Non-classical point of view of the Brownian motion generation via Fractional deterministic model

TL;DR

This paper introduces a deterministic model for Brownian motion using fractional derivatives in a modified Langevin equation, capturing key statistical properties of stochastic Brownian motion without randomness.

Contribution

It proposes a novel deterministic fractional differential system that replicates Brownian motion's statistical features, eliminating the need for stochastic terms.

Findings

System exhibits linear mean square displacement growth.

System produces Gaussian-distributed positions.

Detrended fluctuation analysis confirms Brownian behavior.

Abstract

In this paper we present a dynamical system to generate Brownian motion based on the Langevin equation without stochastic term and using fractional derivatives, i.e., a deterministic Brownian motion model is proposed. The stochastic process is replaced by considering an additional degree of freedom in the second order Langevin equation. Thus it is transformed into a system of three first order linear differential equations, additionally $\alpha$-fractional derivative are considered which allow us obtain better statistical properties. Switching Surfaces are established as a part of fluctuating acceleration. The final system of three $\alpha$-order linear differential equations does not contain a stochastic term, so the system generates motion in a deterministic way. Nevertheless, from the time series analysis, we found that the behavior of the system exhibits statistics properties of Brownian motion, such as, a linear growth in time of the mean square displacement, a Gaussian distribution. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.