Phase transition behavior of a greater membrane model

Abstract

The electrical dipole model that has been proposed by Grodsky to describe the properties of the greater membrane of a nerve cell is investigated. The model Hamiltonian is derived here. It is shown that a generalization of Grodsky's model is necessary for physical reasons. The mathematical structure of the final Hamiltonian resembles that of a nonlinearly coupled spin-phonon system. On the basis of the Bogoliubov inequality, a mean field theory is developed in order to study the phase transition behavior of the model system. The mean field equations determining the macroscopic order parameters are solved under the approximation of weak phonon renormalization. Zero absolute temperature (T = 0) is considered first. Depending on the values of the parameters involved, one finds two situations which possess different types of phase transitions. The solutions of the mean field equations are then treated for finite temperatures. A detailed examination of the thermodynamic stability of these solutions enables the determination of the complete phase diagram of the model system. The results of this analysis show that the antiferroelectric phase is not thermodynamically stable whereas Grodsky, using an approximate zeroth-order Hamiltonian, was led to the conclusion that such a phase should exist. Finally, the biological relevance of the model is discussed.