A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field

Adriana Ruiz-Silva; Bahia Betzavet Cassal-Quiroga; Rodolfo de Jesus Escalante-Gonzalez; Del-Puerto-Flores, J. Alberto; Gilardi Velázquez, Héctor Eduardo; Eric Campos

doi:10.3390/math13050804

A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field

Journal

Mathematics

ISSN

2227-7390

Publisher

MDPI AG

Date Issued

2025-02-28

Author(s)

Adriana Ruiz-Silva

Bahia Betzavet Cassal-Quiroga

Rodolfo de Jesus Escalante-Gonzalez

Eric Campos

Type

text::journal::journal article

DOI

10.3390/math13050804

URL

https://scripta.up.edu.mx/handle/20.500.12552/12068

Abstract

Brownian motion has been studied since 1827, leading to numerous important advances in many branches of science and to it being studied primarily as a stochastic dynamical system. In this paper, we present a deterministic model for the Brownian motion for a particle in a constant force field based on the Ornstein–Uhlenbeck model. By adding one degree of freedom, the system evolves into three differential equations. This change in the model is based on the Jerk equation with commutation surfaces and is analyzed in three cases: overdamped, critically damped, and underdamped. The dynamics of the proposed model are compared with classical results using a random process with normal distribution, where despite the absence of a stochastic component, the model preserves key Brownian motion characteristics, which are lost in stochastic models, giving a new perspective to the study of particle dynamics under different force fields. This is validated by a linear average square displacement and a Gaussian distribution of particle displacement in all cases. Furthermore, the correlation properties are examined using detrended fluctuation analysis (DFA) for compared cases, which confirms that the model effectively replicates the essential behaviors of Brownian motion that the classic models lose.