  We present a microscopic modeling for a decay of a heavy compound nucleus,
starting from a nucleonic degree of freedom. To this end, we develop an
approach based on a non-equilibrium Green's function, which is combined with a
configuration interaction (CI) approach based on a constrained
density-functional theory (DFT). We apply this approach to a barrier-top
fission of $^{236}$U, restricting the model space to seniority zero
configurations of neutrons and protons. We particularly focus on the
distribution of the fission probability. We find that it approximately follows
the chi-squared distribution with the number of degrees of freedom $\nu$ of the
order of 1, which is consistent with the experimental finding. We also show
that $\nu$ corresponds to the number of eigenstates of the many-body
Hamiltonian whose energy is close to the excitation energy of the system and at
the same time which have significant components on both sides of a fission
barrier.
