  We study the hydrodynamic limits of three kinds of one-dimensional stochastic
log-gases known as Dyson's Brownian motion model, its chiral version, and the
Bru-Wishart process studied in dynamical random matrix theory. We define the
measure-valued processes so that their Cauchy transforms solve the complex
Burgers-type equations. We show that applications of the method of
characteristic curves to these partial differential equations provide the
functional equations relating the Cauchy transforms of measures at an arbitrary
time with those at the initial time. We transform the functional equations for
the Cauchy transforms to those for the $R$-transforms and the $S$-transforms of
the measures, which play central roles in free probability theory. The obtained
functional equations for the $R$-transforms and the $S$-transforms are simpler
than those for the Cauchy transforms and useful for explicit calculations
including the computation of free cumulant sequences. Some of the results are
argued using the notion of free convolutions.
