[Physics] How to understand singularities in physics

phase-transitionregularizationrenormalizationsingularitiesstatistical mechanics

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.

Best Answer

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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