We study the first order phase transition of Euler-Heisenberg-AdS black hole
based on free energy landscape. When the Euler-Heisenberg parameter (quantum
electrodynamics parameter) $a<0$, the small (large) Euler-Heisenberg-AdS black
hole can switch to the large (small) black hole due to the change of the
temperature $T$ or Euler-Heisenberg parameter $a$. The local minimal points
correspond to the local stable black holes, and the system prefers black hole
with the lower Gibbs free energy. Furthermore, we research the probability
distribution of the system states by solving the Fokker-Planck equation. A
higher (lower) $T$ corresponds to a larger probability for a large (small)
black hole. A smaller (larger) $a$ corresponds to a larger probability for a
large (small) black hole. The initial the small (large) black hole state can
have the chance to switch to a large (small) black hole state. The coexistent
small and large black hole states can be acquired for some conditions. We also
consider the first passage process for the small-large black hole phase
transition. A higher peak can be acquired for higher (lower) $T$ or smaller
(larger) $a$ with initial small (large) black hole state. For $0\leq a\leq
\frac{32}{7} Q^2 $, the small-large black hole phase transition can be acquired
with a small $a$. A higher (lower) $T$ corresponds to a larger probability for
a large (small) black hole, which is similar to the case of $a<0$. However, the
probability of small (large) black hole will decrease to zero for a large $a$,
which is different from the case of $a<0$. For a small $a$, a higher peak of
the first passage time can be acquired for higher (lower) $T$ or smaller
(larger) $a$ with initial small (large) black hole state.
