# Thermodynamic geometry of a system with unified quantum statistics

Habib Esmaili  Department of Physics, University of Mohaghegh Ardabili, P.O.
Box 179, Ardabil, Iran  Hosein Mohammadzadeh<EMAIL_ADDRESS>Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179,
Ardabil, Iran  Mehdi Biderang  Department of Physics, University of Toronto,
60 St. George Street, Toronto, Ontario, M5S 1A7, Canada  Morteza Nattagh
Najafi  Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179,
Ardabil, Iran

###### Abstract

We examine the thermodynamic characteristics of unified quantum statistics as
a novel framework that undergoes a crossover between Bose-Einstein and Fermi-
Dirac statistics by varying a generalization parameter $\delta$. We find an
attractive intrinsic statistical interaction when $\delta\leq 0.5$ where the
thermodynamic curvature remains positive throughout the entire physical range.
For $0.5<\delta<1$ the system exhibits predominantly Fermi-like behavior at
high temperatures, while at low temperatures, the thermodynamic curvature is
positive and the system behaves like bosons. As the temperature decreases
further, the system undergoes a transition into the condensate phase. We also
report on a critical fugacity ($z=Z^{*}$) defined as the point at which the
thermodynamic curvature changes sign, i.e. for $z<Z^{*}$ ($z>Z^{*}$), the
statistical behavior resembles that of fermions (bosons). Also, we extract the
variation of statistical behaviour of the system for different values of
generalization parameter with respect to the temperature. We evaluate the
critical fugacity and critical $\delta$ dependent condensation temperature of
the system. Finally, we investigate the specific heat as a function of
temperature and condensation phase transition temperature of the system for
different values of generalization parameter in different dimensions.

##  I Introduction

Quantum distributions are essential for elucidating the statistical behaviour
of particles across a multitude of physical systems. Within this realm, the
Bose-Einstein and Fermi-Dirac distributions hold paramount importance,
extensively researched in scientific literature. These distributions grant
valuable insights into the unique traits of bosons and fermions, two essential
categories of quantum particles. Bosons are quantum entities that can occupy a
quantum level without any constraints on particle numbers, whereas each
quantum level can only host a single fermion due to the Pauli’s exclusion
principle. This principle prohibits the co-occupation of a single quantum
level by multiple fermions, a limitation not applicable to bosons. In the
context of quantum field theory, bosons and fermions are associated with
commutative and anti-commutative algebras for their creation and annihilation
operators, respectively.

Driven by both theoretical insights and empirical observations, the field of
generalized statistics has witnessed diverse developments. Physicists have
proposed various generalizations, encompassing anyons [1], Haldane fractional
exclusion statistics [2], deformed statistics [3, 4], and non-extensive
statistics [5]. A recent progress in this field is the unveiling of a unified
quantum statistics by Yan [6]. This novel approach is founded on the premise
that the quantum state of a system composed of multiple particles is
contingent upon the intrinsic properties of these constituent particles. In
this context, a unified quantum statistics has been postulated, with the
potential to seamlessly interpolate between Bose-Einstein and Fermi-Dirac
statistics.

Gibbs played a pioneering role in initiating the exploration of the geometric
aspects within thermodynamic systems [7, 8]. Subsequently, Ruppeiner and
Weinhold laid the foundation for the development of thermodynamic geometry [9,
10]. Ruppeiner introduced a formalism grounded in fluctuation theory. In doing
so, he established a thermodynamic parameter space and defined an appropriate
metric for that space. The elements of this metric are tied to the second
derivatives of the system’s entropy concerning intrinsic extensive
thermodynamic parameters such as internal energy, system volume, and total
particle count [9, 11]. Similarly, by computing the second derivatives of
internal energy with respect to the relevant intrinsic extensive thermodynamic
parameters, the components of the Weinhold metric can be ascertained [10]. It
has been demonstrated that the Ruppeiner and Weinhold metrics are essentially
equivalent, differing primarily by a conformal factor [12, 13]. For different
thermodynamic systems, alternative metrics can be introduced. For example,
Diósi et al., Janyszek and Mrugała employed derivatives of the logarithm of
the grand canonical partition function concerning the nonextensive parameters
of the system to determine the metric elements [14, 15]. Further variations in
metric formalism for geometric analysis were explored in [11].

The thermodynamic curvatures is a scalar which is derived from metric. This
quantity often referred to as the Ricci scalar plays a pivotal role in the
analysis of thermodynamic systems. To date, many thermodynamic systems have
been analyzed within inspecting the corresponding Ricci scalar.For instance,
it has been established that the thermodynamic curvature of a single-component
ideal gas is zero [9, 16]. Janyszek and Mrugała delved into the investigation
of the thermodynamic curvature for well-known ideal quantum gases like Bose
and Fermi gases [15, 17]. Extensive studies have revealed that the
thermodynamic curvature of an ideal Bose gas consistently displays a positive
value across its entire physical range, while for an ideal Fermi gas, it
consistently exhibits a negative value. It is important to note that the
choice of sign convention for thermodynamic curvature is arbitrary. Indeed,
the sign of the thermodynamic curvature serves as a means of categorizing the
inherent statistical interactions among the particles within the system.
Information geometry of quantum gases was reconsidered in [18] and some
results of [15] about the Fermi gas calculation was modified. Furthermore, the
scalar thermodynamic curvature of ideal quantum gases obeying Gentile’s
statistics has been investigated using the information geometric theory [18].
Recently, the thermodynamic geometry of quantum gases and specially the Bose-
Einstein fluid, with a focus on the strongly degenerate case have been
considered [19, 20].

Thermodynamic curvature exhibits singular behavior at the phase transition
points. Specifically, the thermodynamic curvatures of an ideal Bose gas,
nonextensive boson gas, and deformed boson gas all demonstrate singularities
at the condensation transition point [15, 21, 22, 23]. This singularity
phenomenon has been investigated across a diverse array of systems, including
black holes [24, 25], boundary conformal field theory within gauge/gravity
duality [26], as well as nonextensive and Kanadiakis statistics [27, 28], and
trapped ideal quantum gases [29].

In this paper, we introduce a thermodynamic geometry approach tailored for the
unified quantum statistics proposed by Yan [6]. The investigation of
thermodynamic geometry within ideal gas systems composed of particles adhering
to both conventional quantum statistics and various generalized statistics has
been comprehensively addressed in prior works [17, 15, 30, 21, 30]. Based and
analysis of the thermodynamic curvature we show that there is a rich phase
space for this model including BE-FD cross over and critical BEC transition.

The paper is organized as follows: In Sec. II, we provide a concise
introduction to the recently proposed unified quantum statistics. In Sec. III,
we derive expressions for various thermodynamic quantities, including internal
energy and total particle number, applicable to an ideal gas subject to
unified statistics in arbitrary dimensions. Sec. IV is dedicated to an in-
depth examination of thermodynamic geometry, along with a brief discussion on
the computation of thermodynamic curvature. Sec. V is focused on the
construction of the thermodynamic parameter space for an ideal gas
incorporating quantum unified statistics, along with an investigation into the
thermodynamic curvature. In Sec. VI, we conduct a comprehensive exploration of
the condensation phenomenon in the context of quantum statistics. Lastly, in
Sec.n VII, we present concluding remarks to summarize the key findings of this
paper.

##  II Unified Quantum Statistics

Over time, based on theoretical arguments and experimental evidences
sufficient motivations emerged for the introduction of intermediate statistics
which interpolates between the conventional quantum statistics of Bose-
Einstein and Fermi-Dirac distributions. An important example is the BCS-BEC
crossover phenomenon. In a BCS system, Cooper pairs are formed through the
attractive interaction between fermions. These Cooper pairs exhibit a
collective behaviour and can condense into a state of superfluidity or
superconductivity at low temperatures [31, 32]. On the other hand, BEC occurs
in bosonic systems, such as alkali atoms cooled to very low temperatures,
during which a large number of bosonic particles occupy the quantum ground
state, resulting in a macroscopic coherence [33, 34]. BCS-BEC crossover arises
when there is a gradual transformation between these two phenomena as the
strength of the attractive interaction is varied, during which the system can
transition from a BCS-like superfluid phase, where Cooper pairs dominate, to a
BEC-like phase, where the particles exhibit Bose-Einstein condensation [35].
This crossover is often observed in ultracold atomic gases, where the
interparticle interactions can be controlled [36].

There are several methods to explain the crossover between bosonic and
fermionic statistics, among which the intermediate statistics approach has
been a robust, promising methodology. As an illustration, fascinating
intermediate statistics have been proposed by generalizing the statistical
weights associated with the energy levels [37]. Lately, there has been a
growing perspective on the quantum state of systems composed of multiple
particles, viewing it as a functional defined within the particles’ internal
space. As a result, a unified framework has been developed to encompass both
bosons and fermions under a single exchange statistics paradigm [6]. These are
various extensions of the algebra for creation and annihilation operators,
including $q$ and $qp$-deformed bosons and fermions, f-deformed fermions,
Wignons, and deformed Tamn-Dankoff [21, 38, 39, 40, 41]. A very promising
generalization was made by Yan [6], who introduced the following algebra:

$\displaystyle[a_{i},a^{\dagger}_{j}]_{q}=\mathbb{1}^{q}\delta_{ij},$ (1)
$\displaystyle[a^{\dagger}_{i},a^{\dagger}_{j}]_{q}=[a^{\dagger}_{i},a^{\dagger}_{j}]_{q}=0,$
(2)

where $a_{i}$ ($a^{\dagger}_{i}$) represents the annihilation (creation)
operator and the unite operator is defined as follows [6]

$\displaystyle\mathbb{1}^{q}=\left\\{\begin{array}[]{cc}\mathbb{1},&\text{if~{}all~{}$n_{i}=0$~{}or~{}1,}\\\
\frac{1}{2}(\mathbb{1}+\hat{q}),&\text{otherwise}\\\ \end{array}\right.,$ (5)

with $\hat{q}$ as the exchange statistics factor operator [6]. It has been
argued that for completely indistinguishable particles, the exchange
statistics factor operators are independent of the pairs of particles and
commute with any operator in the system. They neither change the physical
configuration nor mix up internal states further. Furthermore, it has been
demonstrated that the exchange statistics factor operator must be Hermitian
and unitary. The Hamiltonian of an ideal gas of such particles is given by
$H(\hat{q})=\sum_{i}\epsilon_{i}\hat{n}_{i}$, in which $\epsilon_{i}$ denotes
the single particle energy levels and $\hat{n_{i}}$ is the number operator of
levels. The grand partition function of the system has been derived as follows

$\displaystyle\hat{\Xi}(\hat{q})=\prod_{i}\frac{1-\frac{1-\hat{q}}{2}e^{2\beta(\mu-\epsilon_{i})}}{1-e^{\beta(\mu-\epsilon_{i})}},$
(6)

with $\beta=1/k_{B}T$ and $\mu$ is the chemical potential. When the particle’s
internal state is an eigenstate of the operator $\hat{q}$, it manifests itself
either as bosonic or fermionic characteristics. Typically, the internal state
of such a particle embodies an admixture of attributes akin to both bosons and
fermions. Within this composite state, we can derive the grand partition
function as follows:

$\displaystyle\Xi(\delta)=\prod_{i}\frac{1-\delta
z^{2}e^{-2\beta\epsilon_{i}}}{1-ze^{-\beta\epsilon_{i}}},$ (7)

where $z=e^{\beta\mu}$ denotes the fugacity and $\delta$ is a constant value
between $0$ and $1$. It is evident that when $\delta=0$ ($1$), the equation
above simplifies to the partition function of an ideal gas of bosons
(fermions). Within this scenario, the mean occupation number of a single
particle state of energy $\epsilon_{i}$ is expressed as:

$\displaystyle
n_{i}=\frac{1}{\exp(\beta(\epsilon_{i}-\mu))-1}-\frac{2\delta}{\exp(2\beta(\epsilon_{i}-\mu))-\delta}.$
(8)

In the limit of $\delta\\!=\\!0$ and $1$, the well-known Bose-Einstein and
Fermi-Dirac distribution function would be recovered, respectively.

##  III Thermodynamic quantities

Utilizing the distribution function of unified quantum statistics, as briefly
discussed in the preceding section, we can derive several noteworthy
thermodynamic quantities. We consider a D-dimensional ideal gas with the
following energy-momentum dispersion relation:

$\displaystyle\epsilon=\alpha p^{\sigma},$ (9)

where $p$ is particle’s momentum , $\sigma$ and $\alpha$ are dispersion
exponent, and the proportionality constant, respectively. In the non-
relativistic limit ($\epsilon={p^{2}}/{2m}$ , $m$ denoting the particle’s
Mass), we have $\alpha={1}/{2m}$, and $\sigma=2$. Conversely, in the
ultrarelativistic limit ($\epsilon=pc$ , $c$ being the light velocity), we
have $\alpha=c$, and $\sigma=1$. In $D$-dimensions, the single particle
density of states $\Omega(\epsilon)$ is

$\displaystyle\Omega(\epsilon)=\frac{A^{D}}{\Gamma(\frac{D}{\sigma})}\epsilon^{\frac{D}{\sigma}-1},\quad
A=\frac{L\sqrt{\pi}}{(a^{\frac{1}{\sigma}}h)}.$ (10)

Where $A$ is a constant which we set $A=1$ for simplicity. Here, $L^{D}$
represents the volume of a $D$-dimensional box. Using Eq. (8) the internal
energy is given by

$\displaystyle U=$ $\displaystyle\int_{0}^{\infty}\epsilon
n(\epsilon)\Omega(\epsilon)d\epsilon$ (11) $\displaystyle=$
$\displaystyle\frac{D}{\sigma}{\beta^{-(\frac{D}{\sigma}+1)}}\left[\bar{g}_{\frac{D}{\sigma}+1}(z)-2^{-\frac{D}{\sigma}}\bar{g}_{{\frac{D}{\sigma}}+1}(z^{\prime})\right]$
$\displaystyle=$
$\displaystyle\nu{\beta^{-(\nu+1)}}\left[G(\delta,\nu+1;z)\right].$

Where $\frac{D}{\sigma}\equiv\nu$ and $z^{\prime}\equiv\delta z^{2}$ and

$G(\delta,\nu;z)\equiv\left[\bar{g}_{\nu}(z)-2^{(1-\nu)}\bar{g}_{\nu}(z^{\prime})\right].$
(12)

also $\bar{g}_{\nu}(z)$ denotes the standard Bose-Einstein integrals which are
defined as follows

$\bar{g}_{\nu}(z)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{z^{-1}\exp(x)-1}dx.$
(13)

The total particles number is obtained as follows

$\displaystyle N=$
$\displaystyle\int_{0}^{\infty}n(\epsilon)\Omega(\epsilon)d\epsilon=\beta^{-\nu}\left[G(\delta,\nu;z)\right].$
(14)

Using Eqs. (7), and (11) and $P=(-\frac{\partial F}{\partial V})_{N,T}$ ,
where $F$ is the well known Helmholtz free energy ; it can be demonstrated
that, akin to conventional statistics, there exists a standard relationship
between pressure (P), volume (V), and internal energy, which is expressed as
follows:

$\displaystyle PV=\frac{U}{\nu}.$ (15)

##  IV Thermodynamic geometry

Ruppeiner and Weinhold introduced thermodynamic geometry as an innovative
approach to the study of thermodynamic systems [9, 10]. The thermodynamic
parameter space can be conceptualized as a Riemannian space, enabling the
establishment of a suitable metric within this space. The Ruppeiner metric is
constructed by computing the second-order derivatives of entropy concerning
the relevant extensive thermodynamic parameters, encompassing internal energy,
volume, and the total number of particles. Furthermore, Weinhold introduced an
alternative metric in the energy representation, which is defined by
evaluating the second-order derivatives of internal energy concerning the
pertinent extensive thermodynamic parameters. It has been noted that these
metrics exhibit conformal equivalence [13].

Performing a Legendre transformation on either entropy or internal energy with
respect to the extensive parameters results in the derivation of various
thermodynamic potentials, such as Helmholtz and Gibbs free energy. The Fisher-
Rao metric is defined by the second derivatives of the logarithm of the
partition function concerning the non-extensive thermodynamic parameters [15,
11, 42, 43], as shown below:

$\displaystyle g_{ij}=\partial_{i}\partial_{j}\ln{\cal{Z}}.$ (16)

Here, $\partial_{i}$ is a shorthand notation for the derivative with respect
to the non-extensive thermodynamic parameter $i$ and $\cal{Z}$ denotes the
partition function. It has been proposed that the logarithm of the partition
function for an ideal classical or quantum gas is contingent upon both the
volume of the system and parameters characterized as $\gamma=-\mu/k_{B}T$.
Typically, the volume is treated as a fixed thermodynamic parameter, resulting
in a two-dimensional parameter space for thermodynamic fluctuations.

The connection coefficient (Christoffel symbols) is defined by utilizing the
components of the metric tensor in the following manner:

$\displaystyle\Gamma^{i}_{jk}=\frac{1}{2}g^{im}\left(g_{mj,k}+g_{mk,j}-g_{jk,m}\right),$
(17)

wherein the elements of the inverse metric tensor are denoted as $g^{mn}$, and
we have the expression $g_{ij,k}=\partial_{k}g_{ij}$. The elements of the
Riemann tensor are obtained as follows:

$\displaystyle
R^{i}_{jkl}=\partial_{k}{\Gamma^{i}}_{lj}-\partial_{l}{\Gamma^{i}}_{kj}+{\Gamma^{i}}_{km}{\Gamma^{m}}_{lj}-{\Gamma^{i}}_{lm}{\Gamma^{m}}_{kj}.$
(18)

The Riemann tensor provides insights into the curvature of the thermodynamic
space. Furthermore, the Ricci tensor is defined as follows:

$\displaystyle R_{ij}=R^{m}_{imj},$ (19)

which is a second rank tensor. Consequently, the Ricci scalar is given by

$\displaystyle R=g^{ij}R_{ij},$ (20)

and is also known as the thermodynamic curvature. In the case of a two-
dimensional parameter space, R is simplified to

$R=-\frac{2\begin{vmatrix}g_{\beta\beta}&g_{\beta\gamma}&g_{\gamma\gamma}\\\
g_{\beta\beta,\beta}&g_{\beta\gamma,\beta}&g_{\gamma\gamma,\beta}\\\
g_{\beta\beta,\gamma}&g_{\beta\gamma,\gamma}&g_{\gamma\gamma,\gamma}\end{vmatrix}}{\begin{vmatrix}g_{\beta\beta}&g_{\beta\gamma}\\\
g_{\beta\gamma}&g_{\gamma\gamma}\end{vmatrix}^{2}}.$ (21)

Thermodynamic curvature plays a crucial role as a tool in the analysis of the
statistical properties of systems. It has been shown that the thermodynamic
parameter space of an ideal gas, where particles adhere to the classical
Maxwell-Boltzmann distribution, is associated with a flat space exhibiting
zero curvature. Furthermore, it has been noted that in the realm of ideal
quantum gases, thermodynamic curvature acquires positive (negative) value for
bosons (fermions) [15, 11, 18]. Indeed, the thermodynamic curvature’s sign
serves as an indicator for detecting the inherent statistical interactions
within the system. Systems featuring particles with intrinsic attractive
statistical interactions manifest positive curvature, whereas those with
intrinsic repulsive statistical interactions showcase negative curvature.
Apparently, the thermodynamic curvature of non-interacting classical gas would
be zero [12, 15, 11].

Behaviour of thermodynamic curvature enables us to identify the occurrence of
phase transitions. For instance, at the condensation point of an ideal boson
gas, the thermodynamic curvature displays singularity [11, 15, 44]. This
approach has been extensively applied to explore the thermodynamic geometry of
various generalized statistics, including deformed, non-extensive, and dual
statistics. These studies have unveiled that the singular points of the
thermodynamic curvature coincide with the phase transition points of the
systems [22, 45, 46].

##  V Thermodynamic parameters space of quantum unified statistics

In the previous sections, we have established the foundation for building the
parameter space of thermodynamic quantities for an ideal gas composed of
particles that adhere to a unified quantum statistics. We will operate under
the assumption that the volume of the system remains constant throughout the
analysis, serving as a fixed parameter. Consequently, the thermodynamic
parameters under consideration are denoted as $\beta$, and $\gamma$, resulting
in a two-dimensional parameter space for thermodynamics. We denote
$z=e^{-\gamma}$, $z^{\prime}=\delta z^{2}=\delta e^{-2\gamma}$, and using the
chain derivative rule and the definition of the Bose-Einstein integrals we
obtain

$\displaystyle\frac{\partial}{\partial\gamma}\bar{g}_{\nu}(z)=$
$\displaystyle\frac{\partial z}{\partial\gamma}\frac{\partial}{\partial
z}\bar{g}_{\nu}(z)=-z\frac{\partial}{\partial
z}\bar{g}_{\nu}(z)=-\bar{g}_{\nu-1}(z)$ (22)
$\displaystyle\frac{\partial}{\partial\gamma}\bar{g}_{\nu}(z^{\prime})=$
$\displaystyle\frac{\partial
z^{\prime}}{\partial\gamma}\frac{\partial}{\partial
z^{\prime}}\bar{g}_{\nu}(z^{\prime})=-2z^{\prime}\frac{\partial}{\partial
z^{\prime}}\bar{g}_{\nu}(z^{\prime})=-2\bar{g}_{\nu-1}(z^{\prime}).$

Therefore, the following relation is obtained for $G(\delta,\nu;z)$

$\frac{\partial}{\partial\gamma}G(\delta,\nu;z)=-G(\delta,\nu-1;z).$ (23)

Using Eqs. (16), (11), (14) and (23) we can derive the metric elements for the
aforementioned two-dimensional parameter space as follows:

$\displaystyle g_{\beta\beta}=$
$\displaystyle\frac{{{\partial^{2}}\ln{\cal{Z}}}}{{\partial{\beta^{2}}}}=-(\frac{\partial
U}{\partial\beta})_{\gamma}=-\frac{\partial}{\partial\beta}\left[{\beta^{-(\nu+1)}}\nu(G(\delta,\nu+1;z))\right]=\frac{\beta^{-(\nu+2)}}{\Gamma(\nu)}\Gamma(\nu+2)G(\delta,\nu+1;z),$
(24) $\displaystyle g_{\beta\gamma}=$ $\displaystyle
g_{\gamma\beta}=\frac{{{\partial^{2}}\ln{\cal{Z}}}}{{\partial{\beta}\partial{\gamma}}}=-(\frac{\partial
N}{\partial\beta})_{\gamma}=-\frac{\partial}{\partial\beta}\left[\beta^{-\nu}G(\delta,\nu;z)\right]=\frac{\beta^{-(\nu+1)}}{\Gamma(\nu)}\Gamma(\nu+1)G(\delta,\nu;z),$
$\displaystyle g_{\gamma\gamma}=$
$\displaystyle\frac{{{\partial^{2}}\ln{\cal{Z}}}}{{\partial{\gamma^{2}}}}=-(\frac{\partial
N}{\partial\gamma})_{\beta}=-\frac{\partial}{\partial\gamma}\left[\beta^{-\nu}G(\delta,\nu;z)\right]=\beta^{-\nu}G(\delta,\nu-1;z).$

We also use Eqs. (23) and (24) to obtain the derivatives of the metric
elements as follows:

$\displaystyle
g_{\beta\beta,\beta}=\frac{\partial}{\partial\beta}g_{\beta\beta}=-\frac{\beta^{-(\nu+3)}}{\Gamma(\nu)}\Gamma(\nu+3)G(\delta,\nu+1;z),$
(25) $\displaystyle
g_{\beta\beta,\gamma}=g_{\beta\gamma,\beta}=g_{\gamma\beta,\beta}=\frac{\partial}{\partial\gamma}g_{\beta\beta}=-\frac{\beta^{-(\nu+2)}}{\Gamma(\nu)}\Gamma(\nu+2)G(\delta,\nu;z),$
$\displaystyle
g_{\beta\gamma,\gamma}=g_{\gamma\beta,\gamma}=g_{\gamma\gamma,\beta}=\frac{\partial}{\partial\beta}g_{\gamma\gamma}=-\frac{\beta^{-(\nu+1)}}{\Gamma(\nu)}\Gamma(\nu+1)G(\delta,\nu-1;z),$
$\displaystyle
g_{\gamma\gamma,\gamma}=\frac{\partial}{\partial\gamma}g_{\gamma\gamma}=-\beta^{-\nu}G(\delta,\nu-2;z).$

In general, we can write the thermodynamic curvature that we defined in (21)
and the matrix elements from Eqs. (24) and (25) as follows:

$R=\frac{2(\nu+1)\beta^{\nu}\left(-2G(\delta,\nu+1;z)G(\delta,\nu-1;z)^{2}+G(\delta,\nu;z)^{2}G(\delta,\nu-1;z)+G(\delta,\nu-2;z)G(\delta,\nu;z)G(\delta,\nu+1;z)\right)}{\left(\nu
G(\delta,\nu;z)^{2}-(\nu+1)G(\delta,\nu-1;z)G(\delta,\nu+1;z)\right){}^{2}}.$
(26)

The relation (26) for $\delta=0$ is reduced to the thermodynamic curvature of
Bose gas which was obtained in [18].

We visualize the behaviour of thermodynamic curvature through diagrams in the
following.

Figure 1: Thermodynamic curvature of an ideal three dimensional ($D=3$) gas of
particles obeying unified quantum statistics as a function of fugacity for
isothermal processes ($\beta=1$). The particles are supposed to have non-
relativistic dispersion ($\sigma=2$). Solid (Brown) line corresponds to
$\delta=0$ (ideal boson gas) and lower solid line (Blue) represents curvature
of $\delta=1$ (ideal fermion gas). All dashed line corresponds to the values
$\delta=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$ from top to bottom respectively.

(a)

(b)

(c)

(d)

Figure 2: Thermodynamic curvature of an ideal gas with particles obeying
generalized unified quantum statistics as a function of fugacity for
isothermal processes ($\beta=1$). Solid (Brown) line corresponds to $\delta=0$
(ideal boson gas) and dash-dotted line (Blue) represents curvature of
$\delta=0.5$. All dashed lines correspond to the values
$\delta=0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1$. 2(a): $(D=2,\sigma=1)$,
2(b): $(D=2,\sigma=2)$, 2(c): $(D=3,\sigma=2)$ and 2(d): $(D=1,\sigma=2)$.

Fig. 1 illustrates the thermodynamic curvature as a function of fugacity in a
three-dimensional system composed of non-relativistic particles following
unified quantum statistics. It could be observed that the thermodynamic
curvatures exhibit distinct behaviour in special cases: when $\delta=0$
(representing bosons), they consistently display positive values, while for
$\delta=1$ (indicating fermions), they consistently show negative values
across the entire physical range. For all values $\delta\leq 0.5$, the
thermodynamic curvature remains positive, demonstrating an attractive
statistical interaction among the particles, which serves as a distinguishing
characteristic in a bosonic system. Nonetheless, when $0.5<\delta<1$, we
observe that depending on the specific value of $\delta$, it is possible to
identify a corresponding fugacity value, denoted as $z=Z^{*}$, at which the
sign of the thermodynamic curvature undergoes a reversal. This implies that
when $z<Z^{*}$ ($z>Z^{*}$), the intrinsic statistical interaction is repulsive
(attractive). Fig. 2 illustrates that the same principles hold true for other
dimensions and dispersion relations. We observe that the thermodynamic
curvature exhibits singularity for all values of $\delta$ except for
$\delta=1$ when $z=1$. It is widely recognized that the thermodynamic
curvature of an ideal boson gas experiences singularity at the critical
fugacity. Indeed, the highest permissible value for the fugacity that allows
for a non-negative Bose-Einstein distribution function is $z=1$, corresponding
to the critical value of the fugacity at the condensation temperature [47].
Studies have emphasized that the thermodynamic curvature of a conventional
boson gas displays singularity at the critical fugacity value. The
thermodynamic curvature of a unified quantum gas for any given $\delta$
appears to possess a unique characteristic at $z=1$, a feature that is known
to be temperature-dependent. We postulate that as the fugacity ($z$) of the
unified quantum gas approaches 1, it tends toward the condensate phase, in
which the fugacity of the system is fixed at $z=1$. Hence, the sole remaining
variable subject to fluctuations in the system is $\beta$. Consequently, the
thermodynamic parameter space at the condensate phase reduces to a one-
dimensional space, which, by definition, is flat. The thermodynamic parameter
space for the normal phase is determined by the fugacity and temperature
variables. All the quantities examined in this study exhibit this behavior,
including the thermodynamic curvature, which can be expressed as functions of
$\beta$ and $z$ in the form $\beta^{\alpha}F(z)$. The exponent $\alpha$ and
the function $F(z)$ vary across different quantities within distinct
dimensions and dispersion relations. To derive the thermodynamic curvature of
a system at a specific particle count as a function of temperature, we can
utilize Eqs. (14) and (20), which relate the thermodynamic curvature to the
total particle number for a particular value of the latter. Fig. 3 reveals
that the thermodynamic curvature undergoes a discontinuity at the transition
temperature. In the next section, we will consider the condensation
temperature in more details. From Figs. 1, 2, and 3, it is evident that the
thermodynamic curvature remains positive across the entire physical range for
$\delta\leq 0.5$. However, for $\delta>0.5$, two distinct regimes with both
positive and negative curvature can be identified.

Figure 3: Thermodynamic curvature of an ideal gas with particles obeying
generalized quantum statistics as a function of $T/T_{c}$. For $D=3$ and
$\sigma=2$ and $(a)$ $\delta=0$, $(b)$ $\delta=0.2$, $(c)$ $\delta=0.7$, and
$(d)$ $\delta=0.85$.  Figure 4: $Z^{*}$ in terms at $\delta$ for various $D$
and $\sigma$ values $Z>Z^{*}(Z<Z^{*})$ shows Boson(Fermion) like behavior.

(a)

(b)

Figure 5: The $T-\delta$ phase diagram of the system for $(a)~{}D=3,\sigma=2$,
and $(b)~{}D=2$ and $\sigma=1$. In both figures the dashed line separates
fermionic regime from the bosonic one, while the lower bold line denotes the
BEC transition. The vertical (red) line exhibits the asymptotic fermion-boson
transition line given by $\delta=0.5$, so that we do not have a fermionic
phase for $\delta<0.5$.

##  VI Transition temperatures

In this section, we illustrate that the fugacity in unified quantum statistics
must be constrained to ensure a positive distribution function for all values
of $\delta$ akin to the behaviour seen in ordinary boson statistics. The
fugacity is confined to the interval $0\\!\leq\\!z\\!\leq\\!1$, and the
behaviour of the thermodynamic curvature exhibits singularity at the upper
bound, $z_{\text{max}}=1$. The thermodynamic curvature exhibits a
discontinuity at $T=T_{c}$. The singular behaviour of the thermodynamic
curvature in an ideal boson gas, occurring at $z_{\text{max}}=1$, is widely
recognized and is associated with the Bose-Einstein condensation transition.
We anticipate a similar transition for arbitrary values of $\delta$. We
determine the phase transition temperature for a constant particle density.
Using Eq. (14), we deduce the phase transition temperature for generalized
statistics in the following manner:

$\displaystyle
K_{B}T_{c}=\left(\frac{N}{A[\zeta(D/\sigma)-2^{(1-D/\sigma)}g_{D/\sigma}(\delta)]}\right)^{\sigma/D}.$
(27)

Since $g_{D/\sigma}(0)=0$, the BEC temperature for special case ($\delta=0$)
can be expressed as:

$\displaystyle
K_{B}T_{c}^{B}=\left(\frac{N}{A\zeta(D/\sigma)}\right)^{\sigma/D},$ (28)

where $T_{c}^{B}$ denotes the condensation critical temperature of an ideal
boson gas. We establish a straightforward relationship that relates the ratio
of the condensation temperature of generalized unified quantum statistics to
the conventional Bose-Einstein condensation temperature, as follows:

$\displaystyle\frac{T_{c}}{T_{c}^{B}}=\left(\frac{\zeta(D/\sigma)}{\zeta(D/\sigma)-2^{(1-D/\sigma)}g_{D/\sigma}(\delta)}\right)^{\sigma/D}.$
(29)

Therefore, we calculate the condensation phase transition temperature for
unified quantum statistics, regardless of the specific value of $\delta$.

Now, we determine the temperature at which the thermodynamic curvature
undergoes a change in sign. Indeed, for a particular parameter value $\delta$,
we extract the value of $Z^{*}$ and, by utilizing Eq. (14), determine the
temperature at which the sign change in thermodynamic curvature occurs for a
constant particle number density. Fig. 4 shows that the value of $Z^{*}$
depends on the parameters $\delta$ and $D/\sigma$. Obviously, when
$\delta<0.5$, the thermodynamic curvature remains positive throughout the
entire physical range. On the other hand, when $\delta>0.5$, the thermodynamic
curvature is positive for $z>z$, indicating a boson-like system behavior.
Conversely, for $z<z$, the intrinsic statistical interaction within the system
becomes repulsive, leading to behavior reminiscent of fermions. We determine
the temperature at which the sign change in curvature occurs for a fixed
density value. We have represented the condensation temperature and the
temperature at which the sign change occurs in Figure 5.

It is a widely recognized fact that thermodynamic response functions exhibit
divergence, discontinuity, or non-differentiability at phase transition
points. Specifically, the heat capacity of an ideal boson gas displays non-
differentiability at the phase transition temperature. Using Eqs. (11), and
(14), we derive the heat capacity of an ideal unified quantum statistics with
an arbitrary value of $\delta$ as follows:

$\displaystyle C_{V}=(\frac{\partial U}{\partial T})_{V}$
$\displaystyle=\frac{15}{4}\frac{\bar{g}_{\tfrac{5}{2}}(z)-2^{\tfrac{-3}{2}}\bar{g}_{\tfrac{5}{2}}(\delta
z^{2})}{\bar{g}_{\tfrac{3}{2}}(z)-2^{\tfrac{-1}{2}}\bar{g}_{\tfrac{3}{2}}(\delta
z^{2})}$ (30)
$\displaystyle-\frac{9}{4}\frac{\bar{g}_{\tfrac{3}{2}}(z)-2^{\tfrac{-1}{2}}\bar{g}_{\tfrac{3}{2}}(\delta
z^{2})}{\bar{g}_{\tfrac{1}{2}}(z)-2^{\tfrac{1}{2}}\bar{g}_{\tfrac{1}{2}}(\delta
z^{2})}.$

Figure 6: Heat capacity at fixed volume as a function of temperature. $(a)$
$\delta=0.2$, $(b)$ $\delta=0.4$, $(c)$ $\delta=0.6$ and $(d)$ $\delta=0.8$.

Fig. 6 shows the behaviour of heat capacity with respect to temperature for
different value of $\delta$. First, we observe that the heat capacity exhibits
non-differentiability at $T=T_{c}$. Furthermore, at extremely high
temperatures, it converges to the heat capacity of an ideal classical gas,
denoted as $C_{V}=3NK/2$. When $\delta$ is less than or equal to 0.5, the
system’s heat capacity exceeds that of an ideal classical gas for temperatures
exceeding $T_{c}$. As temperature increases, it progressively approaches the
heat capacity of the ideal classical gas in an asymptotic fashion.
Nevertheless, when $\delta>0.5$, there exists a finite temperature at which
the heat capacity matches that of an ideal classical gas. This temperature
coincides with the thermodynamic curvature’s sign-change temperature. In fact,
there are two distinct regimes: one resembling Bose-like behaviour and the
other Fermi-like behavior, separated by the curve $C_{V}=3NK/2$.

Using the following relation [47]

$\displaystyle\bar{g}_{\nu}(z)-2^{1-\nu}\bar{g}_{\nu}(z^{2})=-\bar{g}_{\nu}(-z)=f_{\nu}(z),$
(31)

we recover the heat capacity of ideal fermion gas for $\delta=1$ as follows:

$\displaystyle C_{v}$
$\displaystyle=\frac{15}{4}\frac{\bar{g}_{\tfrac{5}{2}}(z)-2^{\tfrac{-3}{2}}\bar{g}_{\tfrac{5}{2}}(z^{2})}{\bar{g}_{\tfrac{3}{2}}(z)-2^{\tfrac{-1}{2}}\bar{g}_{\tfrac{3}{2}}(z^{2})}$
(32)
$\displaystyle-\frac{9}{4}\frac{\bar{g}_{\tfrac{3}{2}}(z)-2^{\tfrac{-1}{2}}\bar{g}_{\tfrac{3}{2}}(z^{2})}{\bar{g}_{\tfrac{1}{2}}(z)-2^{\tfrac{1}{2}}\bar{g}_{\tfrac{1}{2}}(z^{2})}$
$\displaystyle=\frac{15}{4}\frac{f_{\tfrac{5}{2}}(z)}{f_{\tfrac{3}{2}}(z)}-\frac{9}{4}\frac{f_{\tfrac{3}{2}}(z)}{f_{\tfrac{1}{2}}(z)},$

where $f_{n}(x)$ denotes the well-known Fermi-Dirac function which is defined
as follows:

$\displaystyle
f_{v}(z)=\frac{1}{\Gamma(v)}\int_{0}^{\infty}\frac{x^{v-1}}{z^{-1}\exp(x)+1}dx.$
(33)

(a) Figure 7: Heat capacity at fixed volume as a function of temperature for
$(\delta=1)$ (fermions).

We investigate the phase transition temperature across different spatial
dimensions and diverse dispersion relations. It’s worth noting that as
$\lim_{x\rightarrow 1}\zeta(x)\longrightarrow\infty$, for $D/\sigma\leq 1$,
the transition temperature approaches absolute zero. This implies that finite-
temperature condensation in quantum unified statistics only occurs when
$D/\sigma>1$ similar to ordinary bosons.

##  VII Conclusion

We conducted a comprehensive examination of the thermodynamic geometry on a
novel generalized unified quantum statistics. In fact, this generalization is
related to the assumption that the quantum state of a multi particle system is
a functional on the internal space of the particles and is capable of
providing a smooth interpolation between Bose-Einstein and Fermi-Dirac
statistics. We established the thermodynamic parameter space for an ideal gas
composed of particles adhering to unified quantum statistics. In this
framework, we determined the metric elements for the parameter space and
subsequently derived the affine connections and the Ricci scalar for the
thermodynamic parameter space.

The thermodynamic curvature possesses an inherent interpretation related to
statistical interactions. The sign of the thermodynamic curvature is directly
linked to the intrinsic statistical interactions within the thermodynamic
system. The generalization parameter $\delta$ of unified quantum statistics,
as per our argument, plays a different role in shaping the statistical
interactions. For all cases where $\delta\leq 0.5$, the thermodynamic
curvature remains positive, indicating an attractive statistical interaction,
and the overall behaviour closely resembles that of bosons throughout the
entire physical range. On the other hand, for the cases with $\delta>0.5$, the
thermodynamic curvature is either positive or negative depending on the system
temperature. We showed that in the high-temperature limit, the dominant
statistical interaction closely is fermion-like for when $\delta>0$. As the
temperature decreases, the repulsive interaction gradually diminishes and
eventually disappears at a specific fugacity value, $z=Z^{*}$, or
equivalently, at a certain temperature threshold. At temperatures below the
aforementioned threshold, the thermodynamic curvature remains positive, and
the statistical behaviour is best described by Boson statistics.

The singular points of thermodynamic curvature serve as indicator of phase
transitions. We demonstrated that, the thermodynamic curvature is singular at
a critical fugacity ($z_{c}=1$). In fact, a Bose-Einstein condensation takes
place for quantum unified statistics across all values of $\delta$ except when
$\delta=1$ (corresponding to an ordinary fermion gas). For a fixed particle
density, we calculated the condensation temperature, which depends on the
parameter $\delta$. We found that the BEC phase transition temperature for any
arbitrary value of $\delta$ exceeds the BEC temperature.

Finally, we determined the heat capacity as a function of temperature and
illustrated that it exhibits a singular behaviour at the critical condensation
temperature. We investigated condensation phenomena across different spatial
dimensions and various dispersion relations. We showed that for $D/\sigma\leq
1$, the phase transition temperature is zero. In other word, similar to
ordinary ideal boson gas, no finite temperature condensation occurs in low
dimensions [48].

## Acknowledgments

We acknowledge K. Monkman for helpful conversations.

## References

  * Wilczek [1982] F. Wilczek, Physical review letters 49, 957 (1982).
  * Haldane [1991] F. D. M. Haldane, Physical review letters 67, 937 (1991).
  * Macfarlane [1989] A. Macfarlane, Journal of Physics A: Mathematical and general 22, 4581 (1989).
  * Lavagno and Swamy [2002] A. Lavagno and P. N. Swamy, Physical Review E 65, 036101 (2002).
  * Tsallis [1988] C. Tsallis, Journal of statistical physics 52, 479 (1988).
  * Yan [2021] C.-T. Yan, Physical Review E 104, 064118 (2021).
  * Gibbs [1873] J. W. Gibbs, Transactions of Connecticut Academy of Arts and Sciences , 382 (1873).
  * Gibbs [1948] J. Gibbs,  _The Collected Works. Vol. 1\. Thermodynamics_ (Yale University Press, 1948).
  * Ruppeiner [1979] G. Ruppeiner, Physical Review A 20, 1608 (1979).
  * Weinhold [1975] F. Weinhold, The Journal of Chemical Physics 63, 2479 (1975).
  * Ruppeiner [1995] G. Ruppeiner, Reviews of Modern Physics 67, 605 (1995).
  * Salamon  _et al._ [1984] P. Salamon, J. Nulton, and E. Ihrig, The Journal of chemical physics 80, 436 (1984).
  * Mrugała [1984] R. Mrugała, Physica A: Statistical Mechanics and its Applications 125, 631 (1984).
  * Diósi  _et al._ [1984] L. Diósi, G. Forgács, B. Lukács, and H. Frisch, Physical Review A 29, 3343 (1984).
  * Janyszek and Mrugaa [1990] H. Janyszek and R. Mrugaa, Journal of Physics A: Mathematical and General 23, 467 (1990).
  * Ingarden [1978] R. S. Ingarden, Tensor, ns 33, 347 (1978).
  * Janyszek [1986] H. Janyszek, Reports on mathematical physics 24, 11 (1986).
  * Oshima  _et al._ [1999] H. Oshima, T. Obata, and H. Hara, Journal of Physics A: Mathematical and General 32, 6373 (1999).
  * Pessoa and Cafaro [2021] P. Pessoa and C. Cafaro, Physica A: Statistical Mechanics and its Applications 576, 126061 (2021).
  * López-Picón and López-Vega [2021] J. López-Picón and J. M. López-Vega, Physica A: Statistical Mechanics and its Applications 580, 126144 (2021).
  * Mirza and Mohammadzadeh [2011a] B. Mirza and H. Mohammadzadeh, Journal of Physics A: Mathematical and Theoretical 44, 475003 (2011a).
  * Adli  _et al._ [2019a] F. Adli, H. Mohammadzadeh, M. N. Najafi, and Z. Ebadi, Physica A: Statistical Mechanics and its Applications 521, 773 (2019a).
  * Mohammadzadeh  _et al._ [2016] H. Mohammadzadeh, F. Adli, and S. Nouri, Physical Review E 94, 062118 (2016).
  * Babaei-Aghbolagh  _et al._ [2022] H. Babaei-Aghbolagh, H. Mohammadzadeh, D. M. Yekta, and K. B. Velni, Physical Review D 106, 024044 (2022).
  * Mohammadzadeh  _et al._ [2021] H. Mohammadzadeh, M. Rastkatr, and M. N. Najafi, International Journal of Geometric Methods in Modern Physics 18, 2150023 (2021).
  * Rafiee  _et al._ [2022] M. Rafiee, S. A. H. Mansoori, S.-W. Wei, and R. B. Mann, Physical Review D 105, 024058 (2022).
  * Adli  _et al._ [2019b] F. Adli, H. Mohammadzadeh, M. N. Najafi, and Z. Ebadi, International Journal of Geometric Methods in Modern Physics 16 (2019b).
  * Mehri-Dehnavi and Mohammadzadeh [2020] H. Mehri-Dehnavi and H. Mohammadzadeh, Journal of Physics A: Mathematical and Theoretical 53, 375009 (2020).
  * Ebadi and Mohammadzadeh [2022] Z. Ebadi and H. Mohammadzadeh, International Journal of Geometric Methods in Modern Physics 19, 2250088 (2022).
  * Mirza and Mohammadzadeh [2009] B. Mirza and H. Mohammadzadeh, Physical Review E 80, 011132 (2009).
  * Nozieres and Schmitt-Rink [1985] P. Nozieres and S. Schmitt-Rink, Journal of Low Temperature Physics 59, 195 (1985).
  * Yerin  _et al._ [2019] Y. Yerin, H. Tajima, P. Pieri, and A. Perali, Physical Review B 100, 104528 (2019).
  * Anglin and Ketterle [2002] J. R. Anglin and W. Ketterle, Nature 416, 211 (2002).
  * Burt  _et al._ [1997] E. Burt, R. Ghrist, C. Myatt, M. Holland, E. A. Cornell, and C. Wieman, Physical Review Letters 79, 337 (1997).
  * Zwerger [2011] W. Zwerger,  _The BCS-BEC crossover and the unitary Fermi gas_ , Vol. 836 (Springer Science & Business Media, 2011).
  * Regal and Jin [2007] C. Regal and D. Jin, Advances In Atomic, Molecular, and Optical Physics 54, 1 (2007).
  * Chung and Algin [2017] W. S. Chung and A. Algin, Physics Letters A 381, 3266 (2017).
  * Mohammadzadeh  _et al._ [2017] H. Mohammadzadeh, Y. Azizian-Kalandaragh, N. Cheraghpour, and F. Adli, Journal of Statistical Mechanics: Theory and Experiment 2017, 083104 (2017).
  * Chung and Hassanabadi [2017] W. S. Chung and H. Hassanabadi, International Journal of Theoretical Physics 56, 1746 (2017).
  * Chung  _et al._ [2019] W. S. Chung, H. Hassanabadi, and S. Sargolzaeipor, Physics Letters A 383, 1989 (2019).
  * Algin and Ilik [2013] A. Algin and E. Ilik, Physics Letters A 377, 1797 (2013).
  * Crooks [2007] G. E. Crooks, Physical Review Letters 99, 100602 (2007).
  * Crooks [2011] G. E. Crooks,  _Fisher information and statistical mechanics_ , Tech. Rep. (Tech. Rep, 2011).
  * Johnston  _et al._ [2003] D. Johnston, W. Janke, and R. Kenna, arXiv preprint cond-mat/0308316 (2003).
  * Mirza and Mohammadzadeh [2011b] B. Mirza and H. Mohammadzadeh, Physical Review E 84, 031114 (2011b).
  * Mirza and Mohammadzadeh [2010] B. Mirza and H. Mohammadzadeh, Physical Review E 82, 031137 (2010).
  * Pathria and Beale [1996] R. Pathria and P. Beale,  _Statistical Mechanics_ (Elsevier Science, 1996).
  * May [1964] R. M. May, Physical Review 135, A1515 (1964).