Fractional motions

• Published in: Physics reports Vol. 527; no. 2; pp. 101 - 129
• Main Authors: Eliazar, Iddo I., Shlesinger, Michael F.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 10.06.2013
• Subjects: Brownian motion; Fractal trajectories; Fractional Brownian motion; Fractional Lévy motion; Hurst exponent; Joseph effect; Joseph exponent; Langevin’s equation; Long-range correlations; Lévy motion; Noah effect; Noah exponent; Random walks; Scaling limits; Selfsimilarity; Short-range correlations; Sub-diffusion; Super-diffusion; Universality; Long-range correlations; Langevin’s equation; Super-diffusion; Fractional Lévy motion; Lévy motion; Noah exponent; Hurst exponent; Short-range correlations; Universality; Joseph exponent; Selfsimilarity; Brownian motion; Sub-diffusion; Random walks; Noah effect; Joseph effect; Fractional Brownian motion; Scaling limits; Fractal trajectories
• ISSN: 0370-1573; 1873-6270
• DOI: 10.1016/j.physrep.2013.01.004

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.