Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime

Reissner-Nordstr\"{o}m-Anti-de Sitter (RNAdS) black holes in the extended phase space exhibit critical behavior analogous to the liquid-gas system, with critical exponents matching those of van der Waals-type phase transitions. However, the kinetics of these transitions near spinodal and critical points remain poorly understood. We demonstrate that both the autocorrelation time and the variance of trajectories increase significantly as the system approaches these special points, signaling critical slowing down. This behavior is driven by the flattening of the free energy landscape, as further confirmed by the lowest eigenvalue of the Fokker-Planck equation. Moreover, we uncover a clear dynamical crossover separating gas-like and liquid-like regimes in the supercritical region. This kinetic crossover defines the Widom line that closely matches the thermodynamic one obtained from the maxima of the isobaric heat capacity. These findings contribute to a deeper understanding of the kinetics of RNAdS black holes in the vicinity of spinodal and critical points.