  We study equilibrium density and spin density profiles for a model of cold
one-dimensional spin 1/2 fermions interacting via inverse square interaction
and exchange in an external harmonic trap. This model is the well-known
spin-Calogero model (sCM) and its fully nonlinear collective field theory
description is known. We extend the field theory description to the presence of
an external harmonic trap and obtain analytic results for statics and dynamics
of the system. For instance, we find how the equilibrium density profile
changes upon tuning the interaction strength. The results we obtain for
equilibrium configurations are very similar to the ones obtained recently by Ma
and Yang [1] for a model of fermions with short ranged interactions. Our main
approximation is the neglect of the terms of higher order in spatial
derivatives in equations of motion - gradientless approximation [2]. Within
this approximation the hydrodynamic equations of motion can be written as a set
of decoupled forced Riemann-Hopf equations for the dressed Fermi momenta of the
model. This enables us to write analytical solutions for the dynamics of spin
and charge. We describe the time evolution of the charge density when an
initial non-equilibrium profile is created by cooling the gas with an
additional potential in place and then suddenly removing the potential. We
present our results as a simple "single-particle" evolution in the phase-space
reminiscing a similar description of the dynamics of non-interacting
one-dimensional fermions.
