We generalize Ehrenfests equations to systems having two work terms, i.e. systems with three degrees of freedom. For black holes with two work terms we obtain nine equations instead of two to be satisfied at the critical point of a second order phase transition. We finally generalize this method to a system with an arbitrary number of degrees of freedom and found there is $frac{N(N+1)^{2}}{2}$ equations to be satisfied at the point of a second order phase transition where $N$ is number of work terms in the first law of thermodynamics.
We investigate equations of motion and future singularities of $f(R,T)$ gravity where $R$ is the Ricci scalar and $T$ is the trace of stress-energy tensor. Future singularities for two kinds of equation of state (barotropic perfect fluid and generalized form of equation of state) are studied. While no future singularity is found for the first case, some kind of singularity is found to be possible for the second. We also investigate $f(R,T)$ gravity by the method of dynamical systems and obtain some fixed points. Finally, the effect of the Noether symmetry on $f(R,T)$ is studied and the consistent form of $f(R,T)$ function is found using the symmetry and the conserved charge.
We consider the critical behaviors and phase transitions of Gauss Bonnet-Born Infeld-AdS black holes (GB-BI-AdS) for $d=5,6$ and the extended phase space. We assume the cosmological constant, $Lambda$, the coupling coefficient $alpha$, and the BI parameter $beta$ to be thermodynamic pressures of the system. Having made these assumptions, the critical behaviors are then studied in the two canonical and grand canonical ensembles. We find reentrant and triple point phase transitions (RPT-TP) and multiple reentrant phase transitions (multiple RPT) with increasing pressure of the system for specific values of the coupling coefficient $alpha$ in the canonical ensemble. Also, we observe a reentrant phase transition (RPT) of GB-BI-AdS black holes in the grand canonical ensemble and for $d=6$. These calculations are then expanded to the critical behavior of Born-Infeld-AdS (BI-AdS) black holes in the third order of Lovelock gravity and in the grand canonical ensemble to find a Van der Waals behavior for $d=7$ and a reentrant phase transition for $d=8$ for specific values of potential $phi$ in the grand canonical ensemble. Furthermore, we obtain a similar behavior for the limit of $beta to infty$, i.e charged-AdS black holes in the third order of the Lovelock gravity. Thus, it is shown that the critical behaviors of these black holes are independent of the parameter $beta$ in the grand canonical ensemble.
As an extension to our earlier work cite{Mirza2}, we employ the Nambu brackets to prove that the divergences of heat capacities correspond to their counterparts in thermodynamic geometry. We also obtain a simple representation for the conformal transformations that connect different thermodynamics metrics to each other. Using our bracket approach, we obtain interesting exact relations between the Hessian matrix with any number of parameters and specific heat capacities. Finally, we employ this approach to investigate some thermodynamic properties of the Meyers-Perry black holes with three spins.
A helicity entangled tripartite state is considered in which the degree of entanglement is preserved in non-inertial frames. It is shown that Quantum Entanglement remains observer independent. As another measure of quantum correlation, Quantum Discord has been investigated. It is explicitly shown that acceleration has no effect on the degree of quantum correlation for the bipartite and tripartite helicity entangled states. Geometric Quantum Discord as a Hilbert-Schmidt distance is computed for helicity entangled states. It is shown that living in non-inertial frames does not make any influence on this distance, either. In addition, the analysis has been extended beyond single mode approximation to show that acceleration does not have any impact on the quantum features in the limit beyond the single mode. As an interesting result, while the density matrix depends on the right and left Unruh modes, the Negativity as a measure of Quantum Entanglement remains constant. Also, Quantum Discord does not change beyond single mode approximation.
The effects of a running gravitational coupling and the entropic force on future singularities are considered. Although it is expected that the quantum corrections remove the future singularities or change the singularity type, treating the running gravitational coupling as a function of energy density is found to cause no change in the type of singularity but causes a delay in the time that a singularity occurs. The entropic force is found to replaces the singularity type $II$ by $bar{III}$ ($a=$const., $H=$const., $dot{H} to infty$, $p to infty$, $rho to infty$) which differs from previously known type $III$ and to remove the $w$-singularity. We also consider an effective cosmological model and show that the types $I$ and $II$ are replaced by the singularity type $III$.
