Linear-Quadratic GUP and Thermodynamic Dimensional Reduction

Since Planck announced his solution to the black body radiation, a fundamental and very important quantity, namely the Planck constant ($\hbar\simeq{10^{-34}}\textrm{J}.\textrm{s}$) entered physics. Today, this fundamental constant plays a very crucial and consequential role in science, especially in theoretical high energy physics [1, 2, 3]. For example, in quantum gravity phenomenology theories (in fact, approaches to quantum gravity proposal) such as string theory, loop quantum gravity, doubly special relativity (DSR), and also Noncommutative geometry, new quantities were introduced into physics that represent Planck’s constant dependence of the structure of spacetime at high energy level regime [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. One of these new quantities is the fundamental length scale called the Planck Length $l_{{}_{\rm Pl}}$, which is defined by combining $\hbar$ with the gravitational constant $G$ and the constant speed of light $c$ , that is ($l_{{}_{\rm Pl}}=\sqrt{\hbar G/{c^{3}}\;}\approx 1.6\times 10^{-35}$ m). Planck mass, that is, $M_{{}_{\rm Pl}}={\sqrt{\hbar c/{G}}\approx 2.17\times 10^{-8}}$kg, is another such a quantity. Combining quantum field theory with gravity leads naturally to an effective cutoff (a minimal measurable length preferably of the order of the Planck length) in the ultraviolet regime. Indeed, the high energies used to probe small distances significantly disturb the space-time structure by their strong gravitational effects. Thus one common feature of all approaches to Quantum Gravity (QG) proposal is the existence of a minimal measurable length. The existence of a minimal measurable length modifies the Heisenberg uncertainty principle (HUP) to the so called generalized uncertainty principle (GUP) (and also extended uncertainty principle (EUP)). GUP can be understood in the context of string theory and supports the existence of a minimal length/maximal momentum (energy) (UV cutoff) [20, 21, 22, 23, 24, 25]. In the HUP framework, there isn’t any essential limitation on the accuracy of measuring the position of test particles. Therefore, the minimal uncertainty in position measurement ($\Delta{\rm q_{0}}$) can be arbitrarily reduced even to a value of zero. On the other hand, there is a limitation in the GUP framework due to the minimal uncertainty in position measurement. This mentioned phenomenological model, as a flat limit of quantum gravity proposal, represents the deformation for the density of states of the system at very high temperature regimes, which typically leads to self-exclusive measurements over microstates. This property leads to the reduction of the degrees of freedom which may reflect in essence a reduction of the dimensions of the space for the desired system [26, 27, 28, 29, 30]. For a recent and elegant work on dimensional reduction in quantum gravity, see [31, 32]. There are several versions of GUP. The first one, quadratic GUP, modifies the standard Heisenberg Uncertainty Principle (HUP) by including an additional quadratic term in momentum. This suggests that gravity may exhibit different behavior at the minimal length scale than it does under general relativity [33]. Another version, the linear and quadratic GUP (LQGUP ), incorporates both the maximum momentum and minimum length, includes both linear and quadratic terms for momentum, and is compatible with Doubly Special Relativity (DSR) theories [34, 35, 36, 37, 38, 39, 40].
In this interesting and challenging streamline, we are going to study the issue of thermodynamic dimensional reduction through the statistical mechanics of a $3$-D harmonic oscillator system within a version of the GUP model; the Linear-quadratic GUP, in the semiclassical regime. We use the deformed partition functions and study some thermodynamic properties of a system of three-dimensional harmonic oscillators. We’ll see that the number of microstates decreases strongly in the high temperature regime (or equivalently at the low thermal de Broglie wavelength), and this may be effectively related to the reduction of the space dimensions at high energy regime. As an important result, a trace of the fractional degrees of freedom (and possibly space dimensions) will be observed in this setup for a system of $3$-D harmonic oscillators.

In this paper, in Section 2, the statistical mechanics of the LQGUP framework is studied and the corresponding partition function is formulated. In section 3, the thermodynamics of a system of $3$D harmonic oscillators is studied analytically (approximated solutions) and also numerically (exact solutions) within the framework of LQGUP. Section 4 provides an analysis of a 2-dimensional harmonic oscillator thermodynamics in LQGUP setup. Section 5 is devoted to the summary and conclusions.

The classical system’s kinematics and dynamics on the phase space offer a framework for formulating statistical mechanics in the semiclassical regime. The Liouville volume is the crucial factor that determines the density of states, which can be utilized to derive all of a system’s thermodynamic properties.

In this section, we study the thermodynamics of a system of 3-dimensional harmonic oscillators in the LQGUP framework, which accounts for minimal length and maximal momentum effects. The Hamiltonian function for a single harmonic oscillator is then $H=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}q^{2}$, and by substituting it in Eq. (15), the corresponding partition function for a single particle in semi-classical regime can be derived as follows

$$\begin{gathered}{\mathcal{Z}}_{1}=\frac{16\pi^{2}}{h^{3}}\int_{q_{min}}^{\infty}\int_{0}^{p_{{}_{max}}}\frac{\exp\big{[}-\frac{p^{2}}{2mT}\big{]}\exp\big{[}-\frac{m{w^{2}}q^{2}}{2T}\big{]}q^{2}p^{2}dpdq}{\big{(}1-\kappa p+\kappa^{2}p^{2}\big{)}^{4}}.\end{gathered}$$

(16)

The partition function obtained above cannot be solved analytically. Therefore, we must rely on numerical calculation methods to determine this function and other related thermodynamic variables. However, we can approximate the answer by expanding the deformed phase space volume as follows

$$\frac{1}{\big{(}1-\kappa p+\kappa^{2}p^{2}\big{)}^{4}}=1+4\kappa p+6\kappa^{2}p^{2}+O\left(\kappa^{3}\right),$$

(17)

Thus, the partition function (16) can be approximated as follows

$$\begin{gathered}{\mathcal{Z}}_{1}\simeq\frac{16\pi^{2}}{h^{3}}\int_{0}^{\infty}\int_{0}^{\frac{1}{2\kappa}}\exp\big{[}-\frac{p^{2}}{2mT}\big{]}\exp\big{[}-\frac{m{w^{2}}q^{2}}{2T}\big{]}q^{2}p^{2}\big{(}1+4\kappa p+6\kappa^{2}p^{2}\big{)}dpdq\end{gathered}$$

(18)

After integration of the above relation, we’ll have

$$\begin{gathered}{\mathcal{Z}}_{1}[T]=\Big{(}\frac{T}{w}\Big{)}^{3}\Bigg{[}\sqrt{\frac{128mT\kappa^{2}}{\pi}}-\frac{1}{\sqrt{8\pi mT\kappa^{2}}}\Big{(}9+68mT\kappa^{2}\Big{)}e^{\frac{-1}{8mT\kappa^{2}}}\\ +\Big{(}1+18mT\kappa^{2}\Big{)}\text{erf}\Big{(}\frac{1}{\sqrt{8mT\kappa^{2}}}\Big{)}\Bigg{]}\,.\\ \end{gathered}$$

(19)

In this relation, “erf“ is the error function, defined by: $\text{erf}(x)=(2/\sqrt{\pi})\int_{0}^{x}e^{-t^{2}}dt$. To better understand the qualitative features of the partition function (19), it is convenient to rewrite it in terms of the thermal de Broglie wavelength $\lambda$, defined by: $\lambda=\frac{h}{\sqrt{2\pi{m}T}}$ as

$$\begin{gathered}{\mathcal{Z}}_{1}[\lambda]\approx\Big{(}\frac{2\pi\ell^{2}}{\lambda^{2}}\Big{)}^{3}\bigg{[}\frac{4\lambda_{{}_{\rm Pl}}}{\sqrt{\pi}\lambda}-\frac{\lambda}{\sqrt{\pi}\lambda_{{}_{\rm Pl}}}\Big{(}9+\frac{17{\lambda_{{}_{\rm Pl}}}^{2}}{2\lambda^{2}}\Big{)}\exp\Big{[}\frac{-\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}+\Big{(}1+\frac{9{\lambda_{{}_{\rm Pl}}}^{2}}{4\lambda^{2}}\Big{)}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\bigg{]}.\end{gathered}$$

(20)

Where $\ell=\sqrt{\frac{1}{m\omega}}$ is the characteristic length of the harmonic oscillator system. In this formula, ${\lambda_{{}_{\rm Pl}}}$ is the thermal de Broglie wavelength in the LQGUP framework, which is of the order of the Planck scale thermal de Broglie wavelength,

$$\lambda_{{}_{\rm Pl}}=4\sqrt{\pi}\kappa=4\sqrt{\pi}{\kappa_{0}}{{l_{{}_{\rm pl}}}}=(2\kappa_{0})\frac{h}{\sqrt{2\pi{m_{{}_{\rm Pl}}}T_{{}_{\rm Pl}}}}\,.$$

(21)

By expanding the partition function (20) for very high and low temperature regimes, we find

$${\mathcal{Z}}_{1}[\lambda]=\left\{\begin{array}[]{ll}\begin{gathered}(8{\pi}^{3})\frac{{\ell}^{6}}{\lambda^{6}}=\Big{(}\frac{T}{\hbar\omega}\Big{)}^{3}\end{gathered}&\hskip 14.22636pt\lambda\gg\lambda_{{}_{\rm Pl}},\\ \\ \begin{gathered}\Big{(}\frac{544\pi^{5/2}}{15}\Big{)}\frac{\ell^{6}}{\lambda^{3}\lambda_{{}_{\rm Pl}}^{3}}\end{gathered}&\hskip 14.22636pt\lambda\sim\lambda_{{}_{\rm Pl}}\,,\end{array}\right.$$

(22)

where $\mathcal{Z}_{0}=\big{(}\frac{T}{\hbar\omega}\big{)}^{3}$ is the non-deformed partition function for a three dimensional single-harmonic oscillator. As observed from the above relation, the influence of quantum gravity (minimal length and maximal momentum) can be disregarded at the low temperature limit $\lambda\gg\lambda_{{}_{\rm Pl}}\,(T\ll{T}_{{}_{\rm Pl}})$, and the classical outcome for the harmonic oscillator partition function is restored. Additionally, it is evident from Eq.(22) that three degrees of freedom will freeze ($\lambda^{6}\rightarrow\lambda^{3}$) at the Planck length scale $\lambda\sim\lambda_{{}_{\rm Pl}}$ in the LQGUP framework, and the partition function behaves as if it were effectively a partition function of a $1.5$D harmonic oscillator. Therefore, it is possible to conclude an effective reduction in the degrees of freedom from $6$ to $3$ and therefore in the space dimensions from $3$ to $1.5$ in the GUP framework for harmonic oscillator systems. This feature evidently signals the existence of fractional degrees of freedom and possibly space dimensions for such a system  [49, 50].
It is important to note that in these processes of decreasing number of degrees of freedom, at intermediate temperatures in all the figures presented in this paper, we observe a fractional nature of the number of the degrees of freedom. In fact, it would be more accurate to state that the number of the degrees of freedom of the deformed models mentioned are not completely integers at high temperatures; rather, they are indeed fractional (see also figure 3).

From Eq.(20), the total partition function of the system of harmonic oscillators in the LQGUP framework can be expressed as follows

$$\begin{gathered}{\mathcal{Z}}_{N}[\lambda]\approx\Big{(}\frac{2\pi\ell^{2}}{\lambda^{2}}\Big{)}^{3N}\bigg{[}\frac{4\lambda_{{}_{\rm Pl}}}{\sqrt{\pi}\lambda}-\frac{\lambda}{\sqrt{\pi}\lambda_{{}_{\rm Pl}}}\Big{(}9+\frac{17{\lambda_{{}_{\rm Pl}}}^{2}}{2\lambda^{2}}\Big{)}\exp\Big{[}\frac{-\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}+\Big{(}1+\frac{9{\lambda_{{}_{\rm Pl}}}^{2}}{4\lambda^{2}}\Big{)}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\bigg{]}^{N}.\end{gathered}$$

(23)

Now, utilizing Eq. $(\ref{ho3.5})$, we can derive all thermodynamic quantities using the standard definitions. First, we begin with the Helmholtz free energy $F$, which is expressed as follows

$$\begin{gathered}F=-NT\ln\Bigg{[}\Big{(}\frac{2\pi\ell^{2}}{\lambda^{2}}\Big{)}^{3}\bigg{[}\frac{4\lambda_{{}_{\rm Pl}}}{\sqrt{\pi}\lambda}-\frac{\lambda}{\sqrt{\pi}\lambda_{{}_{\rm Pl}}}\Big{(}9+\frac{17{\lambda_{{}_{\rm Pl}}}^{2}}{2\lambda^{2}}\Big{)}\exp\Big{[}\frac{-\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}+\Big{(}1+\frac{9{\lambda_{{}_{\rm Pl}}}^{2}}{4\lambda^{2}}\Big{)}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\bigg{]}\Bigg{]}.\end{gathered}$$

(24)

Having equipped with Helmholtz free energy, we can obtain other thermodynamic quantities as follows.

In this section we consider a 2D harmonic oscillator to see the reduction of the degrees of freedom in this system and whether the results are as expected or not. The 2D LQGUP Liouville volume, $d^{2}\omega$, is obtained from equation (8) as $d^{2}\omega=\frac{d^{2}{q}d^{2}{p}}{\big{[}1-\kappa p+\frac{7}{6}\kappa^{2}p^{2}\big{]}^{3}}$. Hence, for 2D harmonic oscillators in LQGUP phase space, the partition function of a single particle in the semi-classical regime can be derived as follows

$\displaystyle\begin{gathered}Z_{1}(T,V,\kappa)=\frac{1}{h^{2}}\int_{\Gamma}d^{2}\omega\exp[-H/T]=\frac{1}{{{h^{2}}}}\int{{d^{2}}q\int{\frac{{{d^{2}}p}}{\Big{[}1-\kappa p+\frac{7}{6}\kappa^{2}p^{2}\Big{]}^{3}}\exp\left({-\frac{{H(q,p)}}{T}}\right)}}\,.\end{gathered}$

(33)

Expanding the deformed phase space volume yields

$$\frac{1}{\Big{[}1-\kappa p+\big{(}\frac{7}{6}\big{)}\kappa^{2}p^{2}\Big{]}^{3}}=1+3\kappa p+\frac{5\kappa^{2}p^{2}}{2}+O\left(\kappa^{3}\right).$$

(34)

So, the partition function (33) can be approximated as follows

$$\begin{gathered}{\mathcal{Z}}_{1}\simeq\frac{4\pi^{2}}{h^{2}}\int_{0}^{\infty}\int_{0}^{\frac{1}{2\kappa}}\exp\big{[}-\frac{p^{2}}{2mT}\big{]}\exp\big{[}-\frac{m{w^{2}}q^{2}}{2T}\big{]}qp\big{(}1+3\kappa p+\frac{5\kappa^{2}p^{2}}{2}\big{)}dpdq\end{gathered}$$

(35)

Upon integration of the above relation, we obtain

$$\begin{gathered}{\mathcal{Z}}_{1}[T]=\Big{(}\frac{T}{w}\Big{)}^{2}\Bigg{[}1+5mT\kappa^{2}-\Big{(}\frac{25}{8}+5mT\kappa^{2}\Big{)}\exp\Big{[}{\frac{-1}{8mT\kappa^{2}}}\Big{]}+\frac{3}{4}\sqrt{\pi}\sqrt{8mT\kappa^{2}}\text{erf}\Big{(}\frac{1}{\sqrt{8mT\kappa^{2}}}\Big{)}\Bigg{]}\,.\end{gathered}$$

(36)

Expressing it in terms of the thermal de Broglie wavelength $\lambda$, we get

$$\begin{gathered}{\mathcal{Z}}_{1}[\lambda]\approx\Big{(}\frac{2\pi\ell^{2}}{\lambda^{2}}\Big{)}^{2}\bigg{[}[1+\frac{5{\lambda_{{}_{\rm Pl}}}^{2}}{8\lambda^{2}}-\Big{(}\frac{25}{8}+\frac{5{\lambda_{{}_{\rm Pl}}}^{2}}{8\lambda^{2}}\Big{)}\exp\Big{[}\frac{-\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}+\frac{3\sqrt{\pi}{\lambda_{{}_{\rm Pl}}}}{4\lambda}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\bigg{]}.\end{gathered}$$

(37)

Expanding the partition function (37) at very high and low temperatures gives

$${\mathcal{Z}}_{1}[\lambda]=\left\{\begin{array}[]{ll}\begin{gathered}(4{\pi}^{2})\frac{{\ell}^{4}}{\lambda^{4}}=\Big{(}\frac{T}{\omega}\Big{)}^{2}\end{gathered}&\hskip 14.22636pt\lambda\gg\lambda_{{}_{\rm Pl}},\\ \\ \begin{gathered}\Big{(}\frac{37\pi^{2}}{4}\Big{)}\frac{\ell^{4}}{\lambda^{2}\lambda_{{}_{\rm Pl}}^{2}}\end{gathered}&\hskip 14.22636pt\lambda\sim\lambda_{{}_{\rm Pl}}\,,\end{array}\right.$$

(38)

It is evident from Eq. (38) that two degrees of freedom will freeze ($\lambda^{4}\rightarrow\lambda^{2}$) at the Planck length scale $\lambda\sim\lambda_{{}_{\rm Pl}}$ in the LQGUP framework, and the partition function behaves as if it were effectively a partition function of a $1$D harmonic oscillator.
The internal energy of the system can be modified as follows

$\displaystyle U=NT\left(\frac{\begin{gathered}-25-\frac{{61\lambda_{{}_{\rm Pl}}}^{2}}{\lambda^{2}}-\frac{15{\lambda_{{}_{\rm Pl}}}^{4}}{\lambda^{4}}+\frac{2{\lambda_{{}_{\rm Pl}}}^{2}}{\lambda^{2}}\exp\Big{[}\frac{\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}\bigg{(}8+\frac{15{\lambda_{{}_{\rm Pl}}}^{2}}{2\lambda^{2}}+\frac{15\sqrt{\pi}{\lambda_{{}_{\rm Pl}}}}{2\lambda}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\bigg{)}\end{gathered}}{\begin{gathered}\frac{{\lambda_{{}_{\rm Pl}}}^{2}}{\lambda^{2}}\bigg{(}-5(5+\frac{{\lambda_{{}_{\rm Pl}}}^{2}}{\lambda^{2}})+4\exp\Big{[}\frac{\lambda^{2}}{\lambda_{{}_{\rm Pl}}^{2}}\Big{]}\Big{(}2+\frac{5{\lambda_{{}_{\rm Pl}}}^{2}}{4\lambda^{2}}+\frac{3\sqrt{\pi}{\lambda_{{}_{\rm Pl}}}}{2\lambda}\text{erf}\Big{[}\frac{\lambda}{\lambda_{{}_{\rm Pl}}}\Big{]}\Big{)}\bigg{)}\end{gathered}}\right).$

(41)

Expanding the relation (41) for both of the high and low temperature regimes, we have.

$$U=\left\{\begin{array}[]{ll}2\,NT&\hskip 14.22636pt\lambda\gg\lambda_{{}_{\rm Pl}},\\ \\ \,NT&\hskip 14.22636pt\lambda\sim\lambda_{{}_{\rm Pl}}\,.\end{array}\right.$$

(42)

Finally, expanding the specific heat which is defined through the relation (42) is as follows

$$C_{{}_{V}}=\left\{\begin{array}[]{ll}2\,N&\hskip 14.22636pt\lambda\gg\lambda_{{}_{\rm Pl}},\\ \\ \,N&\hskip 14.22636pt\lambda\sim\lambda_{{}_{\rm Pl}}.\end{array}\right.$$

(43)

These results for a system of 2-dimensional harmonic oscillator are in agreement with the general prescription of the reduction of the degrees of freedom as expected.

In this paper, we have investigated the thermostatistics of a harmonic oscillator system in the Linear Quadratic GUP phase space model. Our aim was to gain further insight into the topic of the reduction of the degrees of freedom in high energy regimes. The calculations have been based on the existence of an ultraviolet cutoff (or minimal measurable length and maximal measurable momentum). Referring to the equipartition theorem of energy, we have clearly shown that as the temperature increases towards the Planck temperature in the mentioned framework, the three degrees of freedom freeze out. Such freezing (reduction) of the degrees of freedom has been observed for the Snyder Phase Space framework in a previous work by our research group [27]. The most important observation here is the fact that, while in the Snyder phase space scheme, the reduction of the number of degrees of freedom at high temperatures is $2$-folds, in the GUP framework, this reduction is $3$-folds.
In this direction, we note the following important points:
$\bullet$ First, we note that the reduction of the degrees of freedom is essentially continuous, which means the possibility of having a non-integer degrees of freedom (see also figures 3). This may signal a fractal space dimension in the presence of the phenomenological quantum gravity effect encoded in the presence of a minimal measurable length.
$\bullet$ Second, we considered the classical equipartition theorem for our treatment. It is in fact necessary to consider a modified equipartition theorem and even a Planck scale thermodynamics that may be in dramatic digression with the standard thermodynamics. But these are the issues of the yet-unsolved quantum gravity problem.
In the absence of a well-formulated quantum gravity theory, such phenomenological-based studies may shed light on the issue of the final quantum gravity proposal. So, we emphasize that in general, the deformed spaces in different phenomenological approaches to quantum gravity proposal all address a reduction of the degrees of freedom in high temperature regimes from a thermodynamic point of view. All of these theories coincide with their classical counterparts at low temperatures, as required. The possibility of realizing the fractal degrees of freedom in these setups is also an important outcome of this study.
The behavior shown in the diagrams arises in the field of quantum gravity, where the effects of the generalized uncertainty principle (GUP) lead to changes in the structure of phase space. The study of the LQGUP deformation scenario reveals a remarkable aspect: a significant deviation from classical behavior in the number of microstates at high energies. This deviation, characterized by the blocking of degrees of freedom, underlines a complex interplay between the basic principles of quantum mechanics and gravity. The shift towards a low-dimensional phase space at high energies emphasizes the intricate correlation between the dynamics of microstates and the inherent quantum essence of spacetime. These insights provide valuable perspectives on the nature of spacetime in a realm where the influences of quantum gravity are significant and provide a deeper understanding of the fundamental structure of the universe.
Finally we have checked the case with 2-dimensional harmonic oscillator to see the results which are in agreement with the general prescription: a reduction of the degrees of freedom from $4$ to $2$, as expected.

Acknowledgement
We appreciate the referee for his/her insightful and constructive comments.