  We calculate various thermodynamic quantities of vortex liquids in a layered
superconductor by using the nonperturbative parquet approximation method, which
was previously used to study the effect of thermal fluctuations in
two-dimensional vortex systems. We find there is a first-order transition
between two vortex liquid phases which differ in the magnitude of their
correlation lengths. As the coupling between the layers increases,the
first-order transition line ends at a critical point. We discuss the possible
relation between this critical end-point and the disappearance of the
first-order transition which is observed in experiments on high temperature
superconductors at low magnetic fields.
