  When the density of the fluid surrounding suspended Brownian particles is
appreciable, in addition to the forces appearing in the traditional Ornstein
and Uhlenbeck theory of Brownian motion, additional forces emerge as the
displaced fluid in the vicinity of the randomly moving Brownian particle acts
back on the particle giving rise to long-range force correlations which
manifest as a ``long-time tail'' in the decay of the velocity autocorrelation
function known as hydrodynamic memory. In this paper, after recognizing that
for Brownian particles immersed in a Newtonian, viscous fluid, the hydrodynamic
memory term in the generalized Langevin equation is essentially the 1/2
fractional derivative of the velocity of the Brownian particle, we present a
rheological analogue for Brownian motion with hydrodynamic memory which
consists of a linear dashpot of a fractional Scott-Blair element and an
inerter. The synthesis of the proposed mechanical network that is suggested
from the structure of the generalized Langevin equation simplifies appreciably
the calculations of the mean-square displacement and its time-derivatives which
can also be expressed in terms of the two-parameter Mittag--Leffler function.
