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Abstract

In supernovae neutrinos are emitted from a region with a width $r_{\rm eff}$
of a few kilometers (rather than from a surface of infinitesimal width). We
study the effect of integration (averaging) over such an extended emission
region on collective oscillations. The averaging leads to additional
suppression of the correlation (off-diagonal element of the density matrix) by
a factor $ \sim 1/r_{\rm eff} V_e \sim 10^{-10}$, where $V_e$ is the matter
potential. This factor enters the initial condition for further collective
oscillations and, consequently, leads to a delay of the strong flavour
transitions. We justify and quantify this picture using a simple example of
collective effects in two intersecting fluxes. We have derived the evolution
equation for the density matrix elements integrated over the emission region
and solved it both numerically and analytically. For the analytic solution we
have used linearized equations. We show that the delay of the development of
the instability and the collective oscillations depends on the suppression
factor due to the averaging (integration) logarithmically. If the instability
develops inside the production region, the integration leads not only to a
delay but also to a modification of the exponential grow.