Phenomenology of first-order phase transitions

A phenomenological description of first-order phase transitions is proposed in terms of the solution of a certain linear, first-order, partial differential equation suggested by Fisher. In the asymptotic region of the critical point, the coefficient functions of the equation are uniquely determined by the scaling laws. The imposition of the further requirement that the phase boundary coincide with a characteristic curve of this equation is shown to lead to solutions that can reproduce the singular behavior—for example, the density and entropy discontinuities for the case of the liquid-gas transition—along the entire vapor-pressure curve. The important role played by the characteristic curves is stressed and confirmed for the special case of the ideal, $d$-dimensional, Bose gas. New relations involving discontinuities in the second derivatives of the thermodynamic functions are derived and shown to be useful in obtaining the coefficient functions near the phase boundary. Details are presented for the case of ferromagnetic and ferroelectric systems and include a comparison with experiment.