# Why are phase transitions discrete?

It seems like there is no "in-between" for the phases of matter; it can be "solid" or "liquid", but what about the in-between?

Why is there no spectrum of matter between the phases (e.g. a range of states between fully liquid and fully solid)? Why are the phase transitions discrete?

• First order phase transitions are. Second order phase transitions are discrete. – Jon Custer Jul 20 '16 at 3:45
• Phase transitions are very complicated. The discrete nature only appears in sufficiently large systems and is even then very hard to measure with precision. Theoretically the sharp transition only occurs for infinite system size. At the transition point fluctuation will have very long correlation lengths and start interacting with the system boundaries. – CuriousOne Jul 20 '16 at 4:24
• @JonCuster what are first order transitions? Your sentence is incomplete – domj33 Apr 27 '17 at 13:24
• Actually, I'm a little confused about the question as modified by the bounty. Right at a second-order phase transition, the magnetization (or whatever order parameter you like) goes from $M=0$ to $M=0 + \epsilon$. In what sense is this 'discrete,' and what concievable behavior could go 'in between' these two points? – Rococo Apr 27 '17 at 15:31
• @Rococo, even for second order transitions, there is an intuitive sense that those are two sharply different "phases", otherwise there wouldn't be a reason to identify a single point as a transition point. – Martino May 3 '17 at 11:53

This is a very interesting question. My first instinct is to say that they are, by definition, sharp changes in the form of the distribution of probability of microstates, that occur even for small changes in the macrostate. So, there are plenty of examples of physical systems where changes happen gradually. Phase transitions are the ones where they don't, by definition.

## The Ising example

I'm guessing this answer won't satisfy you, and it doesn't satisfy me either. So, let's have a look at the simplest example I can think of, the all-to-all Ising model without an external field. Here, every spin $i$ has a lower energy if it's aligned to average field of the others, and a higher one if it's anti-parallel to it. $$H_i = -Jm\sigma_i$$ where $\sigma = \pm 1$ is the spin direction, $J$ is a positive constant, and $m$ is the average field that the spin "feels". Now, the point is that this $m$ is itself generated by the other spins: $m = \sum_j \sigma_j.$ You can see, then, that there is a non-linear self interaction of the group of spins. Without going into the details (you probably know them already), it turns out that there's an equation for $m$: $$m = \tanh(Jm).$$ It's easy to see that this has 0 as a trivial solution; however, it also has two additional solutions (as you can see by plotting both sides of the equation) when $J>1$, and these happen to be stable. This shows why there is a phase transition in this system. For $J\leq 1$, there is one solution. For $J>1$, there are two. There is no such a thing as "one solution and a half", of course, so this is necessarily a discontinuous difference. So my second answer is: it descends from the weirdness of certain nonlinear equations that govern the system. This is also related to the mathematical concept of bifurcation.

## Landau theory

Having a look at Landau theory helps us understand how something similar happens in general. Consider a physical system described by an order parameter $m$ (corresponding to the magnetisation in the case above). We can write its free energy as a function of $m$ and of the temperature $T$. Furthermore, let's assume that we can approximate it as follows (it's just a fourth order in $m$ around 0, and $\alpha$ is expanded at first order, details here): $$f(T) = f_0(T) + \alpha(T-T_c)m^2 + \frac{\beta}{2}m^4.$$ The equilibrium $m$ is given by the minimum of free energy: $$\alpha(T-T_c)m + \beta m^2 = 0,$$ which has one real solution when $T>T_c$: zero; and three solutions when $T<T_c$, analogously to what I showed above.

The picture below shows how one minimum of the free energy turns into two:

## Symmetries

Another way of putting it is connected with the concept of symmetry. The disordered phase has a higher degree of symmetry than a state of the ordered phase: an Ising model with zero magnetisation is invariant under up/down flipping of the whole system; a non-synchronised Kuramoto model has undefined average phase of its oscillators, which gives it $U(1)$ symmetry.

However, once the oscillators synchronise, they will have a given global phase. When the spins align, they may randomly align in the "up" or the "down" direction, but they have to collectively choose one. This is referred to as spontaneous symmetry breaking. As far as I know, symmetries can't be half-broken.

## Gas/liquid/solid

I don't know enough about these transitions to tell you about the physical details, but my guess is that something analogous to what I described above also happens in this case. However, these are first order transition, which follow different formalisms, and I'm not familiar with their Landau theory. Some interesting points about those are made in this answer to a different question.

I hope this gives you some intuition.

• This is a description of phase transitions, but it does not answer the question of why second order phase transitions are discrete. The explanation for spins does explain it, but I cannot understand how this applies generally. It's not true that phase transitions are only defined by the fact that they have a very fast change, all the systems that go though phase transitions have similar characteristics, how do these characteristics cause phase transitions? And by the way, Landau theory can describe both first and second order phase transitions... – Adi Ro Apr 27 '17 at 14:17
• Updated, expanding on Landau theory – Martino Apr 27 '17 at 14:35
• I never used the phrase "fast change". What do you mean by "similar characteristics"? – Martino Apr 27 '17 at 14:36
• I think actually showing some plots of the usual Landau free energy above and below Tc would be very useful here and make this more comprehensible to laymen. – Rococo Apr 27 '17 at 15:18
• I have now linked one. There were two linked articles about it, anyway. – Martino Apr 27 '17 at 15:26

Phase transitions represent transitions between different phases of matter and these phases are distinct. Unless one considers the Kosterlitz-Thouless phase transitions, phase transitions separate phases that have different symmetries. As a result a phase transition represent the point where symmetries become broken.

Symmetries are either present or absent. There is no spectrum between the two where they are only partially present. That is why the phases on the opposite sides of a phase transition are distinct.

The way we see this is to look at a particular quantity that does not respect the symmetry. This is called an order parameter. On the one side of the transition where the symmetry is present the order parameter is zero. On the other side the order parameter is not zero, indicating the presence of this quantity (a nonzero vacuum expectation value), which breaks the symmetry (or at least shows that the symmetry cannot exist).

It doesn't matter whether one considers a first order or second order phase transition. The only difference in this case is that the order parameter either jumps from zero to some nonzero value at the phase transition (first order), or it grows from zero starting at the phase transition (second order). In both cases, the function of the order parameter as a function of the control parameter is non-analytic at the phase transition point. The nonzero value of the order parameter indicates that the symmetry is broken, making it distinct from the situation on the other side where the symmetry is in tact.

• This is a better way of saying what I wrote in the Symmetries section of my answer. Nice. – Martino Apr 27 '17 at 15:06

The important point is whether you look locally or globally on a phase transition.

For instance, if you only look locally on different phase transitions, it is always distinct. From the conventional phase transition theory, different phases have different symmetries, e.g. solid breaks continuous translation symmetry, while liquid doesn't break it. So a state can either break it or not, which means locally it can either be solid or liquid.

But if one looks globally on a physical system, the story is a bit different. It relates to the correlation length of the phase transition. For first order phase transition, the correlation length is finite, i.e. solid-liquid transition is first order, so there exists mixture of solid and liquid during phase transition. For second order phase transition, the correlation length is infinite, i.e. paramagnet-ferromagnet transition is second order, so a material can either be ferromagnet below Curie temperature and paramagnet above Curie temperature.

Of course it is a very complicated matter at the transition's critical point, e.g. at Curie temperature. A gapped system would be gapless at critical point, bringing conformal field theory relevant.