Harmonic oscillator Brownian motion: Langevin approach revisited

O. Contreras-Vergara, N. Lucero-Azuara, N. Sánchez-Salas, J. I. Jiménez-Aquino


The original strategy applied by Langevin to Brownian movement problem is used to solve the case of a free particle under a harmonic potential. Such straightforward strategy consists in separating the noise term
in the Langevin equation in order to solve a deterministic equation associated
with the Mean Square Displacement (MSD). In this work, to achieve our goal we first calculate the variance for the stochastic harmonic oscillator and then the MSD appears immediately. We study the problem in the damped and lightly damped cases and show that, for times greater than the relaxation time, Langevin's original strategy is quite consistent with the exact theoretical solutions reported by Chandrasekhar and Lemons, these latter obtained using the statistical properties of a Gaussian white noise. Our results for the MSDs are compared  with the exact theoretical solutions as well as with the numerical simulation.


Langevin Equation, Brownian motion, Harmonic Oscillator Brownian motion, MSD

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DOI: https://doi.org/10.31349/RevMexFisE.18.97


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REVISTA MEXICANA DE FÍSICA E, year 18, issue 1, January-June 2021. Semiannual Journal published by Sociedad Mexicana de Física, A. C. Departamento de Física, 2º Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Apartado Postal 70-348. Tel. (+52)55-5622-4946, https://rmf.smf.mx/ojs/rmf-e, rmf@ciencias.unam.mx. Chief Editor: José Alejandro Ayala Mercado.  INDAUTOR Certificate of Reserve: 04-2019-080216402500-203, ISSN: 2683-2216 (on line), 1870-3542 (print), both granted by Instituto Nacional del Derecho de Autor.  Responsible for the last update of this issue, Technical Staff of Sociedad Mexicana de Física, A. C., Fís. Efraín Garrido Román, 2º. Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Date of last modification, January 04, 2021.

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