# Why the non-analyticity of free energy function implies phase transition? And what's its connection with other 'higher level' free energies?

I have seen 'free energy' arising from several contexts in very different forms, and each contains different amount of information.

For example

1. free energy is defined as the logarithm of the partition function (which is a function of $T$), and the non-analyticity of the free energy at one point means phase transition happening at the corresponding temperature. Here for a given temperature free energy is a number.

2. free energy is defined as Helmholtz or Gibbs free energy, here for a given temperature free energy is a function of some macroscopic parameter such as volume, pressure, etc. According to equilibrium phase transition theory, phase transition happens when two potential wells change stability (at some critical temperature $T_c$).

3. In the field of protein folding simulation, there is also the so called 'free energy landscape', which is a 2D surface on some reaction coordinates (at a given temperature). Here the value of the function at certain point is given by the logarithm of the number of corresponding states counted during sampling.

My intuition for free energy is that it measures the stability of a configuration according to the combined effect of energy and entropy (number of possible states compatible with this configuration). But depending on how detailed information we want for a system, we can map it to a number (where all the possible stats of a system is viewed as one degenerate configuration), a 1D function, a 2D surface, etc. Is this the right way to understand the seemingly very different free energies in the above cases? Also, can anyone help to draw the connection between the phase transition in (1) and (2)? In particular, why the non-analyticity of the free energy implies phase transition?

Introducing free energy can be done in many ways, and somewhat confusingly. To answer your questions, I think a possible way is to start from the Gibbsian statistics (rather than the thermodynamics definition). The Gibbs measure is a family that appears very frequently (c.f. Hammersley-Cliford theorem) where you end up writing the probability of states as:

$P(S=s) = \frac{e^{-\beta E(s)}}{Z}$

where $S$ is the state variable, $\beta$ the inverse temperature and $Z$ the normalization constant, called the partition function in statistical physics. The above distribution describes your system jumping from one state to another, say, at thermal equilibrium.

Due to the exponential, the thermal averages of the observables can usually be written in terms of derivatives of $Z$ such as:

$\langle E\rangle = -\frac{1}{Z} \frac{\partial Z}{\partial \beta} = \frac{\partial \log Z}{\partial \beta}$

You end up seeing the free energy $f =- \frac{1}{\beta} \log Z$, which can be written as a balance between the average energy and the entropy of your system. Your intuition of it is perfectly correct. Because the observables are derivatives of it, and phase transitions correspond to abrupt changes of behaviour of those observables (diverging susceptibility), you naturally link it to non-analyticity of $f$ (which here means "non-differentiability" most of the time). It was even used to classify all the phase transitions (Ehrenfest), although it is not in favour anymore.

Now I think some confusion stems from the fact that the energy $E(s)$ depends on some external factors that are not usually written. The probability distribution above describes the random wandering due to thermal excitations. But as soon as you can play with parameters that change your energy function $E(s)$, those also become parameters of your free energy, which becomes a function over a multi-dimensional space. They are called control parameters and the transition occurring when you vary them is phenomenologically explained by a Landau-Lifshitz approach (the two wells you mentionned).

Finally, your last point gives an example that belongs to disordered systems, where in addition, you have a external disorder (not to confuse with the thermal excitations). When proteins fold, they adopt, say, a given configuration, in which each amino-acid feels a potential dependent of the chosen configuration. Once this configuration is chosen, thermal noise makes it vibrate around this position of, sometimes metastable, equilibrium. The free energy landscape is a way to represent all those positions of equilibrium in a abstract space.

But you have to keep in mind that there are two sources of disorders in those systems, the one stemming from the geometrical configuration, and the more standard one coming from temperature. A common approach (in spin glasses for ex., for which protein folding is part) is to average over both thermal and external (coined "quenched") disorders. Quite often, the quenched average of $Z$ does not exist, but the average of $f$ does (although it is much more difficult to compute).