Evolution in a nutshell

an alternative outline on evoution

and some consequences concerning valuations

by

Gregor Kjellström

            References          

4.2 Disorder and biological diversity

To show that average information is consistent with disorder we may use an example from physics by Reif. Suppose that a large number (a) of molecules of a certain gas enters into a room. Further assume that the room may be divided into a large number (r) of cells numbered by i = 1, 2, ..., r, and that the number of molecules in the cell no i is a_i.

Let us use a simple example with six molecules and three cells to see how many possible placements there are of a₁ molecules in cell no 1, a₂ molecules in cell no 2 etc.

One way to solve the problem is to first see in how many ways 6 molecules may be arranged in a row. If we have 2 molecules (numbered 1 and 2) there are two possible placements namely (1,2) and (2,1). If a third molecule is entered it can be entered before, in the middle or after the two existing molecules, so in this case we get the placements (3,1,2), (3,2,1), (1,3,2), (2,3,1), (1,2,3) and (2,1,3). So if we have three molecules there are 3_*2_*1 placements. In the same manner we get 6_*5_*4_*3_*2_*1 = 6! if we have 6 molecules.

Since we do not consider the placement of molecules inside a cell, the number of possible placements of 6 molecules in 3 cells so that we have 3 molecules in cell number one, 2 in cell number two and 1 molecule in cell number three becomes

     6!/(3!2!1!) = 60.

The table below shows the number of micro states corresponding to some different macro states:

     macro state     number of

                           micro states

     (6, 0, 0)          1

     (5, 1, 0)          6

     (4, 2, 0)          15

     (3, 3, 0)          20

     (3, 2, 1)          60

     (2, 2, 2)          90

More generally, the number (G) of possible placements is

     G = a!/(a₁!a₂!...a_r!) 

According to the second law of thermodynamics, stating that the disorder always increases in all isolated systems, we may expect that the gas will, because of the random collisions between molecules, expand to a uniform distribution over the room. But what actually happens is that the system occupies the macro states in proportion to their probability of occurrence. In the small example with six molecules, this means that the state (2, 2, 2) is 90 times more probable than the state (6, 0, 0).

By introducing the probabilities p_i = a_i/a and using the Stirling approximation of a!, we find that log(G) is proportional to

     H = - å p_i log( p_i )    (equivalent to average information)

where H is usually referred to as the disorder (entropy) of the system.

Since an alternative way of looking at this is that the molecules occupy a diversity of cells in the room and that this diversity tends to increase because of the random changes in the system, there is no reason to distinguish between disorder and biological diversity. The difference is that natural systems may occupy a diversity of cells in a space spanned by phenotypes.