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 \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.