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When the density of the fluid surrounding suspended Brownian particles isappreciable, in addition to the forces appearing in the traditional Ornsteinand Uhlenbeck theory of Brownian motion, additional forces emerge as thedisplaced fluid in the vicinity of the randomly moving Brownian particle actsback on the particle giving rise to long-range force correlations whichmanifest as a ``long-time tail'' in the decay of the velocity autocorrelationfunction known as hydrodynamic memory. In this paper, after recognizing thatfor Brownian particles immersed in a Newtonian, viscous fluid, the hydrodynamicmemory term in the generalized Langevin equation is essentially the 1/2fractional derivative of the velocity of the Brownian particle, we present arheological analogue for Brownian motion with hydrodynamic memory whichconsists of a linear dashpot of a fractional Scott-Blair element and aninerter. The synthesis of the proposed mechanical network that is suggestedfrom the structure of the generalized Langevin equation simplifies appreciablythe calculations of the mean-square displacement and its time-derivatives whichcan also be expressed in terms of the two-parameter Mittag--Leffler function.

A Rheological Analogue for Brownian Motion with Hydrodynamic Memory