Randomness in process of nature and Computer Sciences

An overview to define a metaphoric association between Nature and digital randomness to simulate natural selection.

What is Random ?

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or “trials”) is predictable. For example, when throwing two dices, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.

Computers are fundamentally deterministic. So, it is, in theory impossible to generate numbers that are really random. However, it is possible to define algorithms that generate sequences of numbers whose properties are very close to the properties of theoretical random numbers sequences. In particular, computers are able using pseudorandom generator, to generate a sequence of numbers that approximatively follows a uniform random distribution between 0 and 1. The uniform case is very simple one upon which more complex random variables can be built in different ways.

Randomness intersected with Anthropology:

Concepts of chance and randomness were intertwined with that of fate. Many ancient people threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.

The Chinese of 3000 years ago were perhaps the earliest people to formalize odds and chance. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a positive impact on the formal study of randomness. The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid- to late-20th century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.

Stochastic Process

A stochastic process is any process which can describe the evolution in time of random phenomenon. As well as, modeling random phenomenon to get properties that characterizes the process.

Natural selection is a key mechanism of evolution, could be studied in species level or in lower level which is genes.

Differences between Monte Carlo Simulation and Stochastic simulation:

here is a GIF of computing/approximating pi using Monte Carlo simulation

• The blue circle square has an area of πr².
• The red square has an area of (2 × r)² = 4r².
• The ratio of the circle area to square area is p = πr²/4r² = π/4.
• Therefore the value of π is 4×p.
• The red square contains ¼ of blue and red points so the ratio of points is the same (p).

The implementation takes a random point inside the unit square of size 1×1 and checks if that point lies within the blue circle. The ratio of points inside the circle to all generated points is p. Multiplying p by 4 gives the approximate value of π.

you can find python code for approximating pi here:

Monte Carlo simulations use random numbers in some way, in order to solve a model that is deterministic. Take for instance the classic example of Monte carlo: calculating surface of a circle (image above). The problem is clearly well defined and deterministic, there is only one possible outcome. Yet by sampling randomly you can approximate the answer.

Stochastic simulations are simulations of a model that is inherently random. An example would be a random number generator, giving you a number between 1 and 6 representing the number of eyes on a die. In other words, you are simulating a random throw of the die by generating a random number.

One more thing though: I can already hear you thinking that throwing a die is actually a deterministic process, but that does not make the simulation Monte Carlo. You see, you are not necessarily simulating a physical system, just a model of the physical system. And that model is free to be stochastic, even if the real physical system is deterministic.

Association between Stochastic Process and Natural Selection

Evolution is a stochastic process of change in gene frequencies in natural populations. Since the populations making up a species consist of many individuals and since evolution extends over enormous periods of time, laws which govern the process of change are inevitably statistical.

The process of genie selection under random drift due to small population number, as well as the process of change in gene frequency due to random fluctuation could be modeled as Stochastic process. The essential idea is that the population structure provides stochastic differentiation of local groups.

Concluding

I constantly find it spectacular to simulate every single phenomena in Maths. Mathematics invariably is the best description of any process, as it has crowded subfields to generously help to find the finest analysis, modeling and prediction. Mathematics intersects almost with everything. On the point of applying mathematical modeling to natural selection, Stochastic process suits the series of randomness of this phenomenon to model the natural selection, which is one of the terrific ways that humans found to describe evolution.