Generalized free energy and dynamical state transition of the dyonic AdS black hole in the grand canonical ensemble

We study the generalized free energy of the dyonic AdS black hole in an ensemble with varying electric charge qE and fixed magnetic charge qM. When we adjust the temperature T and the electric potential ΦE of the ensemble, the Ricci scalar curvature R and electromagnetic potential Au usually diverge at the horizon. We regularize them and incorporate the off-shell corrections into the Einstein-Hilbert action. Alternatively, we find that the off-shell corrections can also be obtained by adding a boundary near the horizon to exclude the singularities. Ultimately, we derive the generalized free energy which is consistent with the definition of the thermodynamic relations. Based on the generalized free energy landscape, we can describe the dynamics of state transition as a stochastic process quantified by the Langevin equation. The path integral framework can be formulated to derive the time-dependent trajectory of the order parameter and the time evolution of the transition probability. By comparing the probability with the result of the classical master equation, we attribute the contribution to the probability of one pseudomolecule or antipseudomolecule (the instanton and anti-instanton pairs) to the rate of state transition. These results are consistent with the qualitative analysis of the free energy landscape.