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 AdS: 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. *

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In this paper we discuss a formulation of extended phase space thermodynamics of black holes in Anti de Sitter (AdS) space-time, 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-Nordstrom 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 modelled as a collection of independent quantum harmonic oscillators. We derive the closed form expressions for the mean kinetic and potential energies of the charged-dissipativemagneto-oscillator in the form E k = E k and Ep = Ep respectively, where E k and Ep denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is two-fold, the first one being over the Gibbs' canonical state for the thermostat, giving E k and Ep and the second one denoted by . being over the frequencies ω of the bath oscillators which contribute to E k and Ep according to probability distributions P k (ω) and Pp(ω) 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 Ohmic, Drude and radiation bath spectral density functions, on the typical properties of P k (ω) and Pp(ω). 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.

For a thermodynamic system, apart from thermal fluctuations, there are also fluctuations in thermodynamic volume when the system is in contact with a volume reservoir. For the case of black holes in anti de Sitter spacetime, the effect of thermal fluctuations on the entropy is well studied. The aim of this work is to compute novel logarithmic corrections to black hole entropy coming from simultaneous fluctuations in energy and volume. We work in the isothermal-isobaric ensemble and first obtain a general form of corrections to entropy which are valid for any thermodynamic system. Applying the formalism to Kerr black holes in AdS reveals that entropy gets corrected as: \(\mathcal{S} = S_0 - k \ln S_0 + \cdots\) where \(S_0\) is given by the Bekenstein-Hawking formula and \(k = - 1\). The same leading coefficient is also obtained in the canonical ensemble, i.e. by considering energy fluctuations alone. This coefficient is found to be unaltered in the slowly rotating and high temperature limits.