In this paper, we study the thermodynamic geometry of charged Gauss-Bonnet black holes (and Reissner-Nordström black holes, for the sake of comparison) in anti–de Sitter spacetimes in both ($T$, $V$) and ($S$, $P$) planes. The thermodynamic phase space is known to have an underlying contact and metric structure; Ruppeiner geometry then naturally arises in this framework. Sign of Ruppeiner curvature can be used to probe the nature of interactions between the black hole microstructures. It is found that there are both attraction and repulsion dominated regions which are in general determined by the electric charge, Gauss-Bonnet coupling, and horizon radius of the black hole. The results are physically explained by considering that these black hole systems consist of charged as well as neutral microstructures much like a binary mixture of fluids.

In this work, we present a study to probe the nature of interactions between black hole microstructures for the case of the Banados, Teitelboim, and Zanelli (BTZ) black holes. Even though BTZ black holes without any angular momentum or electric charge thermodynamically behave as an ideal gas, i.e., with noninteracting microstructures; in the presence of electric charge or angular momentum, BTZ black holes are associated with repulsive interactions among the microstructures. We extend the study to the case of exotic BTZ black holes with mass $M=\alpha m+\gamma \frac{j}{l}$ and angular momentum $J=\alpha j+\gamma lm$, for arbitrary values of $(\alpha ,\gamma )$ ranging from purely exotic $(\alpha =0,\gamma =1)$ to slightly exotic $(\alpha >\frac{1}{2},\gamma <\frac{1}{2})$ to highly exotic $(\alpha <\frac{1}{2},\gamma >\frac{1}{2})$. We find that, unlike the normal BTZ black holes [the case ($\alpha =1$, $\gamma =0$)], there exist both attraction as well as repulsion dominated regions in all the cases of exotic BTZ black holes.

In this paper we discuss a formulation of extended phase space thermodynamics of black holes in anti–de Sitter (AdS) spacetimes from the contact geometry point of view. Thermodynamics of black holes can be understood within the framework of contact geometry as flows of vector fields generated by Hamiltonian functions on equilibrium submanifolds in the extended phase space that naturally incorporates the structure of a contact manifold. Deformations induced by the contact vector fields are used to construct various maps among thermodynamic quantities. Thermodynamic variables and equations of state of Schwarzschild black holes are mapped to that of Reissner-Nordström black holes in AdS, with charge as the deformation parameter. In addition, the equations of state of general black holes in AdS are shown to emerge from the high-temperature ideal gas limit equations via suitable deformations induced by contact vector fields. The Hamilton-Jacobi formalism analogous to mechanics is set up, and the corresponding characteristic curves of contact vector fields are explicitly obtained to model thermodynamic processes of black holes. Extension to thermodynamic cycles in this framework is also discussed.

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms $E_k=\langle {\mathcal{E}}_k\rangle$ and $E_p=\langle {\mathcal{E}}_p\rangle$, respectively, where ${\mathcal{E}}_k$ and ${\mathcal{E}}_p$ denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving ${\mathcal{E}}_k$ and ${\mathcal{E}}_p$, and the second one, denoted by $\langle ·\rangle$, is over the frequencies $\omega$ of the bath oscillators which contribute to $E_k$ and $E_p$ according to probability distributions ${\mathcal{P}}_k\left(\omega \right)$ and ${\mathcal{P}}_p\left(\omega \right)$, respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of ${\mathcal{P}}_k\left(\omega \right)$ and ${\mathcal{P}}_p\left(\omega \right)$. Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.