Kramer's Escape Rate and Phase Transition Dynamics in AdS Black Holes

Traditional static methods in phase transition studies, provide good insights into the thermodynamics of black holes. However, they practically lose sight of the dynamic aspects and temporal sequence of events. The Kramer's escape rate, central to our research, offers a somewhat dynamic approach to phase transition. We examine the free energy landscapes for black holes under the influence of 'dark' and 'stringy dark' structures, assessing how additional parameters affect the escape rates and dynamics of the transition during the first-order phase transition from small to large black holes. In our analysis, we consider the escape rate as a function of the black hole radius and study its variations. We will observe that, on one hand, the escape rate well represents our assumption based on the movement from zero, increasing to a maximum point, and then decreasing back to zero as reactive structures become active during the phase transition interval. However, the critical point in this method is the encounter with a specific and distinct point. This is where the diagram of the direct process (escape rate from small to large black holes) intersects with the reverse process (large to small black holes), becoming equally probable (contact point). The point, which seems improbable at the onset of the phase transition or very negligible, gains more significance as the process progresses. This increase indicates the dominance of a region where the escape rate from larger black holes to smaller ones prevails. The predominance of the reverse process, which increases as we approach the end of the process and is necessarily accompanied by a variation in radius, may be considered as a natural reaction of the black hole against the 'phase change' action. A reaction which attempting to prevent any uncontrolled radial growth that could jeopardize the stability of the black hole.