[1] Tian Ma and Shouhong Wang, Phase Transition Dynamics, Springer, pp 555, 2013

This book synthesizes the mathematical and physical theories for a phase transition phenomena, established by the authors. This blog post intends to give a brief introduction to the theories developed in this book, and to point out some main differences between the transition theory and the classical bifurcation theory.

**1. Mathematical theory**

The mathematical theory is also called dynamic transition theory, and we summarize hereafter the main ingredients of the theory.

First, we have proved a general transition classification theorem, which states that all dynamic transitions are classified into three categories: the Type-I, Type-II and Type-III.

Second, one important tool of the dynamic transition theory is the central manifold reduction, and we have derived for the first time approximate central manifold reduction formulas, which are crucial for many applications of the theory to physical problems. Also, these formulas has been generalized to random dynamical systems by Chekroun et al.

Third, we have systematically developed theorems and criteria for different transition types. For example, for the first time, we introduced the concept of attractor bifurcation, and proved a general attractor bifurcation theorem, which can be used to handle most Type-I transitions. A sequence of theorems are established also for the Type-II, and Type-III transitions. These theorems are easy to use in applications and are crucial in deriving the needed physical theory for the related physical problems.

**2. Physical theory**

Our physical theory of transition dynamics involves a wide range of scientific fields, including statistical physics, fluid dynamics, atmospheric and ocean physics, and biology and chemistry. Hereafter we present a few physical theory we have derived, and we refer interesting readers to the book [1] for more details and for other physical applications.

*Principle of phase transition dynamics:* We discovered a general principle of phase transitions for dissipative physical systems, which we call principle of phase transition dynamics. Namely, phase transitions of all dissipative physical systems are classified into three categories: continuous, catastrophic, and random.

*General dynamic model for equilibrium phase transitions:* We introduced a unified dynamical model for equilibrium phase transitions, based on the Le Chatelier principle and the Ginzburg-Landau mean field theory.

*Discovery of third-order phase transition*: It is well-known that the gas-liquid coexistence curve terminates at a critical point, also called the Andrews critical point, and gas-liquid transition is of first order before the critical point and of the second-order at the critical point. Going beyond the critical point, physical phenomena indicates that a high-order phase should occur. However, it is a longstanding open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. For the first time, 1) we derived the gas-liquid co-existence curve beyond the Andrews critical point, and 2) we show that the transition is *first order* before the critical point, *second-order* at the critical point, and *third order* beyond the Andrews critical point. This gives rise to the mechanism of the Andrews critical point, and the reason why it is hard to observe the gas-liquid phase transition beyond the Andrews critical point.

*Prediction of a new superfluid phase in liquid helium-3:* We have derived new dynamical models for liquid helium-3, helium-4 and their mixture, leading to various physical predictions, such as the existence of a new phase for helium-3. Although these predictions need yet to be verified experimentally, they certainly offer new insights to both theoretical and experimental studies for a better understanding of the underlying physical problems.

*New mechanism of El Nino Southern Oscillation (ENSO)*: We discovered a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled oscillatory processes: 1) the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and 2) the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO.

**3. Differences between the dynamic transition theory and classical bifurcation theory**

It is important to emphasize the main differences between the dynamical transition theory and the classical bifurcation theory.

First the key difference is that the transition states derived in our dynamic transition theory are physical, and the bifurcation states derived from the classical bifurcation theory may not be physical.

In fact, the classical bifurcation theory first seeks bifurcation solutions and then decides the stability of the bifurcated solutions. The main drawback for this approach is that there is no way to know if the bifurcated solutions represent all transition physical states. In addition, it is always technically difficult to derive the stability of the bifurcated solution.

Instead, our dynamic transition theory finds all physical phase transition states.

Second, our theory indicates that transition always happen at the critical point, but bifurcation may not occur.

Third, it is clear that the general physical principle for phase transitions can only be discovered by using the dynamic transition theory.

**4. Summary**

The dynamic transition theory can be viewed as a true mathematical representation of a physical theory. The general principle of phase transition dynamics clearly offers guidance to the understanding of dissipative physical systems.