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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"> </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> </span></span><span style="font-size:10pt;font-family:Arial">Newton</span><span style="font-size:10pt;font-family:Arial">’s classical Law of Universal gravitation is understood, and relatively simple.<span> </span>Attraction is directly proportional to mass and inversely proportional to distance.<span> </span>In the case of two bodies, the motion of the two bodies is readily determined and we quickly are led to Kepler’s laws of orbital motion.<span> </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> </span>The situation continues as more bodies are added to the scenario.<span> </span>We can find approximate solutions, very good approximations, using computers and simulation techniques for many special cases.<span> </span>But there are questions that still defy solution.</span><span style="font-size:10pt;font-family:Arial"> </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> </span>That question might be of interest to most people, scientists and non-scientists alike.<span> </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 – with obvious consequences in terms of major, abrupt global warming or cooling.<span> </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> </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> </span>The analysis of orbital motion of smaller bodies subject to perturbations from larger ones is a more difficult and perhaps pressing problem – 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> </span>Computer codes have been devised to solve this equation and they are generally successful in providing useful answers to scientists and engineers.<span> </span>There are, however, exceptions to this success.<span> </span>Deep questions remain in modeling turbulent flows.<span> </span>Practical problems also remain in solving large-scale problems.<span> </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> </span>Weather prediction seems to be such a problem.<span> </span>I’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. </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> </span>Systems of interest to biologists, climatologists, and ecologists involve coupled effects of heat transfer, fluid dynamics, population dynamics and chemistry.<span> </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> </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> </span><span> </span>This is not to say that models of such complex systems should be dismissed.<span> </span>We need those models and the insight that they provide.<span> </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 “chaos” or even more misleadingly as “chaos theory”.<span> </span>There really is no chaos theory.<span> </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> </span>Complex systems are, well, complex.<span> </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> </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 “All difficult problems have simple, easy-to-understand, wrong answers.”</span><span style="font-size:10pt;font-family:Arial"> </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> </span>Sometimes action or no action must be decided upon in the face of massive uncertainty.<span> </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> </span>That final decision properly involves the population as a whole.<span> </span></span></p> <div class="Discussion_UserSignature"> </div>