  Using It\^o's calculus and the mass optimal transportation theory, we study
the generalized Dyson Brownian motion (GDBM) and the associated McKean-Vlasov
evolution equation with an external potential $V$. Under suitable condition on
$V$, we prove the existence and uniqueness of strong solution to SDE for GDBM.
Standard argument shows that the family of the process of empirical measures
$L_N(t)$ of GDBM is tight and every accumulative point of $L_N(t)$ in the weak
convergence topology is a weak solution of the associated McKean-Vlasov
evolution equation, which can be realized as the gradient flow of the
Voiculescu free entropy on the Wasserstein space over $\mathbb{R}$. Under the
condition $V''\geq -K$ for some constant $K\geq 0$, we prove that the
McKean-Vlasov equation has a unique solution $\mu(t)$ and $L_N(t)$ converges
weakly to $\mu(t)$ as $N\rightarrow \infty$. For $C^2$ convex potentials, we
prove that $\mu(t)$ converges to the equilibrium measure $\mu_V$ with respect
to the $W_2$-Wasserstein distance on $\mathscr{P}_2(\mathbb{R})$ as
$t\rightarrow \infty$. Under the uniform convexity or a modified uniform
convexity condition on $V$, we prove that $\mu(t)$ converges to $\mu_V$ with
respect to the $W_2$-Wasserstein distance on $\mathscr{P}_2(\mathbb{R})$ with
an exponential rate as $t\rightarrow \infty$. Finally, we discuss the
double-well potentials and raise some conjectures.
