  We obtain rates of convergence to stationarity in L^1-Wasserstein distance
for a d-dimensional reflected Brownian motion (RBM) in the nonnegative orthant
that are explicit in the dimension and the system parameters. The results are
then applied to a class of RBMs considered in Blanchet-Chen (2016) and to
rank-based diffusions including the Atlas model. In both cases, we obtain
explicit rates and bounds on relaxation times. In the first case we improve the
relaxation time estimates of O(d^4(log d)^2) obtained in Blanchet-Chen (2016)
to O((log d)^2). In the latter case, we give the first results on explicit
parameter and dimension dependent rates under the Wasserstein distance. The
proofs do not require an explicit form for the stationary measure or
reversibility of the process with respect to this measure, and cover settings
where these properties are not available. In the special case of the standard
Atlas model, we obtain a bound on the relaxation time of O(d^6(log d)^2).
