  Chiral pair fluctuation are considered near the phase boundary of the
inhomogeneous chiral phase (iCP). The fluctuations are then bosonized and an
effective action for the chiral pair fluctuation is basically constructed by
considering the ring diagram of the polarization function. We can evaluate the
self-energy and effective four point interaction among fluctuations in a
consistent way. The peculiar dispersion of the fluctuation, reflecting the
spatially inhomogeneous transition, gives rise to interesting and qualitative
results. Thermal fluctuations prohibit the second-order transition, while the
effect of the quantum fluctuations is rather modest. Quantum and thermal
fluctuations changes the second-order transition to the first one by changing
the sign of the effective four-point interaction between effective mesons.
These features may be observed by relativistic heavy-ion collisions through the
analysis of the thermodynamic observables. Distinct from the second-order phase
transition, the first moment such as entropy production exhibits an anomalous
behavior due to fluctuations, which is one of the signals of the phase
transition to iCP. Some similar aspects are also remarked between iCP and the
FFLO state in superconductivity.
