  We study the generalized Dyson Brownian motion (GDBM) of an interacting
$N$-particle system with logarithmic Coulomb interaction and general potential
$V$. Under reasonable condition on $V$, we prove the existence and uniqueness
of strong solution to SDE for GDBM. We then prove that the family of the
empirical measures of GDBM is tight on $\mathcal
  {C}([0,T],\mathscr{P}(\mathbb{R}))$ and all the large $N$ limits satisfy a
nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and
Speicher, Carrillo, McCann and Villani, we prove that the McKean-Vlasov
equation is indeed the gradient flow of the Voiculescu free entropy on the
Wasserstein space of probability measures over $\mathbb{R}$. Using the optimal
transportation theory, we prove that if $V"\geq K$ for some constant $K\in
\mathbb{R}$, the McKean-Vlasov equation has a unique weak solution. This proves
the Law of Large Numbers and the propagation of chaos for the empirical
measures of GDBM. Finally, we prove the longtime convergence of the
McKean-Vlasov equation for $C^2$-convex potentials $V$.
