
 ~~Variable~~ & ~~Meaning~~ & ~~Value~~ \\
 \hline
$\varrho$ & Density of the medium & $10^{12}$~g/cm$^3$ \\
$T$ & Temperature of the medium & 7 MeV \\
$\mu_e$ & Electron chemical potential & 20 MeV \\
$Y_e$ & Electron fraction & 0.13 \\
$E_\nu$ & Neutrino energy & 20 MeV \\
$\delta m^2$ & Mass-squared splitting & $2.4 \times 10^{-3}$~eV$^2$ \\
$\theta$ & Matter-suppressed mixing angle & $10^{-6}$ \\
$\omega$ & Vacuum oscillation frequency & 0.3 km$^{-1}$ \\
$n_{\nu_e} (0)$ & Initial number density of $\nu_e$ & $3.0 \times 10^{33}$~cm$^{-3}$ \\
$n_{\bar{\nu}_e} (0)$ & Initial number density of $\bar{\nu}_e$ & $2.5 \times 10^{33}$~cm$^{-3}$ \\
$n_{\nu_x} (0)$ & Initial number density of $\nu_x$ & $1.0 \times 10^{33}$~cm$^{-3}$ \\
$\mu | \mathbf{D}_0 (0) |$ & Neutrino--neutrino forward-scattering potential & $3 \times 10^5$ km$^{-1}$ \\ 
$\Gamma$ & Neutrino scattering rate & 26 km$^{-1}$ \\ 
$\bar{\Gamma}$ & Antineutrino scattering rate & 2.6 km$^{-1}$ \\ 
 \hline
\end{tabular}
\caption{Parameters adopted for the numerical calculations. The number density of $\bar{\nu}_x$ is taken to be the same as that of $\nu_x$. The strength of the initial-time neutrino--neutrino forward-scattering potential is presented as $\mu | \mathbf{D}_0 (0)|$, $\mu$ is the usual self-coupling parameter and $\mathbf{D}_0 (0)$ is the the $l=0$ difference vector at $t = 0$ [Eq.~\eqref{eq:D0}]. Because of the initial symmetry between $\nu_x$ and $\bar{\nu}_x$, the forward-scattering potential is equal to $\sqrt{2} G_F [ n_{\nu_e} (0) - n_{\bar{\nu}_e} (0) ]$. See the main text and Fig.~\ref{ang_distrs} for specification of the angular distributions. \label{paramtab}}
\end{table*}





\begin{figure}
\begin{subfigure}{
\centering
\includegraphics[width=.42\textwidth]{ang_distrs_noFFC.pdf}
}
\end{subfigure}

\begin{subfigure}{
\centering
\includegraphics[width=.42\textwidth]{ang_distrs.pdf}
}
\end{subfigure}
\caption{Neutrino angular distributions used in this study: $n_{\nu_e, v}$ (solid), $n_{\bar{\nu}_e, v}$ (dashed), $n_{\nu_x, v} = n_{\bar{\nu}_x, v}$ (dotted), where $v = \cos\theta$ and $\theta$ is the neutrino propagation angle. The lower panel shows the same distributions used in Ref.~\cite{johns2022}. The distributions in the upper panel are identical but with the $l = 2$ Legendre moments zeroed out. Although the two panels look very similar and both contain angular crossings, only the lower panel is unstable to fast flavor conversion \cite{johns2020}. \label{ang_distrs}}
\end{figure}




In order to observe collisionally unstable evolution both with and without FFC, we compare two sets of angular distributions. In the case where the system is unstable to collisional and fast instabilities, the angular distributions are taken to be
\begin{equation}
n_{\nu_\alpha, v} = \frac{1}{2} n_{\nu_\alpha, 0} L_0 (v) + \frac{3}{2} n_{\nu_\alpha, 1} L_1 (v) + \frac{5}{2} n_{\nu_\alpha, 2} L_2 (v), \label{eq:ang_ffc}
\end{equation}
with $n_{\nu_\alpha, l}$ the $l$th Legendre moment of the angular distribution of flavor $\alpha$, $L_l (v)$ the $l$th Legendre polynomial, and $n_{\nu_\alpha, 0} = n_{\nu_\alpha}$ as specified above. For $\nu_e$ we use the number-density moments
\begin{align}
&n_{\nu_e, 1} = 0.050 n_{\nu_e} \notag \\
&n_{\nu_e, 2} = 0.002 n_{\nu_e},
\end{align}
for $\bar{\nu}_e$ we use
\begin{align}
&n_{\bar{\nu}_e, 1} = 0.100 n_{\bar{\nu}_e} \notag \\
&n_{\bar{\nu}_e, 2} = 0.008 n_{\bar{\nu}_e}, 
\end{align}
and for $\nu_x$ we use
\begin{align}
&n_{\nu_x, 1} = 0.150 n_{\nu_x} \notag \\
&n_{\nu_x, 2} = 0.017 n_{\nu_x}.
\end{align}
The $\bar{\nu}_x$ distribution is the same as that of $\nu_x$. All higher moments are set to zero. In the other case, where the system is stable to fast oscillations, we use
\begin{equation}
n_{\nu_\alpha, v} = \frac{1}{2} n_{\nu_\alpha, 0} L_0 (v) + \frac{3}{2} n_{\nu_\alpha, 1} L_1 (v), \label{eq:ang_noffc}
\end{equation}
which is equivalent to dropping the $l=2$ moments, keeping the other moments the same. The distributions are plotted in Fig.~\ref{ang_distrs}. Although zeroing out the $l=2$ moments only changes the distributions slightly, and does not change the presence of an angular crossing, in a homogeneous setting it makes the difference between being fast-stable and fast-unstable \cite{johns2020}. This is related to the fact that homogeneity isolates the $K = 0$ Fourier mode of $\rho$---all $K \neq 0$ being forced to vanish by symmetry---and limits the range of fast instabilities that can be expressed by the system.

Following Ref.~\cite{johns2022}, we consider three different collisional prescriptions, implementing them one at a time and comparing the results. The first is
\begin{equation}
\textbf{\textrm{NC (isotropizing):}} ~~~ i\mathcal{C} =  \Gamma \left( \rho_\textrm{NC} - \rho \right),
\end{equation}
where
\begin{equation}
\rho_\textrm{NC} = \langle \rho \rangle = 
\begin{pmatrix}
\langle \rho \rangle_{ee} & \langle \rho \rangle_{ex} \\
\langle \rho \rangle_{xe} & \langle \rho \rangle_{xx}
\end{pmatrix}
\end{equation}
and
\begin{equation}
\langle \rho \rangle = \frac{1}{2} \int_{-1}^{+1} dv \rho_v.
\end{equation}
This is a simplified way of describing, for example, neutrino--nucleon scattering. The second prescription is
\begin{equation}
\textbf{\textrm{CC (isotropizing):}} ~~~ i\mathcal{C} =  \Gamma \left( \rho_\textrm{CC} - \rho \right),
\end{equation}
where
\begin{equation}
\rho_\textrm{CC} = 
\begin{pmatrix}
\langle \rho \rangle_{ee} & 0 \\
0 & \langle \rho \rangle_{xx}
\end{pmatrix}.
\end{equation}
This collisional form approximates the effect of CC neutrino--electron scattering. The third prescription is
\begin{equation}
\textbf{\textrm{CC (non-isotropizing):}} ~~~ i\mathcal{C} =  - \Gamma \rho_T,
\end{equation}
where
\begin{equation}
\rho_T = \begin{pmatrix}
0 & \rho_{ex} \\
\rho_{xe} & 0
\end{pmatrix}.
\end{equation}
This form is not meant to describe any particular process, but rather to isolate the effect of collisional decoherence on the flavor evolution.

By comparing \textit{NC (isotropizing)} and \textit{CC (isotropizing)}, we establish whether the