Hello. Today is July 23rd, 2011. I have moved into my new place and less some space the home is coming together nicely. I decided I would start my Quantum search at the very beginning with a book by David J. Griffiths titled, "Introduction to Quantum Mechanics." But this is not the subject of the blog today. I haven't actually started reading it except for the first couple chapters but from what I've read this gets straight into the meat of Quantum Mechanics with the tools that one uses to perform simple calculations. From these initial tools I may try and explore philosophies about meaning. I hope though that we do not extend ourselves too far and get lost in the translation between the language of mathematics and English as there is no official English to Math dictionary or even an unofficial one. My biggest concern is that reality is obscured in the formalism of mathematics and what may make sense mathematically is not a true representation of the world. One of my other goals is to use LaTeX to express the mathematical relationships for you the reader. This may take me a bit to hash out so this blog will be a snails pace until I develop the means to communicate with you effectively.
Instead, today I will try and talk about Chaos in a very limited sense and hopefully add an entry to your Mathematical Dictionary. I have been reading a book titled, "The Essence of Chaos," by Edward N. Lorenz. He was very concerned with predicting the weather and is inadvertently the father of the "Butterfly Effect," from a paper named, Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Lorenz is quick to point out however that the likelihood that butterfly would cause a Tornado is equal to the likelihood that a butterfly would prevent a tornado in Texas. This illustrates on of Lorenz's big ideas which is Sensitive Dependence. Sensitive Dependence is the idea that beginning from an initial state looking at the same system later it will look nothing alike. A state has sensitive dependence if it can change by orders of magnitude very quickly. If you are not looking closely then you might interpret this as chaos when really it is sensitive dependence.
Imagine standing on a balcony in Grand Central Station and taking a picture of the lobby then waiting one minute and taking another picture. Continue in this fashion until you have one hundred pictures. Examining one state to the next there may be no coherence or regularity in the positions of people. This could be seen as chaos but we're really seeing a system that changes rapidly in short periods of time. If you were to instead, take pictures ever ten seconds, you may be able to track the motions of people and rather than concluding that the system is chaotic, you may discover that the system is merely volatile, or sensitive.
Despite making this discovery about Grand Central Station, we still cannot predict what the system will look like in another ten seconds. There are way too many factors in this case to track the thoughts, plans and movements of everyone who is in or will be inside Grand Central Station. If we observe long enough, we can begin to see behavior that is periodic however; there are more people in the Station in the morning and evening when people go to work or leave the city to go home. Though the system is unpredictable we can look at it and infer patterns. What then should we label this system, is it inherently chaotic or can we track everyone who uses the Station.
This example seems to have a few consequences first and foremost; chaos seems to be defined by our capacity to track changes. Spotting the difference between a ruler that is 1000 millimeters (mm) long and one that is 1005 mm long would be challenging but we assume that every ruler we use is exact despite this inability. Our inability to perceive differences becomes an assurance that everything is the same. Lorenz illuminates this idea perfectly showing that our ability to predict is limited by our ability to measure. Imagine a meter stick making machine that used the last stick as a guide but had an error of 5%. After three iterations a meter stick could be (1.05*1.05*1.05)=1.15763 meters long. The inability of our machine to detect a 5% difference led to a false prediction of the size of 1 meter.
As this post is called Chaos Experiment Number 1 I have a fun and easy project that I pulled from the pages of "The Essence of Chaos." Leave the cell A1 empty. In A2 input the equation ((A1^2)-A1). That is the quantity in A1 squared minus A1. Coy this cell and paste it as far as your curiosity allows. I made my spread sheet as columns of 100 up to 19,400. This will iteratively square the quantity before it then subtract the squared quantity. This is our system.
Now experiment. Input any number you want into the initial box. Once you've tried a few things read further and see a few of the things I discovered.
My favorite thing to do is always start with 0 as 0 has a tendency to test the limits of any system. Obviously every entry will consequently be 0. Our system, from an initial state of 0, is perfectly stable. Nothing will perturb it from being 0 and we can predict to the nth value of the system for any value of n where n greater than or equal to 1. This is will be our rock as we will always know where 0 takes us.
Next let us plug in 1. This leads us to the the exact same result as 0. -1 however provides an infinite string of 2.
My favorite is plugging in 2 which returns 2 over and over again to infinity. -2 however explodes and we end up in the billions after 6 iterations.
Positive 3 interestingly enough brings us to the exact same numbers as -2. If we continue this method you will discover than n and -n+1 will produce the exact same results. We have found that a relationship exists between the integers.
The next thing that caught my attention is that anything greater than 2 is explosive and outstrips the capabilities of my computer easily. Putting in a number between -1 and 2 that is not the number 1 or 0 will ultimately lead to a pattern. The values will oscillate between positive and negative values continually getting closer to 0. This pattern is consistent for all values that I checked. If you find something different please share it in the comments =).
Chaos. Based on a few examples it seems as if we defined some rules of this system. We can predict what will happen based on the number we put in but can we easily predict what those numbers will be? If we didn't know the algorithm before hand how long would it take for this system to cease being chaotic to us? Chaos is matter of what we are perceiving. Ultimately it seems as if Chaos is a necessary generalization that can qualify a great deal of information quickly.
I will say that Chaos is indeed something real. It exists just outside our perception and is manifest when we can see how systems change. We can be aware of our uncertainty but that does not eliminate it. This brings up something to ponder: If we could somehow track ever single variable, every single motion and every single fluctuation in existence, have we eliminated Chaos? Can we ever eliminate uncertainty in measurement?
Thanks much. Don't hesitate to point out mistakes in my logic or errors in my grammar and feel free to share your thoughts or ideas.