Phase transition for the system of finite volume in the ϕ4 theory in the Tsallis nonextensive statistics

Abstract

We studied the effects of nonextensivity on the phase transition for the system of finite volume [Formula: see text] in the [Formula: see text] theory in the Tsallis nonextensive statistics of entropic parameter [Formula: see text] and temperature [Formula: see text], when the deviation from the Boltzmann–Gibbs (BG) statistics, [Formula: see text], is small. We calculated the condensate and the effective mass to the order [Formula: see text] with the normalized [Formula: see text]-expectation value under the free particle approximation with zero bare mass. The following facts were found. The condensate [Formula: see text] divided by [Formula: see text], [Formula: see text], at [Formula: see text] ([Formula: see text] is the value of the condensate at [Formula: see text]) is smaller than that at [Formula: see text] for [Formula: see text] as a function of [Formula: see text] which is the physical temperature [Formula: see text] divided by [Formula: see text]. The physical temperature [Formula: see text] is related to the variation of the Tsallis entropy and the variation of the internal energies, and [Formula: see text] at [Formula: see text] coincides with [Formula: see text]. The effective mass decreases, reaches minimum, and increases after that, as [Formula: see text] increases. The effective mass at [Formula: see text] is lighter than the effective mass at [Formula: see text] at low physical temperature and heavier than the effective mass at [Formula: see text] at high physical temperature. The effects of the nonextensivity on the physical quantity as a function of [Formula: see text] become strong as [Formula: see text] increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.