A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field
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 o...
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2025-02-01
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| author | Adriana Ruiz-Silva Bahia Betzavet Cassal-Quiroga Rodolfo de Jesus Escalante-Gonzalez José A. Del-Puerto-Flores Hector Eduardo Gilardi-Velazquez Eric Campos |
| author_facet | Adriana Ruiz-Silva Bahia Betzavet Cassal-Quiroga Rodolfo de Jesus Escalante-Gonzalez José A. Del-Puerto-Flores Hector Eduardo Gilardi-Velazquez Eric Campos |
| author_sort | Adriana Ruiz-Silva |
| collection | DOAJ |
| description | 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. |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-1ab22ea68f574e5cb55be6fbbdf9a5692025-08-20T02:04:36ZengMDPI AGMathematics2227-73902025-02-0113580410.3390/math13050804A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force FieldAdriana Ruiz-Silva0Bahia Betzavet Cassal-Quiroga1Rodolfo de Jesus Escalante-Gonzalez2José A. Del-Puerto-Flores3Hector Eduardo Gilardi-Velazquez4Eric Campos5Programa de Ingeniería Biomédica, Universidad Estatal de Sonora, Unidad Hermosillo, Ley Federal del Trabajo, Col. Apolo, Hermosillo 83100, Sonora, MexicoFacultad de Ciencias, Universidad Autónoma de San Luis Potosí, Av. Parque Chapultepec 1570, Privadas del Pedregal, San Luis Potosí 78295, San Luis Potosí, MexicoElectrical, Electronic and Mechatronics Department, Technological Institute of San Luis Potosí, Tecnológico Avenue, Soledad de Graciano Sánchez 78437, San Luis Potosí, MexicoFacultad de Ingeniería, Universidad Panamericana, Álvaro del Portillo 49, Zapopan 45010, Jalisco, MexicoFacultad de Ingeniería, Universidad Panamericana, Josemaría Escrivá de Balaguer 101, Aguascalientes 20290, Aguascalientes, MexicoDivision 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, MexicoBrownian 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.https://www.mdpi.com/2227-7390/13/5/804deterministic Brownian motionharmonic oscillatorDFA analysisjerk system |
| spellingShingle | Adriana Ruiz-Silva Bahia Betzavet Cassal-Quiroga Rodolfo de Jesus Escalante-Gonzalez José A. Del-Puerto-Flores Hector Eduardo Gilardi-Velazquez Eric Campos A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field Mathematics deterministic Brownian motion harmonic oscillator DFA analysis jerk system |
| title | A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field |
| title_full | A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field |
| title_fullStr | A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field |
| title_full_unstemmed | A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field |
| title_short | A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field |
| title_sort | comparative study of brownian dynamics based on the jerk equation against a stochastic process under an external force field |
| topic | deterministic Brownian motion harmonic oscillator DFA analysis jerk system |
| url | https://www.mdpi.com/2227-7390/13/5/804 |
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