  Using the test-particle method to solve the transport equation derived from
the Nambu-Jona-Lasino (NJL) model, we study how phase separation occurs in an
expanding quark matter like that in a heavy ion collision. To test our method,
we first investigate the growth rates of unstable modes of quark matter in a
static cubic box and find them to agree with the analytical results that were
previously obtained using the linear response theory. In this case, we also
find the higher-order scaled density moments to increase with time and saturate
at values significantly larger than one, which corresponds to a uniform density
distribution, after the phase separation. The skewness of the quark number
event-by-event distribution in a small sub-volume of the system is also found
to increase, but this feature disappears if the sub-volume is large. For the
expanding quark matter, two cases are considered with one using a blast-wave
model for the initial conditions and the other using initial conditions from a
mulple-phase transport (AMPT) model. In both cases, we find the expansion of
the quark matter is slowed down by the presence of a first-order phase
transition. Also, density clumps appear in the system and the momentum
distribution of partons becomes anisotropic , which can be characterized by
large scaled density moments and non-vanishing anisotropic elliptic and
quadrupolar flows, respectively. The large density fluctuations further lead to
an enhancement in the dilepton yield. In the case with the AMPT initial
conditions, the presence of a first-order phase transition also results in a
narrower rapidity distribution of partons after their freeze out. These effects
of density fluctuations can be regarded as possible signals for a first-order
phase transition that occurs in the baryon-rich quark matter formed in
relativistic heavy ion collisions.
