  Traditional static methods in phase transition studies, such as the
swallowtail bifurcation diagram, provide good insights into the thermodynamics
of black holes. However, they practically lose sight of the dynamic aspects and
temporal sequence of events. The Kramers 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 âdarkâ and
âstringy+darkâ 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 âphase changeâ action. A
reaction which attempting to prevent any uncontrolled radial growth that could
jeopardize the stability of the black hole.
