One of the greatest shift in physics happened after the famed Isaac Newton shaped our conception of nature to a very strict mechanistic one. The universe as a whole was understood to be operating with some strict laws like a clock. If you know everything there is to know about an event now you can predict the next instant with reasonable accuracy. Much like a billiard board, if you know what angle your cue will shoot at what speed you can map out the trajectories billiard balls will take after being hit before the impact. Such ideas gave rise to the sentiment that we can predict the next moment just by solving some math and thus free will as a faculty made no sense. The word chance was attributed to willful or otherwise errors in measurement that can be perfected or adjusted within limits to make a sensible prediction of the future. A bleak concept that the happening in the universe is already determined by its naturalistic process from the get go and we are just unaware gears and pulleys doing our parts as we live and breathe. Then came chaos!
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Chaos in a system is characterized by the manner in which error accumulates from the prediction and the reality. To understand this first we need to understand what is meant by error. Imagine a simple physical system that can be well characterized by Newtonian mechanics, say a pendulum (a ball held by a string swinging). We can calculate out the position of the ball at a specific time if we know the initial position of the ball and certain other variables that effects the movement of the ball during swinging. To make a general point here in a simple mathematical model of a system there are 3 things that we need to know to have a sound predictive power: 1) A coherent mathematical model, 2) the initial condition (needed if the system is time-dependent) 3) the boundary conditions (the extra variables that I mentioned that affects the system during the process). In short if we have these for the pendulum we can trace the balls movement just from mathematics. Discrepancy in the inputs of the initial and boundary condition will create a difference between what was predicted by math to the real case scenario. As time passes the error accumulates and in simple system as described the accumulation is almost linearly proportional. That is the rate of accumulation of the discrepancy between calculated and measured is proportional to the amount of time passed. Thus for a short period of time, the errors won’t matter. Systems with this makeup are called non-chaotic systems and they provide us with manageable predictive capabilities. As the errors vary linearly the predictive capability still hold. For a chaotic system the situation becomes very complicated.
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To imagine a chaotic system you have to expand your imagination of that very pendulum to another level. Imagine you have 2 pendulum the pendulum A followed by pendulum B. It’s not hanging side by side rather one pendulum forms the base of another that is it’s a double pendulum scenario. So pendulum A s ball is fitted with the string of pendulum B and they are allowed to swing. The input of the position and speed for pendulum A will have an effect on B s motion and B s inertia will, in turn, affect the pendulum A. even a slight variation of A s position in our equation will generate a drastic change in the position and pathway of the system. The calculated results and the measured will diverge to such a length that it will be impossible to even call it a decent model of reality. To note, the problem isn’t really about the model. The chaotic system doesn’t really work against the law of nature rather it works very much in line with the laws of nature. However unlike the non-chaotic system the error of measured and calculated doesn’t pile up linearly it piles up exponentially. This feature makes it such that a slight difference in the input of the initial and boundary condition can generate a massive difference in the final state of the system compared to the predicted. There comes the famous example of a butterfly flapping its wing in one end of the world can potentially create a tornado/cyclone in another part: The Butterfly effect. Slight act in the world can magnify to mammoth proportion and have an overreaching effect. To get an illustration of what I am talking about just check the video of a simulation done on 3 such double pendulums each have a very small difference in mass from the other(very small but different boundary condition). Imagine that you want to chart out the path of the red pendulum with your mathematical model. The model is accurate there is no error in the model but your inputs of initial and boundary condition is off just by 0.5 i.e. the inputs you are giving is either of the blue or the green pendulum. The difference in the pathway and the speed of the pendulum as you can see will be drastic. The error accumulating per time is so huge that prediction will lose its meaning any more. Now imagine that the output of your mathematics serves as an input to another system. Can you see the massive difference it will cause? The predictive power, if any, will completely lose its mark. The study in the field of exponential error piling up is what’s known as the chaos theory.
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By now I think the problem for the excessively deterministic universe is clear to all. Our knowledge of a system’s initial and boundary conditions can never be 100 % accurate. If anyone studied instrumentation and measurement he/she will acknowledge that every measurement of ours are accompanied by a certain level of errors. Thus, for a chaotic system, the errors would indicate that prediction of the future becomes extremely difficult to account. The universe we live in is a fine amalgamation of chaotic and non-chaotic processes. With a lack of a physical model for the whole reality coupled with the impossibility of knowing our input parameters down to the very last digit tells us a profound fact: the universe is the fastest simulator of itself. While the universe behaves deterministically in the sense that it operates in accordance with the laws of physics, the moment to come can be dramatically different from what we can expect given even the tiniest of uncertainty in our knowledge of the present. The conclusion from that gives us the sense that the universe is still somewhat open. Knowing and characterizing the laws of nature that determine the natural process is not good enough to know tomorrow because somewhere far away a tiny butterfly can be flapping its way to its nest, and that innocuous act can bring about a tomorrow that we didn’t even dream of living by knowing just today.
As Salamo Alaikum 