Selection is something like a phase transition at the error threshold, and you have similar problems as in solid state physics where you also talk about phase transitions and ferromagnetism and so on. If you go through theory you come to infinities, you come to singularities and so... so you have to do something which physicists nowadays call renormalisation of theory, and John McCaskill did this for this evolutionary theory I talked about. So he did the renormalisation and could show that you can solve all these problems, can get rid of infinities and... that's one aspect. The other aspect is he also looked at stochastic theory.  Why is stochastic theory important? At equilibrium, stochastic theory is not so important. Why? Because if you talk about a chemical equilibrium, given Avogadro's number – Avogradro's number of 10²⁴ – it means fluctuations go with the square root of that number. So one in 10¹² is something nobody can measure. So fluctuation theory of equilibrium systems only is of importance if you are dealing really with only a few molecules. We will do so later, but generally what a chemist is doing, he always works with grams or milligrams and these are still a large number of molecules. That's due to the fact that you have fluctuations on the molecule level, but they average out, and then appear finally with the square root of the number of molecules and that's a very small relative fluctuation.

In evolutionary theory this is different. Let's say you get an advantageous mutant, just one molecule.  You have a big fluctuation for this to come about, you cannot predict it and everything on the molecular scale is probabilistic. So you have the very big fluctuation for this molecule to come about, but once it is there, and it is of advantage, it will double before it decays. So you have two of them, and the two will double before they decay, you have four, eight, and you get this exponential increase and now you see the fluctuation of that one molecule to appear will very soon map on the macroscopic scale, because of that amplification method. So stochastic theory is very important in evolutionary behaviour.

[Q] In explaining, also, evolutionary behaviour.

Yes. At the beginning the favourable molecule can always decay, can just happen that right after it came about it is decomposed. But in this fluctuation theory shows that, like every air pilot knows, there is a point of no return when he accelerates his plane and if he reaches a certain speed he has to take off otherwise he would crash, he could not stop it any more. And so it is here, once after a few replications of a favourable molecule there is a point of no return and it will build up. And here you map your elementary phenomenon of one single molecule to come about, and one new one, if it is advantage, can do so, and that's very important to know, that otherwise evolution wouldn't be that efficient.