all vectors $\mathbf{n} =
(n_\ell)_{\ell=1}^\infty$, in which $n_\ell\in \mathds{N}_0$ is the
occupation number of $\ell$-th cell.

The role of the condition in (\ref{16a}) is to yield the possibility
to apply Laplace's method for asymptotic calculating the integral in
(\ref{13}). By direct calculations it follows that
\begin{eqnarray}
  \label{17}
& & E_2 (y, p , \mu) = - \frac{1}{p}  + \frac{1}{2\left[K (y,
p , \mu)\right]^2}\\[.2cm] \nonumber & & \quad \times \sum_{n_1, n_2 =0}^\infty
\frac{\upsilon^{n_1+n_2}}{n_1! n_2!} (n_1 - n_2)^2 \exp\bigg{(} (y+
\mu)(n_1+n_2) - \frac{ap}{2}( n_1^2 + n_2^2) \bigg{)}.
\end{eqnarray}
In dealing with the equation in (\ref{16c}) we will fix $p>0$ and
consider $E_1$ as a function of $y\in \mathds{R}$ and $\mu\in
\mathds{R}$. Then, for a given $\mu_0$, we solve (\ref{16c}) to find
$\bar{y}_0$ and then check whether it is the unique point of global
maximum and (\ref{16a}) is satisfied, i.e., whether $(p,\mu_0)$
belongs to a single-phase domain. Then we slightly vary $\mu$ and
repeat the same. This will yield a function $\mu \mapsto
\bar{y}(\mu)$ defined in a neighborhood of $\mu_0$, which dependends
on the choice of $p$ and satisfies $\bar{y}(\mu_0) =\bar{y}_0$. In
doing so, we will use the analytic implicit function theorem based
on the fact that, for each fixed $p>0$, the function
$\mathds{R}^2\ni (y, \mu) \mapsto E_1(y,p,\mu)$ can be analytically
continued to some complex neighborhood of $\mathds{R}^2$, see
(\ref{15}) and (\ref{16c}). For the reader's convenience, we present
this theorem here in the form adapted from \cite[Section 7.6, page
34]{CC}. For some $p_0>0$, let $\mathcal{B}\subset \mathds{C}^2$ be
a connected open set containing $\mathds{R}^2$ such that the
function $(y, \mu) \mapsto E_1(y,p_0,\mu)$ be analytic in
$\mathcal{B}$.
\begin{proposition}[Implicit function theorem]
  \label{0pn}
Let $p_0$ and $(y_0,\mu_0)$  be such that $E_1 (y_0,p_0,\mu_0) =0$
and $E_2 (y_0,p_0,\mu_0) \neq 0$. Let also $\mathcal{B}\subset
\mathds{C}^2$ be as just described. Then there exist open sets
$\mathcal{D}_1\subset \mathds{C}$ and $\mathcal{D}_2\subset
\mathds{C}$ such that $y_0 \in \mathcal{D}_1$, $ \mu_0 \in
\mathcal{D}_2$, $\mathcal{D}_1\times \mathcal{D}_2 \subset
\mathcal{B}$, and an analytic function $\bar{y}: \mathcal{D}_2 \to
\mathcal{D}_1$, for which the following holds
 \begin{equation*}

\{ (y,\mu)\in \mathcal{D}_1 \times \mathcal{D}_2 : E_1 (y,p_0,\mu) =
0\} = \{ (\bar{y}(\mu),  \mu): \mu\in \mathcal{D}_2\}.
\end{equation*}
The derivative of $\bar{y}$ in $\mathcal{D}_2$ is
\begin{eqnarray}
  \label{SR1}
  \frac{d \bar{y} (\mu)}{d \mu} & = & - \frac{1}{E_2 (\bar{y}(\mu), p_0,
 \mu)} \left[\frac{\partial }{\partial \mu} E_1 (y, p_0, \mu)\right]_{y=\bar{y}(\mu)}.
\end{eqnarray}
\end{proposition}
\begin{remark}
  \label{Luftrk}
In the sequel, we will also use the version of the implicit function
theorem in which we do not employ the analytic continuation of $E_1$
to complex values of $p$. Let the conditions of Proposition
\ref{0pn} regarding $E_1$ and $E_2$ be satisfied. Then there exist
open sets $\mathcal{D}_i \subset \mathds{R}$, $i=1,2,3$, and a
continuous function $\bar{y}:\mathcal{D}_3 \times \mathcal{D}_2 \to
\mathcal{D}_1$ such that $p_0 \in \mathcal{D}_3$, $\mu_0 \in
\mathcal{D}_2$ and the following holds
\begin{eqnarray*}

& & \{ (y,p,\mu)\in \mathcal{D}_1 \times \mathcal{D}_2
\times\mathcal{D}_3 : E_1 (y,p_0,\mu) = 0\} \\[.2cm] & &  \qquad \qquad = \{ (\bar{y}(p,\mu),
p, \mu): p\in \mathcal{D}_3, \ \mu\in \mathcal{D}_2\}.
\end{eqnarray*}
\end{remark}

The partial derivative of $\bar{y} (p, \mu)$ over $\mu\in
\mathcal{D}_2$ is given by the right-hand side of (\ref{SR1}) In the
sequel, by writing $\bar{y} (\mu)$ we assume both the function as
Proposition \ref{0pn}, defined for a fixed $p$ known from the
context, and that as in Remark \ref{Luftrk} with the fixed value of
$p$.

For a fixed $p_0>0$, assume that $(p_0, \mu_0)$ belongs to a
single-phase domain. By Proposition \ref{0pn} there exists
$\varepsilon >0$ such that the function $(\mu_0 - \varepsilon, \mu_0
+ \varepsilon)\ni \mu \mapsto \bar{y}(\mu)$ can be defined by the
equation $E_1 (y, p_0, \mu)=0$. Its continuation from the mentioned
interval is related to the fulfilment of the condition $E_2
(\bar{y}(\mu), p_0 , \mu) <0$, cf. (\ref{16a}), which may not be the
case. At the same time, by (\ref{17}) we have that
\begin{equation*}
\frac{\partial }{\partial \mu} E_1 (y, p, \mu)= E_2 (y, p, \mu) +
\frac{1}{p}
>0,
\end{equation*}
holding for all $y\in\mathds{R}$, $p>0$ and $\mu \in\mathds{R}$. In
view of this and  (\ref{SR1}), it might be more convenient to use
the inverse function $y\mapsto \bar{\mu}(y)$ since the
$\mu$-derivative of $E_1$ is always nonzero. Its properties are
described by the following statement obtained from the analytic
implicit function theorem mentioned above. Recall that only positive
$y$ solve the equation in (\ref{16c}).
\begin{proposition}
  \label{01pn}
Given $p_0$, let $\mathcal{B}$ be as in Proposition \ref{0pn}. Then
there exist open connected subsets $\mathcal{D}_i\subset
\mathds{C}$, $i=1,2$, and an analytic function $\mathcal{D}_1\ni y
\mapsto \bar{\mu}(y)\in \mathcal{D}_2$ such that $\mathcal{D}_1$
contains $\mathds{R}_{+}$, $\mathcal{D}_1 \times \mathcal{D}_2
\subset \mathcal{B}$, and the following holds
 \begin{equation*}

\{ (y,\mu)\in \mathcal{D}_1 \times \mathcal{D}_2 : E_1 (y,p_0,\mu) =
0\} = \{ (y,  \bar{\mu}(y)): y\in \mathcal{D}_1\}.
\end{equation*}
The derivative of $\bar{\mu}$ in $\mathcal{D}_1$ is
\begin{eqnarray*}

  \frac{d \bar{\mu} (y)}{d y} & = & - \frac{E_2 (y, p_0,
\bar{ \mu}(y))}{E_2 (y, p_0, \bar{ \mu}(y)) + \frac{1}{p}}.
\end{eqnarray*}
\end{proposition}
\begin{proposition}
  \label{1pn}
Each single-phase domain, $\mathcal{R}$, has the following
properties: (a) it is an open subset of $\mathds{R}_{+} \times
\mathds{R}$; (b) for each $(p_0, \mu_0)\in \mathcal{R}$, the
function $\mathcal{I}_{p_0}:=\{ \mu \in \mathds{R}: (p_0, \mu )\in
\mathcal{R}\}\ni \mu \mapsto \bar{y}(\mu)$ as in Proposition
\ref{0pn} is continuously differentiable on $\mathcal{I}_{p_0}$.
Moreover,
\begin{equation}
  \label{Luft}
\frac{d\bar{y}(\mu)}{d \mu}  >0 , \qquad {\rm for} \ {\rm all} \
\mu\in \mathcal{I}_{p_0}.
\end{equation}
\end{proposition}
\begin{proof}
For a single-phase domain, $\mathcal{R}$, take $(p_0, \mu_0)\in
\mathcal{R}$. By Remark \ref{Luftrk} the function $(p,\mu) \mapsto
\bar{y}(p,\mu)$, defined by the equation $E_1(y,p,\mu)=0$ is
continuous in some open subset of $\mathcal{R}$ containing $(p_0,
\mu_0)$. By the continuity of $E_2 (\bar{y}(p,\mu), p, \mu)$ and the
fact that $E_2 (\bar{y}(p_0,\mu_0), p_0, \mu_0)<0$ (since $(p_0,
\mu_0)\in \mathcal{R}$) we get that $E_2 (\bar{y}(p,\mu), p, \mu)<0$
for $(p,\mu)$ in some open neighborhood of $(p_0, \mu_0)$. Hence,
$\mathcal{R}$ contains $(p_0, \mu_0)$ with some neighborhood and
thus is open. The continuous differentiability of $\bar{y}$ and the
sign rule in (\ref{Luft}) follow by Proposition \ref{0pn} and
(\ref{SR1}), respectively.
\end{proof}

By (\ref{19}) and  (\ref{16c}) we get the $Q_{p,\mu}$-mean value
$\bar{n} = \bar{n}(p,\mu)$ of the occupation number of a given cell
in the form
\begin{equation}
 \label{oct}
 \bar{n}(p, \mu) = \sum_{n=0}^\infty n Q_{p,\mu}(n) =
\frac{K_1(\bar{y}(p,\mu),p,\mu)}{K(\bar{y}(p,\mu),p,\mu)}
 =  \frac{\bar{y}(p , \mu)}{p}.
\end{equation}
Note that, up to the factor $\upsilon^{-1}$, $\bar{n}(p, \mu)$ is
the particle density in phase $\mathbf{Q}_{p, \mu}$. For a fixed
$p$, by Proposition \ref{1pn} $\bar{n}(p, \cdot)$ in an increasing
function on $\mathcal{I}_p$, which thus can be inverted to give
$\bar{\mu}(p, \bar{n})$. By Laplace's method we then get the
following corollary of Proposition \ref{1pn}.
\begin{proposition}
  \label{2pn}
Let $\mathcal{R}$ be a single-phase domain. Then, for each $(p ,
\mu) \in \mathcal{R}$, the limiting pressure $P (p , \mu) =
\lim_{N\to +\infty} P_N(p , \mu)$, see (\ref{15a}), exists and is
continuously differentiable on $\mathcal{R}$. Moreover, it is given
by the following formula
\begin{equation}
  \label{21}
  P(p, \mu) = \upsilon^{-1} E(\bar{y}(p , \mu), p ,
  \mu) .
\end{equation}
\end{proposition}
Let $\mathcal{N}_p$ be the image of $\mathcal{I}_p$ under the map
$\mu\mapsto \bar{n}(p, \mu)$. Then the inverse map $\bar{n}\mapsto
\bar{\mu}(p , \bar{n})$ is continuously differential and increasing
on $\mathcal{N}_p$. By means of this map, for a fixed $p$, the
pressure given in (\ref{21}) can be written as a function of
$\bar{n}$
\begin{equation}
  \label{22}
P = \bar{P}(\bar{n}) = \upsilon^{-1} E(p\bar{n}, p ,
 \bar{ \mu}(p , \bar{n})) , \qquad \bar{n}\in \mathcal{N}_p,
\end{equation}
which is the equation of state.

\subsection{The phase transition}

Recall that the notion of the single-phase domain was introduced in
Definition \ref{1df}, and each such a domain is an open subset of
the open right half-plane $\{(p,\mu): p>0, \ \mu \in \mathds{R}\}$,
see  Proposition \ref{1pn}. With this regard we have the following
possibilities: (i) the whole half-plane $\{(p,\mu): p>0, \ \mu \in
\mathds{R}\}$ is such a domain; (ii) there exist more than one
single-phase domains. In case (i),  for all $(p,\mu)$ there exists
one phase (\ref{19a}). In the context of this work, a phase
transition is understood as the possibility of having different
phases at the same value of the pair $(p,\mu)$. If this is the case,
$(p,\mu)$ is called a \emph{phase coexistence point}. Clearly, such
a point should belong to the common topological boundary of at least
two distinct single-phase domains. That is, to prove the existence
of phase transitions we have to show that possibility (ii) takes
place and that these single-phase domains have common boundary. We
do this in Theorems \ref{4pn} and \ref{1tm} below.

Let $\mathcal{R}$ be a single-phase domain. Take $(p_0,\mu_0)\in
\mathcal{R}$ and consider the line $l_{p_0}=\{(p_0,\mu): \mu\in
\mathds{R}\}$. If the  whole line lies in $\mathcal{R}$, by
Proposition \ref{1pn} $\bar{y}(\mu)$ is a continuously
differentiable and increasing function of $\mu \in \mathds{R}$. In
our first theorem, we prove that this is the case for small enough
$p_0$.
\begin{theorem}
  \label{4pn}
There exists $p_0>0$ such that the set $\mathcal{R}(p_0):=\{ (p,
\mu): p\in(0,p_0]\}$ is a single-phase domain.
\end{theorem}
\begin{proof}
In view of Remark \ref{1rk}, we have to show that, for fixed $p\leq
p_0$ and all $\mu\in \mathds{R}$, $E(y,p,\mu)$ has exactly one local
maximum such that (\ref{16a}) holds. For $x\in \mathds{R}$, we set,
cf. (\ref{15}),
\begin{eqnarray}
  \label{23}
\phi(x,p) &=& \ln \sum_{n=0}^\infty \frac{\upsilon^n}{n!}
\exp\bigg{(} x n - \frac{ap}{2} n^2\bigg{)}, \\ \nonumber \phi_k
(x,p) &=& \frac{\partial^k}{\partial x^k}\phi(x,p), \qquad k=1,2.
\end{eqnarray}
Similarly as in (\ref{16}) we then get
\begin{equation}
  \label{24}
\phi(x,p) \leq \upsilon + \frac{x^2}{2ap}.
\end{equation}
Note also that, for  the functions defined in (\ref{23}), we have
\begin{equation}
  \label{23a}
\lim_{p\to 0} \phi(x,p) = \lim_{p\to 0} \phi_1(x,p) = \lim_{p\to 0}
\phi_2(x,p) = \upsilon e^x, \qquad x\in \mathds{R}.
\end{equation}
By means of (\ref{23}) we rewrite the equation in (\ref{16c}) in the
following form
\begin{equation}
  \label{25}
\left\{ \!
\begin{array}{ll}
& x = y + \mu,\\[.3cm]
& \mu = x -  p \phi_1 (x, p).
\end{array}
\right.
\end{equation}
Our aim is to show that there exists $p_0>0$ such that, for each
$p\in (0,p_0]$, the following holds: (a) the second line in
(\ref{25}) defines an increasing unbounded function $\tilde{\mu}
(x)$, $x\in \mathds{R}$; (b) $p \phi_2 (x,p) \leq 1-\delta$ for some
$\delta\in (0,1)$ and all $x\in \mathds{R}$. Indeed, the function
mentioned in (a) can be inverted to give an unbounded increasing
function $\tilde{x} (\mu)$, $\mu \in \mathds{R}$, such that the
solution of (\ref{16c}) is $\bar{y}(\mu) = \tilde{x}(\mu ) - \mu$.
Then by (\ref{23}) and (\ref{17}) we get
\[
E_2 (\bar{y}(\mu), p, \mu) = -\frac{1}{p} + \phi_2 (\tilde{x}(\mu),
p) <0,
\]
where the latter inequality follows by (b). Thus, to prove both (a)
and (b) it is enough to show that there exists positive $p_0$ such
that
\begin{equation}
  \label{26}
p \phi_2 (x,p) \leq \frac{1}{a}, \qquad {\rm for} \ \ {\rm all} \ \
x \in \mathds{R} \ \ {\rm and} \ \ p\in (0,p_0].
\end{equation}
By (\ref{23}) we have
\begin{eqnarray}
\label{d} \phi_2 (x,p) & = & \frac{1}{2\Phi(x,p)} \sum_{n_1, n_2
=0}^\infty
\frac{\upsilon^{n_1+n_2}}{n_1! n_2!} (n_1-n_2)^2 \\[.2cm] & & \qquad \qquad \quad \times  \exp \bigg{(} x
(n_1 + n_2) - \frac{ap}{2} (n_1^2 + n_2^2)\bigg{)}, \nonumber \\[.2cm]
\label{d1} \Phi(x,p) & = & \left( \sum_{n=0}^\infty
\frac{\upsilon^n}{n!} \exp\left(x n - \frac{ap}{2} n^2
\right)\right)^2 \\[.2cm]
& = & \sum_{n_1, n_2 =0}^\infty \frac{\upsilon^{n_1+n_2}}{n_1! n_2!}
\exp \bigg{(} x (n_1 + n_2) - \frac{ap}{2} (n_1^2 + n_2^2)\bigg{)}.
\nonumber
\end{eqnarray}
Since $\Phi(x,p) \geq 1$, we  get from the latter
\begin{eqnarray}
  \label{27}
\phi_2 (x,p) & \leq & \frac{1}{2} \sum_{n_1, n_2 =0}^\infty
\frac{\upsilon^{n_1+n_2}}{n_1! n_2!} (n_1-n_2)^2 \exp \bigg{(} x
(n_1 + n_2) - \frac{ap}{2} (n_1^2 + n_2^2)\bigg{)} \qquad \\[.2cm]
\nonumber & \leq & \frac{1}{2} \sum_{n_1, n_2 =0}^\infty
\frac{\upsilon^{n_1+n_2}}{n_1! n_2!} (n_1-n_2)^2 \exp \bigg{(} x
(n_1 + n_2)\bigg{)} \\[.2cm] \nonumber & = &
\bigg{(}\sum_{n=1}^\infty \frac{n^2}{n!} \left(\upsilon e^x
\right)^n \bigg{)} \cdot \bigg{(}\sum_{n=0}^\infty \frac{1}{n!}
\left(\upsilon e^x \right)^n \bigg{)} - \bigg{(}\sum_{n=1}^\infty
\frac{n}{n!} \left(\upsilon e^x \right)^n \bigg{)}^2 \\[.2cm]
\nonumber & = &
 \upsilon \exp \bigg{(} x+ 2 \upsilon e^x \bigg{)}.
\end{eqnarray}
Fix some $x_0>0$ and then set
\begin{equation}
  \label{27a}
  p_{01} = \frac{1}{a\upsilon} \exp \bigg{(}- x_0 - 2 \upsilon e^{x_0} \bigg{)}.
\end{equation}
Then by (\ref{27}) we get
\begin{equation}
  \label{29a}
p \phi_2 (x,p) \leq \frac{1}{a}, \qquad {\rm for} \ \ {\rm all} \ \
x \leq x_0 \ \ {\rm and} \ \ p\leq p_{01}.
\end{equation}
By (\ref{23}) and (\ref{23a}) we see that the function
\begin{equation}
  \label{28}
  \psi(p) = \frac{\upsilon - \phi(0,p)}{\phi_1(0,p)}
\end{equation}
continuously depends on $p>0$ and $\psi(p) \to 0$ as $p\to 0$. For
each $x>0$, one finds $\xi \in (0,1]$, dependent on $x$ and $p$,
such that
\begin{equation*}

\phi (x,p) = \phi(0,p) + \phi_1 (0,p) x + \frac{1}{2} \phi_2(\xi
x,p) x^2.
\end{equation*}
From this and (\ref{24}) we get
\begin{equation}
  \label{29}
\phi (\xi^{-1} x,p) = \phi(0,p) + \phi_1 (0,p) \xi^{-1} x +
\frac{1}{2} \phi_2( x,p)\xi^{-2} x^2 \leq \upsilon + \frac{\xi^{-2}
x^2}{2ap}.
\end{equation}
For the function defined in (\ref{28}) and $x_0$ as in (\ref{27a}),
we pick $p_{02}$ such that $\psi(p) \leq x_0$ for all $p\leq
p_{02}$. For such values of $p$, this yields
\[
\phi_1(0,p) \xi^{-1} x \geq \phi_1(0,p)  x_0  \geq \upsilon -
\phi(0,p), \qquad {\rm for}\ \ {\rm all} \ \ x\geq x_0.
\]
We apply this in (\ref{29}) and get
\begin{equation}
  \label{30}
p \phi_2 (x, p) \leq \frac{1}{a}, \qquad {\rm for} \ \ {\rm all} \ \
x \geq x_0 \ \ {\rm and} \ \ p\leq p_{02}.
\end{equation}
This and (\ref{29a}) then yields (\ref{26}) with $p_0 =
\min\{p_{01}; p_{02}\}$, which completes the proof.
\end{proof}
\begin{theorem}
  \label{1tm}
For each $a>1$, there exists $p_1 = p_1 (a) >0$ such that, for each
$p\geq p_1$, the line $l_{p}=\{(p, \mu):\mu \in \mathds{R}\}$
contains at least one phase-coexistence point.
\end{theorem}
\begin{proof}
Let $\bar{\mu}(y)$ be the function as in Proposition \ref{01pn}. By
(\ref{23}) and (\ref{25}) we have that
\begin{equation}
  \label{s}
\bar{\mu}(y) = \bigg{[}x - p \phi_1 (x,p) \bigg{]}_{x=\bar{x}(y)},
\end{equation}
and
\begin{equation*}

\lim_{y\to 0} \bar{\mu} (y) = - \infty, \qquad \lim_{y\to +\infty}
\bar{\mu} (y) = + \infty.
\end{equation*}
In (\ref{s}), $\bar{x}(y)$ is the inverse of the function
$\mathds{R}\ni x \mapsto y= p \phi_1 (x,p)$. Note that
\begin{equation}
  \label{Naha1}
  \frac{d \bar{x}(y)}{d y} = \left[ \frac{1}{ p \phi_2
  (x,p)}\right]_{x=\hat{x}(y)},
\end{equation}
hence $(0, +\infty) \ni y \mapsto \bar{x}(y) \in \mathds{R}$ is
increasing. By (\ref{s}) and (\ref{Naha1}) it follows that
\begin{equation}
  \label{s1}
\frac{d \bar{\mu}(y)}{ d y} = \frac{ d \bar{x}(y)}{ d y} \bigg{[}1 -
p \phi_2 (x,p) \bigg{]}_{x=\bar{x}(y)} =  \left[ \frac{1}{ p \phi_2
  (x,p)}\right]_{x=\bar{x}(y)} - 1.
\end{equation}
For a given $p>0$, pick $x^p>0$ such that $\psi(p)$ defined in
(\ref{28}) satisfies $\psi(p) \leq x^p$. Then as in (\ref{30}) we
obtain $p \phi_2 (x, p) \leq 1/a$ for all $x\geq x^p$. By (\ref{s1})
this yields that
\begin{equation}
  \label{Naha2}
\frac{d \bar{\mu}(y)}{ d y} \geq a -1 , \qquad {\rm for} \ \ y \in
[y^p, +\infty),
\end{equation}
where  $y^p= p\phi_1(x^p,p)$. For the same $p$, let $x_p$ be such
that $a\upsilon p = \exp\left( - x_p -2 \upsilon e^{x_p}\right)$,
cf. (\ref{27a}). Then by (\ref{27}) and (\ref{s1}) we conclude that
the inequality in (\ref{Naha2}) holds also for  $y\in (0,y_p]$,
$y_p:= p\phi_1(x_p,p)$. As we have seen in the proof of Theorem
\ref{4pn}, the mentioned two intervals may overlap, i.e., it may be
that $y_p > y^p$, if $p$ is small enough. Let us prove that this is
not the case for big $p$. That is, let us show that there exists
$p_1>0$ such that, cf. (\ref{s1}) and (\ref{30}), for all $p\geq
p_1$, there exists $x\in \mathds{R}$ such that following holds
\begin{equation}
  \label{s2}
p_1 \phi_2 (x, p_1) \geq 1 , \qquad {\rm and} \quad p \phi_2 (x, p)
> 1, \quad {\rm for} \quad p>p_1.
\end{equation}
To this end we estimate the denominator of (\ref{d}) from above and
the numerator from below. For $x = a p/2$, by (\ref{d1})  it follows
that
\begin{eqnarray}
  \label{s3}
\Phi (x, p) =  \left( \sum_{n=0}^\infty \frac{\upsilon^{n}}{n!}
\exp\left[  - \frac{ap}{2}n(n-1) \right] \right)^2 \leq \left(
\sum_{n=0}^\infty \frac{\upsilon^{n}}{n!}\right)^2 = e^{2\upsilon},
\end{eqnarray}
where we used the fact that $p>0$. In the sum in the numerator of
(\ref{d}), we take just two summands corresponding to $n_1=1$,
$n_2=0$ and $n_1=0$, $n_2=1$, and obtain by (\ref{s3}) the following
estimate
\begin{equation}
  \label{s4}
p \phi_2 (x, p) \geq p \upsilon e^{- 2 \upsilon},
\end{equation}
holding for $x=ap/2$.  Then we set $p_1= \upsilon^{-1}e^{2
\upsilon}$. For this $p_1$ and $x =  ap_1/2$, by (\ref{s4}) we
obtain (\ref{s2}). Clearly,  for $p> p_1$, $x_p$ and $x^p$
introduced above satisfy
\[
x_p < ap/2 < x^p.
\]
For $p>p_1$, let $({x}_p, {x}^p)$ be the biggest interval which
contains $x=ap/2$ and is such that $p \phi_2 (x, p) >1$ for each
$x\in ({x}_p, {x}^p)$. Then $p \phi_2 (x, p) =1$ for $x= {x}_p$ and
$x ={x}^p$. Set
\[
{y}_p =p \phi_1 ({x}_p, p), \qquad {y}^p =p \phi_1 ({x}^p, p).
\]
Then by (\ref{s1}) and (\ref{Naha2}) it follows that
\begin{eqnarray}
  \label{s5}
  \frac{d\bar{\mu}(y)}{d y} & = & 0, \qquad {\rm for} \quad
  y = {y}_p , \ {y}^p, \\[.2cm] \nonumber
 \frac{d\bar{\mu}(y)}{d y} & <  & 0, \qquad {\rm for} \quad
 y\in ({y}_p , {y}^p),
 \\[.2cm] \nonumber
 \frac{d\bar{\mu}(y)}{d y} & \geq & a-1> 0, \qquad {\rm for} \quad
 y< {y}_p  \ \ {\rm and} \ \ y> {y}^p.
\end{eqnarray}
From this we see that ${y}_p$ (resp. ${y}^p$) is the first maximum
(resp. the last minimum) of $\bar{\mu}(y)$. Let $\hat{y}^p$ be the
first minimum of $\bar{\mu}(y)$. Set $\hat{\mu}^p = \bar{\mu}
(\hat{y}^p)$. Now we pick $y_2 > \hat{y}^p$ such that: (a) either
$\bar{\mu}(y_2) = \bar{\mu}(y_p)$; (b) or $y_2$ is the second
maximum of $\bar{\mu}(y)$ if $\bar{\mu}(y_2) \leq \bar{\mu}(y_p)$.
Then set $\hat{\mu}_p = \bar{\mu}(y_2)$. Clearly, $\hat{\mu}_p >
\hat{\mu}^p$. In case (a), we have $\hat{\mu}_p = \bar{\mu}(y_p)$;
and  $\hat{\mu}_p \leq \bar{\mu}(y_p)$ in case (b). Now we pick $y_1
\in (0, y_p)$ such that $\bar{\mu}(y_1) = \bar{\mu}(\hat{y}^p)$. By
(\ref{s5}) and the above construction  the function $\bar{\mu}(y)$
is increasing on $[y_1, y_p)$ and $(\hat{y}_p, y_2)$, and decreasing
on $(y_p, \hat{y}^p)$, see Fig. \ref{fig_3}. Let $\bar{y}_1$ and
$\bar{y}_2$ be the inverse functions to the restrictions of
$\bar{\mu}(y)$ to $[y_1, y_p)$ and $(\hat{y}_p, y_2)$, respectively.
Let also $\bar{y}_3$ be the inverse function to the restrictions of
$\bar{\mu}(y)$ to the interval $(y_p, \hat{y}^p)$. All the three
functions are defined on $M_p:=(\hat{\mu}^p, \hat{\mu}_p)$ and are
continuously differentiable thereon.
 Note that
$\bar{y}_2 (\mu) > \bar{y}_3 (\mu)
> \bar{y}_1 (\mu)$ for all $\mu\in (\hat{\mu}^p, \hat{\mu}_p)$ and
\begin{equation}
  \label{Nah}
\bar{y}_1 (\hat{\mu}_p ) = \bar{y}_3 (\hat{\mu}_p ), \qquad
\bar{y}_2 (\hat{\mu}^p ) = \bar{y}_3 (\hat{\mu}^p ).
\end{equation}
Moreover, all the three $\bar{y}_i (\mu)$, $i=1,2,3$, satisfy
(\ref{16c}) and, for $\mu \in (\hat{\mu}^p, \hat{\mu}_p)$,
$E(y,p,\mu)$ has local maxima at $y=\bar{y}_i(\mu)$, $i=1,2$ and a
local minimum at $y=\bar{y}_3(\mu)$. This follows from the fact that
$E_2 (\bar{y}_i (\mu), p, \mu)< 0$ for $\mu\in M_p$ and $i=1,2$, and
from  $E_2 (\bar{y}_3 (\mu), p, \mu)> 0$ for $\mu\in M_p$, see
(\ref{s5}).

Set
\begin{equation}
  \label{31} D(\mu) = E(\bar{y}_2 (\mu), p, \mu) - E(\bar{y}_1 (\mu), p,
  \mu), \qquad \mu \in M_p.
\end{equation}
If $D(\mu) < 0$, then $\bar{y}_1(\mu)$ is the point of global
maximum of $E(y,p,\mu)$ and hence $(p,\mu)$ lies in a single-phase
domain, say $\mathcal{R}_1$. If $D(\mu) > 0$, then the same holds
for $\bar{y}_2(\mu)$ and $\mathcal{R}_2$. If
\begin{equation}
  \label{Na}
D(\mu_1) < 0, \qquad {\rm and} \qquad D(\mu_2) > 0,
\end{equation}
for some $\mu_1,\mu_2 \in M_p$, then there should exist $\mu_c$ in
between where $D$ vanishes. Thus, $(p,\mu_c)$ belongs to the
boundaries of both $\mathcal{R}_1$ and $\mathcal{R}_2$, and hence is
a phase coexistence point, if $\mu_c$ is an isolated zero of
(\ref{31}). The phases are then given in (\ref{19}) and (\ref{19a})
with $\bar{y}_1(\mu_c)$ and $\bar{y}_2(\mu_c)$, respectively. Note
that the vanishing of $D$ at $\mu_c$ corresponds to the Maxwell
rule, cf. \cite{LebP}, and to the existence of two global maxima of
$E(y , p, \mu)$. Since both $\bar{y}_i(\mu)$ are differentiable, by
(\ref{14}), (\ref{15}), (\ref{16b}), and (\ref{17}) we have
\begin{eqnarray*}

\frac{d D (\mu)}{d  \mu} & = & E_1 (\bar{y}_2 (\mu), p , \mu)
\frac{d \bar{y}_2(\mu)}{ d \mu}  + \frac{K_1 (\bar{y}_2 (\mu), p ,
\mu)}{K
(\bar{y}_2 (\mu), p , \mu)} \\[.2cm] \nonumber & - & E_1 (\bar{y}_1 (\mu), p , \mu) \frac{d
\bar{y}_1(\mu)}{ d \mu}  - \frac{K_1 (\bar{y}_1 (\mu), p , \mu)}{K
(\bar{y}_2 (\mu), p , \mu)} \\[.2cm] \nonumber & = & \frac{K_1 (\bar{y}_2 (\mu), p ,
\mu)}{K (\bar{y}_2 (\mu), p , \mu)} - \frac{K_1 (\bar{y}_1 (\mu), p
, \mu)}{K (\bar{y}_2 (\mu), p , \mu)} \\[.2cm] \nonumber & = & E_2 (\bar{y}_* (\mu), p , \mu) + \frac{1}{p} >0
\end{eqnarray*}
for some $\bar{y}_* (\mu)\in [\bar{y}_1 (\mu), \bar{y}_2 (\mu)]$.
Note that $E_1 (\bar{y}_i (\mu), p , \mu)=0$, $i=1,2$, cf.
(\ref{16b}). Therefore, $D$ can hit the zero level at most once. Let
us show that (\ref{Na}) does hold. If $\mu_1\in M_p$ is close enough
to $\hat{\mu}^p$, then by (\ref{Nah}) and the mentioned continuity
we have that $E(\bar{y}_2 (\mu_1), p, \mu_1)$ is close to
$E(\bar{y}_3 (\mu_1), p, \mu_1)$, and hence $\bar{y}_2$ cannot be
the global maximum of $E(y , p, \mu_1)$. Therefore, $D(\mu_1) < 0$
for such $\mu$. Likewise we establish the existence of $\mu_2$ such
that $D(\mu_2) > 0$. Now the existence of $\mu_c\in (\mu_1, \mu_2)$
follows by the continuity and (\ref{Na}).
\end{proof}

\section{Numerical Results}

Here we present the results of numerical calculations of the
functions which appear in the preceding part of the paper.

We begin by considering the extremum points of the functions which
appear in Section 2.4. According to Definition \ref{1df}, the line
$l_p =\{ (p,\mu): \mu\in \mathds{R}\}$ lies in a single-phase
domain, if the function $\mathds{R}_{+} \ni y \mapsto E(y, p, \mu)$,
see (\ref{14}), has a unique non-degenerate global maximum for all
$\mu\in \mathds{R}$. The corresponding condition in (\ref{16c})
determines an increasing function $\bar{y}(\mu)$, see Proposition
\ref{1pn}, which can be inverted to give $\bar{\mu}(y)$, see
(\ref{s}). In Theorem \ref{4pn}, we show that this holds for
sufficiently small $p$. Fig. \ref{fig_1} presents the results of the
calculation of $\bar{\mu}(y)$ for

\begin{figure}[t!]
\begin{centering}
a)\includegraphics[width=150pt]{F1a.eps} b)
\includegraphics[width=150pt]{F1b.eps}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c)
\includegraphics[width=150pt]{F1c.eps}
d)
\includegraphics[width=150pt]{F1d.eps}
\caption{Plot of $\bar\mu(y)$ defined in (\ref{s}) for $a=1.2$,
$\upsilon = 12$ and various values of the attraction parameter:
$p=2$ (curve a), $p=3$ (curve b), $p=4$ (curve c), $p=6$ (curve d).
} \label{fig_1}
\end{centering}
\end{figure}
\begin{equation}\label{42m}
a = 1.2,\qquad \upsilon = 12,
\end{equation}
and $p=2,3,4,6$ --- curves (a), (b), (c) and (d), respectively. In
the first two cases $\bar{\mu}$ is an increasing function, which
corresponds to the situation described in Theorem \ref{4pn}. That
is, the values of $p=2$ and $3$ are below the critical value $p_c =
p_c(a)$. For $a$ and $\upsilon$ as in (\ref{42m}), our calculations
yield
\begin{equation*}\label{45m}
p_{c} = 3.928235(8).
\end{equation*}
For $p=p_c$, the function $y\mapsto E(y, p_c, \mu)$ still has a
unique global maximum, which gets degenerate, i.e., $E_2 (\bar{y},
p_c, \mu)=0$, cf. (\ref{16a}). The value of $\bar{y}=\bar{y}_c$ at
which this occurs gives the value of the critical density $\bar{n}_c
= \bar{y}_c / p_c$, see (\ref{22}). For various values of the
parameter $a$, see (\ref{7a}), the values of $p_c(a)$, $\bar{y}_c
(a)$ and $\bar{n}_c (a)$ are given in  following table.
\begin{table}[h!]
  \centering
  \caption{Values of $p(a)$, $\bar y (a)$, $ \bar n (a)$ in the critical point for $v = 12$}
  \label{tab:table1}
  \begin{tabular}{c|c|c|c|c|c}
    $a$ \ \  & \  \ 1.0001 \ \ & \ \ 1.2 \ \ & \ \ 1.5 \ \ & \ \ 2 \  \ & \ \ 10 \ \ \\
    \hline
   $ p_c(a)$  &  3.8255  &   3.9282  &  3.9796  &  3.9973  &  4.0000  \\
    \hline
  $\bar y_c (a)$ \ \  & \  \ 2.0485 \ \ & \ \  2.0187  \ \ & \ \ 2.0052 \ \ & \ \ 2.0007 \  \ & \ \ 2.0000 \ \ \\
    \hline
    $\bar n_c (a)$ \ \  & \  \ 0.5355 \ \ & \ \  0.5139 \ \ & \ \ 0.5038 \ \ & \ \ 0.5005 \  \ & \ \ 0.5000 \ \ \\
  \end{tabular}
\end{table}

The values of $p=4$ and $6$ are above the critical point $p_c$,
which can clearly be seen from the curves (c) and (d) of Fig.
\ref{fig_1}. In this case, one deals with the situation described by
Theorem \ref{1tm}. Fig. \ref{fig_3} presents in more detail the
curve plotted in Fig. \ref{fig_1} (d), i.e., corresponding to $p=6$
and $a=1.2$. It provides a good illustration to the proof of Theorem
\ref{1tm}. Here we have $\bar{\mu}(y_2) = \bar{\mu}(y_p)$.

\begin{figure}[t!]
\begin{centering}
\includegraphics[width=200pt]{F2n.eps}
\caption{ Plot of $\bar{\mu}(y)$ for $a=1.2$ and $p=6$ in more
detail. } \label{fig_3}
\end{centering}
\end{figure}

Let us now turn to the maximum points of $E(y,p, \mu)$. For $a$ as
in (\ref{42m}) and $p=6$, Fig \ref{fig_4} presents the dependence of
$E$ on $y$ for $\mu_1 = - 2.3080$ (curve a), and $\mu_2 = - 1.4700$
(curve b). This provides an illustration to (\ref{Na}). The curve
plotted in Fig \ref{fig_5} corresponds to the critical value of
$\mu$ defined by the condition $D(\mu)=0$. That is, $(p,\mu_c)$ with
$p=6$ and $\mu_c=-1.890291$ is the phase coexistence point the
existence of which was proved in Theorem \ref{1tm}. Fig \ref{fig_6}
presents the dependence of $E(\bar{y}_1(\mu), p, \mu)$ (line 1, red)
and $E(\bar{y}_2(\mu), p, \mu)$ (line 2, blue) on $\mu\in M_p$,
$p=6$, cf. (\ref{31}). Their intersection occurs at
$\mu=\mu_c=-1.890291$.

\begin{figure}[t!]
\begin{centering}
a)\includegraphics[width=150pt]{F3a.eps} b)
\includegraphics[width=150pt]{F3b.eps}
\caption{ Plot of the function $E(y, p,\mu)$ for $p=6$, $a=1.2$ and
$\mu_1 = -2.3080$ (curve a), $\mu_2=-1.4700$ (curve b). }
\label{fig_4}
\end{centering}
\end{figure}

\begin{figure}[h]
\begin{centering}
\includegraphics[width=150pt]{F4.eps}
\caption{ The same as in Fig \ref{fig_4} for $\mu = \mu_c = -
1.890291$. } \label{fig_5}
\end{centering}
\end{figure}

\begin{figure}[t!]
\begin{centering}
\includegraphics[width=150pt]{F5.eps}
\caption{Plot of the functions $M_p \ni \mu \mapsto E(y_i(\mu), p,
\mu)$, $i=1,2$, see (\ref{31}), and $p=6$, $a =1.2$. } \label{fig_6}
\end{centering}
\end{figure}

\begin{figure}[h]
\begin{centering}
\includegraphics[width=250pt]{F7.eps}
\caption{Plot of the dependence of the pressure on $\bar y = p
\bar{n}$ (isotherms) see (\ref{21}) and (\ref{22}).  Curve 1
corresponds to $p=3.8<p_c$. The curves 2--9 correspond to $p\geq
p_c$:
 $p=p_c$  (curve 2),
$p=4$ (curve 3),  $p=4.135$ (curve 4), $p=4.3647$  (curve 5),
$p=4.5824$  (curve 6), $p=4.8$ (curve 7),  $p=5$ (curve 8), $p=6$
(curve 9). } \label{fig_7}
\end{centering}
\end{figure}

The curves plotted in Fig. \ref{fig_7} present the isotherms --- the
dependence of the pressure on $\bar{y}$, which is equivalent to the
dependence on the density $\bar{n}$, see (\ref{oct}), calculated
from (\ref{22}) for a number of fixed values of $p$.

\newpage

\section{Concluding Remarks}

In this work, we proved the existence of multiple thermodynamic
phases at the same values of the extensive model parameters --
temperature and chemical potential. In contrast to the approach of
\cite{LebP}, we deal directly with thermodynamic phases in the grand
canonical setting. To the best of our knowledge, this is the first
result of this kind.

redpajama_set_name: RedPajamaArXiv


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text:
Solo en Inglès
This audio guide highlights selected works by artists in America Is Hard to See. Curators, scholars, and artists provide additional commentary.
Liliana Porter, Untitled (with Cube – tall), 1974
Donna De Salvo: I'm Donna De Salvo, Chief Curator and Deputy Director for Programs at the Whitney Museum of American Art.
This is a work by Liliana called Untitled (with Cube – tall), 1974. When you look at this work, you see what appears to be a photograph of a cube sitting on a form. In addition, there are a series of drawn lines. So on one level, the photograph, which always implies something that lives in the world, that's realistic. And then the drawn line, which is an allusion to something that is not there but added on. And it's how she brings these two together, and if you really—this work forces you to study it, to engage with it, to really meditate, to a certain extent on the tension that exists between these two kinds of forms and the way it plays with perceptual reality. What's real? Is it the photograph of the cube? Is it the line that's drawn, that's added to the print? It's these two sort of systems of representation that she brings together in this work, in an extremely beautiful way.
000 Introduction to America Is Hard to See
523 Sherrie Levine, Knot Paintings, 1987–2002
524 Jean-Michel Basquiat, Hollywood Africans, 1987
530 Martin Wong, Big Heat, 1988
541 Paul Pfeiffer, Fragment of a Crucifixion, 1999
544 David Hammons, Untitled, 1992
550 Paul Chan, 1st Light, 2005
551 RH Quaytman, Distracting Distance, Chapter 16, 2010
565 Adam McEwen, Untitled (Jeff), 2004
602 Ad Reinhardt, Abstract Painting, 1960–66
611 Bruce Conner, PORTRAIT OF ALLEN GINSBERG, 1960
621 Marisol, Women and Dog, 1964
793 Roy Lichtenstein, Little Big Painting, 1965
627 Jasper Johns, Three Flags, 1958
625 John Baldessari, An Artist is Not Merely the Slavish Announcer, 1966–68
645 Anne Truitt, Triad, 1977
701 Alexander Calder, Calder's Circus, 1926–31
741 Georgia O'Keeffe, Summer Days, 1936
720 Edward Hopper, Early Sunday Morning, 1930
722 Jared French, State Park, 1946
740 Willem de Kooning, Woman and Bicycle, 1952–53
714 Arshile Gorky, The Artist and His Mother, c. 1926–36
742 Arshile Gorky, The Betrothal II, 1947
745 David Smith, Hudson River Landscape, 1951
743 Mark Rothko, Four Darks in Red, 1958
811 Charles Burchfield, Cricket Chorus in the Arbor, 1917
820 Joseph Stella, The Brooklyn Bridge: Variation on an Old Theme, 1939
821 Man Ray, New York, 1917
822 Charles Demuth, My Egypt, 1927
823 Jacob Lawrence, War Series, 1946–47
102 Edward Hopper, Life Studies
103 John Sloan, Backyards, Greenwich Village, 1914
705 George Bellows, Dempsey and Firpo, 1924
703 Reginald Marsh, Twenty Cent Movie, 1936
733 Ben Shahn, The Passion of Sacco and Vanzetti, 1931–32
721 George Tooker, The Subway, 1950
703 Marsden Hartley, Painting, Number 5, 1914–1915
612 Lee Bontecou, Untitled 1961, 1961
642 Richard Serra, Prop, 1968, refabricated 2007
502 Chuck Close, Phil, 1969
613 Louise Nevelson, Dawn's Wedding Chapel II, 1959
723 Man Ray, La Fortune, 1938
561 Richard Serra, Abu Ghraib, 2004
810 Georgia O'Keeffe, Music, Pink and Blue No 2, 1918
101 Robert Henri, Gertrude Vanderbilt Whitney, 1916
511 Liliana Porter, Untitled (with Cube – tall), 1974
514 Hans Haacke, Shapolsky et al. Manhattan Real Estate Holdings, a Real-Time Social system, as of May 1, 1971, 1971
540 Karen Kilimnik, The Hellfire Club Episode of the Avengers, 1989
601 Frank Stella, Die Fahne Hoch!, 1959
646 Robert Morris, Untitled (3 Ls), 1965
503 Philip Guston, Cabal, 1977
702 Thomas Hart Benton, Poker Night (from A Streetcar Named Desire), 1948
521 Jeff Koons, New Hoover Convertibles Green, Blue, New Hoover Convertibles Green, Blue Doubledecker, 1981–87
560 Glenn Ligon, Rückenfigur, 2009
542 Mike Kelley, More Love Hours than Can Ever Be Repaid and Wages of Sin, 1987
532 Robert Gober, Untitled, 1991
510 Joan Jonas, Vertical Roll, 1972
732 Louis Lozowick, Hooverville, 1932
610 Jay DeFeo, The Rose, 1958–66
632 Milton Glaser, Don't Eat Grapes, 1969
643 Richard Tuttle, Grey Extended Seven, 1967
647 Tony Smith Die, 1962
520 Richard Prince, Spiritual America, 1983
531 David Wojnarowicz, Untitled, 1989
823 Elsie Driggs, Pittsburgh, 1927
824 Alexander Calder, Hanging Spider, c. 1940
704 Paul Cadmus, Sailors and Floosies, 1938
713 Elizabeth Catlett, Head, 1947
746 David Smith, Running Daughter, 1956
622 Yayoi Kusama, Air Mail Stickers, 1962
620 Andy Warhol, Before and After, 4, 1962
603 Carmen Herrera, Blanco y Verde, 1959
543 Fred Wilson, Guarded View, 1991
Liliana Porter, Untitled (with Cube – tall), 1974. Photoetching and graphite pencil: sheet (irregular), 31 3/4 × 24 3/8 in. (80.6 × 61.9 cm); image (composition), 19 1/2 × 12 1/2 in. (49.5 × 31.8 cm); image, 8 7/8 × 11 5/8 in. (22.5 × 29.5 cm). Whitney Museum of American Art, New York; purchase, with funds from the Print Committee 2011.89 © Liliana Porter courtesy of Barbara Krakow Gallery, Boston, MA

redpajama_set_name: RedPajamaCommonCrawl