  We study the long-range asymptotic behavior for an out-of-equilibrium
countable one-dimensional system of Brownian particles interacting through
their rank-dependent drifts. Focusing on the semi-infinite case, where only the
leftmost particle gets a constant drift to the right, we derive and solve the
corresponding one- sided Stefan (free-boundary) equations. Via this solution we
explicitly determine the limiting particle-density profile as well as the
asymptotic trajectory of the leftmost particle. While doing so we further
establish stochastic domination and convergence to equilibrium results for the
vector of relative spacings among the leading particles.
