  We study interacting systems of linear Brownian motions whose drift vector at
every time point is determined by the relative ranks of the coordinate
processes at that time. Our main objective has been to study the long range
behavior of the spacings between the Brownian motions arranged in increasing
order. For finitely many Brownian motions interacting in this manner, we
characterize drifts for which the family of laws of the vector of spacings is
tight, and show its convergence to a unique stationary joint distribution given
by independent exponential distributions with varying means. We also study one
particular countably infinite system, where only the minimum Brownian particle
gets a constant upward drift, and prove that independent and identically
distributed exponential spacings remain stationary under the dynamics of such a
process. Some related conjectures in this direction have also been discussed.
