Transformation of random distributions into power-like distributions due to non-linearities: application to geophysical phenomena

Details Transformation of random distributions into power-like distributions due to non-linearities: application to geophysical phenomena Transformation of random distributions into power-like distributions due to non-linearities: application to geophysical phenomena

Jan 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001geoji.144..197c&link_type=abstract

Geophysical Journal International, Volume 144, Issue 1, pp. 197-205.

Astronomy and Astrophysics

Astronomy

1

Crack Tip Propagation, Fractals, Non-Linear Transformations, Power Distribution, Stochastic Model

Scientific paper

In analyses of phenomena leading to fractal distributions there prevail opinions that the causes of such distributions are critical phenomena in phase transitions or SOC models. Power distributions are also associated with deterministic chaos generated by low-dimensional non-linear systems. Non-linearity is considered to be a necessary condition for scale invariance and fractal statistics. However, an explanation of what kinds of non-linearities lead to fractal distributions and how they do it is lacking. In this contribution, the influence of the non-linearity of a model on the output behaviour is investigated. In many cases the relation between the `input' x and the `output' y of a system can be expressed by a non-linear transformation y=g(x). The kind of non-linearity of a model that transforms an input random variable with exponential distribution into a variable with a long-tail distribution over a wide range of scales is analysed. A very wide class of non-linearly increasing functions g leads to power-like output distributions. Non-linear relations of the type y=g(x) can be solutions of random differential equations or of a system of random differential equations that describe some physical phenomena. Various kinds of random differential equations are analysed and discussed. The contribution shows in an elementary way the causes of universality of fractal distributions in many branches of science. A simple geophysical example of crack tip propagation is included.

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