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The question is probably two-folded and I will try not to make it too vague, but nonetheless the question remains general.

First fold:

In most physical laws, that we have analytic mathematical expressions for, one comes across functions that diverge at a given point, typical examples would be the Coulomb or the gravitational forces being $\propto 1/r^2,$ clearly they diverge at $r=0.$

  • Physically it is obvious that if by distance $r$ we mean the distance between center of masses of the objects, then $r=0$ is trivially excluded (for macroscopic objects at least) because they have well defined excluded volumes and cannot occupy the same space at the same time, hence one may argue that the divergence at $r=0$ case is a mathematical artifact and is to be ignored, but is this really the case or do we have an explanation for such extreme cases?

  • Are most singularities met in classical physics just reminders of the fact that within classical models, not all can be explained, and one has to turn to more general frameworks such as QM, where then the singularities would be resolved?

Second fold:

The second type of singularity that one comes across, is in statistical mechanics or thermodynamics, namely the association of phase transitions to singularities of the free energy of the system. We know that if the nth order derivative of the free energy becomes singular then the system must at some critical point exhibit an nth order phase transition, or conversely if the free energy never becomes singular, e.g. if $F(T) \propto \frac{1}{T},$ then there can be no phase transition that depends on temperature as such function would only diverge at $T=0 K$ which physically is impossible anyway.

Typical examples would be second order phase transition in the Ising ferromagnet system, where the second derivative of the free energy with respect to $T$ diverges at the critical temperature $T_c,$ at which point the system transitions from a paramagnet to a ferromagnet or the other way around. An example for first order transition would be liquid water into ice, where the transition is first order because the first order derivative of the free energy becomes singular. Furthermore there are also cases that the free energy derivatives diverge on change of density of the system instead of temperature.

  • What is the main difference between such type of singularities met in phase transitions, compared to the previous ones mentioned in the first part?

  • Finally why should a phase transition correspond to a singularity in the free energy or entropy at all? What is the physical intuition here?

Feel free to use any mathematical argumentation you find necessary, or other examples that may find more illustrative.

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    $\begingroup$ Not a duplicate, but very closely related: physics.stackexchange.com/q/167529 $\endgroup$
    – tpg2114
    Mar 26 '15 at 13:48
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    $\begingroup$ First fold. You are mostly right. A singularity is a huge red flag suggesting that your theory is not completely correct: it's an approximation, or something is left out, or \emph{etc.}. However, a singularity might be an indication that your mathematical approach is not up to the task. $\endgroup$
    – garyp
    Mar 26 '15 at 14:02
  • $\begingroup$ @Qmechanic thanks for your help with tags, it was difficult for me to figure it out. $\endgroup$
    – user929304
    Mar 26 '15 at 14:25
  • $\begingroup$ @tpg2114 and garyp, thank you both for the link and comment, quite helpful. $\endgroup$
    – user929304
    Mar 26 '15 at 14:26
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    $\begingroup$ Not sure about your second part. Phase transitions are defined as singularities of thermodynamic potentials, because this is what one seems to observe experimentally. Statistical mechanics then tells you that these singularities are in fact only approximate (they only exist in the ideal limit of infinite systems). The "real" functions remain smooth, although they mimic very closely singular behaviour. $\endgroup$ Mar 26 '15 at 14:38
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As noted already, within classical physics, singularities such as $1/r^2$ signal a break down of the theory. If we are really interested in what is happening at the point of the singularity, we should use quantum physics. You can think of $1/r^2$ as the asymptotic scaling form of the quantum theory for large $r$. The actual singularity is not physical.

On the other hand, the singularities of thermodynamics are a direct result of the thermodynamic limit. When you have many particles, they may all work together to make physical quantities (typically susceptibilities) very large. In the limit of infinite particles, the corresponding quantity diverges. In practice these singularities are not realised for two reasons. First, you never really are in the thermodynamic limit. This is however not a real limitation because atoms are so small that you can easily have $10^{23}$ of them. Such a big number is indistinguishable from infinity. The real reason is that in order to find a such a divergence, you usually have to fine tune some parameter of the system to make it sit exactly at the critical point. You need the temperature and pressure to be mathematically equal to their critical values. You can never do it.

It is actually natural that you find something non-analytic at a phase transition. Physically, a phase transition is a point in the phase space where the properties of the system change abruptly. You go from water to ice. The system is either liquid or solid, there is no interpolating state in between where the system soft. Mathematically, this manifests its self as a non-analytic change of the thermodynamic potential, i.e. a divergence of it's derivative (of sufficiently high order).

I would conclude that these two types of singularities are unrelated. There is however a connection in the theoretical tool that one uses to solve these problems: Renormalisation.

On the $1/r^2$ side, the quantum field theory tells us that actually, the particles do interact with them selves and that this leads to divergences in theories that are defined on a continuous space. These divergences can be re-absorbed into the microscopic (and unobservable) parameters of the system which diverge in such a way that all infinities cancel out. See this article.

On the thermodynamics side, critical points are associated with fixed points of the renormalisation group. There the system is invariant under the combined coarse graining of it's fine details and zoom out. Then we find scale invariance and the power laws that one can observe at a phase transition.

Even though these procedures have completely different interpretations, they are technically extremely similar and contain the same ideas. On the quantum field theory side, you want the space to be continuous. You use renormalisation to make the space-time grid infinitely small without generating divergences. On the other hand, at critical point of statistical systems, the correlation length of the system is so big that the spatial grid (for example in a crystal) is irrelevant and your theory is effectively continuous.

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... r=0 is trivially excluded (for macroscopic objects at least) because they have well defined excluded volumes and cannot occupy the same space at the same time, hence one may argue that the divergence at r=0 case is a mathematical artifact

Radius of elementary particles can be 0 if they are point particles (electrons are so far best thought of as point particles). If so, they are physical mass points, with finite charge and mass. There is no problem with such physical singularity, as long as we know its properties and laws of its behaviour in given conditions.

That $r=0$ does not happen is true when $r$ stands for distance of two electrons. Imagine two point electrons. They can very well have zero dimensions and volume, as long as they have positive mutual distance. They cannot approach each other up to distance 0, since that would require, per Coulomb's law, infinite energy.

Are most singularities met in classical physics just reminders of the fact that within classical models, not all can be explained, and one has to turn to more general frameworks such as QM, where then the singularities would be resolved?

It depends. If some model breaks down at certain point where we know the correct answer exists and is quantifiable, then the model is wrong at this point and there is a good reason to look for a better model.

If the singularity is physical ( point particles ) and we can use it and calculate with it consistently ( distance never gets to be 0 in practice ) that kind of singularity is OK and has its place in physics.

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Singularity in Force Laws

If force laws were fundamental to nature, this would be a serious problem. Imagine, for example, the gravitational energy between photons. They are Bosons and can hence occupy the same quantum state; crucially, more than one of them can be and stay in the same position where the gravitational force (they have energy and hence, relativistically, mass) and erergy diverges.

In fact, the situation is even worse: Even if we somehow found a loophole around the divergence when interacting particles are in the exact same place, there is still a problem with a single particle. For a point-like (or even just nearly point-like) electron, just the self-energy from the electric repulsion of its charge acting on itself (imagine it assembled by shrinking a spatially extended charge distribution) easily exceeds its rest mass. Where could this energy come from?

The truth is that forces are just a useful simplification of something more fundamental. Virtual particles describe the interaction between particles which (only) for low energies (with correspondingly limited resolution in momentum and hence position) becomes indistinguishable from force laws.

Singularity in Phase Transitions

A phase transition is the sudden change in something, usually the arrangement or behavior of an ensemble of particles. That typically corresponds to changes in just about any property of the collective system. The definition you use attempts to be as broad as possible whilst limiting itself to considering one quantity, free energy. To be more general than just prescribing a sudden change in the free energy itself, it includes the concept of $n$-th order phase changes where the sudden change only occurs in a (possibly higher) derivative of the free energy. But the important point is simply that (usually) almost any quantity will change similarly (although possibly in a different minimum derivative).

The main difference of this divergence to the kind encountered in force laws is that the existence of this divergence, the sudden change, is central to the physics described. If it were not there, there would not be a phase change. In the force laws, the divergences occur in the mathematical idealization or simplification of reality whereas reality is subtly different (or more complicated, if you like).

This also explains your final question: Why should a phase transition correspond to a divergence or mathematical singularity? It is because it corresponds to a change that is not gradual in a key parameter (e.g. temperature). Hence it (or its derivative or the derivative of that, etc.) must make a sudden rather than a smooth jump. You may be able to mathematically make the transition smooth in some way; for example, if you parametrize it by the entropy rather than by temperature, a $0$-th order phase transition often could be seen as a first order transition because to bring it about, at a defined (constant) temperature, entropy must be added to or removed from the system to complete the transition. But you will still encounter your discontinuity (or divergence or singularity if you like), if possibly only in some higher derivative.

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I can only offer a partial answer to the first portion of your question.

one comes across functions that diverge at a given point, typical examples would be the Coulomb or the gravitational forces being ∝ 1/r², clearly they diverge at r=0...

Gravitational force isn't actually proportional to 1/r². Take a look at the plot of gravitational potential on Wikipedia.

enter image description here

CCASA image by AllenMcC, see Wikipedia Commons

The slope of the plot depicts the force of gravity for an equatorial slice through the Earth and the surrounding space. And whilst the slope initially increases as r decreases, it ends up decreasing. So the infinity at r=0 is a mathematical fiction.

Are most singularities met in classical physics just reminders of the fact that within classical models, not all can be explained...

I would say no, but that some singularities are the result of "non-real solutions". For example, gravitational potential could be plotted using light-clock rates dotted throughout the equatorial slice - clocks go slower when they're lower. Then when you did this for a black hole, the light-clock rate at the event horizon is zero. And it can't go lower than this. So your plot looks like the picture below, without a point-singularity in the middle.

enter image description here

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I can't speak to singularities in the sense of general relativity, but your example of $1/r^2$ laws in classical physics is actually solved most of the time by internal structure. One of my physics professors used to always say that nature solves infinities with internal structure.

For example, for a charged sphere of uniform charge density, the electric field goes like $r$ for $r<R$ (the radius of the sphere) and like $1/r^2$ for $r>R$. The same is true for a spherical gravitational body of uniform mass density. So the internal structure helps you avoid infinities.

Now obviously this doesn't take into account the fact that electrons and quarks as we know, so far, are point like objects, but that is just because or precision and accuracy on experiments is limited. We may find that they do have internal structure, but nevertheless quantum mechanics must come into play when dealing with these.

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