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Messages - beret

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Studying and Exam Taking / Eternal Recurrence
« on: December 12, 2008, 01:56:53 PM »

Interesting username, butterfly! It reminded me right away the technical notion of sensitive dependence on initial conditions in Chaos Theory I read. Small variations of the initial condition of a non-linear dynamical system that may produce large variations in the long term behavior of the system. So this is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position. The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear (or prevent a tornado from appearing). The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

The idea that one butterfly could have a far-reaching ripple effect on subsequent events seems first to have appeared in a 1952 short story by Ray Bradbury about time travel, although Lorenz made popular the term. In 1961, Lorenz was using a numerical computer model to rerun a weather prediction, when, as a shortcut on a number in the sequence, he entered the decimal .506 instead of entering the full .506127 the computer would hold. The result was a completely different weather scenario. Lorenz published his findings in a 1963 paper for the New York Academy of Sciences noting that "One meteorologist remarked that if the theory were correct, one flap of a seagull's wings could change the course of weather forever." Later speeches and papers by Lorenz used the more poetic butterfly. According to Lorenz, upon failing to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly's wings in Brazil set off a tornado in Texas as a title.



These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t=30. Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather).


Do I have a Choice, a Chance, to Make Each Return a Little Different? A Little Better?

No choice, no chance, no different, no better . . . (and no worse).

Each time you return it's exactly the same you. If there were the slightest infinitesimal difference, the butterfly effect would not only radically change your entire life and the future course of human history but radically change the whole future of the universe such that at the incredibly large number of years after which you are supposed to return, you wouldn't be there since the universe would be radically different.

That's how well connected you are to the universe.

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