The renormalization group and fractional Brownian motion

TL;DR

This paper demonstrates how fluctuations and interactions in field theories lead to probability distributions resembling fractional Brownian Motion, using the Renormalization Group to analyze deviations from classical Brownian motion.

Contribution

It introduces a novel application of the Renormalization Group to show how non-linearities induce fractional Brownian Motion in field theories beyond the classical regime.

Findings

Fluctuations and interactions cause deviations from classical Brownian motion.

The probability distribution transitions to fractional Brownian Motion due to non-linear effects.

Results are applicable to non-equilibrium systems and critical phenomena.

Abstract

We find that in generic field theories the combined effect of fluctuations and interactions leads to a probability distribution function which describes fractional Brownian Motion (fBM) and ``complex behavior''. To show this we use the Renormalization Group as a tool to improve perturbative calculations, and check that beyond the classical regime of the field theory (i. e., when no fluctuations are present) the non--linearities drive the probability distribution function of the system away from classical Brownian Motion and into a regime which to the lowest order is that of fBM. Our results can be applied to systems away from equilibrium and to dynamical critical phenomena. We illustrate our results with two selected examples: a particle in a heat bath, and the KPZ equation.