Dynamic phase transition of charged dilaton black holes

The dynamic phase transition of charged dilaton black holes is investigated in this paper. The Gibbs free energy landscape is introduced, and the corresponding begin{document}$G_{rm L}$end{document} is calculated for the dilaton black hole. We numerically solve the Fokker-Planck equation constrained by only the reflecting boundary condition. The effects of dilaton gravity on the probabilistic evolution of dilaton black holes are explored. Firstly, the horizon radius difference between a large dilaton black hole and a small dilaton black hole increases with the parameter begin{document}$alpha$end{document}. Secondly, with increasing begin{document}$alpha$end{document}, the system needs much more time to achieve a stationary distribution. Finally, the values attained for begin{document}$rho(r_{rm l},t)$end{document} and begin{document}$rho(r_{rm s},t)$end{document} vary with begin{document}$alpha$end{document}. Additionally, by resolving the Fokker-Planck equation constrained by both the reflecting boundary condition and absorbing boundary condition, we investigate the first passage process of dilaton black holes. The initial peak decays more slowly with increasingbegin{document}$alpha$end{document}, which can also be observed via the slowing decay of begin{document}$Sigma(t)$end{document} (the sum of the probability of the black hole system not having completed a first passage by time t). Moreover, the time corresponding to the single peak of the first passage time distribution is found to increase with the parameter begin{document}$alpha$end{document}. Considering these observations, the dilaton field is found to slow down the dynamic phase transition process between a large black hole and a small black hole.