## Dynamical Theory of Thermodynamical Phase Transitions

Tian Ma & Shouhong Wang, Dynamical Theory of Thermodynamical Phase Transitions, Preprint, July 14, 2017

This blog post is on the above paper. The goals of this paper, as well as all other phase transition theories, are

1. to determine the definition and types of phase transitions,
2. to derive the critical parameters,
3. to obtain the critical exponents, and
4. to understand the mechanism and properties of phase transitions, such as supercooled and superheated states and the Andrews critical points, etc.

There are two routes for studying phase transitions: the first is Landau’s approach using thermodynamic potentials, and the second is a microscopic approach using statistical theory. These routes are complementing to each other. Our dynamical transition theory follows the Landau’s route, but is not a mean field theory approach. It is based, however, on the thermodynamical potential.

1. For a thermodynamic system, there are three different levels of physical quantities: control parameters ${\lambda}$, the order parameters (state functions) ${u=(u_1, \cdots, u_N)}$, and the thermodynamic potential ${F}$. These are well-defined quantities, fully describing the system. The potential is a functional of the order parameters, and is used to represent the thermodynamic state of the system.

In a recent paper [Tian Ma & Shouhong Wang, Dynamic Law of Physical Motion and Potential-Descending Principle, Indiana University, The Institute for Scientific Computing and Applied Mathematics Preprint \#1701] and a related previous blog, we postulated a potential-descending principle (PDP) for statistical physics, which give rise to a dynamic equation of thermodynamical systems: Also importantly, based on PDP, the dynamic equations of a thermodynamic system in non-equilibrium state take the form

$\displaystyle \frac{\text{d} u }{\text{d} t } = - A \delta F(u, \lambda), \ \ \ \ \ (1)$

which leads to an equation for the deviation order parameter ${u}$ from the basic equilibrium state ${\bar u}$:

$\displaystyle \frac{du}{dt} = L_\lambda u +G(u,\lambda), \ \ \ \ \ (2)$

where ${L_\lambda}$ is a linear operator, and ${G(u,\lambda) }$ is the nonlinear operator.

We obtain in this paper three theorems, providing a full theoretical characterization of thermodynamic phase transitions.

2. The first theorem states that as soon as the linear instability occurs, the system always undergoes a dynamical transition, to one of the three types: continuous, catastrophic and random. This theorem offers the detailed information for the phase diagram and the critical threshold ${\lambda_c}$ in the control parameter ${\lambda=(\lambda_1, \cdots, \lambda_N) \in \mathbb R^N}$. They are precisely determined by the following equation:

$\displaystyle \beta_1(\lambda_1, \cdots, \lambda_N)=0. \ \ \ \ \ (3)$

Here ${\beta_1}$ is the first eigenvalue of the linear operator ${L_\lambda}$ given by

$\displaystyle L_\lambda w \equiv -\left(\frac{\partial^2F(\bar u; \lambda)}{\partial u_i, \partial u_j}\right) w = \beta_1(\lambda) w, \ \ \ \ \ (4)$

where ${F}$ is the potential functional.

3. The second theorem states that there are only first, second and third-order thermodynamic transitions; and establishes a corresponding relationship between the Ehrenfest classification and the dynamical classification. The Ehrenfest classification offers clear experimental quantities to determine the types of phase transition, while the dynamical classification provides a full theoretical characterization of the transition.

4. The last theorem states that both catastrophic and random transitions lead to saddle-node bifurcations, and both latent heat, superheated and supercooled states always accompany the saddle-node bifurcation associated with the first order transitions.

5. We emphasize that the three theorems lead to three important diagrams: the phase diagram, the transition diagram and the dynamical diagram. These diagrams appear only to be derivable by the dynamical transition theory presented in this paper. In addition, our theory achieves the fourth goal of phase transition theories as stated in the beginning of this blog post, which is hardly achievable by other existing theories.

Tian Ma & Shouhong Wang