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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. [ABSTRACT FROM AUTHOR]

Based on the equations of state, one can infer the underlying interaction potentials among the black hole molecules in the case of Schwarzschild-AdS and charged AdS black holes. The microscopic molecules with the interaction potential arrange in a specific way to form the mesostructure, whose size is between the macro (black hole system) and the micro (black hole molecules). As a result, the mesostructure leads to the emergence of the macroscopic phase. However, the information about the mesostructure of the AdS black hole are still elusive. In this paper, the radial distribution function is introduced to probe the mesostructure of the AdS black hole. We find that the mesostructure of the Schwarzschild-AdS black hole behaves as the ideal gas when the temperature is high. Furthermore, we find the mesostructure for the liquid-like (gas-like) phase of the small (large) charged AdS black hole. A sudden change of the mesostructure emerges from the liquid-like phase to the gas-like phase when the charged AdS black hole undergoes a phase transition from the small to large black hole, consistent with the viewpoint that the phase transition of the charged AdS black hole is reminiscent of that of the vdW fluid. This study provides a new angle towards understanding the black hole from its mesostructure. [ABSTRACT FROM AUTHOR]

Phosphorus-modified ZSM-5 zeolites were prepared with a novel method, hydrothermal dispersion. XRD showed that the catalysts prepared by the hydrothermal dispersion had better hydrothermal stability than that by impregnation. At the same time, more pronounced cracking activity and higher yield of ethylene plus propylene were obtained on the phosphorus-modified ZSM-5 catalysts prepared by hydrothermal dispersion. Highest yield of ethylene plus propylene was obtained when the loading of phosphorus was 0.68%. The higher amount of phosphorus fixed on the pores of zeolite, higher hydrothermal stability and appropriate acid amount were the possible reasons for obtaining higher yield of ethylene plus propylene. [ABSTRACT FROM AUTHOR]