Consideration of a phase transition in a one-dimensional gas

Abstract

A phase transition in a one-dimensional gas is treated based on a linear-potential model. In this model, the molecules interact with each other with a long-range attractive potential plus a hard-sphere repulsive potential. Because of the simple potential, the phase-space integral can be evaluated rigorously in an elementary way. The role of the Van der Waals limit is discussed, and the Van der Waals equation is obtained for the pressure in this limit. When 1 > 8 aβ p, where a is the hard-sphere diameter, β = 1/ kT and p is the pressure, and with the Maxwell rule of equal areas, a first-order phase transition appears as in the case of Kac, Uhlenbeck and Hemmer. The scaling law and other details of the equation of state are discussed. Different from the exponential-potential model, the linear-potential model is applicable to a wider class of potentials that can be expanded in a Taylor series. In fact, it will be shown that the higher-order terms in the potential constant γ in the expansion of the Kac exponential potential do not contribute to the Van der Waals equation of state.