Chaos Theory - Pretty interesting stuff and somehow very profound. It is yet another of those mathematical models which goes against the very grain of mathematics itself (well the 19th century view of mathematics to be precise).
Mathematics has always been regarded as the holiest of the scientific subjects owing to its inherent properties of being testable, predictable and deterministic. It was natural to assume that if something could be described mathematically then there would remain no surprises and it made perfect sense. Then maths got complicated!
1) Probability Theory - Here, after rigorous mathematical somersaults all you would end up with is the probability of the occurrence of an event not a certainty. Though it can be argued that it does limit the range of possible outcomes and hence is deterministic in some sense.
2) Quantum Theory - My favorite ! The uncertainty principle virtually guarantees that no matter what instrument is used it is impossible to determine the position and momentum simultaneously. Then of course there is the "Measurement problem" the mention of which will send shivers down any Physicists spine. To put it bluntly, if you set up an experiment to measure something like a Photon as a particle then you will detect it as a particle. An experiment set up to detect it as a wave will detect it as a wave. Funny isnt it? :-)
Well thats how the collapsible wave function of gauge boson behaves..deal with it. So essentially here mathematics has determined that you cannot determine.
3) Chaos Theory - Perhaps the most counter intuitive of all theories. Seemingly innocuous linear equations display behavior that is very surprising. I was trying to model the one dimension logistic map equation which is incidentally one of the good example of deterministic chaos. This equation is also very intriguing because this has wide uses in fields as diverse as ecology! ( Population Growth model) . In essence this equation has a very high dependency on the initial conditions and the behavior changes drastically with changes in initial conditions.
The linked spreadsheet has the equation embedded and plots a graph of 200 points based on the input value of the scaling parameter k ( Sheet: Input, Cell: A8). I have put down the ranges of k that induces different behavior from the same equation. Have fun trying it out for yourselves.
Also, Wolframs Mathematica has done a better job of modeling this equation so if you want to try out a widget which allows a greater control of parameters as well as a mean square minimized error then please refer the Wolfram Demonstration Project
And to think, our universe Probably started from Chaotic conditions on a scale that necessitates the existence of Quantum effects.
If God exists he must be ROFLing now.. couldnt have made it more puzzling. :-)
3 comments:
Interesting stuff :)
Interesting stuff :)
From day one, probability always made me think the same.
After Baye's theorem, stochastic processes and what not, probability still cannot tell me whether India is gonna win the next match.
Education needs to change.
And to think of it, probability wasn't the only crap I learned!!
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