  Weakly interacting Fermi gases exhibit rich collective dynamics in
spin-dependent potentials, arising from correlations between spin degrees of
freedom and conserved single atom energies, offering broad prospects for
simulating many-body quantum systems by engineering energy-space "lattices,"
with controlled energy landscapes and site to site interactions. Using quantum
degenerate clouds of $^6$Li, confined in a spin-dependent harmonic potential,
we measure complex, time-dependent spin-density profiles, varying on length
scales much smaller than the cloud size. We show that a one-dimensional mean
field model, without additional simplifying approximations, quantitatively
predicts the observed fine structure. We measure the magnetic fields where the
scattering lengths vanish for three different hyperfine state mixtures to
provide new constraints on the collisional (Feshbach) resonance parameters.
