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

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 $\mathit{\text{Euler-MacLaurin}}$ 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 $\omega$, $\text{Tilde}\omega$, ${\omega }_{\eta }$, where $\omega$ is the frequency of the oscillator, $\text{Tilde}\omega$ 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 $\omega$, $\text{Tilde}\omega$, ${\omega }_{\eta }$. 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 $\mathit{\text{Dulong-Petit}}$ law. Lastly, we also note that the electric field does not influence the properties in any way.