††thanks: Corresponding author

# Bose and Fermi gases in metric-affine gravity and linear Generalized
Uncertainty Principle

Aneta Wojnar  0000-0002-1545-1483<EMAIL_ADDRESS>Department of Theoretical
Physics & IPARCOS, Complutense University of Madrid, E-28040, Madrid, Spain
Débora Aguiar Gomes  0000-0003-0079-7323<EMAIL_ADDRESS>Laboratory
of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi
1, 50411 Tartu, Estonia

###### Abstract

We examine the relationship between Palatini-like theories of gravity and
models incorporating linear generalized uncertainty principles. Additionally,
we delve into the thermodynamics of systems comprising both Bose and Fermi
gases. Our analysis encompasses the equations of state for various systems,
including general Fermi gases, degenerate Fermi gases, Boltzmann gases, Bose
gases such as phonons and photons, as well as Bose-Einstein condensates and
liquid helium.

##  I Introduction

The consistency of equations within Modified Gravity (MG) is brought into
question by numerous indications from prior investigations. For instance, the
dependence of chemical potential on gravity suggests that modifications in the
description of gravitational fields would have an impact Kulikov and Pronin
(1995). Modified Gravity has been demonstrated to alter the geodesic deviation
equation on the surface of stars, resembling Hook’s law and introducing
corrections to the polytropic equation of state Kim (2014). Moreover,
microscopic quantities such as opacity exhibit modifications, implying a need
for an effective treatment Sakstein (2015a). Thermodynamic laws, stellar
stability criteria, and properties of Fermi gases also display corrections
stemming from gravitational proposals Wojnar and Velten (2016); Wojnar (2018);
Sarmah  _et al._ (2022); Wojnar (2023a).

Theoretical descriptions of thermonuclear processes within stars’ interiors
undergo changes under modified gravity, thereby affecting computations of
energy generation rates Sakstein (2015b); Olmo  _et al._ (2019); Crisostomi
_et al._ (2019); Rosyadi  _et al._ (2019); Wojnar (2021). Some gravity
theories introduce a dependence of elementary particle interactions on local
energy-momentum distributions Delhom-Latorre  _et al._ (2018). Additionally,
specific heats, Debye temperatures, and crystallization processes in white
dwarfs are influenced by the gravity model Kalita  _et al._ (2023); Wojnar
_et al._ (2024). Furthermore, chemical reaction rates, which are influenced by
gravity Lecca (2021), are expected to change with modifications to this
interaction.

Relativistic effects in equations of state, when neglected in Tolman-
Oppenheimer-Volkoff equations derived from General Relativity (GR), result in
an underestimation of compact star limiting masses. Equations of state in
curved spacetime for degenerate stars explicitly depend on metric components,
leading to alterations in chemical potentials and temperatures Hossain and
Mandal (2021a, b); Li  _et al._ (2022); Chavanis (2004). Moreover,
thermodynamic quantities and equations of state are further modified when
(pseudo-)scalar fields, such as axions, are considered Sakstein  _et al._
(2022).

The emerging link between MG and Generalized Uncertainty Principle (GUP), as
shown in Wojnar (2023b); Ali and Wojnar (2024), opens avenues for testing
gravitational theories within Earth-based laboratories. Additionally,
techniques advanced by either community can be utilized to evaluate proposals
concerning MG or GUP interchangeably Wojnar (2024). Notice that such a
correspondence is also expected; it is believed that MG is an effective theory
of some Quantum Gravity. Many proposals emphasize integrating the quantum
structure of space-time and deforming associated quantum phase spaces, leading
to the generalization of the Heisenberg uncertainty principle. This emphasis
is underscored for the potential measurable effects it offers Pachoł and
Wojnar (2023a, b); Kozak  _et al._ (2023a). GUP has proven valuable in
predicting quantum gravity effects, as evident in various models found in
Kempf  _et al._ (1995); Maggiore (1993, 1994); Chang  _et al._ (2002a, b);
Moussa (2015). Despite variations in mathematical structures, a shared
characteristic among most of these models is the existence of a minimum length
scale, anticipated to be approximately the Planck length,
$L_{P}\sim\sqrt{\frac{\hbar G}{c^{3}}}$, as detailed in Bishop  _et al._
(2020, 2022); Segreto and Montani (2023).

The GUP models introduce modifications to equations of state and microscopic
variables, stemming from the interplay between special relativity and gravity.
This proposition incorporates a dispersion relation involving energy, mass,
and momentum within Heisenberg’s Uncertainty Principle, incorporating
constants such as the speed of light and the gravitational constant Moussa
(2015); Rashidi (2016); Belfaqih  _et al._ (2021); Mathew and Nandy (2021);
Hamil and Lütfüoğlu (2021); Gregoris and Ong (2022).

Below, we will delve into a review of previous findings regarding ideal gases,
derived from the connection between Ricci-based gravity and linear GUP. Before
proceeding, we will provide a brief recap of this correspondence.
Additionally, we introduce, for the first time, a theoretical framework for
ideal and Boltzmann gases, incorporating Fermi statistics in both idealized
and realistic scenarios, along with a simplified portrayal of Bose gases, such
as photon and phonon gases, within the context of Ricci-based and linear GUP
models.

##  II Metric-affine gravity and linear GUP correspondence

The class of metric-affine gravity theories under consideration is defined by
the action:

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\mathcal{L}_{G}(g_{\mu\nu},R_{\mu\nu})+\mathcal{S}_{m}(g_{\mu\nu},\psi_{m})\
.$ (1)

Here, $g$ denotes the determinant of the space-time metric $g_{\mu\nu}$, and
$R_{\mu\nu}$ represents the symmetric Ricci tensor, which relies solely on the
affine connection $\Gamma\equiv\Gamma_{\mu\nu}^{\lambda}$. To construct the
gravitational Lagrangian $\mathcal{L}_{G}$ as a scalar function using powers
of traces of ${M^{\mu}}{\nu}\equiv g^{\mu\alpha}R_{\alpha\nu}$, the object
${M^{\mu}}_{\nu}$ is introduced.

The matter action is described by:

$\mathcal{S}_{m}=\int d^{4}x\sqrt{-g}\mathcal{L}_{m}(g_{\mu\nu},\psi_{m}).$
(2)

In this framework, the matter action couples minimally to the metric,
neglecting the torsion (the antisymmetric part of the connection), akin to
minimally coupled bosonic fields. This simplification extends to fermionic
particles, such as degenerate matter, effectively described by a fluid
approach represented by the perfect fluid energy-momentum tensor Alfonso  _et
al._ (2017). Focusing on the symmetric part of the Ricci tensor avoids
potential ghostlike instabilities Borowiec  _et al._ (1998); Allemandi  _et
al._ (2004); Beltrán Jiménez and Delhom (2019); Jiménez and Delhom (2020).
This framework encompasses various gravity theories, including GR, Palatini
$f(R)$ gravity, Eddington-inspired Born-Infeld gravity Vollick (2004), and its
extensions Jiménez  _et al._ (2018).

The gravitational action encapsulates theories that, despite complex field
equations, can be conveniently reformulated Jiménez  _et al._ (2018):

${G^{\mu}}_{\nu}(q)=\frac{\kappa}{|\hat{\Omega}|^{1/2}}\left({T^{\mu}}_{\nu}-\delta^{\mu}_{\nu}\left(\mathcal{L}_{G}+\frac{T}{2}\right)\right)\
.$ (3)

Here, $|\hat{\Omega}|$ denotes the determinant of the deformation matrix, and
$T$ represents the trace of the energy-momentum tensor of matter fields. The
Einstein tensor ${G^{\mu}}_{\nu}(q)$ is linked to a tensor $q_{\mu\nu}$, where
the connection $\Gamma$ adopts the Levi-Civita connection of $q_{\mu\nu}$:

$\nabla_{\mu}^{\Gamma}(\sqrt{-q}q^{\alpha\beta})=0.$ (4)

In this formalism, the tensor $q_{\mu\nu}$ is related to the space-time metric
$g_{\mu\nu}$ through:

$q_{\mu\nu}=g_{\mu\alpha}{\Omega^{\alpha}}_{\nu},$ (5)

where the deformation matrix ${\Omega^{\alpha}}_{\nu}$ is theory-dependent and
determined by the gravitational Lagrangian $\mathcal{L}_{G}$. These theories
yield second-order field equations, reducing to GR counterparts in vacuum
(${T_{\mu}}^{\nu}=0$), implying no extra degrees of freedom propagate in these
theories beyond the usual two polarizations of the gravitational field.

The nonrelativistic limit of the field equations (3) is particularly
intriguing. In Palatini $f(R)$ Toniato  _et al._ (2020) and EiBI Banados and
Ferreira (2010); Pani  _et al._ (2011) gravities, the Poisson equation assumes
the form:

$\nabla^{2}\phi=\frac{\kappa}{2}\Big{(}\rho+\bar{\alpha}\nabla^{2}\rho\Big{)}.$
(6)

Here, $\phi$ denotes the gravitational potential, $\kappa=8\pi G$, and
$\bar{\alpha}$ represents a theory parameter. In Palatini $f(R)$,
$\bar{\alpha}=2\bar{\beta}$, with $\bar{\beta}$ accompanying the quadratic
term, while in EiBI, $\bar{\alpha}=\epsilon/2$, where $\epsilon=1/M_{BI}$ and
$M_{BI}$ denotes the Born-Infeld mass. The resemblance in the Poisson equation
between these two gravity proposals is not coincidental; the EiBI gravity, in
the first-order approximation, reduces to Palatini gravity with the quadratic
term Pani and Sotiriou (2012). Furthermore, only the quadratic term $R^{2}$
influences the non-relativistic equations, as higher curvature scalar terms
enter the equations at the sixth order Toniato  _et al._ (2020).

As indicated in Wojnar (2023b), the augmentation in the Poisson equation (6)
can be interpreted as a modification to the Fermi gas at finite temperature.
This modification arises when considering a deformation of the phase space
represented by the integral:

$\frac{1}{(2\pi\hbar)^{3}}\int\frac{d^{3}xd^{3}p}{(1-\sigma p)^{d}},$ (7)

where $d=1$ corresponds to Palatini-like gravity theories. The connection
between the deformation parameter $\sigma$ and the Palatini parameter
$\bar{\beta}$ is established as:

$\sigma=\frac{4\pi
G}{K_{2}}\bar{\beta}\,\,\,\,\text{and}\;\;\;K_{2}=\frac{3}{\pi}\frac{h^{3}N_{A}^{2}}{m_{e}\mu_{e}^{2}},$
(8)

with $m_{e}$ being the electron mass, $\mu_{e}$ the mean molecular weight per
electron, and other constants holding their usual definitions.

This correspondence enables the expression of a general partition function in
three dimensions within a large volume

$\mathrm{ln}Z=\frac{V}{(2\pi\hbar)^{3}}\frac{g}{a}\int\mathrm{ln}\left[1+aze^{-E/k_{B}T}\right]\frac{d^{3}p}{(1-\sigma
p)^{d}}\ ,$ (9)

where $V:=\int d^{3}x$ denotes the volume of the cell in configuration space,
and setting $a=1$ ($a=-1$) corresponds to a system of fermionic (bosonic)
particles with energy states $E_{p}$. The fugacity is represented by
$z=e^{\mu/k_{B}T}$, with $\mu$ as the chemical potential, and $g$ indicating
the spin of a particle.

Similar to the GUP incorporating linear $p$-modifications Cortes and Gamboa
(2020); Ali  _et al._ (2009); Ali (2011); Abac  _et al._ (2021); Vagenas  _et
al._ (2019a); Tawfik and Diab (2014), our approach integrates a deformed phase
space measure characterized by the parameter $\sigma$. In the context of GUP,
the deformation is derived through the utilization of the Liouville theorem
Vagenas  _et al._ (2019b). Consequently, the effective $\hbar$ depends on the
momentum $p$ in the generalized uncertainty relation, leading to a momentum-
dependent size of the unit cell for each quantum state in phase space.

With this modified partition function, one can straightforwardly derive
thermodynamic variables for the requisite statistics, focusing primarily on
pressure, number of particles, internal energy, and specific heat, which are
respectively given by:

$\displaystyle P=$ $\displaystyle\,k_{B}T\frac{\partial}{\partial
V}\mathrm{ln}Z,$ (10) $\displaystyle n=$
$\displaystyle\,k_{B}T\frac{\partial}{\partial\mu}\mathrm{ln}Z\mid_{T,V},$
(11) $\displaystyle U=$ $\displaystyle\,k_{B}T^{2}\frac{\partial}{\partial
T}\mathrm{ln}Z\mid_{z,V}$ (12) $\displaystyle C_{V}=\,$
$\displaystyle\frac{\partial U}{\partial T}\mid_{V}.$ (13)

##  III Boltzmann gas

In the case of high temperature, thermodynamic systems described by Bose or
Fermi statistics behave as the Boltzmann one. The Maxwell-Boltzmann statistics
corresponds to the limit $a\rightarrow 0$, from which the general partition
function (9) acquires the form:

$\mathrm{ln}Z=\frac{V}{(2\pi\hbar)^{3}}\int
ze^{-E/k_{B}T}\frac{d^{3}p}{(1-\sigma p)^{d}}\ .$ (14)

Moreover, the distribution function for this statistics is given by

$f_{\text{MB}}(E)=ze^{-E/k_{\text{B}}T},$ (15)

corresponding to the regime where Fermi-Dirac, Bose-Einstein and Maxwell-
Boltzmann statistics become identical, i.e., $(E-\mu)/k_{\text{B}}T>>1$.

Taking the partition function (14), we can easily obtain

$P=\frac{1}{2\pi^{2}\bar{h}^{3}}\int\frac{1}{3}p^{3}\,_{2}F_{1}\left(3,d,4,p\sigma\right)f_{\text{MB}}(E)\frac{c^{2}p}{E}dp,$
(16) $n=\frac{V}{2\pi^{2}\hbar^{3}}\int
f_{\text{MB}}(E)\frac{p^{2}dp}{(1-\sigma p)^{d}}$ (17)
$U=\frac{V}{2\pi^{2}\hbar^{3}}\int Ef_{\text{MB}}(E)\frac{p^{2}dp}{(1-\sigma
p)^{d}}.$ (18)

##  IV Fermi statistics

In this extended scenario, utilizing (9) and (10) with $a=1$ for fermions and
$g=2$ for electrons, we derive the microphysical description of the system
employing the Fermi-Dirac distribution $f(E)$:

$f(E)=\left(1+z^{-1}e^{E/k_{B}T}\right)^{-1},$ (19)

resulting in the expression for pressure as:

$P=\frac{1}{\pi^{2}\hbar^{3}}\int\frac{1}{3}p^{3}\,_{2}F_{1}\left(3,d,4,p\sigma\right)f(E)\frac{c^{2}p}{E}dp,$
(20)

where ${}_{2}F_{1}$ represents the hypergeometric function. Meanwhile, the
particle number density and internal energy are given by:

$n=\frac{V}{\pi^{2}\hbar^{3}}\int f(E)\frac{p^{2}dp}{(1-\sigma p)^{d}}$ (21)
$U=\frac{V}{\pi^{2}\hbar^{3}}\int Ef(E)\frac{p^{2}dp}{(1-\sigma p)^{d}}.$ (22)

For the case when $|\sigma p|<1$, the general form of the pressure (20) can be
written as series

$P=\frac{1}{\pi^{2}\hbar^{3}}\int\frac{p^{3}}{3}\left(\sum_{k=0}^{\infty}\frac{(d)_{k}\left(\frac{3}{2}\right)_{k}(\sigma
p)^{k}}{\left(\frac{5}{2}\right)_{k}k!}\right)f(E)\frac{c^{2}p}{E}dp,$ (23)

while taking into account only the first two terms of the series (as we
consider the gravity deformation only up to linear terms in $\sigma$), we have

$P=\frac{1}{\pi^{2}\hbar^{3}}\int\frac{1}{3}p^{3}\left(1+\frac{3d}{4}\sigma
p\right)f(E)\frac{c^{2}p}{E}dp.$ (24)

We will now discuss its form in more detail.

###  IV.1 Non-relativistic and relativistic degenerate Fermi gas

We will now shift our focus to a specific form of matter that holds
significant importance in stellar physics: degenerate gases. These gases are
utilized to characterize the dense cores of stellar and substellar objects, as
well as degenerate matter in compact stars. In the toy model that we are
employing here, all states with energies lower than the Fermi energy level are
occupied, while all states with higher energies remain empty at absolute zero
temperature ($T\rightarrow 0$). Consequently, the chemical potential $\mu$
takes on the value of the Fermi energy $E_{F}$. In such scenarios, the Fermi-
Dirac distribution takes the form of a step function:

$f(E)=\left\\{\begin{array}[]{ll}1&\mbox{if }E\leq E_{F}\\\
0&\mbox{otherwise.}\\\ \end{array}\right.$

Consequently, in equation (24), integration is carried out up to the Fermi
energy $E_{F}$:

$P=\frac{1}{\pi^{2}\hbar^{3}}\left[\frac{(2m_{e}E_{F})^{5/2}}{6m_{e}}+\frac{\sigma
d}{3}\frac{(2m_{e}E_{F})^{3}}{8m_{e}}\right],$ (25)

which corresponds to the effective pressure in Palatini theories when $d=1$
(see Wojnar (2023b)). Using the definition of the electron degeneracy

$\psi=(\beta
E_{F})^{-1}=\frac{k_{\text{B}}T}{A}\left(\frac{\mu_{e}}{\rho}\right)^{2/3}$
(26)

we can rewrite it as a function of the density such that one deals with the
non-relativistic polytrope EoS. Let us write it in a more convenient form for
the further analysis:

$P=K\rho^{\frac{5}{3}}+\sigma K_{2}\rho^{2},$ (27)

where $K_{2}$ is given by (8) while

$K=\frac{1}{20}\left(\frac{3}{\pi}\right)^{\frac{2}{3}}\frac{h^{2}}{m_{e}(\mu_{e}m_{u})^{\frac{5}{3}}}$

refers to the standard polytropic parameter for $n=3/2$.

On the other hand, the relativistic polytropic EoS is given simply by
inserting $E=pc$ in (24) and using the relation between the pressure and
density (recall that for electrons $g=2$)

$p=\left(\frac{6\pi^{2}\hbar^{3}\rho}{gm_{e}}\right)^{\frac{1}{3}},$ (28)

providing

$P=\frac{\hbar
c}{4}\left(\frac{3\pi^{2}}{m_{e}^{4}}\right)^{1/3}\rho^{4/3}+\frac{9}{20}d\sigma\hbar^{2}c\left(\frac{3^{2}\pi^{4}}{m_{e}^{5}}\right)^{1/3}\rho^{5/3}.$
(29)

Note that the phase-space deformation provides a mixture of two polytropes;
that is, in the case of non-relativistic case, the modification is in the form
of the rigid polytrope while relativistic one has a term resembling the non-
relativistic polytrope ($n=3/2$).

##  V Bose statistics

Let us briefly recall the basic equations describing a system of $N$ spinless,
non-interacting particles in Palatini gravity Wojnar (2024). The Hamiltonian
is given by $H=\sum^{N}_{i=1}\frac{p_{i}^{2}}{2m}$, with
$p_{i}^{2}=\mathrm{p}_{i}\cdot\mathrm{p}_{i}$ and $\mathrm{p}_{i}$ being the
momentum operator of the single-particle with energy $E_{p}=p^{2}/2m$. Since
the grand partition function of an ideal Bose gas for such a system is given
by ($\beta=:(k_{B}T)^{-1}$)

$Z=\prod_{p}\frac{1}{1-ze^{-\beta E_{p}}},$ (30)

an average occupation number for a state $p$ is (the total number of particles
$N=\sum_{p}\langle n_{p}\rangle$)

$\langle n_{p}\rangle=-\frac{1}{\beta}\frac{\partial}{\partial
E_{p}}\mathrm{ln}Z=\frac{ze^{-\beta E_{p}}}{1-ze^{-\beta E_{p}}},$ (31)

while the equation of state

$\beta{PV}=-\sum_{p}\mathrm{ln}(1-ze^{-\beta E_{p}}).$ (32)

Considering $V\rightarrow\infty$ and defining the specific volume $v=V/N$, the
above series can be written as

$\displaystyle\beta P=$
$\displaystyle-\frac{4\pi}{(2\pi\hbar)^{3}}\int_{0}^{\infty}\frac{dpp^{2}}{1-\sigma
p}\mathrm{ln}\left[1-ze^{-\beta\frac{p}{2m}}\right]-\frac{\mathrm{ln}(1-z)}{V},$
(33) $\displaystyle\frac{1}{v}=$
$\displaystyle\frac{4\pi}{(2\pi\hbar)^{3}}\int_{0}^{\infty}\frac{dpp^{2}}{1-\sigma
p}\frac{1}{z^{-1}e^{\beta\frac{p}{2m}}-1}+\frac{1}{V}\frac{z}{1-z}.$ (34)

Taking into account only the terms linear in the parameter
$|\alpha|=:|\sigma|\sqrt{2mk_{B}T}$, one writes111To see how the series
converge, see Wojnar (2024).

$\displaystyle\beta P$
$\displaystyle=\frac{1}{\lambda^{3}}\left[g_{5/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{3}(z)\right]-\frac{\mathrm{ln}(1-z)}{V},$
(35) $\displaystyle\frac{1}{v}$
$\displaystyle=\frac{1}{\lambda^{3}}\left[g_{3/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{2}(z)\right]+\frac{1}{V}\frac{z}{1-z},$
(36)

where $\lambda=\sqrt{\frac{2\pi\hbar^{2}}{mk_{B}T}}$ is the thermal
wavelength, $g_{m}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{m}}$ and
$\mathrm{Li}_{n}(z)$ is the polylogarithm function which can be expressed for
$|z|<1$ as $\mathrm{Li}_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}.$ The
internal energy is given simply by

$\frac{U}{V}=-\frac{1}{V}\frac{\partial}{\partial\beta}\mathrm{ln}Z=\frac{3k_{B}T}{2\lambda^{3}}\left[g_{5/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{3}(z)\right].$
(37)

Comparing it with (35) with the assumption that its last term can be
neglected, we deal with a well known relation between the internal energy and
temperature $U=\frac{3}{2}PV$. Notice that in order to study the further
properties of ideal Bose gas, one needs to know the fugacity $z$ dependence on
the temperature and specific volume $v$. We will come back to this issue in
the subsection V.3.

###  V.1 Photon gas

Since photons are massless, their energy will be simply given by $E=cp$. Its
wave number and frequency are related by $p=\hbar k=\hbar\omega/c$ and there
are two propagating modes, which are taken into account by multiplying the
partition function (9) by a factor of 2. The internal energy will be given by

$U=\frac{V}{\pi^{2}}\left[\frac{\pi^{4}(k_{\text{B}}T)^{4}}{15(\hbar
c)^{3}}+24\sigma d\frac{(k_{\text{B}}T)^{5}\zeta(5)}{c^{4}\hbar^{3}}\right],$
(38)

where the first term corresponds to Stefan’s law. Accordingly, the specific
heat will be given by

$C_{\text{V}}=\frac{V}{\pi^{2}}\left[\frac{4\pi^{4}k_{\text{B}}^{4}T^{3}}{15(\hbar
c)^{3}}+120\sigma
d\frac{k_{\text{B}}^{5}T^{4}\zeta(5)}{c^{4}\hbar^{3}}\right]$ (39)

while pressure is simply given by $PV=U/3.$

###  V.2 Phonon gas

In spite of not being particles, the vibrations in lattices and in particular
its normal modes, phonons, can be mathematically described as bosons. This
treatment is specially relevant when considering a solid subject to low
temperatures, where phonon interactions are negligible. In this case, we have
three propagation modes, in contrast with photons which only have two, and the
allowed frequencies are bounded. The energy per atom is

$\frac{U}{N}=3\kappa_{\text{B}}TD_{3}(\beta\hbar\omega)+\frac{9}{4c}\kappa_{\text{B}}Td\sigma\hbar\omega_{m}D_{4}(\beta\hbar\omega),$
(40)

where $\omega_{m}=c\left(\frac{6\pi^{2}N}{V}\right)^{1/3}$ is the maximum
frequency and

$D_{n}(x)=\frac{n}{x^{n}}\int_{0}^{x}\frac{t^{n}}{e^{t}-1}dt$ (41)

is the Debye function. By defining the Debye temperature via
$\kappa_{\text{B}}T_{D}=\hbar\omega_{m}$, and using the solutions for the
Debye functions, we can rewrite (40) as

$\frac{U}{N}=3\kappa_{\text{B}}T\left(1-\frac{3}{8}\frac{T_{D}}{T}\right)+\frac{9}{4c}\kappa_{\text{B}}^{2}T_{D}Td\sigma\left(1-\frac{2}{5}\frac{T_{D}}{T}\right)$
(42)

for $T>>T_{D}$ and

$\frac{U}{N}=3\kappa_{\text{B}}T\left[\frac{\pi^{4}}{5}\left(\frac{T}{T_{D}}\right)^{3}\right]+216\zeta(5)\kappa_{\text{B}}Td\sigma(\kappa_{\text{B}}T_{D})\left(\frac{T}{T_{D}}\right)^{4}$
(43)

for $T<<T_{D}$.

The specific heat, for high and low temperatures, will be

$\frac{C_{\text{V}}}{\kappa_{\text{B}}N}=3\left[1-\frac{1}{20}\left(\frac{T_{D}}{T}\right)^{2}\right]+\frac{9}{4c}d\sigma\kappa_{\text{B}}T_{D}\left[1-\frac{1}{18}\left(\frac{T_{D}}{T}\right)^{2}\right],\,T>>T_{D}$
(44)

and

$\frac{C_{\text{V}}}{\kappa_{\text{B}}N}=\frac{12\pi^{4}}{5}\left(\frac{T}{T_{D}}\right)^{3}+1080\zeta(5)\frac{d\sigma}{c}\kappa_{\text{B}}T_{D}\left(\frac{T}{T_{D}}\right)^{4},\,T<<T_{D}.$
(45)

The first term in (44) corresponds to the Dulong-Petit law, namely
$C_{v}=3N\kappa_{\text{B}}$. However, it should be emphasized that although
$T>>T_{D}$, the temperatures considered here are low enough so that we can
consider the phonons to be non-interactive. At very high temperatures, the
phonons interaction cannot be overlooked and the current approach will no
longer be valid.

###  V.3 Bose-Einstein condensate

The behaviour of the fugacity $z$ depends on the properties of the the
functions $g_{3/2}(z)$ and $\mathrm{Li}_{2}(z)$. Analyzing the equation (36),
one can derive a modified condition which the ideal Bose gas has to satisfy to
become the Bose-Einstein condensate ( $g_{3/2}(1)\approx 2.612)$):

$\displaystyle\frac{\lambda^{3}}{v}-\frac{2\alpha}{\pi}\zeta(2)=g_{3/2}(1).$
(46)

It clearly provides the critical value for the specific volume (or critical
density $n_{c}=1/v_{cr}$):

$n_{c}=\left(\frac{1}{4\pi\hbar^{2}}\right)^{\frac{3}{2}}\left[\zeta(3/2)(2mk_{B}T)^{\frac{3}{2}}+\sigma\frac{\pi}{3}(2mk_{B}T)^{2}\right].$
(47)

Moreover, the fugacity dependence on the temperature $T$ and specific volume
$v$ is

$z=\begin{cases}1\mbox{ for
}\;\;\;\;\;\;\;\;\;\frac{\lambda^{3}}{v}-\frac{\pi\alpha}{3}\geq
g_{3/2}(1)&\\\ \mbox{solution of
}\frac{\lambda^{3}}{v}=\left[g_{3/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{2}(z)\right]&\mbox{otherwise.
}\end{cases}$

Hence, the fugacity remains fixed at 1 throughout the Einstein-Bose
condensate, indicating a zero chemical potential. In other words, within the
region where $\frac{\lambda^{3}}{v}-\frac{\pi\alpha}{3}\leq g_{3/2}(1)$, we
are dealing with the gas phase.

The equation of state and other thermodynamic functions in both regions can be
also derived and have the following forms:

$\beta
P=\begin{cases}\frac{1}{\lambda^{3}}\left[g_{5/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{3}(z)\right]&\mbox{if
}v>v_{cr},\\\
\frac{1}{\lambda^{3}}\left[g_{5/2}(1)+\frac{2\alpha}{\pi}\zeta(3)\right]&\mbox{if
}v<v_{cr},\end{cases}$

$\frac{U}{N}=\frac{3}{2}Pv=\begin{cases}\frac{3}{2}\frac{k_{B}Tv}{\lambda^{3}}\left[g_{5/2}(z)+\frac{2\alpha}{\pi}\mathrm{Li}_{3}(z)\right]&\mbox{if
}v>v_{cr},\\\
\frac{3}{2}\frac{k_{B}Tv}{\lambda^{3}}\left[g_{5/2}(1)+\frac{2\alpha}{\pi}\zeta(3)\right]&\mbox{if
}v<v_{cr},\end{cases}$

$\frac{C_{V}}{Nk_{B}}=\begin{cases}\frac{15}{4}\frac{v}{\lambda^{3}}h_{1}(T)+\frac{3}{2}\frac{Tv}{\lambda^{3}}h_{2}(T)\frac{dz}{dT}&\mbox{if
}v>v_{cr},\\\ \frac{15}{4}\frac{v}{\lambda^{3}}g_{5/2}(1)+\sigma
f_{1}(T)&\mbox{if }v<v_{cr},\end{cases}$

where the functions $h_{1}\,,h_{2},\,f_{1}$ were defined in Wojnar (2024). The
vapor pressure takes the following form

$P_{0}(T)=\frac{k_{B}T}{\lambda^{3}}\left[g_{5/2}(1)+\frac{2\sigma\sqrt{2mk_{B}T}}{\pi}\zeta(3)\right],$
(48)

while a modified latent heat resulting from the Clapeyron equations derived
from (48) is222The function $f_{2}$ is given by Wojnar (2024)
$f_{2}(T)=\frac{48\sqrt{\frac{\hbar^{2}\pi}{k_{B}}}}{5g_{5/2}(1)(g_{3/2}(1)v_{cr})^{1/3}}\zeta(3)-\frac{\sqrt{8k_{B}mT}\zeta(2)}{g_{3/2}(1)}.$
(49)

$L=\frac{5}{2}\frac{k_{B}Tg_{5/2}(1)}{g_{3/2}(1)}\left(1+\frac{\sigma}{\pi}f_{2}(T)\right).$
(50)

The Bose-Einstein condensation is then also a first-order phase transition in
the considered models of gravity if $L\neq 0$.

##  VI More physical models

Building upon the discoveries outlined in the preceding section of the paper,
one can delve into studying toy-model systems. However, it is crucial to
handle the conclusions drawn from such investigations with utmost care, as
they have the potential to foster misconceptions. Therefore, overly optimistic
bounds on a theory parameter, as discussed in Wojnar (2024), or ”no-go
theorems” for models of gravity with toy-model assumptions Barausse  _et al._
(2008a, b); Pani and Sotiriou (2012); Casado-Turrión  _et al._ (2022) should
not be decisive in assessing the plausibility of a theory (see e.g. Kim
(2014); Olmo (2008); Wojnar (2023a); Olmo and Rubiera-Garcia (2020)).

Therefore, in the following section, we will derive a Fermi EoS with finite
temperature corrections and revisit a simple Landau model of liquid helium.
This model realistically describes its behavior at low temperatures, in
contrast to treating it as an ideal Bose-Einstein condensate.

###  VI.1 Fermi gas with finite temperature corrections

In the non-relativistic limit, the energy becomes $E=p^{2}/2m$ and,
consequently, the pressure (24) becomes

$P=a_{1}\left[\int_{0}^{\infty}\frac{E^{3/2}}{1+e^{\beta(E-\mu)}}dE+a_{2}\int_{0}^{\infty}\frac{E^{2}}{1+e^{\beta(E-\mu)}}dE\right]\,,$
(51)

where $a_{1}=\frac{(2m)^{3/2}}{3\pi^{2}\hbar^{3}}$ and
$a_{2}=\frac{3}{4}\sigma d(2m)^{1/2}$. The two integrals above are Fermi
integrals with the following solutions:

$\displaystyle\int_{0}^{\infty}\frac{E^{3/2}}{1+e^{\beta(E-\mu)}}dE=$
$\displaystyle\frac{2}{5}\mu^{5/2}-\frac{1}{8}\beta^{-1}\mu^{3/2}\ln(1+e^{-\beta\mu})$
(52) $\displaystyle+$
$\displaystyle\frac{\pi^{2}}{4}\beta^{-2}\mu^{1/2}+\frac{3}{4}\beta^{-2}\mu^{1/2}Li_{2}(-e^{-\beta\mu})+...$
$\displaystyle\int_{0}^{\infty}\frac{E^{2}}{1+e^{\beta(E-\mu)}}dE=$
$\displaystyle\frac{\mu^{3}}{3}+\frac{\pi^{2}}{3}\beta^{-2}\mu^{1}+...$ (53)

Then, by setting $\mu=E_{F}$ we can rewrite the pressure as

$\displaystyle P_{F}$ $\displaystyle=$ $\displaystyle
a_{1}\left\\{\frac{2A^{5/2}}{5}\left(\frac{\rho}{\mu_{e}}\right)^{5/3}\left[1-\frac{5}{16}\psi\ln(1+e^{-1/\psi})\right.\right.$
(54) $\displaystyle+$
$\displaystyle\left.\frac{5\pi^{2}}{8}\psi^{2}+\frac{15}{8}\psi^{2}Li_{2}(e^{-1/\psi})\right]$
$\displaystyle+$
$\displaystyle\left.\frac{a_{2}}{3}A^{3}\left(\frac{\rho}{\mu_{e}}\right)^{2}(1+\pi^{2}\psi^{2})\right\\}\,,$

where $A=\frac{(3\pi^{2}\hbar^{3}N_{A})^{2/3}}{2m_{e}}$ is a constant and
$\psi$ is the degeneracy parameter defined in (26). Notice that the first
terms of the EoS (54) corresponds to the pressure for $T=0$ in a non-deformed
phase space, i.e., the polytrope equation for $n=3/2$, discussed already in
IV.1. The first line in the same equation corresponds to the pressure for a
finite temperature $T$ in a non-deformed phase space.

Such an equation of state is used to describe the matter properties in low
mass stars and brown dwarfs, in which the finite temperature corrections,
providing the time evolution of electron degeneracy, are important, see e.g.
Auddy  _et al._ (2016); Benito and Wojnar (2021); Kozak  _et al._ (2023b).

###  VI.2 Liquid helium He 4

The Landau model Landau (2018); Tisza (1947) offers a comprehensive
microscopic depiction of a two-fluid model near absolute zero. As $T$
approaches zero, the specific heat of liquid helium behaves as $T^{3}$,
characteristic of a phonon gas and experimentally confirmed. However, at
finite temperatures, an additional term emerges. Thus, the dispersion relation
of quasiparticles as a function of wave number $k$ for 4He can be described
by:

$\hbar\omega=\begin{cases}\hbar ck&\mbox{if }k<<k_{0},\\\
\Delta+\frac{\hbar^{2}(k-k_{0})^{2}}{2\gamma}&\mbox{if }k\approx
k_{0},\end{cases}$

where $c$ denotes the sound velocity, while $\Delta$, $k_{0}$, and $\gamma$
are experimental constants333We will use the data from Yarnell  _et al._
(1959) $\displaystyle c=239\text{ m s}^{-1},\,\,\rho=144\text{ kg
m}^{-3},\,\,\Delta/k_{B}=8.65\text{K},$ $\displaystyle k_{0}=1.92\times
10^{10}\text{ m}^{-1},\,\,\gamma=1.07\times 10^{-27}\text{ kg}.$ . In the
Landau theory, it is posited that the quantum states of 4He near the ground
state can be treated as those of a non-interacting gas with energy levels
given by:

$\displaystyle U=E_{0}+\sum_{k}\hbar\omega_{k}\langle
n_{k}\rangle=E_{0}+\frac{V}{2\pi^{2}}\int^{\infty}_{0}\frac{k^{2}\hbar\omega_{k}}{e^{\beta\hbar\omega_{k}}-1}\frac{dk}{(1-\sigma\hbar
k)}.$ (55)

Here, $\hbar\omega_{k}$ represents the elementary excitation energy with wave
vector $k$ and occupation number $\langle n_{k}\rangle$. In the second
equality, the deformation of the phase space has been taken into account. Now,
let us compute the internal energy and its gravity corrections at low
temperatures. In this context, only contributions from the phonon and roton
components Cohen and Feynman (1957); Yarnell  _et al._ (1959) affect the
energy in Eq. (55). They are given, respectively, as:

$\displaystyle E_{\text{phonon}}$
$\displaystyle=\frac{V}{2\pi^{2}}\left(\frac{\pi^{4}(k_{B}T)^{4}}{15\hbar^{3}c^{3}}+24\sigma\frac{(k_{B}T)^{5}\zeta(5)}{c^{4}\hbar^{3}}\right),$
(56) $\displaystyle\frac{E_{\text{roton}}}{V}$
$\displaystyle\approx\frac{k_{0}^{2}\Delta}{\pi}\sqrt{\frac{\gamma
k_{B}T}{2\pi\hbar^{2}}}e^{-\frac{\Delta}{k_{B}T}}(1+\sigma\hbar k_{0}).$ (57)

Using the expression (13), we derive the specific heat for liquid helium in
low temperature

$\displaystyle C_{\text{He}^{4}}$ $\displaystyle=20.7T^{3}+\frac{387\times
10^{3}}{T^{3/2}}e^{-8.85/T}+\sigma(5.73\times 10^{-24}T^{4}$
$\displaystyle+\frac{7.83\times 10^{-19}}{T^{3/2}}e^{-8.85/T}).$ (58)

##  VII Discussion and conclusions

In this paper, our focus was to investigate the impact of Ricci-based gravity
theories, such as Palatini $f(R)$ and Eddington-inspired Born-Infeld models,
in conjunction with linear Generalized Uncertainty Principle (GUP) models, on
systems governed by Bose and Fermi statistics. By leveraging the recently
established correspondence between modified gravity and GUP models, we
developed a formalism to analyze ideal Bose and Fermi gases. As anticipated,
the inclusion of modified gravity or linear GUP resulted in the introduction
of additional terms to the familiar expressions. We outlined these
modifications for general Fermi gases, degenerate Fermi gases, Boltzmann
gases, as well as Bose gases including phonons, photons, Bose-Einstein
condensates, and liquid helium.

It’s worth noting that the thermodynamic framework derived from this
correspondence is best suited for non-relativistic systems or those
characterized by low curvature regimes, such as planets, brown dwarfs, active
stars, and white dwarfs. Even white dwarfs, despite their dense nature, can
still be considered within a non-relativistic regime due to their size.
Consequently, the compactness criterion applicable to our formalism is
$\mathcal{C}<<1$.

This intriguing connection between Modified Gravity and GUP models opens up
avenues for testing gravitational proposals through various tabletop
experiments, some of which have already been explored and documented in the
literature. Ongoing research in this direction aims to delve deeper into the
implications of Modified Gravity on the microscopic properties of matter,
seeking further validation and insights.

## Acknowledgements

AW acknowledges financial support from MICINN (Spain) Ayuda Juan de la Cierva
- incorporación 2020 No. IJC2020-044751-I.

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