## Sunday, December 09, 2012

### Config Space via Mathematical Impressionism. Part 1

In my posts on evolution I have often talked about “configuration space”. By “configuration space” I am referring the set of all possible configurations of atoms/particles consistent with our physical regime. In this post I’m going to attempt to give this concept further clarification; further clarification but not complete clarification. This post is an impressionist’s “first parse” based on mathematical intuition rather than mathematical rigor, with the hope that in time a fuzzy picture will eventually give way to something of higher definition. I have doubts, however, that the picture will sharpen up much more because I have feeling that we are dealing with a subject that will not fully reduce to mathematics human beings find tractable; biology, I feel, is destined to remain narrative intense.

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Given a class of configurations then the mathematical problem can be posed of how to count these configurations. Counting objects requires one to organize these objects into a sequence. In fact many computational problems involve tracking through a set of objects in sequence and this raises the question of how one constructs this sequence. A very natural way of organizing configurations is to juxtapose configurations that are separated by a small increment of change. For example, two binary configurations that only have a one bit difference would be adjacent to one another. This idea of bringing together configurations associated by incremental differences underlines an important fact: The operation of counting imposes a one dimensional sequence on a naturally multidimensional object: Configuration’s separated by small increments of change form a network of objects where each configuration is linked to a large number of near neighbours and not just two neighbours as is imposed by a simple counting operation. It is with this network view of configuration space in mind that I will later be adding a cautionary note when we use my proposed categorization of configurations based on the graph I present and explain below.

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In the above graph the horizontal axis, S, represents some kind of size dimension of a configuration; for example, for a binary sequence this would simply be the number of cells in the sequence.  The vertical axis, Z, represents the number of physically possible configurations for a given size. Because Z is a number that, generally speaking, is going to be immense I've carried out the common practice of taking the Log of Z. This practice also has the useful side effect of turning products into sums so that the addition of parts to a system is also additive as far as the value of Log Z is concerned.

The line I have labelled  L0 represents the increase in the total number of logically possible configurations as we increase the size of the system. I have shown L0 as a straight line because on taking the Log of Z we usually find that Log Z, as a function of S, is approximately linear (The actual expression is often “S Log S” which is approximately linear). Superimposed on this graph are other straight lines, labeled L1 L2,…Ln whose meaning I will explain below.

The line L0, as I have said above, is the Log of all the logically possible configurations as a function of configuration size S. In fact it is possible to arrange a 1 to 1 map between each point below the Lline in the Log Z-S plane and each configuration represented by a unique point below L0. Thus our graph above effectively counts and arranges the possible configurations by means of this 1 on 1 map.

The line, L0 enumerates configurations of all types. What we now need to do is to further organise this counting method using the concept of "disorder". In what follows I’m assuming we know what “disorder” means. (I have written a lengthy private paper on this subject, a subject I will not be going into here) . To this end we start by introducing the line L1, such that the points between L1 and the S axis map on a 1 to 1 basis to the configurations that are classed as very highly ordered (that is, of very low disorder). For example, simple periodic configurations would classify as highly ordered; a configuration with a repeating pattern such as 10010010001…. would be included in the enumeration defined by L1.  I have shown L1  as a straight line that increases with S; an indication that the number of highly ordered configurations increases exponentially with S.

High order isn't something that suddenly cuts out; rather it fades as configurations become more complex in pattern. To represent this we imagine the Log Z-S plane be divided up into bands using lines L2, L3, ….Ln as shown above, where each line is separated from its two neighbours by the same increment in Log Z. The bands bounded by  L2, L3, ….Ln respectively represent regions of configurations of increasing disorder. Using this banding system means that the Log Z axis doubles up to give an indication of disorder as well as numbers of configurations.  Therefore, if we take a particular configuration of a given size it can, by virtue of its level of disorder, be placed somewhere below the L0 line, in one of the regions bounded by the lines L2, L3, ….Ln.

The vertical width between the bands demarked by L1,L2, ….L is a log value of the number of configurations of a given size in the respective band. Because this value is a logarithm of a configuration count, it implies that when  this value is translated to a literal count by taking the inverse log, it increases exponentially as we move through the bands L1,L2, to Ln. This fact brings out an important feature of disorder; namely, that the number of configurations associated with a particular value of disorder increases steeply with increasing disorderAn interesting corollary of the steep increase in configuration count as a function of disorder is that when disorder is at a maximum then as system size increases the number of maximally disordered configurations tends toward the total possible configurations as expressed by  L0
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Some Quasi-Axioms
I will be using the Log Z-S plane to consider evolution, but I can’t take this consideration much further without some further assumptions. The following assertions are too high level to be called axioms, but as we are dealing with a very high level phenomenon we can get a good head start by making some shrewd high level guesses as per the 4 points below:

Definition

Living structures have powers of self-repair and reproduction; or using a catch-all term powers of “self-perpetuation”.

"Axioms"

1. The level of complexity required for self-perpetuation is going to position living configurations in a band intermediate between high and low order.

2. The set of self-perpetuating structures is going to be of “vanishingly” small size when compared to the set of all configurations

3. The larger a configuration becomes (i.e. increasing S) then the greater the improbability of it forming spontaneously; (that is, of it forming without precursors). This looks to be a consequence of the assumption of equal a-priori probabilities amongst configs. In fact the probability of spontaneous formations is likely to be some decaying exponential term that looks something like  "A exp [-B S]" where A and B are constants

4. If we take a given configuration C1 separated from another configuration C2 by d changes then the probability of C1 morphing spontaneously into C2 is likely to be a decaying exponential of d; that is,      the probability of a “saltation” leap being made from one to the other will have a mathematical form that looks something like the expression "A exp[-B d]" where A and B are constants.  This assumption is closely related to assumption 3.

The foregoing, then, summarizes the model I use and will be using when discussing the evolutionary question.