Chaos Theory

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CHAOS CHAOS CHAOS CHAOS CHAOS

Introduction:

Introduction Before getting on with the “Chaos” subject, there are some points in which you have to be informed: The laws of nature, are basically the way our minds tend to describe things. As far as many scientists are concerned, they may be entirely wrong (like the belief that people had long ago that the Earth was flat) And so, to know the nature of the laws of physics, you’d better have an understanding of the human mind. Our minds like to use 2 main ways of thinking: 1.deductive reasoning: Which is basically the famous “cause and effect” 2.inductive reasoning: The prediction that the sun will rise tomorrow is an example of inductive reasoning based on the fact that the sun has faithfully risen everyday so far in our experience.

Itroduction(2):

Itroduction (2) Of the most famous and evolutionary of theses laws, are The Laws Of Thermodynamics: 1. The First Law basically says that energy or matter can neither be created nor destroyed. In terms of the machine, this meant that the total energy output (work by the machine) is equal to the heat supplied. In other words, the change in the internal energy of a closed system is equal to the heat added to the system minus the work done by the system.

Introduction(3):

Introduction(3) 2. The Second Law essentially says that it is impossible to obtain a process where the unique effect is the subtraction of a positive heat from a reservoir and the production of a positive work. Energy exhibits entropy. It moves away form its source. In this sense, energy or heat cannot flow form a colder body to a hotter body. You cannot keep a continual flow of heat to work to heat to work without adding energy to the system. In machine terms, you have to add energy to get more work, and the ratio of heat to work will never equal 100% due to energy expanding away from its source.

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Introduction(4) 3.The Third Law explains this further. It says that all processes cease as temperature approaches absolute zero. This is the temperature at which molecules cease movement, cease producing kinetic energy. In other words, there is no energy. 4.There’s a so-called Zeroth law as well! It essentially says that if each of two systems is in equilibrium with a third system, the first two are in equilibrium with one another. For example, John weighs the same as Bill. Sam weighs the same as Bill. Therefore John and Sam weigh the same. Now, back to the main subject:

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In physics, chaos is a word with a specialized meaning, one that differs from the everyday use of the word. To a physicist, the phrase "chaotic motion" really has nothing with whether or not the motion of a physical system is frenzied or wild in appearance. In fact, a chaotic system can actually evolve in a way which appears smooth and ordered. So, chaos refers to the issue of whether or not it is possible to make accurate long-term predictions about the behaviour of the system. Before we go on, let’s take a look at some phrases:

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Determinism: Determinism is the philosophical belief that every event or action is the inevitable result of preceding events and actions. Thus, in principle at least, every event or action can be completely predicted in advance, or in retrospect. As a philosophical belief about the material world, determinism can be traced as least as far back as the time of Ancient Greece, several thousand years ago. When the idea of cause and effect started to rise. Thus until recently, it was assumed that it was always possible to make accurate long-term predictions of any physical system so long as one knows the starting conditions well enough.

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According to the deterministic model of science, the universe unfolds in time like the workings of a perfect machine, without a shred of randomness or deviation from the predetermined laws.

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The person most closely associated with the establishment of determinism at the core of modern science is Isaac Newton. Newton demonstrated that his three laws of motion, combined through the process of logic, could accurately predict the orbits in time of the planets around the sun, the shapes of the paths of projectiles on earth, and the schedule of the ocean tides throughout the month and year, among other things.

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Newton's laws are completely deterministic because they imply that anything that happens at any future time is completed determined by what happens now. Newton's three laws were so successful that for several centuries after his discovery, the science of physics consisted largely of demonstrating how his laws could account for the observed motion of nearly any imaginable physical process. Although Newton's laws were superseded around the year 1900 by a larger set of physical laws, determinism remains today as the core philosophy and goal of physical science.

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One of the important innovations that created modern science around the year 1500 A.D. was the idea that the laws of the material universe could be understood meaningfully only by expressing physical properties as quantified measurements, that is, in numerical terms and not just in words. For example, although Newton's laws are expressible in words, in order to apply the laws to study a particular system, it is necessary to employ the laws in their form as mathematical equations.

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Newton's laws are perhaps the most important examples of dynamical laws, which means that they connect the numerical values of measurements at a given time to their values at a later or earlier time. The measurements that appear in Newton's laws depend on the particular system being studied, but they typically include the position, speed, and direction of motion of all the objects in the system, as well as the strength and direction of any forces on these objects, at any given time in the history of the system.

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In expressing the measurements appropriate for a given system, the values of the measurements at a given starting time are called the initial conditions for that system. As dynamical laws, Newton's laws are deterministic because they imply that for any given system, the same initial conditions will always produce identically the same outcome.

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The Newtonian model of the universe is often depicted as a billiard game, in which the outcome unfolds mathematically from the initial conditions in predetermined fashion, like a movie that can be run forwards or backwards in time. The billiard game is a useful analogy, because on the microscopic level, the motion of molecules can be compared to the collisions of the balls on the billiard table, with the same dynamical laws invoked in both cases.

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Uncertainty: One of the fundamental principles of experimental science is that no real measurement is infinitely precise, but instead must necessarily include a degree of uncertainty in the value. This uncertainty which is present in any real measurement arises from the fact that any imaginable measuring device--even if designed and used perfectly---can record its measurement only with a finite precision .

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One way to understand this fact is to realize that in order to record a measurement with infinite precision, the instrument would require an output capable of displaying an infinite number of digits. By using more accurate measuring devices, uncertainty in measurements can often be made as small as needed for a particular purpose, but it can never be eliminated completely , even as a theoretical idea.

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In dynamics, the presence of uncertainty in any real measurement means that in studying any system, the initial conditions cannot be specified to infinite accuracy. But for example, in studying the motion of a rocket, one could know the final position of the rocket ten times as accurately by specifying the initial conditions at launch ten times as accurately. It is important to remember that the uncertainty in the dynamical outcome does not arise from any randomness in the equations of motion--since they are completely deterministic--but rather from the lack of the infinite accuracy in the initial conditions .

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Having understood what is meant by determinism, initial conditions, and uncertainty of measurements, you can now learn about dynamical instability, which to most physicists is the same in meaning as chaos. Dynamical instability refers to a special kind of behaviour in time found in certain physical systems and discovered around the year 1900, by the physicist Henri Poincaré. Poincaré was a physicist interested in the mathematical equations which describe the motion of planets around the sun.

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The equations of motion for planets are an application of Newton's laws, and therefore completely deterministic. That these mathematical orbit equations are deterministic means, of course, that by knowing the initial conditions---in this case, the positions and velocities of the planets at a given starting time---you find out the positions and speeds of the planets at any time in the future or past.

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Of course, it is impossible to actually measure the initial positions and speeds of the planets to infinite precision, even using perfect measuring instruments, since it is impossible to record any measurement to infinite precision. Thus there always exists an imprecision, however small, in all astronomical predictions made by the equation forms of Newton's laws. Up until the time of Poincaré, the lack of infinite precision in astronomical predictions was considered a minor problem, however, because of a tacit assumption made by almost all physicists at that time. The assumption was that if you could shrink the uncertainty in the initial conditions---perhaps by using finer measuring instruments---then any imprecision in the prediction would shrink in the same way.

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In other words, by putting more precise information into Newton's laws, you got more precise output for any later or earlier time. Thus it was assumed that it was theoretically possible to obtain nearly-perfect predictions for the behaviour of any physical system. But Poincaré noticed that certain astronomical systems did not seem to obey the rule that shrinking the initial conditions always shrank the final prediction in a corresponding way. By examining the mathematical equations, he found that although certain simple astronomical systems did indeed obey the "shrink-shrink" rule for initial conditions and final predictions, other systems did not.

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The astronomical systems which did not obey the rule typically consisted of three or more astronomical bodies with interaction between all three. For these types of systems, Poincaré showed that a very tiny imprecision in the initial conditions would grow in time at an enormous rate. Thus two nearly-indistinguishable sets of initial conditions for the same system would result in two final predictions which differed vastly from each other. Poincaré mathematically proved that this "blowing up" of tiny uncertainties in the initial conditions into enormous uncertainties in the final predictions remained even if the initial uncertainties were shrunk to smallest imaginable size. That is, for these systems, even if you could specify the initial measurements to a hundred times or a million times the precision, etc., the uncertainty for later or earlier times would not shrink, but remain huge.

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The gist of Poincaré's mathematical analysis was a proof that for these "complex systems," the only way to obtain predictions with any degree of accuracy at all would entail specifying the initial conditions to absolutely infinite precision. For these astronomical systems, any imprecision at all, no matter how small, would result after a short period of time in an uncertainty in the deterministic prediction which was hardly any smaller than if the prediction had been made by random chance.

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The extreme "sensitivity to initial conditions" mathematically present in the systems studied by Poincaré has come to be called dynamical instability, or simply chaos . Because long-term mathematical predictions made for chaotic systems are no more accurate that random chance, the equations of motion can yield only short-term predictions with any degree of accuracy. Although Poincaré's work was considered important by some other foresighted physicists of the time, many decades would pass before the implications of his discoveries were realized by the science community as a whole.. One reason was that much of the community of physicists was involved in making new discoveries in the new branch of physics called quantum mechanics, which is physics extended to the atomic realm.

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One of the most important discoveries was made in 1963, by the meteorologist Edward Lorenz, who wrote a basic mathematical software program to study a simplified model of the weather. Specifically Lorenz studied a primitive model of how an air current would rise and fall while being heated by the sun. Lorenz's computer code contained the mathematical equations which governed the flow the air currents. Since computer code is truly deterministic, Lorentz expected that by inputing the same initial values, he would get exactly the same result when he ran the program. Lorenz was surprised to find, however, that when he input what he believed were the same initial values, he got a drastically different result each time.

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By examining more closely, he realized that he was not actually inputting the same initial values each time, but ones which were slightly different from each other. He did not notice the initial values for each run were different because the difference was incredibly small, so small as to be considered microscopic and insignificant by usual standards.

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The mathematics inside Lorenz's model of atmospheric currents was widely studied in the 1970's. Gradually it came to be known that even the smallest imaginable discrepancy between two sets of initial conditions would always result in a huge discrepancy at later or earlier times, the hallmark of a chaotic system, of course. Scientists now believe that like Lorenz's simple computer model of air currents, the weather as a whole is a chaotic system. This means that in order to make long-term weather forecasts with any degree of accuracy at all, it would be necessary to take an infinite number of measurements.

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Because the atmosphere is chaotic, the uncertainties, no matter how small, would eventually overwhelm any calculations and defeat the accuracy of the forecast. This principle is sometimes called the "Butterfly Effect." In terms of weather forecasts, the "Butterfly Effect" refers to the idea that whether or not a butterfly flaps its wings in a certain part of the world can make the difference in whether or not a storm arises one year later on the other side of the world.

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And so, it is now believed that long-term predictions of weather is impossible because of the uncertainties that the measurements bring. The discovery of chaos seems to imply that randomness lurks at the core of any deterministic model of the universe. One of the most interesting issues in the study of chaotic systems is whether or not the presence of chaos may actually produce ordered structures and patterns on a larger scale. Some scientists have speculated that the presence of chaos---that is, randomness operating through the deterministic laws of physics on a microscopic level---may actually be necessary for larger scale physical patterns to arise.

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Recently, some scientists have come to believe that the presence of chaos in physics is what gives the universe its "arrow of time," the irreversible flow from the past to the future. As the study of chaos in physics enters its second century, the issue of whether the universe is truly deterministic is still an open question, and it will undoubtedly remain so, even as we come to understand more and more about the dynamics of chaotic systems.

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Check these sites out for more information: http://order.ph.utexas.edu/chaos/index.html http://www.physicsplanet.com/articles/three-laws-of-thermodynamics http://www.entropylaw.com/

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