Title: Kramer's Escape Rate and Phase Transition Dynamics in AdS Black Holes with Dark Structures

Abstract: We explore free energy landscapes for black holes influenced by 'dark' structure, assessing how additional parameters affect escape and transition rates during phase transitions. In our dual-perspective analysis, we first consider the Kramers' rate as a function of the black hole radius. From this viewpoint, the escape rate from smaller to larger black holes will consistently be greater than the reverse. From the second perspective, we consider the escape rate as a function of temperature (pressure). In both models under study, initially, the behavior of the probability diagrams is akin to the radius-based behavior. However, with an increase in temperature (and corresponding pressure), we reach a specific and distinct point, where the systems will move towards either smaller or larger black holes with the same probability. Beyond this temperature (pressure), the behavior of the models diverges. In the nonlinear model, an inversion occurs where the escape rate from larger black holes to smaller ones becomes predominant. This inversion area seems to be significantly reduced compared to the charged black hole model, although it is still discernible in the temperature-based escape rate diagram. In other words, within the temperature range studied, where the likelihood of radiative conditions and movement from a large black hole to a small black hole is less, It seems that the addition of dark parameters minimizes the occurrence of this transition, indicating the subtle influence of dark structures on the phase transition dynamics between black holes. In the string model, although the trend of increasing pressure in the transition from a large black hole to a smaller one is greater than the reverse, it seems that the transition process from a small black hole to a larger one, with decreasing pressure, completely prevents the 'low probable' transition from a large black hole to a smaller one.