Linear-Quadratic GUP and Thermodynamic Dimensional Reduction

TL;DR

The paper analyzes semiclassical statistical mechanics in the Linear-Quadratic GUP (LQGUP) framework, introducing a deformed phase-space measure that encodes a minimal length and maximal momentum. By applying this to a system of harmonically-confined particles, it demonstrates a progressive reduction of effective degrees of freedom at high temperatures, with a fractional effective dimension emerging near the Planck scale; specifically, a 3D harmonic oscillator transitions from 6 to 3 DOF (U≈(3/2)NT, C_V≈(3/2)N) and a 2D oscillator from 4 to 2 DOF (U≈NT, C_V≈N). The results, obtained via analytic expansions and exact numerical calculations, indicate a continuous, fractal-like dimensional reduction and align with low-temperature classical thermodynamics. These findings offer a phenomenological bridge between quantum gravity-induced phase-space modifications and macroscopic thermodynamic behavior, highlighting possible fractal spacetime signatures at high energies.

Abstract

In this paper we investigate the statistical mechanics within the Linear-Quadratic GUP (LQGUP, i.e, GUP with linear and quadratic terms in momentum) models in the semiclassical regime. Then, some thermodynamic properties of a system of 3-dimensional harmonic oscillators are investigated by calculating the deformed partition functions. According to the equipartition theorem, we show that the number of accessible microstates decreases sharply in the very high temperatures regime. When the thermal de Broglie wavelength is of the order of the Planck length, three degrees of freedom are frozen in this setup. In other words, it is observed that there is an effective reduction of the degrees of freedom from 6 to 3 for a system of 3D harmonic oscillators in this framework. The calculations are carried out using both approximate analytical and exact numerical methods. The results of the analytical method are also presented in the form of thermal wavelengths for better understanding. Finally, the case of a 2-dimensional harmonic is treated as another example to comprehend the results, leading to a reduction of the degrees of freedom from 4 to 2.

Figures (6)

• Figure 1: The number of microstates vs. momentum $p_\ast$ for a single-particle phase space are illustrated. The red dot-dashed and blue solid lines represent microstate numbers in the non-deformed and LQGUP-deformed phase spaces, respectively. These curves coincide at low energy regime $\kappa{p_\ast}\ll{1}$ ($p_\ast\ll{ p_{_{\rm Pl}}}$) and diverge at high energy regime $\kappa{p_\ast}\sim{1}$ ($p_\ast\sim{p_{_{\rm Pl}}}$). In the LQGUP-deformed scenario, microstates behave like a zero-dimensional phase space, signifying a freezing of three degrees of freedom at high energy due to quantum gravity effects. This suggests a reduction of the degrees of freedom at high energy. The figure is plotted in units of $\kappa=\frac{\kappa_0}{p_{_{\rm Pl}} }=0.004$ with $\kappa_0=1$ and $p_{_{\rm Pl}}=250$.
• Figure 2: The internal energy of a $3$-D harmonic oscillator is depicted in Figure (a) as a function of temperature. It is observed that at high temperatures (Planck temperature), the system's internal energy deviates significantly from its classical state. However, as the temperature decreases, this deviation gradually diminishes until it eventually aligns with the classical state. Also, it can be observed that the approximate solution and the exact numerical solution are very close together. Figure (b) displays the internal energy in terms of thermal wavelength. It is evident that at extremely small wavelengths (minimal length effects), the deviation from the classical state intensifies. Conversely, as the thermal wavelength increases, the behavior of the internal energy approaches its classical state. Figures are plotted in units of $N=5, m=100, \omega=10, \kappa = 0.004$.
• Figure 3: The number of degrees of freedom of a 3D harmonic oscillator is plotted in Figure (a) as a function of temperature. According to the classical equipartition theorem of energy, the number of degrees of freedom of a system can be determined by the formula $\frac{(U/N)}{(T/2)}$. It is observed that in the GUP model, this value is temperature-dependent and decreases from $6$ to $3$ at extremely high temperatures for a 3D harmonic oscillator system. Figure (b) shows the number of degrees of freedom in terms of thermal wavelength. It is evident that at extremely small wavelengths, the system's three degrees of freedom are reduced.
• Figure 4: The specific heat of a 3D harmonic oscillator is shown in Figure (a) as a function of temperature. The system's specific heat is temperature-dependent for the LQGUP model and asymptotically approaches $1.5$ at the very high temperature regime, which signals the effective reduction of the degrees of freedom from $6$ to $3$ in this setup. It is also clear from the figure that the specific heat is bounded as $1.5 \leq\frac{C_{_V}}{N}\leq 3$ in the LQGUP framework. also, as can be seen in this diagram, the values of the numerical and analytical solutions are very close together. Figure (b) displays the specific heat in terms of thermal wavelength. It is observed that at very small wavelengths, the deviation from the classical state intensifies and finally reaches the value of $3/2$.
• Figure 5: The entropy of a 3D harmonic oscillator is shown in Figure (a) as a function of temperature. It is evident that, in the LQGUP model with minimal length and maximal momentum, the entropy increases at a slower rate than in the standard non-deformed case. This is due to the reduction in the number of accessible microstates in the high temperature regime, caused by the effects of quantum gravity. Figure (b) illustrates the entropy in relation to the thermal wavelength. It is observed that at extremely small wavelengths, the entropy demonstrates a slower growth rate compared to the classical (HUP) scenario.