We study the fully nonlinear fast flavor evolution of neutrinos in 1 + 1 dimensions. Our numerical analysis shows that at late times, the system reaches an approximately steady state. Using the steady-state approximation, we analytically show that the spatial variation of the polarization vectors is given by their precession around a common axis, which itself has a motion reminiscent of a gyroscopic pendulum. We then show that the steady-state solution to the equations of motion cannot be separated in position and velocity—that is, the motion is not collective in the usual sense. However, the fast evolution allows spectral-swap-like dynamics leading to partial decoherence over a range of velocities, constrained by the conservation of lepton number(s). Finally, we numerically show that at late times, the transverse components of the polarization vectors become randomly oriented at different spatial locations for any velocity mode and lepton asymmetry.
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