Is it possible to mathematically explain why solids go under mollification when heated?

Question

Well, I'm sure that many people on MSE might object that this is not a math question, however, I think that there might be a well-posed mathematical answer to this question, or at least I hope so.

We all have seen in our every day life that if we heat a heat conducting solid it starts to get smoother like in this picture:

Is it possible to mathematically explain why the solid tends to get a smoother shape after it's heated?

I'm asking this question because I found the idea similar to what mollifiers do in distribution theory.

Answer 1

My totally disinformed opinion: when the body starts to melt the relevant forces are surface tension and gravity. Surface tension wants to minimize the surface and kills the edges. Gravity pulls down. I would be surprised if the result could be expressed as a convolution in a natural way.

EDIT: expanding the commeny by Zack Li, in the case of temperature the answer is yes. The temperature is the convolution of initial temperature with the heat kernel/fundamental solution.

Answer 2

Disclaimer: I know nothing of mollifiers but for what I could grasp from a brief look at wikipedia.

You ask for a possible mathematical description of the smoothness under temperature rise in solids. I'm not very clear about what you mean, so I'll give an interpretation and then comment on it.

If you concede that solid-state physics, which is based on quantum mechanics and therefore could be based in as much mathematics as desired, correctly describes every behavior of solids one could arguee that there is indeed such explanation, although it may be complicated. But what I think you want is a model of the smoothing process under temperature variation, in the sense that the heat equation models temperature transfer.

Notice that the heat equation is derivable from Newton's Law of Cooling, an empirical equation for heat transfer in solids. In the same sense the one dimensional wave equation can be obtained by applying Newton's Laws (from classical mechanics) to an elastic rope. The secret is that we obtain simple PDEs, not to mention linear ones, that, I must stress the word, model these systems. In principle one can describe the rope as a chain of linear oscillators and get the wave equation. Heat conduction on the other hand is a stochastic process and not strictly derivable from statistical mechanics. In fact we're not even sure about the experimental limits of the heat equation (I'm not in my office rigth now, but could reference this facts if someone asks for). So the idea is to get a very simple description of empirical based models.

Now, is there such empirical description for the process you mention? I've never seen it (though my ignorance should not be evidence against it) but since it's a phenomenon similar for metals and rubber, which have widely different thermodynamical descriptions, I would guess such model is not simply derivable from this and should be based entirely on elasticity theory, which according to wikipedia is a rather hard task when accounting for temperature changes.

For the record, the smoothness of temperature distributions is a interesting property of the heat equation. Is not difficult to show that if the initial function is square-integrable all its moments decay with time except for the first. Translational invariance of the heat equation then shows that the solution "rounds the edges" as time goes by.