Quantum fluctuation in an inhomogeneous background and its influence on the phase transition in a finite volume system

We have studied the grand potential and phase transitions of an inhomogeneous finite volume spherical quark system. First the finite volume effects are considered by applying the multiple reflection expansion method which is an approximation for the density of states of the momentum in the grand potential of the finite size system. Then, the density of states of momentum is further modified by the thermal fluctuations in the thermal system with the inhomogeneous field background. The modification of the density of states is calculated by the scattering phase shift from the Dirac equation of quarks in the inhomogeneous field background. By the numerical calculation we show how the phase transition is changed by varying the configuration of the inhomogeneous field background and find that the strong first order phase transition could be weakened or even changed to a crossover as a result.