Phase-transition temperature in finite systems.

Abstract

Pathria's approach has been used to deal with the properties of the finite cubic system of $^{4}\mathrm{He}$ under constant pressure. The analytic expressions for the total number of particles N and the total pressure p near the critical point are obtained for mixture, antiperiodic, Neumann, and periodic boundary conditions. Influences of various boundary conditions upon low-temperature and critical characters of finite systems are discussed. For five boundary conditions, the relationship between the superfluid transition temperature T' and the size of the finite system ${\mathit{L}}_{0}$ has been obtained. The results of this paper can be used in the case of superconductivity. Besides, from this work and others, we have obtained the formula for the phase-transition temperature T' in a finite system, t=\ensuremath{\sigma}${\mathit{L}}_{0}^{\mathrm{\ensuremath{-}}\mathit{b}}$; here b is described as the finiteness constant, and \ensuremath{\sigma} is determined by the boundary conditions, the properties of the system, and the interaction between the system and the walls of the container.