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

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  • Adriana Ruiz-Silva

    (Programa de Ingeniería Biomédica, Universidad Estatal de Sonora, Unidad Hermosillo, Ley Federal del Trabajo, Col. Apolo, Hermosillo 83100, Sonora, Mexico)

  • Bahia Betzavet Cassal-Quiroga

    (Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, Av. Parque Chapultepec 1570, Privadas del Pedregal, San Luis Potosí 78295, San Luis Potosí, Mexico)

  • Rodolfo de Jesus Escalante-Gonzalez

    (Electrical, Electronic and Mechatronics Department, Technological Institute of San Luis Potosí, Tecnológico Avenue, Soledad de Graciano Sánchez 78437, San Luis Potosí, Mexico)

  • José A. Del-Puerto-Flores

    (Facultad de Ingeniería, Universidad Panamericana, Álvaro del Portillo 49, Zapopan 45010, Jalisco, Mexico)

  • Hector Eduardo Gilardi-Velazquez

    (Facultad de Ingeniería, Universidad Panamericana, Josemaría Escrivá de Balaguer 101, Aguascalientes 20290, Aguascalientes, Mexico)

  • Eric Campos

    (Division of Control and Dynamical Systems, Instituto Potosino de Investigación Científica y Tecnológica A. C., Camino a la Presa San José 2055, Col. Lomas 4 Sección, San Luis Potosí 78216, San Luis Potosí, Mexico)

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.

Suggested Citation

  • Adriana Ruiz-Silva & Bahia Betzavet Cassal-Quiroga & Rodolfo de Jesus Escalante-Gonzalez & José A. Del-Puerto-Flores & Hector Eduardo Gilardi-Velazquez & Eric Campos, 2025. "A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field," Mathematics, MDPI, vol. 13(5), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:804-:d:1602012
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    References listed on IDEAS

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    1. Beck, Christian, 1996. "Dynamical systems of Langevin type," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 233(1), pages 419-440.
    2. Caldeira, A.O. & Leggett, A.J., 1983. "Path integral approach to quantum Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 121(3), pages 587-616.
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