  Searching for the effect of quintessence dark energy on the kinetics of black
hole phase transition, we investigate in detail the dynamic phase transition of
charged AdS black holes surrounded by quintessence in this paper. Based on the
Gibbs free energy landscape, we obtain the analytic expression of the
corresponding Gibbs free energy. As shown in $G_L-r_+$ curve at the phase
transition temperature, there exist double wells with the same depth, providing
further support on the finding in the former literature. By numerically solving
the Fokker-Planck equation with both the initial condition and reflecting
boundary condition imposed, we probe the probabilistic evolution of charged AdS
black holes surrounded by quintessence. The peak denoting the initial black
hole state gradually decreases while the other peak starts to grow from zero,
approaching to be a stationary distribution in the long time limit with two
peaks denoting the large and small black holes respectively. We also study the
first passage process of charged AdS black holes surrounded by quintessence and
discuss the relevant quantities. We resolve the Fokker-Planck equation by
adding the absorbing boundary condition for the intermediate transition state.
It is shown intuitively that the peaks located at the large (small) black hole
decay very rapidly, irrespective of the initial black hole state. In all the
procedures above, we have compared the cases with different choices of the
state parameter of quintessence dark energy $\omega_q$. The larger $\omega_q$
is, the faster the initial black hole state decays, showing the effect of
quintessence dark energy. To the best of our knowledge, it is the first probe
on the influence of dark energy on the dynamic phase transition of charged AdS
black hole.
