  We perform systematic simulation experiments on model systems with
soft-sphere repulsive interactions to test the predicted dynamic equivalence
between soft-sphere liquids with similar static structure. For this we compare
the simulated dynamics (mean squared displacement, intermediate scattering
function, {\alpha}-relaxation time, etc.) of different soft-sphere systems,
between them and with the hard-sphere liquid. We then show that the referred
dynamic equivalence does not depend on the (Newtonian or Brownian) nature of
the microscopic laws of motion of the constituent particles, and hence, applies
independently to colloidal and to atomic simple liquids. In addition, we verify
another more recently-proposed dynamic equivalence, this time between the
long-time dynamics of a Brownian fluid and its corresponding atomic liquid
(i.e., the atomic system with the same interaction potential).
