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Discontinuities in self-affine functions lead to multiaffinity.
1Schuit Institute for Catalysis and Department of Chemical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. smitchell@physast.uga.edu
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 21, 2006
Summary
Introducing vertical discontinuities into self-affine functions creates multiaffine scaling. This finding applies to systems exhibiting complex scaling behavior, offering insights into phenomena like surface morphology and noise.
Area of Science:
- Physics
- Complex Systems
- Applied Mathematics
Background:
- Many theoretical and applied systems exhibit multiaffine scaling at small scales.
- Self-affine functions are common in modeling natural phenomena.
Purpose of the Study:
- To demonstrate analytically and numerically how introducing vertical discontinuities into self-affine functions leads to multiaffine behavior.
- To investigate the role of discontinuity distribution in generating multiaffinity.
Main Methods:
- Analytical derivation of scaling forms for functions with discontinuities.
- Numerical examination of two distinct functions with varying discontinuity distributions.
- Comparison of analytical predictions with numerical results.
Main Results:
- Vertical discontinuities transform self-affine functions into multiaffine functions.
- The distribution of discontinuities directly influences the resulting multiaffinity.
- An analytic scaling form for the function of discontinuities at small scales was derived.
Conclusions:
- Multiaffine scaling in physical systems can arise from the presence of vertical discontinuities.
- The study provides a framework for understanding multiaffinity in terms of underlying discontinuous processes.
- Findings are relevant for modeling systems with surface overhangs or discontinuous noise.