Chaos and Order
Chaos and order, as used in chaos theory, are terms used to describe conditions of complex systems in which, out of seemingly random, disordered (e.g. aperiodic) processes there arise processes that are deterministic and predictable. Accordingly, despite its name, chaos theory attempts to identify and quantify order in apparently unpredictable systems.
Along with quantum and relativity theories, chaos theory—with its inclusive concepts of chaos and order--is widely regarded as one of the great intellectual leaps of the twentieth century. The modern physical concepts of chaos and order, however, actually trace their roots to classical mechanical concepts introduced in English physicist Sir Isaac Newton's 1686 work, Philosophy Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). It was Newton, one of the inventors of the calculus, who revolutionized astronomy and physics by showing that the behavior of all bodies, celestial and terrestrial, was governed by the same laws of motion, which could be expressed as differential equations. These differential equations relate the rates of change of physical quantities to the values of those quantities themselves. Such calculated predictability of physical phenomena led to the concept of a mechanistic, clockwork universe that operated according to deterministic laws. The idea that the universe operated in strict accord with physical laws was profoundly influential on science, philosophy and theology.
Most physical models are devoted to the understanding of simple systems (e.g., kinetic molecular theories often rely on concepts related to a ball bouncing in a box). From fundamental laws, using easily quantifiable behavior of such simple systems, theorists often attempt to project the behavior of more complex systems (e.g., the collision and dynamics of hundreds of balls bouncing in a box). It was long thought by physicists that, with regard to these types of models, the complexity of a system simply veiled an underlying fundamental simplicity.
For example, according to classical deterministic concepts, the accurate analysis and prediction of complex systems (e.g., the determination of the momentum of a particular ball among hundreds of other balls bouncing and colliding in a box) could be calculated if the initial or starting conditions were accurately known. The fact that is usually impossible to predict the exact condition or behavior of a system (especially considering that such interactions or measurements of a systems must also alter the system itself) is explained away as the result of a lack of knowledge regarding starting conditions or a lack of calculating vigor (e.g., inadequate computing power).
Complex phenomena are those generally regarded as having too many variables (or too many possible conditions or states) to yield to conventional quantitative analysis. The motion of molecules in swirling smoke or the turbulent hydraulics of a river current, for example, are systems that exhibit such chaotic complexity.
Used together, the terms chaos and order fundamentally describe conditions related to the thermodynamic concept of entropy. Entropy is a measure of thermodynamic equilibrium used to explain irreversibility in physical and chemical processes. The second law of thermodynamics specifies that in an isolated system, increasing entropy corresponds to changes in the system over time and that entropy tends toward (a statistical mechanical concept) maximization. The second law of thermodynamics dictates that in natural processes, without work being done on a system, there is a movement from order to disorder.
According to the laws of thermodynamics, in all natural processes there is—when considering both system and surroundings--tendency a movement from the ordered state to a more the chaotic (disordered) state. Conversely, according to some chaos theory models, chaotic, unpredictable, and irreversible processes may, evolve into or produce ordered states.
Because entropy in natural processes increases over time, even very straightforward linear-type relationships must eventually take on a degree of irregularity (i.e., of seemingly disordered complexity). In accord with the second law of thermodynamics, apparently chaotic phenomenon arise from initially ordered (i.e., lower entropy) systems. This dual tendency toward increasing entropy and chaos from an initially stable state can take place spontaneously. Small perturbations in initial conditions intensify these tendencies. Chaos theorists describe such departures as the Butterfly Effect.
The study of such mathematical irregularities involving chaos and order remained a relatively unnoticed corner of advanced mathematics until the advent of the digital computer. In 1956, Edward Lorenz, a professor of meteorology at the Massachusetts Institute of Technology was studying the numerical solution to a set of three differential equations in three unknowns, a highly simplified version of the type of equations meteorologists were then using to describe atmospheric phenomena. Lorenz came to the conclusion that his set of differential equations displayed a sensitive dependence on initial conditions, a sensitivity of the same type that French mathematician Jules-Henri Poincaré had discovered for the Newtonian equations when those equations were applied to celestial dynamics. Lorenz, however, gave this phenomenon a new and highly appealing name, the butterfly effect, suggesting that, in the extreme, the flapping of a butterfly's wings in Kansas might be responsible for a monsoon in India a month later.
Precise definitions of chaos and order are hotly debated among physicists. The terms themselves presuppose definitions for entropy, complexity, and irreversibility that are anything but well-settled in the modern physics community. Regardless, physicists generaly agree that the discovery of underlying order--instead of complete randomness--in seemingly chaotic systems is perfectly consistent with the proper application of the laws of thermodynamics.
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