On the Hydrodynamic Equilibrium of aRod in a Lattice

We model the behavior of a big (Brazil) nut in a medium of smaller nuts with a stochastic asymmetric simple exclusion dynamics of a polymer-monomer lattice system. The polymer or`rod' can move up or down in an external negative eld, occupying N horizontal lattice sites where the monomers cannot enter. The monomers (at most one per site) or``uid particles' are moving symmetrically in the horizontal plane and asymmetrically in the vertical direction, also with a negative eld. For a xed position of the rod, this lattice uid is in equilibrium with a vertical height proole reversible for the monomers' motion. Upon`shaking' (speeding up the monomers) the motion of thèrod' dynamically decouples from that of the monomers resulting in a reversible random walk for the rod around an average height proportional to log N.