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.

In this article, we have examined the kinetics of state switching associated with black hole phase transitions using the path integral approach in the phase space. Based on the assumption that the state switching process can be described by a stochastic Langevin equation and the free energy landscape, we have derived the Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) functional and obtained the path integral expression of the transition probability.

The MSRJD functional allows us to extract the effective Hamiltonian for the dynamics of the state switching process. By solving the Hamiltonian equations of motion, we have obtained the kinetic path in the phase space using the RNAdS black hole as an example. Additionally, we have calculated the dominant kinetic path within the configuration space and discussed the kinetic rate using the functional formalism.

Furthermore, we have examined two additional examples: the Hawking-Page phase transition and the Gauss-Bonnet black hole phase transition at the triple point. Our analysis has shown that in the Hawking-Page phase transition, there exists a dominant kinetic path in the phase space from the large SAdS black hole to the thermal AdS space. However, there is no kinetic path for the inverse process.

For the Gauss-Bonnet black hole phase transition at the triple point, we have found that the state switching processes between the small, intermediate, and large Gauss-Bonnet black holes form a chemical reaction cycle.

## Future Roadmap

Building on our current findings, there are several potential directions for future research:

**Exploring Different Black Hole Systems:**While we have focused on the RNAdS black hole in this study, it would be valuable to investigate other black hole systems and observe if similar kinetic paths and phase transitions occur.**Investigating Quantum Effects:**The path integral approach used in this study provides a classical description of the state switching process. Extending this analysis to incorporate quantum effects could uncover new insights into black hole dynamics.**Examining the Role of Entropy:**The concept of entropy plays a crucial role in black hole thermodynamics. Investigating how entropy influences the kinetics of state switching and phase transitions could shed light on the underlying mechanisms.

**Potential Challenges:**

**Complexity of Calculations:**Solving Hamiltonian equations of motion and evaluating kinetic rates can be computationally intensive. Developing efficient numerical methods or analytical techniques to handle complex calculations will be a challenge.**Quantum Gravity and Black Hole Thermodynamics:**Incorporating quantum gravity into black hole thermodynamics is a longstanding challenge. Bridging the gap between classical and quantum descriptions of black holes will require innovative approaches and theoretical frameworks.

**Potential Opportunities:**

**Applications in Astrophysics and Cosmology:**Understanding the kinetics of black hole state switching and phase transitions could have implications for astrophysical and cosmological phenomena, such as the evolution of galaxies and the early universe.**Advancing Fundamental Physics:**Investigating the dynamics of black hole phase transitions contributes to our understanding of fundamental physics, including gravity, quantum mechanics, and the nature of spacetime.**Potential Technological Applications:**Insights gained from studying black hole kinetics could lead to advancements in areas such as information theory, quantum computing, and thermodynamic systems.

In conclusion, our study has provided an analysis of the kinetics of state switching in black hole phase transitions using the path integral approach. We have identified dominant kinetic paths, calculated kinetic rates, and examined specific examples. The roadmap for future research involves exploring different black hole systems, investigating quantum effects, and examining the role of entropy. While challenges such as complexity of calculations and incorporating quantum gravity exist, the opportunities for advancing astrophysics, fundamental physics, and potential technological applications are exciting.