  We study in this article the hydrodynamic limit in the macroscopic regime of
the coupled system of stochastic differential equations, \begin{equation}
d\lambda_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(\lambda_t^i) dt+ \frac{\beta}{2N}
\sum_{j\not=i} \frac{dt}{\lambda^i_t-\lambda^j_t}, \qquad i=1,\ldots,N,
\end{equation} with $\beta>1$, sometimes called generalized Dyson's Brownian
motion, describing the dissipative dynamics of a log-gas of $N$ equal charges
with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently
regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a
mean-field Mac-Kean-Vlasov equation. We prove that, for suitable initial
conditions, fluctuations around the limit are Gaussian and satisfy an explicit
PDE.
  The proof is very much indebted to the harmonic potential case treated in
Israelsson \cite{Isr}. Our key argument consists in showing that the
time-evolution generator may be written in the form of a transport operator on
the upper half-plane, plus a bounded non-local operator interpreted in terms of
a signed jump process. As an essential technical argument ensuring the
convergence of the above scheme, we give an $N$-independent large-deviation
type estimate for the probability that $\sup_{t\in[0,T]}\max_{i=1,\ldots,N}
|\lambda_t^i|$ is large, based on a multi-scale argument and entropic bounds.
