Restricted Phase Space Thermodynamics of 4D Dyonic AdS Black Holes: Insights from Kaniadakis Statistics and Emergence of Superfluid $λ$-Phase Transition

Abstract: We study the thermodynamics of $4D$ dyonic AdS black hole in the Kaniadakis statistics framework using the Restricted Phase Space (RPST) formalism. This framework provides a non-extensive extension of classical statistical mechanics, drawing inspiration from relativistic symmetries and presenting a fresh perspective on black hole thermodynamics. Our study analyzes how including Kaniadakis entropy modifies the phase transition of the dyonic black holes. We consider the central charge $C$ and its conjugate chemical potential $\mu$ as the thermodynamic variable along with others except the pressure and volume. Due to the addition of the magnetic charge $\tilde{Q}_m$, the study of the phase transition becomes much richer by obtaining a non-equilibrium phase transition from an unstable small black hole to a stable large black hole along with the Van der Waals phase transition in the $T-S$ processes. In the $F-T$ plot, we get an extra Hawking-Page phase transition. Including the deformation parameter $\kappa$ introduces an unstable (ultra-large BH) branch seen in almost all the plots. Turning off the magnetic charge flips the direction of the phase transition seen during its presence. We observe a novel phenomenon that is the superfluid $\lambda$ phase transition in the mixed $(\tilde{\Phi}_e,\tilde{Q}_m)$ which is due to the additional $\tilde{Q}_m$ inclusion. Also, in the plots varying $\kappa$ match with the plot varying $C$ which underlines some sort of correspondence in its meaning which is not possible to observe in Gibbs-Boltzmann statistics. As the entropy models change the homogeneity is not lost where mass is of the first order and the rest is zeroth order. Finally, the $\mu-C$ processes in quite similar across black hole systems and entropy formulation marking some kind of universality of this process.