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We employ the approach of path integral in the phase space to study the kinetics of state switching associated with black hole phase transitions. Under the assumption that the state switching process of the black hole is described by the stochastic Langevin equation based on the free energy landscape, we derived the Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) functional and obtained the path integral expression of the transition probability. The MSRJD functional inherently represents the path integral in the phase space, allowing us to extract the effective Hamiltonian for the dynamics of state switching process. By solving the Hamiltonian equations of motion, we obtain the kinetic path in the phase space using an example of the RNAdS black hole. Furthermore, the dominant kinetic path within the configuration space is calculated. We also discuss the kinetic rate by using the functional formalism. Finally, we examine two further examples: Hawking-Page phase transition and Gauss-Bonnet black hole phase transition at the triple point. Our analysis demonstrates that, concerning the Hawking-Page phase transition, while a dominant kinetic path in the phase space from the large SAdS black hole to the thermal AdS space is present, there is no kinetic path for the inverse process. For the Gauss-Bonnet black hole phase transition at the triple point, the state switching processes between the small, the intermediate and the large Gauss-Bonnet black holes constitute a chemical reaction cycle., Comment: More discussions on generalized free energy from York's approach and phase portrait at the critical point, and a brief review of time-dependent Ginzburg-Landau model are added

Bone -borne ra pid ma xilla ry expansion appliances can achieve skeletal expansion while avoiding the undesirable dental side effec ts caused by a conventional rapid palatal expansion appliance. Typically, t hese (bone-bo rne appliances) included prefabricated devices, which can have limitations such as inadequate palatal adaptation leading to anch orage los s. In addit ion, a s bone thickness is not accounted for, prefabricated expanders cannot ensure the primary stability of the mini-implants. These disadvantages can be overcom e by customisation. This repor t aims to describe the digital design and three-dimensional printing workflow for constructing a personali sed M iniscrewassisted rapid palatal expansion (pMARPE) and present a case depicting its application in a 27-year-old female with 5.0 mm t ransverse discrepancy b etween the maxilla and the mandible. The result demonstrated that the pMARPE could be manufactured without the need for conventional impre s sion or laborator y p rocedures and effec tively e xpanded the palate of an ad ult pat ient with maxillar y transverse deficiency.