In the first part to this article we have covered the basics of phase transitions. Let us continue the discussion here.
After reviewing some general classification of phases and transitions, it is time to turn to models. Let us first discuss why it makes sense to look at models.
In a model a physical system is broken down using a crude description that neglects most of the messy details. For example a solid might become an infinitely large collection of atoms on a regular lattice that only interact with their nearest neighbour. One thus neglects long-range interactions, lattice defects, etc. A gas might be described by a large number of hard balls in a box and one often assumes their density is low and they only interact through collisions (i.e. the gas is ideal). There are many more examples and in light of our knowledge of the details of any such system it is valid to raise the question of why crude models even yield any insight at all. Should the many details we ignored not change just about anything?
A short answer would be the following: Since we are concerned with the systems behaviour on large scales (roughly the size of the entire system which is much larger than the individual atoms) many of the details “average out”. Most microscopic details of a system become irrelevant on large scales - this is one of the key takeaways from the Renormalisation Group which provides a more detailed answer. Without going into detail, it often suffices to regard the simplest interactions on a microscopic level to understand the general structure of a systems phase diagram. In a sense one has to leave out the right details.
One of the most common examples of a phase transition model is the (ferromagnetic) Ising model. It serves as a starting point to describe real magnetism since it is simple yet shows a transition between a ferromagnetic state at low temperatures and a paramagnetic state at high temperatures.
Let us start with the simple case of spins1
The system’s Hamiltonian in an external magnetic field
where the coupling
and the free energy by
as it couples to the thermodynamic energy by
It is easy to see that at temperature
Since
The two ground states for
The qualitative dependence of M on T with no external field B = 0. We have argued that M = ± 1 at T = 0 and it turns out the phase transition persists for positive T < TC in dimensions d > 1. Which of the legs of the graph the sytem chooses depends on boundary conditions or the way in which B → 0. Either way below TC the system shows a net magnetisation M ≠ 0 which flips sign as B does. The qualitative function here is derived in the Mean Field approximation. Taken from Tong, David (2017). Statistical Field Theory. https://www.damtp.cam.ac.uk/user/tong/sft.html.
The phase diagram for d > 1-dim. Ising Model. For transitions from B > 0 to B < 0 and vice versa there is a phase transition below the critical temperature TC. As one goes from T > TC to T < TC at B = 0 the system will spontaneously choose a ground state with positive or negative m. The point at B = 0, T = TC is a second order phase transition. From Tong, David (2017). Statistical Field Theory. https://www.damtp.cam.ac.uk/user/tong/sft.html.
Let us take a closer look at the liquid-gas transition for e.g. water. We focus on the region of the phase diagram close to the critical point.
There is an obvious similarity with the Ising model’s phase diagram if we replace
Let us assume the same lattice as for the Ising model to describe the possible positions of gas atoms/molecules. Assuming discrete spatial positions seems a crude approximation but with a lattice spacing that is much smaller than the observational scales (so
This is just the Ising model Hamiltonian (restricted to nearest neighbour-interactions) with the identification
We have established that many liquid-gas transitions behave very similarly to the Ising model and both types of systems undergo a
In the liquid-gas system light passing through is absorbed and refracted differently depending on its wavelength. Very shortly speaking only light with wavelengths that roughly correspond to length scales in the system (particle size, size of fluctuations in the density distribution, ...) can be scattered. If at the critical point density fluctuations on all length scales (up to the size of the sample) exist then light of all colours will be scattered and the system becomes opaque.
Below
Example for an experimental realisation of critical opalescence. Some see-through liquid at T < TC (left) is heated to the critical temperature TC (middle) and becomes opalescent. It is then heated above TC (right) and the liquid and gaseous phase start to separate. Both of those are see-through again. Cropped from M. Buklesk, V. Petruševski on www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0187-893X2019000100111 (accessed 2023-12-03) under CC BY-NC-ND 4.0 Deed.
The phase transitions we have discussed so far are referred to as thermal phase transitions. We have neglected quantum fluctuations completely. This is reasonable since at positive temperature they are heavily suppressed relative to thermal fluctuations. While it is not theoretically impossible to find quantum fluctuations on small scales their effect on long-range order (that is created or destroyed by a phase transition) vanishes in the thermodynamic limit
There is however a class of transitions at
We mentioned why phase transitions are important from both practical and theoretical considerations and discussed the Ehrenfest and Landau classifications. Both are sensible approaches to classifying phase transitions that also come with problems. We have discussed the liquid-gas transition in water and the Ising model phase transition in detail. We have also hinted at how microscopic behaviour drives macroscopic, unexpected behaviour and how phase transitions connect to Quantum Field Theories. There is much more to phase transitions than water and the Ising model. Let us mention as one area of current research the exact structure of the phase diagram of Quantum Chronodynamics (QCD)-matter. Let us close on the following remark: Despite being motivated by everyday experience, our investigation of phase transitions quickly leads to deep theoretical insights and connects many areas of physics.