A Perspective on Science 4 -- More on Complexity

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DrRocket

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<p><span style="font-size:10pt;font-family:Arial">A Perspective on Science 4 -- More on Complexity</span><span style="font-size:10pt;font-family:Arial">&nbsp;</span></p><p><span style="font-size:10pt;font-family:Arial">There are problems for which the fundamental principles are well understood but for which the application of those principles to the systems involved do not yield ready solutions.</span></p><p><span style="font-size:10pt;font-family:Arial">A very simple example is the three-body problem.<span>&nbsp; </span></span><span style="font-size:10pt;font-family:Arial">Newton</span><span style="font-size:10pt;font-family:Arial">&rsquo;s classical Law of Universal gravitation is understood, and relatively simple.<span>&nbsp; </span>Attraction is directly proportional to mass and inversely proportional to distance.<span>&nbsp; </span>In the case of two bodies, the motion of the two bodies is readily determined and we quickly are led to Kepler&rsquo;s laws of orbital motion.<span>&nbsp; </span>One then tries to apply similar techniques to a system with three point masses, and one finds oneself unable to formulate and solve the problems in closed form.<span>&nbsp; </span>The situation continues as more bodies are added to the scenario.<span>&nbsp;&nbsp; </span>We can find approximate solutions, very good approximations, using computers and simulation techniques for many special cases.<span>&nbsp; </span>But there are questions that still defy solution.</span><span style="font-size:10pt;font-family:Arial">&nbsp;</span></p><p><span style="font-size:10pt;font-family:Arial">A case in point is the question of the stability of our solar system.<span>&nbsp; </span>That question might be of interest to most people, scientists and non-scientists alike.<span>&nbsp; </span>The question is whether the observed orbits of the planets are essentially fixed, or whether one or more, say the third rock from the Sun, might suddenly depart from its usual orbit and go into some other trajectory &ndash; with obvious consequences in terms of major, abrupt global warming or cooling.<span>&nbsp; </span>We can do numerical simulations of the motions of the planets, and the result is that the orbits do not show instability for a long period of time into the future, so there is no need to panic.<span>&nbsp; </span>But we cannot solve the complete problem and there is reason to believe that the orbits may become unstable in the far distant future.<span>&nbsp; </span>The analysis of orbital motion of smaller bodies subject to perturbations from larger ones is a more difficult and perhaps pressing problem &ndash; see the thread on Apophis.</span></p><p><span style="font-size:10pt;font-family:Arial">Fluid dynamics problems involve the solution of a very difficult partial differential equation, the Navier-Stokes equation.<span>&nbsp; </span>Computer codes have been devised to solve this equation and they are generally successful in providing useful answers to scientists and engineers.<span>&nbsp; </span>There are, however, exceptions to this success.<span>&nbsp; </span>Deep questions remain in modeling turbulent flows.<span>&nbsp; </span>Practical problems also remain in solving large-scale problems.<span>&nbsp; </span>Fluid dynamics is sufficiently challenging that a major reason for the existence of super computers is the solution of flow problems.</span></p><p><span style="font-size:10pt;font-family:Arial">Some problems, even if understood in principle, are sufficiently sensitive to initial conditions or boundary conditions as to defy effective practical analysis.<span>&nbsp; </span>Weather prediction seems to be such a problem.<span>&nbsp; </span>I&rsquo;m sure you have heard it said that a butterfly flaps its wings in </span><span style="font-size:10pt;font-family:Arial">Beijing</span><span style="font-size:10pt;font-family:Arial"> and it hails in </span><span style="font-size:10pt;font-family:Arial">Omaha</span><span style="font-size:10pt;font-family:Arial">, as an illustration of this sort of concern.&nbsp;</span></p><p><span style="font-size:10pt;font-family:Arial">The problems listed above which involve simple mechanics and thermodynamics pale in comparison to the difficulties in modeling truly complex systems.<span>&nbsp; </span>Systems of interest to biologists, climatologists, and ecologists involve coupled effects of heat transfer, fluid dynamics, population dynamics and chemistry.<span>&nbsp; </span>Those coupled effects are not sufficiently well understood and input data sufficiently well determined so as to be amenable to analysis based solely on first principles.<span>&nbsp; </span>Even if such understanding were to become available, the size of the resulting computer models might be beyond practical implementation on available equipment.<span>&nbsp; </span><span>&nbsp;</span>This is not to say that models of such complex systems should be dismissed.<span>&nbsp; </span>We need those models and the insight that they provide.<span>&nbsp; </span>It is to say that before the results are accepted, we should ask the necessary probing questions and be aware of the potential errors involved.</span></p><p><span style="font-size:10pt;font-family:Arial">Sometimes the effects of complexity are discussed under the heading of &ldquo;chaos&rdquo; or even more misleadingly as &ldquo;chaos theory&rdquo;.<span>&nbsp; </span>There really is no chaos theory.<span>&nbsp; </span>There are some useful pieces of knowledge available under the headings of nonlinear dynamics, ergodic theory, and stability theory of differential equations, but even there the results are scattered and not broadly applicable.<span>&nbsp; </span>Complex systems are, well, complex.<span>&nbsp; </span>They are also diverse in nature and require application of techniques tailored to the specific problem at hand, if any effective techniques are available at all.<span>&nbsp; </span>When faced with complex systems one ought to be cognizant of the difficulties involved and be skeptical of hard and fast conclusions in the face of massive uncertainty. Somebody once told me that &ldquo;All difficult problems have simple, easy-to-understand, wrong answers.&rdquo;</span><span style="font-size:10pt;font-family:Arial">&nbsp;</span></p><p><span style="font-size:10pt;font-family:Arial">Science is not always up to the task of providing definitive answers to complex and difficult problems.<span>&nbsp; </span>Sometimes action or no action must be decided upon in the face of massive uncertainty.<span>&nbsp; </span>In that case the best that can be done is to identify the possibilities, the uncertainties, and the potential consequences and then make an informed but non-scientific decision on the course of action based on social and economic considerations.<span>&nbsp; </span>That final decision properly involves the population as a whole.<span>&nbsp; </span></span></p> <div class="Discussion_UserSignature"> </div>
 
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Saiph

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<p>I'd like to elaborate a bit on "Chaos Theory" to help clear up many common misconceptions about the various phenomena attributed to this specific domain of scientific inquiry.</p><p>Whoo boy did my vocabulary show there!&nbsp; Yikes!</p><p>Anyway, one of the common conceptions of Chaos Theory is that it's basically unpredictable.&nbsp; If a butterfly flapping it's wings in china can cause a city destroying tornadoe in nebraska, then how do you predict the weather at all!</p><p>I like the way DrRocket puts it, they're more aptly named complex systems.&nbsp; Each step in a chaotic system is completely deterministic.&nbsp; Take the three body problem for example.&nbsp; You can write code to calculate the position of the three bodies at any given time, and their various properties (velocity, acceleration etc) and predict the next step.&nbsp; The problem in complex systems however, is that they are incredibly delicate as far as inputs are.</p><p>In a standard physics problem, say throwing a baseball, if you change one of the initial conditions slightly, your end result is also only slightly different.&nbsp; You throw the ball a little harder, it goes a little further.&nbsp; However in a chaotic system, this sort of common sense expectation doesn't pan out.&nbsp; If you change the inititial conditions at all, the end result can be drastically, and seemingly disproportionately different.</p><p>I'll go back to the three body problem, because I've actually written a program to demonstrate this.&nbsp; In my example I've got two equal mass objects orbiting eachother, happy as can be.&nbsp; I then throw in a third, equal mass object into the mix on an intersecting trajectory.</p><p>The result is best described as tangle fishing line of orbital paths.&nbsp; In the end, the original two are still orbiting eachother, but are now moving along together (no longer "stationary") and in a different orbital relationship (say further apart).&nbsp; The interloper was ejected from the system, on a different path, never to return.</p><p>Okay, no problem, each step is completely deterministic, if a bit complex.&nbsp; Now, I change the initial velocity of the interloper by 0.001%&nbsp; Shouldn't produce much change right?</p><p>The first quarter of the simulation is basically the same, but as time goes on the differences get more and more pronounced.&nbsp; Halfway through and there's only the most basic similarities, the end result:&nbsp; One of the originals gets ejected, and the interloper is now in a mutual orbit with the remaining object.&nbsp; They're not even traveling along the same final trajectory.</p><p>Change the initial conditions by another little bit:&nbsp; End result (began very similar) is agian completely different.&nbsp; Only the overall characteristics are the same (the entire system is drifting in the same direction due to conservation of momentum).&nbsp; But the final orbits and which one is ejected is almost impossible to predict, without doing a full simulation. &nbsp;</p><p>Now, consider the fact that I changed the initial conditions by only 0.001%...on one variable.&nbsp; This could easily be within the margin of error for say, meteorologists using the most sophisticated equipment.&nbsp; This is why they can't predict to far out into the future.&nbsp; They go much further than 3 days out, and the systems begin looking incredibly dissimilar, even with the best data available.&nbsp; A week out and you get the broadest predictions...cloudy, sunny, hot, cold.&nbsp; Beyond that...and they can't tell you.&nbsp;</p><p>&nbsp;</p><p>So these 'Chaotic' systems are completely deterministic, completely predictable.&nbsp; If you know every variable to the utmost precision.&nbsp; Because small changes (or error) in the initial conditions turn into drastic differences given enough time.&nbsp;</p> <div class="Discussion_UserSignature"> <p align="center"><font color="#c0c0c0"><br /></font></p><p align="center"><font color="#999999"><em><font size="1">--------</font></em></font><font color="#999999"><em><font size="1">--------</font></em></font><font color="#999999"><em><font size="1">----</font></em></font><font color="#666699">SaiphMOD@gmail.com </font><font color="#999999"><em><font size="1">-------------------</font></em></font></p><p><font color="#999999"><em><font size="1">"This is my Timey Wimey Detector.  Goes "bing" when there's stuff.  It also fries eggs at 30 paces, wether you want it to or not actually.  I've learned to stay away from hens: It's not pretty when they blow" -- </font></em></font><font size="1" color="#999999">The Tenth Doctor, "Blink"</font></p> </div>
 
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chembuff1982

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<p>&nbsp; I suppose very futuristic super AI computers could calculate probabilities of many many many many events almost like an if then logic machine, as of now it's almost impossible to calculate all these variables with the means we have and determine what might or might not happen.&nbsp; However like I said, with some type of super intelligent computer way in the future, knowing position, mass, speed, spin, tilt, orbiting objects, collisons, and the list goes on you could probably print out with high odds how likely a certain asteroid is going to hit earth in a billion years or what would happen if Jupiter fell out of orbit, etc.</p><p>&nbsp;</p><p>&nbsp;</p><p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I'd like to elaborate a bit on "Chaos Theory" to help clear up many common misconceptions about the various phenomena attributed to this specific domain of scientific inquiry.Whoo boy did my vocabulary show there!&nbsp; Yikes!Anyway, one of the common conceptions of Chaos Theory is that it's basically unpredictable.&nbsp; If a butterfly flapping it's wings in china can cause a city destroying tornadoe in nebraska, then how do you predict the weather at all!I like the way DrRocket puts it, they're more aptly named complex systems.&nbsp; Each step in a chaotic system is completely deterministic.&nbsp; Take the three body problem for example.&nbsp; You can write code to calculate the position of the three bodies at any given time, and their various properties (velocity, acceleration etc) and predict the next step.&nbsp; The problem in complex systems however, is that they are incredibly delicate as far as inputs are.In a standard physics problem, say throwing a baseball, if you change one of the initial conditions slightly, your end result is also only slightly different.&nbsp; You throw the ball a little harder, it goes a little further.&nbsp; However in a chaotic system, this sort of common sense expectation doesn't pan out.&nbsp; If you change the inititial conditions at all, the end result can be drastically, and seemingly disproportionately different.I'll go back to the three body problem, because I've actually written a program to demonstrate this.&nbsp; In my example I've got two equal mass objects orbiting eachother, happy as can be.&nbsp; I then throw in a third, equal mass object into the mix on an intersecting trajectory.The result is best described as tangle fishing line of orbital paths.&nbsp; In the end, the original two are still orbiting eachother, but are now moving along together (no longer "stationary") and in a different orbital relationship (say further apart).&nbsp; The interloper was ejected from the system, on a different path, never to return.Okay, no problem, each step is completely deterministic, if a bit complex.&nbsp; Now, I change the initial velocity of the interloper by 0.001%&nbsp; Shouldn't produce much change right?The first quarter of the simulation is basically the same, but as time goes on the differences get more and more pronounced.&nbsp; Halfway through and there's only the most basic similarities, the end result:&nbsp; One of the originals gets ejected, and the interloper is now in a mutual orbit with the remaining object.&nbsp; They're not even traveling along the same final trajectory.Change the initial conditions by another little bit:&nbsp; End result (began very similar) is agian completely different.&nbsp; Only the overall characteristics are the same (the entire system is drifting in the same direction due to conservation of momentum).&nbsp; But the final orbits and which one is ejected is almost impossible to predict, without doing a full simulation. &nbsp;Now, consider the fact that I changed the initial conditions by only 0.001%...on one variable.&nbsp; This could easily be within the margin of error for say, meteorologists using the most sophisticated equipment.&nbsp; This is why they can't predict to far out into the future.&nbsp; They go much further than 3 days out, and the systems begin looking incredibly dissimilar, even with the best data available.&nbsp; A week out and you get the broadest predictions...cloudy, sunny, hot, cold.&nbsp; Beyond that...and they can't tell you.&nbsp;&nbsp;So these 'Chaotic' systems are completely deterministic, completely predictable.&nbsp; If you know every variable to the utmost precision.&nbsp; Because small changes (or error) in the initial conditions turn into drastic differences given enough time.&nbsp; <br />Posted by Saiph</DIV><br /></p> <div class="Discussion_UserSignature"> You may be a genius, but google knows more than you! </div>
 
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DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&nbsp; I suppose very futuristic super AI computers could calculate probabilities of many many many many events almost like an if then logic machine, as of now it's almost impossible to calculate all these variables with the means we have and determine what might or might not happen.&nbsp; However like I said, with some type of super intelligent computer way in the future, knowing position, mass, speed, spin, tilt, orbiting objects, collisons, and the list goes on you could probably print out with high odds how likely a certain asteroid is going to hit earth in a billion years or what would happen if Jupiter fell out of orbit, etc.&nbsp;&nbsp; <br />Posted by chembuff1982</DIV></p><p>Actually it may be more difficult than that.&nbsp; Suppose that you had a computer with NO round-off error, and with enough memory and speed to calculate orbits with no errors, given all of the input parameters.&nbsp; Essentially a machine that could calculate precisely in closed form, with no errors due to numerical approximations.&nbsp; You still might not be able to give accurate predictions of events such as you postulate far into the future.</p><p>The reason is that some complex dynamical systems, are extremely sensitive to initial conditions.&nbsp; Very small changes in those conditions can make very large changes in system response as time progresses.&nbsp; This is fundamental characteristic of some systems.&nbsp; The implication is that even with perfect computational capability, the tiniest errors in measurement of the starting conditions can lead to big misses in later predictions.&nbsp; There is reason to think that multi-body orbital systems have this characteristic, though I don't know that this has been proved rigorously.&nbsp; Shorter term models are pretty good, but when you talk about billion year projections, I don't think we really know if such are viable.</p><p>Of course if our ability to make accurate determinations of orbital parameters and to know for sure all of the bodies that are involved progresses to the point of essential certainty, then such predictions could be made.&nbsp; These very sensitive systems are in fact deterministic and with perfect knowledge are in principle perfectly predictable.&nbsp; But you have to be virtually perfect, there is no "close enough".<br /></p> <div class="Discussion_UserSignature"> </div>
 
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