Understanding Chaos is the key to Everything…
Finding Order in Chaos is an important step in understanding the Universe
There is chaos everywhere. How do we bring order to it? How do we understand and predict what comes next? Small changes in measurements will give widely diverging results. So what do we do?
Over the past few decades we have only just begun to understand nature’s lack of structure and unpredictability. After the discovery of deterministic chaos and the dissolution of Laplace’s strict determinism of classical physics. The understanding of chaos allows us to part the veil of apparent “randomness” of a system.
How do we define a dynamical system with apparent randomness, irregularities and laws which are sensitive to initial conditions in mathematical terms? We have to find the underlying pattern in a chaotic complex system which could range from constant feedback loops to self-organization.
How do we define Chaos we can understand? Like in physics we have inertial and non-inertial frames/systems. What about for chaos? There is one generalized system: Deterministic Chaos.
What is Deterministic Chaos?
It is the production of random, chaotic dynamic system based on a simple but non-linear rule.
Chaos theory of deterministic chaos relies on these properties to predict its behavior for a while and then appears to become “random” again. The amount of time that the behavior of a chaotic system can be predicted depends on three things:
- The amount of uncertainty tolerated in computation
- The accuracy of initial measurements
- The Lyapunov time of a chaotic dynamical system
Lyapunov time: Its value shows the limits of predictability of a chaotic dynamical system. It is the time taken for the distance between two nearby trajectories to increase by a factor of e.
How do we define Deterministic Chaos?
- It must be extremely sensitive to initial conditions
Here is an example of the “sensitive to initial conditions”. As you can see here due to infinitesimally small changes in the initial conditions each systems behavior diverged after some time.
This is also known as the butterfly effect. A small change in the initial conditions causes huge diverging results.
- It must be topological transitive
Mathematically this mean a functions f is topologically transitive if fᵏ(U)∩V≠∅ where k is a positive integer and is the iterate of function f while U and V are two sets of intervals. This refers to the mixing using only the topology of the system. So here this defines a topological transitive bounded in a linear operator which defines a continuous linear map on a topological vector space.
In layman terms this means that neighborhoods of points are eventually moved around to big sets so they don’t stick together.
Mixing: it describes irreversible thermodynamic processes of mixing like mixing paint, drinks, etc,
Topology: It is about the geometric properties of an object undergoing continuous deformations but not including tearing or gluing.
Topological vector space: It is a vector space over a set of points and sets of neighborhoods for each points which satisfies Hausdorff axioms.
- It must have dense periodic orbits
The last two properties are inter-related to the first property and implies sensitivity to initial conditions.
Using these basic principles and understanding of chaos, we have to define the chaos — Deterministic Chaos — which we can predict. Now due to our limited understanding — and limited imagination of the human nature — it acts as a bottleneck to our understanding of the universe.
How?
In Astrophysics lets talk about the Interstellar medium. The density structure of the Interstellar medium where stars are formed, energy is released, momentum of particles and heavy metals, and driving the evolution of galaxies. Seeded density variations are then amplified by gas motion across the interstellar medium affecting through unknown spatial scales and galactic environments. As you can understand from this, you can see “sensitivity to initial conditions” playing out here.
The formation of dense star-forming gas emerges from a combination of instabilities. Such as ubiquitous wind at high speeds which carries large amounts of material from star forming galaxies. These winds carries enriching gas — gas which can be used to convert gas into stars — , metals and magnetic fields — depending on the metal — with angular momentum. The stellar winds can affect star formation and galaxy formation while also the amount of magnetic metal can also affect star formation. The power of the magnetic fields can induce Magnetorotationally-Driven Galactic Turbulence and the Formation of Giant Molecular Clouds which allow star and galaxy formation.
We need a better understanding of chaos so we can model chaotic dynamic systems like the Interstellar medium from initial conditions we measure. Kinda like a weather man, we have a better understand how stars and galaxies are forms but also we can predict — in a certain amount of uncertainty — when a star or galaxy is formed.
While another new field surrounding the world of chaos is quantum chaos which aims to connect and understand how classical chaos affects quantum mechanic and the quantum world.
We are only beginning to understand the chaotic nature of various systems and learning how to define them mathematically and using them to predict. Understanding chaos is very important to understand the world. Dimensionality is another big player in the field of chaos. Chaotic behavior can be exhibited in two dimensional continuous systems in non-Euclidean geometry. Chaos can also occur in linear systems provided that they have infinite dimensions.
Chaos is a key to understanding the Universe and an important part in solving various important questions we have about the world we live in.
