Geometric approach to the phenomenological theory of phase transitions of the second kind

TL;DR

This paper introduces a geometric framework for the phenomenological theory of second-kind phase transitions, highlighting the limitations of traditional approaches and proposing higher-dimensional models to better capture critical phenomena.

Contribution

It develops a geometric approach using an extended space including local order parameters to describe non-analytical behaviors near critical points.

Findings

Zero field curve coincides with the critical point.

Higher-dimensional space allows modeling of anomalous specific heat increases.

Traditional Landau theory limitations are discussed.

Abstract

Geometrical approach to the phenomenological theory of phase transitions of the second kind at constant pressure $P$ and variable temperature $T$ is proposed. Equilibrium states of a system at zero external field and fixed $P$ and $T$ are described by points in three-dimensional space with coordinates $\eta$, the order parameter, $T$, the temperature and $\phi$, the thermodynamic potential. These points form the so-called zero field curve in the $(\eta, T, \phi)$ space. Its branch point coincides with the critical point of the system. The small parameter of the theory is the distance from the critical point along the zero-field curve. It is emphasized that no explicit functional dependency of $\phi$ on $\eta$ and $T$ is imposed. It is shown that using $(\eta, T, \phi)$ space one cannot overcome well-known difficulties of the Landau theory of phase transitions and describe non-analytical behavior of real systems in the vicinity of the critical point. This becomes possible only if one increases the dimensionality of the space, taking into account the dependency of the thermodynamic potential not only on $\eta$ and $T$, but also on near (local) order parameters $\lambda_{i}$. In this case under certain conditions it is possible to describe anomalous increase of the specific heat when the temperature of the system approaches the critical point from above as well as from below the critical temperature $T_{c}$.