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Evolution math

Continuing to write about evolution.

So far evolution have been described as a set of statements. This paper tries to introduce math in evolutionary science.

To be absolutely clear, this paper does not dispute an existence of evolution phenomenon and assumes it in full.

Evolution is

  1. the process by which different kinds of living organisms are thought to have developed and diversified from earlier forms during the history of the earth. synonyms: Darwinism, natural selection. Or
  2. the gradual development of something, especially from a simple to a more complex form.

Let me offer another version of this definition:

“The systems that did not survive in a transition from one state to another will not continue its existence.”

Let’s consider what happens when a dynamic system, such as a live organism goes through 2 scenarios. In the first scenario, a living system to be used interchangeably with dynamic system, encounters an adverse change in environment, adapts to it and in the end survives, passing to survivors the changes. In the second scenario the system does not survive and undergoes a transformation typical to second law, where it decays into separate parts increasing the entropy.

The simplest explanation that the first scenario creates more entropy over the long run. The living system will be more organized but during it’s lifespan it will consume and metabolize food and producing heat and waste. In the end it will die, but in perpetuity, assuming it will exist forever it will produce infinite amount of heat.

So second law of thermodynamics should be amended by measuring entropy in the final state.

Let’s imagine a system of evolving rules, kind of enhanced cellular automata, or supercell automata, SCA if I may. It’s similar to a regular cellular automata described by Steve Wolfram ( where cells evolve according to simple rules, with one exception. In this model, rules are also subject to change according to just one simple super rule. This simple super rule states that the rules change from iteration to the next iteration randomly and, most importantly continue to evolve as long as a combination of cells stays alive. In other words, only rules that generate surviving combinations survive.

My question is how to describe this system mathematically. I thought of a more general question, how to describe evolution mathematically. A colleague from Columbia suggested Markov chains. It is a poor fit, because Markov chains don’t preserve state, and evolution in it’s classical darwinian form preserves all the history of all the previous states.
A better fit will be a math behind cellular automata, but it lacks mutations factor. However if we introduce a small mutation factor where transition rules change a little bit, the mathematics behind it has all the right attributes.
The same principle could be applied to other systems, for example to fundamental science. Consider, early universe and a phenomenon of fine tuned universe. Super-inflation have been created to explain the problem fine tuned universe. However if you take a view that the fundamental laws of physics are not only changing, but only the laws that  that lead to surviving combinations stay. That view would explain fine tuned universe perfectly.
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