Why does a good mathematical equation sometimes prevent the deep understanding of a phenomenon?

Some years ago, I became fascinated by the electric properties of liquid water. One year ago, I realized that in order to understand these properties I first needed to investigate them in ice (see last post). But I was lucky. The hard work had already been done. In the sixties, Jaccard developed a complete theory describing how charges are moving through ice. I was surprised. I intensively studied sold states during my studies, but nobody had ever mentioned the peculiarity of electrical conduction in ice. 

Fig. 1 How electric conduction is 

visualized in most solids.

Jaccard understood that there must be a strong coupling between the orientation of the water molecules and the movement of the charges in ice and this really is a rear phenomenon. In most solids charges (like electrons, holes, ions, …) move without changing the structure (see Fig. 1). Not so in ice, however. The mechanism behind this will be addressed in our next post. The theme of this post is not the explanation but the mathematics behind it.

Jaccard succeeded in quantifying the relation between movement of charges in ice and the orientation of the water molecules in a set of equations (see Fig. 2). And even better, the equations had an analytical solution. Because of this success the theory got its name, Jaccard’s theory, and the correspondence with the experimental results were impressive. 

Fig. 2: The most important equations of Jaccard’s theory. 

The moment physicists have a closed set of equations, they feel comfortable and have the feeling that the whole system is understood. But I am not sure this is true. Jaccard derived his equations step by step using the Maxwell equations and on some ideas about the ice structure. And there’s the problem. In his first derivations, Jaccard used only kinetic arguments to describe the movement of the charges. He found the same equations of Fig. 2, but in the second terms there was an extra factor of 2 (so instead of Phi there was 2 Phi). He couldn’t solve the problem and after some years, he changed his approach. He replaced its initial kinetic description by a much more abstract thermodynamic description, now leading to the correct equations. This new approach didn’t give extra insight in the movement of the charges or the reasons why the kinetic approach wasn’t working. A fundamental problem was kept hidden. 

Jaccard’s equations became successful and helped to understand several experimental results. The problems with their derivation were excused because there was a correct thermodynamic derivation. However, my purpose was to apply these equations in the context of liquid water (see last post) so I couldn’t afford not to apprehend why the charges were moving like they did. I had to reinvestigate the crashed kinetic derivation of Jaccard. 

Fig. 3: The law of Gauss in matter. Notice the 

difference between free and bound charges. 

I am still grateful to Professor Pauwels (from the university of Ghent) for giving me, many years ago, a good understanding of the theory of electromagnetism. I found out rather easily that the factor of 2 wasn’t the only weak point in Jaccard’s theory. The theory didn’t make a clear distinction between bound and free charges (see Fig. 3 for the specialists). By consistently keeping this difference in mind, I could rebuild the theory more clearly and solve some of the errors that were in it (like the factor of 2) (click here for a full description of the improved formulation).  

Jaccard’s equations could be derived from straightforward kinetic considerations. The thermodynamic deviation wasn’t necessary. I could now visualize how the charges were really moving (see next post) and I made the mathematics behind these processes more comprehensible. With this knowledge I was ready to tackle the problem of electric conduction in liquid water. 

I needed to tell this story. Not only because it was my first real success after several years of searching, but also because it tells a lot about the meaning of deep understanding. I see the same phenomenon with some of my students. They think they understand a problem if they manage to write the correct equations down, but they are completely missing the story behind these equations. What were the assumptions made? What is the physical meaning of this quantity? What is the influence of the quadratic factor…? Some textbooks and many papers show similar problems. They collect a bunch of equations without sufficient framing or trying to grasp the meaning of the equations. Difficult mathematical equations are a status symbol, that shows how smart the writer of the paper is. But isn’t real insight hidden in the simplicity of the derivations? Doesn’t a difficult derivation imply ‘I do not understand it completely yet.’ 

Fig. 5 ‘Il dialogo’ of Galilei who
popularized his ideas without any equation. 

But don’t get me wrong. Physics needs mathematics. ‘The book of nature is written in the language of mathematics’, dixit Galileo Galilei. But the same person was also brilliant in explaining his mathematical ideas in dialogues – which are still worth reading (see Fig. 5). 

Now that the story behind Jaccard’s theory is clear, we are ready to for the real work. Why is ice a special semiconductor? This will be the subject of my next post. It will be more challenging than this one, but relevant for everybody who is interested in a new understanding of conductivity. I won’t write down any equations, promised!