From Chaos to Order and Back

Since each individual living organism passes the transition from non-existence(chaos), through birth, to existence (order) and then, through death, back to not-existence(chaos), the question is if the entire life as we know it, the entire biosphere, will eventually pass through the same stages as well.

One of the descriptions of the second law of thermodynamics is a room divided in two equal parts, one filled with hot and another with the same amount of cold air. By measuring the temperature it would be possible to know in which part of the room is the thermometer. This is considered to be the state of order. However, after removing the divider, the temperature will gradually equalize in the entire room and on macro level it would not be possible to distinguish any particular point in the room based on its temperature which will be the state of chaos.
Instead of this pure thermodynamic model we could use one based not only on temperature, but on different colors of the particles in the room as well. In one half of the room will be only hot particles colored with white, and in another only the cold particles colored with black. After removing the divider, at some point both black and white particles will be equally distributed and the color in the entire room will become gray. What makes this model interesting are some visual properties that could be identified in it. Namely, a picture of the beginning state will show clear black and white parts of the room as well as the negative of the same picture, but in which black and white parts would switch places. On the other hand, a picture of the final state (gray) and its negative would be indistinguishable. Thus, while the distinct parts of the room colored with black and white representing order are sensitive to the “operation negative”, the final state, state of chaos (entropy) is indifferent to this operation.
However, if one part at the beginning contains, lets say 25% of black and another part 75% of white particles, the end state and its negative will not be the same. While the color of the entire room will be uniformly light-gray, its negative picture would be dark-gray. Thus the neutrality of the final state in relation to positive-negative exists only when the amounts of black and white particles are the same.
Usually the relationship between order and chaos (entropy), is observed within a system with very large number of particles and temperature differences. It would be interesting to try to examine these relationships within a system (universe) that is only visually based and with a small number of elements like, for example, a 3×4 matrix (12 positions) within which are distributed six black and six white elements. Lets name the neighborhoods with the same values (white-white, or black-black) “connections”(C), while the neighborhoods with different values (white-black) would be “junctions”(J) (Fig.1). In these six examples of different distributions of two values, it is possible to identify connections and junctions.


The minimum number of junctions (J=3) and maximum connections (C=14) is in the case with maximum concentrations of both black and white elements that correspond to the highest level of organization for this system/universe(order), while the case with lowest number of connections (C=0) and highest number of junctions (J=17) is the state of the lowest organization(entropy, chaos) for this universe. In this way it seems it is possible to identify the level of entropy for such universe by simply comparing the number of junctions. The higher this number is, the higher the entropy is.

Since this 3×4 universe is already a bit complex case, it would be interesting to observe even simpler cases. For all of them there we assume there is a referential observer that is outside of the system. The simplest case would be a universe consisting of one element and one value (Fig.2a). Within this universe nothing else could be defined. The next would be one consisting of one element and two values: black and white (Fig.2b). This universe could switch from black to white and back, thus enabling the introduction of change (and indirectly-time) as shown in Fig.2c.


Then comes a system of two elements/positions and one value (Fig.2d). Within this universe there is no change, but it would be possible to define the connection which is in this case C=1. Now it is possible to define a universe with two elements/positions an two values (Fig.3a). Here, there are four states, two with both neighboring elements having the same values(C=1, J=0) and two cases in which the neighboring elements have different values(C=0, J=1).


These last two states are in some way unique since at the same time they represent both order and chaos. This simple universe of four states enables generation of various processes (Fig.3b). With three elements/positions there are two configurations possible. One that has one element with two neighbors and two elements with one neighbor each, a configuration that structurally resembles cordon in DNA, and another in which all the elements have two neighbors (Fig.4a). In the first structure and two values (black and white) it is possible to define eight different states(Fig.4b).


Among them it is possible to identify states with a bit higher (C=1 and J=1) and a bit lower organization(C=0, J=2). Because of odd numbers of elements it is not possible to have a state with average total value of 50%. This could be achieved only within a universe with even number of positions, like one resembling two cordons placed one below another forming 2×3 matrix. There are altogether 64 states that can be generated with six positions and two values(Fig.5), out of which 20 are with equal numbers of black and white values (3+3). Among those there are six states with highest organization/order(C=4, J=3) and two cases of lowest organization/entropy(C=0, J=7).


On this micro level the positive-negative relationship is visible for both low and high entropy cases. It would become apparent in a universe with very large number of elements, where there will be many states of order and only one state of entropy. And on that level it is irrelevant if the number of positions is odd or even. But the highest number of junctions is a clear indicator of the state of entropy/disorder/chaos for given initial conditions (number of positions and values and their neighborhoods).
The 3×4 structure introduced in the beginning, as structure built of four cordon like linear structures, could be used for a conversion of RNA and DNA linear strands into 2D images and, among other properties, identify stretches with high organization or states of entropy (chaos). For these initial conditions there are eight types of highly organized state (order) and only one that is of highest entropy (chaos) as shown in Fig.6.


Most of those highly organized I was able to identify while looking for binary A and T stretches in the E.coli K-12 MG1655 strand, some of them multiple times. In this particular representation of DNA the darker color is gray and denotes Adenine(A) while white stands for Thymine(T).
In this universe the transition from order to chaos (which is also another kind of order) can take place in various ways. Unlike in the thermodynamic case where transition from order to chaos is driven by the temperature differences and unfolds in one direction, the transition here is determined by the structure of a DNA sequence and can move in both directions as shown in these cases of the stretches consisting of A and T bases that most likely exist in some concrete DNA strand.


The initial state is defined with a frame of first 12 positions, and the frame moves gradually one position at a time until the last base enters the frame, and can go from order to chaos (Fig.7) and from chaos to order (Fig.8).


Although the notion of entropy and thus order and chaos were introduced in physics observing the behavior of very large numbers of particles of gases and fluids and their relationship to temperature, it might be interesting to observe another kind of particles, this time organized in linear structures, known as bases of RNA and DNA. If we consider a possibility that the earliest living molecules had to preserve their structural integrity, in other words to survive, the main information about the environment necessary for their survival must have been the distinction between what we call “hot” and “cold”, which at that point were synonymous for “light” and “dark”. And in order to be able to distinguish these properties of its surrounding, the “knowledge” about them had to be previously encoded into these molecules. Those earliest living molecules, as proto-observers, which by some chain of events acquired this capacity to distinguish hot from cold  had a better chance to survive and passed this to the next generations. Thus the very basic properties of the world we could distinguish today: hot-cold, dark-light, order-chaos, are most likely first acquired by the earliest living molecules, then encoded and memorized within their molecular structure, and then transmitted to all living matter including us.

Gregor Mobius

New York 2017


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