  We consider a system of diffusing particles on the real line in a quadratic
external potential and with repulsive electrostatic interaction. The empirical
measure process is known to converge weakly to a deterministic measure-valued
process as the number of particles tends to infinity. Provided the initial
fluctuations are small, the rescaled linear statistics of the empirical measure
process converge in distribution to a Gaussian limit for sufficiently smooth
test functions. We derive explicit general formulae for the mean and covariance
in this central limit theorem by analyzing a partial differential equation
characterizing the limiting fluctuations.
