Thermodynamic properties of the noncommutative Dirac oscillator with a permanent electric dipole moment

TL;DR

This paper studies the thermodynamic behavior of a noncommutative Dirac oscillator with an electric dipole moment under electromagnetic fields, revealing how various parameters influence energy, entropy, and free energy at high temperatures.

Contribution

It introduces a detailed analysis of thermodynamic properties of the noncommutative Dirac oscillator with electric dipole moment, including relativistic and nonrelativistic cases, using the Euler-MacLaurin formula.

Findings

Helmholtz free energy decreases with temperature and noncommutative frequency.

Entropy increases with temperature and noncommutative frequency.

Mean energy and heat capacity are twice in relativistic case compared to nonrelativistic.

Abstract

In this paper, we investigate the thermodynamic properties of the noncommutative Dirac oscillator with a permanent electric dipole moment in the presence of an electromagnetic field in contact with a heat bath. Using the canonical ensemble, we determine the properties for both relativistic and nonrelativistic cases through the \textit{Euler-MacLaurin} formula in the high temperatures regime. In particular, the main properties are: the Helmholtz free energy, the entropy, the mean energy, and the heat capacity. Next, we analyze via 2D graphs the behavior of the properties as a function of temperature. As a result, we note that the Helmholtz free energy decreases with the temperature and $\omega_\theta$, and increases with $\omega$, $\Tilde{\omega}$, $\omega_\eta$, where $\omega$ is the frequency of the oscillator, $\Tilde{\omega}$ is a type of cyclotron frequency, and $\omega_\theta$ and $\omega_\eta$ are the noncommutative frequencies of position and momentum. With respect to entropy, we note an increase with the temperature and $\omega_\theta$, and a decrease with $\omega$, $\Tilde{\omega}$, $\omega_\eta$. Now, with respect to mean energy, we note that such property increases linearly with the temperature, and their values for the relativistic case are twice that of the nonrelativistic case. As a direct consequence of this, the value of the heat capacity for the relativistic case is also twice that of the nonrelativistic case, and both are constants, thus satisfying the \textit{Dulong-Petit} law. Lastly, we also note that the electric field does not influence the properties in any way.