# Thermodynamics of interacting hard rods on a lattice

Tounsi Benmessabih<EMAIL_ADDRESS>Department of Physics,
University Mustapha Stambouli of Mascara, 29000 Mascara, Algeria  Benaoumeur
Bakhti<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Department of Physics, University Mustapha
Stambouli of Mascara, 29000 Mascara, Algeria  Reda Chellali
<EMAIL_ADDRESS>Institute of Nanotechnology, Karlsruhe Institute of
Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen,
Germany

###### Abstract

We present an exact derivation of the isobaric partition function of lattice
hard rods with arbitrary nearest neighbor interactions. Free energy and all
thermodynamics functions are derived accordingly and they written in a form
that is a suitable for numerical implementation. As an application, we have
considered lattice rods with pure hard core interactions, rods with long range
gravitational attraction and finally a charged hard rods with charged
boundaries (Bose gas), a model that is relevant for studying several phenomena
such as charge regulation, ionic liquids near charged interfaces, and an array
of charged smectic layers or lipid multilayers. In all cases, thermodynamic
analysis have been done numerically using the Broyden algorithm.

##  I Introduction

The physics of one dimensional structures attracts recently much interest both
in theory and in experiment due to its relevant for nanotechnology Colinge and
Greer (2016); Xu  _et al._ (2011); Dreyfus  _et al._ (2009). With the
emergence of th manipulation and high resolution imaging instruments such as
the Scanning tunneling microscopy (STM), scanning tunneling spectroscopy (STS)
and atom probe tomography Chellali  _et al._ (2020, 2019, 2016, 2012), it is
possible in currents experiments to manipulate one dimensional chains of atoms
or molecules with only one atom or molecules in diameter Flöhr  _et al._
(2012); Iancu  _et al._ (2009); Zhou  _et al._ (2008). The latter have been
shown to have a unique and fascinating properties which does not exist in two
or three dimensions Dubin (1997); Prestipino and Giaquinta (2003); Prestipino
(2003).

One dimensional hard rods played a central role in the development of
classical density functional theory Bakhti  _et al._ (2012, 2013); Bakhti
(2013); Bakhti  _et al._ (2014, 2015); Girardeau and Astrakharchik (2012) and
it is paradigm model for the statistical mechanics of an extended molecules or
polymers. It has been extensively studied and many exact results have been
derived. Beside their mathematical interest, the exactly solvable models in
one dimension are important to test the quality of the simulation experiments
and they provide an insight to the structure of the theory in three dimension
such as the fundamental measure theory Rosenfeld (1989); Lafuente and Cuesta
(2002); Tarazona  _et al._ (2008) which has been inspired in its construction
by the Percus free energy for hard rods in an arbitrary external field Percus
(1976); Robledo and Varea (1981). In addition, if the external potential is
uni-directional, the $2D$ and the $3D$ systems are reduced to an effective
$1D$ systems in which the $1D$ effective density is the planar average of
density of the higher dimensional systems.

Despite its important role in the developement of the classical theory of
fluids, only few results are available for hard rods with interaction beyond
the hard core exclusion Percus (1989); Vanderlick  _et al._ (1989); Brannock
and Percus (1996); Percus (1997); Mortazavifar and Oettel (2017), and the
thermodynamics of these systems have been very rarely invoked in the
literature. Even more, results for lattice systems are scarcity. The lattice
description of classical fluids Lafuente and Cuesta (2004) are shown to be
simple and very useful in many cases such DNA denaturation Azbel (1979),
fluids in porous media Kierlik  _et al._ (2001), glasses Biroli and Mézard
(2001) and roughening phenomena Chui and Weeks (1981). Thus, it is our aim in
this paper set up a gerenal formalism for treating lattice hard rods with
interactions in the isobaric ensemble Bishop and Boonstra (1983) that it
suitable for numerical implementation. Then, a detailed numerical studies of
three cases will be presented. The paper is organized as fellow: we consider
first the general problem of hard rods with arbitrary nearest neighbor and
long range interactions. Then we make applications the pure hard rods system,
hard rods with gravitational attraction and Bose gas of hard rods which plays
a dominant role in the formation of many one dimensional structures. We
conclude with some outlooks for future work.

##  II General Formalism

Consider a lattice system of $N$ hard rods distributed on a one dimensional
lattice of $L$ sites. Each site can be either occupied or empty. We introduce
an occupation number $n_{i}$, that takes the value one if site $i$ is occupied
and zero else. The position of the rod $i$ ($z_{i}$) is labeled by the
position of its left end and the rods are ordered such that
$0<z_{1}<z_{2},\ldots,<z_{N}<L$. Beside the hard core exclusion, the rods
interact through an arbitrary two-body interaction and they are subject to
uniform external field and temperature. To calculate the exact partition
function of this system we adapted a procedure suggested in Ref Bishop and
Boonstra (1983) to calculate the partition function of continuum system of
hard rods. We consider first system of hard rods with nearest neighbor
interaction. The canonical partition function can be written as

$\mathcal{Z}=\sum\limits_{z_{N}=0}^{L}\sum\limits_{z_{N-1}=0}^{z_{N}}\ldots\sum\limits_{z_{1}=0}^{z_{2}}t(z_{1})t(z_{2}-z_{1})\ldots
t(L-z_{N})$ (1)

where $z_{i}$ is a discrete position of rod $i$ on the lattice ($z_{i}$ is
expressed in unit of the lattice parameter $a$ which we have omitted here for
simplicity). The factor $N!$ in the canonical partition function is canceled
by the $N!$ permutation of the rods ordering. The function $t(z)$ is given by

$t(z)=e^{-\beta\phi(z)}$ (2)

$\phi(z)$ is an arbitrary nearest neighbor interaction and $z$ is the discrete
separating distance between the rods. We can prove using the convolution
theorem of the Laplace transform that the Laplace transform of the partition
function Eq. (2) is given by

$\mathcal{\tilde{Z}}_{N}=(\tilde{t})^{N+1}$ (3)

where

$\tilde{t}(s)=\sum\limits_{z=0}^{\infty}e^{-pz}t(z)$ (4)

For application in statistical mechanics $p$ represents the pressure of the
system and $\mathcal{\tilde{Z}}$ is the partition function in the isobaric
ensemble. The canonical partition function is the inverse Laplace transform of
$\mathcal{\tilde{Z}}$. Despite it is not always possible to calculate the
inverse Laplace transform of $\mathcal{\tilde{Z}}$, the later itself is
important because its thermodynamic average agree to $1/N$ with those of the
canonical partition function Lebowitz and Percus (1961). The problem of
calculating the partition function is reduced to calculate the function
$\tilde{t}$ and all thermodynamic functions can be inferred from the latter
via differentiation.

##  III long range interaction

The precedent calculations can be generalized to the case where the
interaction is not limited to the nearest neighbor interaction. In this case,
the potential energy of each particle results from the interaction with all
other particles of the system. We present here a general formalism to deal
with the long range interaction and we present two specific examples in the
next section. The total potential energy is given by

$\displaystyle H=\sum\limits_{i<j}\phi_{i,j}n_{i}n_{j}$ (5)

where $\phi_{i,j}=\phi(x_{i},x_{j})$ is the interparticle interaction, $x_{i}$
the position on the lattice of rod $i$. $n_{i}$ and $n_{j}$ are the
occupations numbers of sites $i$ and $j$. The total interaction potential
between particle $i$ and the rest of the system can be written as

$\displaystyle\phi_{i}(x_{1},x_{2},\ldots,x_{N})=\sum\limits_{j\neq
i}^{N}\phi_{i,j}$ (6)

If the pair interactions depend only on the separating distance between the
rods, .ı.e $\phi(x_{i},x_{j})=\phi(|x_{j}-x_{i}|)$, then the coordinates
$\\{x_{1},x_{2},\ldots,x_{N}\\}$ can be mapped to a new coordinates
representing the separating distance between the rods in addition to the
center of mass, ı.e. $\\{x_{G},z_{1},z_{2},\ldots,z_{N-1}\\}$ where
$z_{i}=x_{i+1}-x_{i}$ and $x_{G}=\sum_{i=1}^{N}x_{i}/N$. In this case, we can
write $\phi(z)=\phi(|x_{j}-x_{i}|)$. For hard rods of size $\sigma$, the
interaction energy Eq. (6) manifests in the hard core exclusion which prevents
overlapping of particles. Because the later extends over the rod diameter
$\sigma$, the function $\tilde{t}$ is given by

$\displaystyle\tilde{t}(p,T)=\sum\limits_{z=\sigma}^{\infty}e^{-\beta(pz+\phi(z))}.$
(7)

Let as introduce the following notations

$e(z)=pz+\phi(z);\hskip 8.53581pt\omega(z)=e^{-\beta e(z)}.$ (8)

The free energy per particle is given by

$\displaystyle\beta
G=-\ln\\!\\!\left(\sum\limits_{z=\sigma}^{\infty}\omega(z)\right),$ (9)

from which all thermodynamic functions can be inferred. Expressed in term of
$e$ and $\omega$, the pressure $p$, entropy $S$, internal energy $U$, heat
capacity $C_{p}$, isothermal compressibility $\chi_{T}$ and expancivityb
$\alpha$ are given respectively by

$\displaystyle\frac{1}{\rho}=\frac{\sum\limits_{z=\sigma}^{\infty}z\omega(z)}{\sum\limits_{z=\sigma}^{\infty}\omega(z)},$
(10)

$\displaystyle\frac{S}{k_{B}}=\ln\\!\\!\left(\sum\limits_{z=\sigma}^{\infty}\omega(z)\right)+\frac{\sum\limits_{z=\sigma}^{\infty}e(z)\omega(z)}{k_{B}T\sum\limits_{z=\sigma}^{\infty}\omega(z)},$
(11)

$\displaystyle
U=\frac{\sum\limits_{z=\sigma}^{\infty}\phi(z)\omega(z)}{\sum\limits_{z=\sigma}^{\infty}\omega(z)},$
(12)

$\frac{C_{p}}{k_{B}\beta^{2}}=\frac{\sum\limits_{z=\sigma}^{\infty}e(z)^{2}\omega(z)}{\sum\limits_{z=\sigma}^{\infty}\omega(z)}+\left[\frac{\sum\limits_{z=\sigma}^{\infty}e(z)\omega(z)}{\sum\limits_{z=\sigma}^{\infty}\omega(z)}\right]^{2},$
(13)

$\chi_{T}=-\frac{1}{\rho}+\beta\frac{\sum\limits_{z=\sigma}^{\infty}z^{2}\omega(z)}{\sum\limits_{z=\sigma}^{\infty}z\omega(z)},$
(14)

$\frac{\alpha_{V}}{k_{B}}=\beta^{2}\frac{\sum\limits_{z=\sigma}^{\infty}e(z)(\rho
z-1))\omega(z)}{\sum\limits_{z=\sigma}^{\infty}\omega(z)}.$ (15)

The previous general formalism for treating the nearest neighbor and long
range interaction can be treated numerically for systems with any kind of
interaction. the advantage of the formalism presented in this work is that it
is more easier for numerical treatment compared to other approaches. In some
cases such as hard rods with only hard core exclusion, all thermodynamic
functions Eqs. (9)-(15) can evaluated analytically and explicitly as a
function of the control parameters $N$, $p$ and $T$. For long range
interactions, variation of thermodynamic function versus density can be still
evaluated with a direct numerical calculation. When varying temperature, we
have solved Eqs. (9)-(15) numerically using the Broyden algorithm.

##  IV Applications

In this section, we consider three cases that are relevant for many
applications such as sedimentation of effective one dimensional colloids under
gravity Bakhti and Müller (2021) or ordering transition in one dimensional
system with coulomb interactions King  _et al._ (2001).

###  IV.1 Hard rods

System of pure hard rods has been investigated in the literature both in
continuum Jang  _et al._ (2017); Champion and Alastuey (2015); Kierlik and
Rosinberg (1992); Zhang (1991); Honnell and Hall (1990); Percus (1976) and on
a lattice (Bakhti  _et al._ , 2012, 2013; Bakhti, 2013; Bakhti  _et al._ ,
2014, 2015) for mono and multicomponent cases. However, to the best of our
knowledge, there is no thermodynamic analysis of this system. Here we show how
the general formalism can be adapted to this system and present numerical
analysis of all the thermodynamics functions, which could be useful for
testing computer simulations. Let us Consider the case of hard rods with size
$\sigma$ (in unit of lattice site $a$). The inter-atomic coupling is given by

$\displaystyle\phi(z)=\left\\{\begin{array}[]{ll}\infty&\mbox{ for
$z<\sigma$},\\\ 0&\mbox{for $z\geq\sigma$}.\end{array}\right.$ (18)

The kinetic energy produces a constant term that is not relevant for
thermodynamics. Inserting the interaction energy into Eq. (4) we get an
analytic expression for the function $\tilde{t}$

$\displaystyle\tilde{t}(p)=\frac{x_{p}^{\sigma}}{1-x_{p}},$ (19)

where the pressure $p$ is absorbed in the quantity

$\displaystyle x_{p}=e^{-\beta p}.$

The free energy per particle is

$\displaystyle\beta
G(p,T)=-\ln\\!\left[\frac{x_{p}^{\sigma}}{1-x_{p}}\right].$ (20)

The equation of state inferred from the free energy reads

$\frac{1}{\rho}=\sigma+\frac{x_{p}}{1-x_{p}},$ (21)

from which we get an explicit expression for the mass density
$\tilde{\rho}=\sigma\rho$ as a function of the pressure

$\tilde{\rho}=\frac{\sigma(1-x_{p})}{\sigma(1-x_{p})+x_{p}}.$ (22)

Other thermodynamics functions follow straightforwardly using standard
thermodynamic relation. Entropy, heat capacity, isothermal compressibility and
thermal expansion coefficient (expansivity) are given respectively by

$S/k_{B}=\frac{x_{p}\ln x_{p}}{x_{p}-1}-\ln(1-x_{p}),$ (23)
$C_{p}=\left(\frac{x_{p}\ln x_{p}}{x_{p}-1}\right)^{2},$ (24)
$\chi_{T}=\frac{\beta x_{p}}{1-x_{p}}\frac{1}{x_{p}+\sigma(1-x_{p})},$ (25)

and

$\alpha_{T}/k_{B}=\frac{1}{1-x_{p}}\frac{\beta x_{p}\ln
x_{p}}{x_{p}+\sigma(1-x_{p})},$ (26)

where all the quantity are referred to its value per particle. The previous
thermodynamic functions can be re-expressed explicitly in term of the mass
density $\tilde{\rho}$ and they are given by

$\displaystyle\beta
G(p,T)\\!=\\!-\\!\ln\\!\\!\left[1+\frac{\sigma(\tilde{\rho}-1)}{\tilde{\rho}}\\!\right]\\!+\\!\sigma\ln\\!\\!\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right],$
(27) $\displaystyle\beta
p=\ln\\!\\!\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right],$ (28)
$\displaystyle\frac{S}{k_{B}}\\!\\!=\\!\\!\ln\\!\\!\left[\\!1\\!+\\!\frac{\sigma(\tilde{\rho}-1)}{\tilde{\rho}}\\!\right]\\!\\!-\\!\\!\left[\\!1\\!+\\!\frac{\sigma(\tilde{\rho}-1)}{\tilde{\rho}}\\!\right]\\!\\!\ln\\!\\!\left[\\!1\\!+\\!\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\\!\right],$
(29) $\displaystyle
C_{p}=\left(\frac{\sigma\left[1-\tilde{\rho}\right]}{\tilde{\rho}}\ln\\!\\!\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right]\right)^{2},$
(30)
$\displaystyle\chi_{T}=\frac{\beta\sigma}{\tilde{\rho}}\left[1-\tilde{\rho}\right]^{2}\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right],$
(31)
$\displaystyle\frac{\alpha_{T}}{k_{B}}\\!=\\!\frac{\beta\sigma}{\tilde{\rho}}\\!\\!\left[1-\tilde{\rho}\right]^{2}\\!\\!\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right]\\!\\!\ln\\!\\!\left[1+\frac{\tilde{\rho}}{\sigma(1-\tilde{\rho})}\right].$
(32)

Variations of the different thermodynamic functions as function of mass
density are depicted in Fig. (1) for different values of the rod diameter.

Figure 1: Free energy $G$, equations of state $p(\tilde{\rho})$, entropy $S$,
heat capacity $C_{p}$, isothermal compressibility $\chi_{T}$ and expansivity
$\alpha_{V}$ of hard rods on with different sizes. All quantities are per
particle. The number of particles is set to $N=50$.

The free energy $G$ increases monotonically with increasing the mass density
reflecting the absence of phase transition, which is a well known results for
$1D$ systems with short range interactions. At low density, increasing the
rods diameter at a given volume reduces the free volume accessible for the
rods and hence the free energy decreases. At high density, when increasing the
rods diameters, the correlation between rods increases and this rises the free
energy. At low density, the pressure rises linearly with temperature as it
does in the classical ideal gas. At high density, the correlation increases
and the variation of the pressure deviates from the ideal gas pressure
profiles. Increasing the density (or decreasing the rods diameter) enhance the
ordering and accordingly the entropy $S$ decreases with both increasing the
mass density or decreasing the rod diameter. This has a direct and similar
consequence on the heat capacity $C_{p}$. Both quantities reflect the athermal
nature of the hard core interactions. Isothermal comressibility $\chi_{T}$ and
thermal expansion coefficient $\alpha_{V}$ express respectively the relative
change in volume due to change in pressure (at fixed temperature) or change in
temperature (at fixed pressure). Panels for compressibility and expansivity
show the lattice fluids of hard rods is very compressible at low density but
at high density where the system is filled, the volume changes slightly with
the applied pressure.

Figure 2: Variation of the different thermodynamic functions versus
temperature for rods with pure hard core interaction. The number of particles
and pressure are set respectively to $N=50$ and $p=0.2$.

Variations of thermodynamic functions versus temperature are shown in Fig.
(2). Free energy decreases with increasing temperature as the later disperse
the rods and it reduce correlations. Rising temperature also reduces order and
hence entropy increases as can be seen in Fig. (2).

###  IV.2 Attractive gravitational interaction

Hard rods in continuum with gravitational attraction has been investigated in
different ensemble in the context of studying gravitational collapse Kumar and
Miller (2017); Champion and Alastuey (2015); Youngkins and Miller (2000) and
it also important for studying effects of gravity on phase transtions in
colloids Bakhti and Müller (2021) where the spherical symmetry of the
gravitational interactions reduce any higher dimensional system to an
effective one dimensional system but with structures different from the true
one dimensional system. Analytical expression of the partition function and
some other thermodynamic quantities have been worked out in Champion and
Alastuey (2015); Jang  _et al._ (2017). Here, we consider the lattice version
of this system where the rods are distributed on a lattice of L sites. The
rods interact through an attractive two-body gravitational potential of the
form

$\displaystyle H=\sum\limits_{i<j}\phi_{i,j}n_{i}n_{j}$ (33)

with

$\displaystyle\phi_{i,j}=\left\\{\begin{array}[]{ll}\infty&\mbox{if
$|z_{i}-z_{j}|<\sigma$ }\\\ Gm^{2}|z_{i}-z_{j}|&\mbox{if
$|z_{i}-z_{j}|\geq\sigma$ }\end{array}\right.$ (36)

where $m$ is the mass of the rod, $G$ is the strength of gravitation and
$z_{i}$ is the position of rod $i$. The total potential energy can be
rewritten as

$\displaystyle\phi(z_{1},z_{2},\ldots,z_{N})$ $\displaystyle=\sum_{1\leq
i<j\leq N}\phi_{i,j}$ $\displaystyle=\sum_{i=1}^{N-1}a_{i}(z_{i+1}-z_{i})$
(37)

with

$\displaystyle a_{i}=Gm^{2}i(N-i)$ (38)

To calculate the partition function, we make a variable change in Eq. (IV.2)
where we switch from using the position of the rods $z_{i}$ as coordinates to
the distance between the rods and the center of gravity as a new coordinates.
Inserting the total interaction energy into Eq. (4) we get an exact expression
for the isobaric partition function

$\displaystyle\mathcal{Z}=\prod_{i=0}^{N}\tilde{t}_{i},$ (39)

where

$\displaystyle\tilde{t}_{i}=\frac{\omega_{i}^{\sigma}}{1-\omega_{i}},\hskip
28.45274pt\omega_{i}=e^{-\beta(p+a_{i})}.$ (40)

The partition function can be simplified into

$\displaystyle\mathcal{Z}=e^{-\beta
p_{0}\sigma}\prod_{i=1}^{N}\frac{1}{1-\omega_{i}}$ (41)

with $p_{0}=[p(N+1)+Gm^{2}N(N^{2}-1)/6]$. Thermodynamics functions inferred
from Eq. (41) are given by

Figure 3: Free energy, pressure, entropy, heat capacity, compressibility and
expansivity of a gas of hard rods on a lattice with attractive gravitational
interactions. The parameter $k$ is $k=\beta m^{2}g\sigma$.

$\displaystyle\beta G=\beta p_{0}\sigma+\sum_{i=1}^{N}\ln(1-\omega_{i}),$ (42)
$\displaystyle\frac{1}{\rho}=\sigma+\frac{1}{N+1}\sum_{i=1}^{N}\tilde{\omega}_{i},$
(43)
$\displaystyle\mathcal{S}=-\sum_{i=1}^{N}\ln(1-\omega_{i})+\beta\sum_{i=1}^{N}\tilde{\omega}_{i}(p+a_{i}),$
(44) $\displaystyle
C_{p}=\beta^{2}\sum_{i=1}^{N}(p+a_{i})^{2}\frac{\tilde{\omega}_{i}^{2}}{\omega_{i}},$
(45)
$\displaystyle\chi_{T}=\frac{\beta\rho}{N+1}\sum_{i=1}^{N}\frac{\tilde{\omega}_{i}^{2}}{\omega_{i}},$
(46)
$\displaystyle\frac{\alpha_{V}}{k_{B}}=\frac{\beta^{2}\rho}{N+1}\sum_{i=1}^{N}(p+a_{i})\frac{\tilde{\omega}_{i}^{2}}{\omega_{i}},$
(47)

where

$\displaystyle\tilde{\omega}_{i}=\frac{1}{\omega_{i}^{-1}-1}.$ (48)

We have evaluated Eq. (42)-(47) numerically using the Broyden algorithm and
they are presented in Fig. (3) for different value of the parameter $k=\beta
Gm^{2}\sigma$, that expresses the ratio the gravitational energy to thermal
energy. For convenience, we have shown variations of all as functions of the
excess volume

$\displaystyle\bar{V}=\frac{1}{\rho}-1,$ (49)

Figure 4: Variation of the different thermodynamic functions versus
temperature for rods with gravitational interaction. The number of particles,
pressure and the strength of the gravitational are set respectively to $N=50$,
$p=0.2$ and $Gm^{2}$. The rod diameter is fixed at $\sigma=5$.

in which the lattice parameter is set to one ($a=1$). Because of the finite
size of the system, no phase transition happens as shown in Fig. (3). As
expected, the free energy $G$ increases with increasing the strength of the
gravitational attraction and this changes is more pronounce at low values of
the excess volume where the the rods are close to each other. This has its
effect on the pressure $p$ in which the pressure increases (decreases) when
increasing the filling fraction (excess volume). At low excess volume and due
to the long range attraction, any small changes in the excess volume strongly
effects the pressure and vice versa. In kinetic theory the distinction is made
between kinetic pressure and interaction pressure. Kinetic pressure is
attributed to elastic collisions between rods and container walls. Attractive
interatomic couplings tend to reduce the total pressure. The entropy $S$
decreases with increasing the attractive forces due to the fact the the later
produces more order in the system. At very strong attraction, all the rods
sticks together to form larger cluster and the entropy is almost zero.
Contrary to the athermal hard core interactions, in the case of attractive
interactions, temperature is involved explicitly in the expression of entropy
and heat capacity $C_{p}$. The effect of correlation induced by the presence
of attraction reduces drastically the heat capacity and also makes the lattice
fluids more sensitive to the effect of reservoir pressure as shown for the
curves of compressibility and expansivity. Variation of all thermodynamics
functions versus temperature are shown in Fig. (4), where the monotonic
variations confirm the absence of phase transitions. The energy transfer
during isobaric expansion is naturally expressed as a change in enthalpy,
$E=U+pV$ (with $U$ is the internal energy). The system performs work on the
environment ($\Delta W<0$) if heat is added ($\Delta Q>0$). Energy
conservation with $U=0$ implies that $\Delta W=-\Delta W$ and we have

$\displaystyle\Delta W=-p\Delta V=-\Delta E,$ $\displaystyle\Delta
Q=\int_{T_{ini}}^{T_{fin}}C_{p}dT=+\Delta E.$

###  IV.3 Bose gas with hard wall

System of charged particles with charged bounderies and confined into one
dimensional channel has been introduced long time ago as a model for one-
component plasma but the model still attracts many interest Dean  _et al._
(2014); Dubin (1997); King  _et al._ (2002); Heo  _et al._ (2019). The model
has been shown to be very useful in studying mechanism of charge regulation
Frydel (2019), ionic liquids near charged interfaces Hayes  _et al._ (2015);
Ivanistsev and Fedorov ; Fedorov and Kornyshev (2014), and array of charged
smectic layers or lipid multilayers Dean  _et al._ (2014). We apply the
general formalism of Sec. (III) to the case of $N$ charged rods in the
presence of charged hard wall (considered as a neutralizing background). The
Hamiltonian is given by

Figure 5: Free energy, pressure, entropy, heat capacity, compressibility and
expansivity of a gas of hard rods with $k=\beta\sigma/N\lambda$.

$H=\sum\limits_{i<j}\phi_{i,j}n_{i}n_{j}$ (50)

where the one dimensional coulomb interactions between rods have the form

$\phi_{i,j}=\left\\{\begin{array}[]{ll}\infty&\mbox{for
$|z_{i}-z_{j}|<\sigma$}\\\ -\frac{|z_{i}-z_{j}|}{N\lambda}&\mbox{for
$|z_{i}-z_{j}|\geq\sigma$}\end{array}\right.$ (51)

with $\lambda$ is the interaction strength and the factor $N$ in the
denominator has been introduced to neutralize the system Frydel (2019). The
wall confinement is represented by the potential energy

$\displaystyle\beta V_{w}(x_{i})=\frac{x_{i}}{\lambda}$ (52)

applied on particle $i$. The combined effects of coloumb interpaticles
interaction and wall poential can be rearanged into Frydel (2019)

$\displaystyle\phi(z)=a_{i}z;\hskip 28.45274pta_{i}=\frac{2N-2i+1}{N\lambda}$
(53)

The function $\tilde{t}$ is given by

$\displaystyle\tilde{t}_{i}(p,T)=\frac{e^{-\beta(p+a_{i})\textasciiacute\sigma}}{1-e^{-\beta(p+a_{i})}}$
(54)

and the free energy per rod is given by

$\beta
G(p,T)=\beta\\!\left(p(N+1)+\frac{N}{\lambda}\right)\\!\sigma+\sum_{i=1}^{N}\ln\\!\left(1-\omega_{i}\right)$
(55)

with $\omega_{i}=e^{-\beta(p+a_{i})}$. Thermodynamic functions have the same
analytical forms as for the gravitational interactions but with a different
$\tilde{\omega}$. Numerical results for the thermodynamic functions are shown
in Fig. (5) for different values of the strength of the electrostatic
interaction $\lambda$ encoded in the parameter $k=\beta\sigma/N\lambda$. The
pressure results from the ideal gas contribution and from the interaction.
Contrary to the attractive gravitational interactions, the repulsive
electrostatic interaction augment the pressure as shown in Fig. (5).
Compressing the gas of lattice hard rods means removing vacant cells. Removing
vacant cells lowers the entropy. Lowering the entropy at $T>0$ (Fig. (6))
implies that heat is expelled. The heat capacity $C_{p}$ rises exponentially
from zero both at low $\bar{V}$ (Fig. (5)) or $T$ (Fig. (6)) and approaches
unity asymptotically as $\bar{V}$ or $T$ becomes large. The combined effect of
electrostatic interaction and charged hard wall makes the lattice hard rods
more sensitive to external pressure and less sensitive to the effect of
temperature as show for the panels of compressibility and expansivity in Fig.
(5).

Figure 6: Variation of the different thermodynamic functions versus
temperature for charger hard rods in neutralizing background and under sem-
confinement. The number of particles, pressure and the strength of the
gravitational are set respectively to $N=50$, $p=0.2$ and $\sigma=5$ and
$\lambda=1.0$.

##  V Conclusion

In this paper we have carried out an exact analytic and numerical studies of
some statistical lattice models in the isobaric ensemble. Despite that the
quantum description is the more natural at the micro and nanoscale, it has
been proved that for system at high temperature where thermal fluctuation is
compared or dominated over the quantum fluctuation, the statistical
description is appropriate and highly justified. First, we have established a
general formalism to calculate the exact partition function and the
corresponding thermodynamics quantities for an arbitrary nearest neighbor
interactions. Calculation is made for the Isobaric ensemble, but an inverse
Laplace transform can be used to get the canonical partition function.
However, in the thermodynamic limit, thermodynamic average in the isobaric
partition function agree to $1/N$ with those of the canonical partition
function Lebowitz and Percus (1961). Applications is made to systems with hard
core exclusion, gravitational attractive interactions and charged hard rods in
semi-confinement with one hard wall. Our study are motivated by the recent
extensive experimental study of the one dimensional atomic chains and their
formation on surface. Extension of this to the case of inhomogeneous system
with an arbitrary external field is currently a work under progress.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author. The data are not publicly available due to privacy.

###### Acknowledgements.

B. B. would like to thank Prof. Gerhard Müller for very valuable discussion.
R. C. acknowledges the financial support received from Karlsruhe Institute of
Technology (KIT).

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