Theory of Phase Transitions/Introduction

Thermodynamics tells us about the macroscopic properties of a system with a large number of particles while taking into account only a very small number of relevant quantities such as energy, temperature, volume, etc. However, as we know from many examples such as an entire lake freezing in cold weather, the macroscopic behavior can change dramatically when one of the relevant quantities changes only very slightly. If such a behavior is observed. we speak of a "phase transition", as the state of matter of the system (also called the "thermodynamic phase") has changed.

The investigation of phase transition is interesting mainly for two reasons. First of all, phase transitions are interesting from a fundamental point of view. A phase transition is a prime example for how systems with relatively simple dynamics on the microscopic level display highly complex behavior that emerges on much larger scales. As a result, concepts and methods first developed for the understanding of phase transitions have found applications in other areas such as particle physics, but even in fields beyond physics such as biology or economics. The other reason why phase transitions are interesting is that certain phases have important technological applications. For instance, a ferromagnet is an excellent permanent memory device used in many computer hard disks. Other examples include liquid crystals used in displays or smart glasses that change its transparency upon a change in temperature or when applying an external voltage.

What is a phase transition?
While we are familiar with the melting of ice from our daily lives, this is less so for the microscopic details of the underlying phase transition. To get a better understanding, it is helpful to first make some very general considerations. During many (but not all) phase transitions, the symmetry of the system will change. For example, a liquid exhibits translational invariance, i.e., the probability of find an atom at some point in space is the same everywhere. However, if the liquid is frozen into a crystalline solid, this is no longer the case, as a crystal has atoms only at certain fixed positions. In this case, we speak of a symmetry-breaking phase transition. Another important example is the ferromagnet-paramagnet transition at the Curie temperature of a magnetic material.

For such phase transitions, it is evident that the symmetry-broken phase exhibits a higher degree of order than the symmetric phase. We can quantify this order in terms of an order parameter, which is the magnetization $$m$$ for the ferromagnet-paramagnet transition. In the paramagnetic phase, $$m$$ is exactly zero, whereas it is finite in the ferromagnetic phase. While this last statement looks pretty harmless, it actually has pretty dramatic consequences for how a phase transitions works. Remarkably, an analytic function (i.e., representable by a power series) that is zero on a finite interval is zero everywhere. This immediately implies that the free energy of a system undergoing a phase transition cannot be analytic everywhere, as the order parameter is some derivative of the free energy. However, we can always find locally valid analytic functions for each of the phases, meaning that the analyticity is broken exactly at the phase transition. This brings us to a definition of a phase transition which actually goes beyond the notion of symmetry-breaking transitions: A phase transition occurs whenever the free energy exhibits nonanalytic behavior.

A consequence of this definition is that phase transitions can only occur in the thermodynamic limit of infinite system sizes. For finite systems, the partition function
 * $$Z = \sum\limits_i\exp(-\beta E_i)$$

of a system at inverse temperature $$\beta$$ with energy levels $$E_i$$ is always analytic, as the sum contains only a finite number of terms. Hence, the free energy $$F=-\beta^{-1}\log Z$$ is always analytic for finite systems.