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Abstract
The subject of deterministic chaos is a new philosophical approach to the study of complex systems. It deals with system behavior that should be predictable by the deterministic laws of physics, but in which the number of variables, and their interaction, are so large and complex that they can be regarded as a chaotic and unpredictable system. In some cases, however, it is possible to observe patterns of probable behavior. These patterns can, in essence, be discovered only experimentally, except that powerful Monte Carlo modeling of the interaction of a set of variables can enable one to anticipate certain patterns. Thus, Monte Carlo routines predict that clustering of pigments in a randomized paint structure is describable by a hyperbolic function.
Probability theory should be regarded as a branch of deterministic chaos. Different chaotic systems encountered in powder technology manifest different distribution statistics describable by Gaussian and log-Gaussian distributions as well as other well-known probability functions. Examples of such systems are described with suggestions as to why a particular probability distribution function, as an appropriate mathematical descriptive function for a specifically interactive system, can be anticipated.
Another relatively new scientific discipline that is having an impact on powder technology is fractal geometry. Applied fractal geometry, a branch of more formal subject developed originally by Mandelbrot, is essentially the geometry of rugged systems. Many powder systems produced by chaotic systems are fractally structured. A prime example of this is fumed fine particles, such as soot and precipitate flocs, produced by interacting variables, which have fractal structure. Furthermore, it is becoming apparent that the fractal dimension of a system, such as carbon black pigment, embodies information on the formation dynamics that created it. This fact is discussed with respect to fumes and fractured material.
Two other branches of deterministic chaos with application to powder technology are critically self-organized systems and catastrophe theory. The use of theorems from these descriptions to characterize and study the flow of powders is discussed and typical systems described.