  We study a longstanding problem in lattice QCD at low temperature and nonzero
quark chemical potential on an onset of the quark number density at
$\mu=m_\pi/2$. We introduce a physical parametrization of the eigenvalues in
the reduction formula of the fermion determinant. It is shown that the
parametrization reduces the quark number density operator to an expression with
the Fermi distribution of the quark. For each configuration, the eigenvalues of
the reduced matrix correspond to one-particle energy states of a quark. The gap
of the eigenspectrum of the reduced matrix corresponds to the gap of the energy
states, which causes the $\mu$-independence of the fermion determinant for
small $\mu$ at T=0. Once $\mu$ exceeds the gap, the quark number density
becomes nonzero for each configuration, which causes the early onset of the
quark number density.
