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

Full Text:



P. Langevin, “Sur la th ́eorie du mouvement brownien”, C. R. Acad. Sci.146, 530–533 (1908);English translation: “On the Theory of Brownian Motion”, D. S. Lemons and A. Gythiel(1997) Am. J. Phys.65, 1079 (1997).

A. Einstein, Ann. Physik17, 549 (1905),Investigations on the theory of the Brownian move-ment, (Dover Publications, Inc. 1956).

R. Brown, “On the Particles contained in the Pollen of Plants; and on the General Existenceof active Molecules in Organic and Inorganic Bodies”, Edinburgh New Philosophical Journal5(1828) 358–371.

N. Wax,Selected papers on noise and stochastic processes, (Dover Publications, 1954); W. T.Coffey, Yu. P. Kalmykov,The Langevin Equation:With Applications to Stochastic Problems inPhysics, Chemistry and Electrical Engineering, (World Scientific Publishing Company, 2004).

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy”, Rev. Mod. Phys.15, 1(1943).

D. S. Lemons,An Introduction to Stochastic Processes in Physics, (The Johns Hopkins Uni-versity Press, 2002).

Daniel T. Gillespie,Markov Processes: An Introduction for Physical Scientists, (Elsevier,1991).

R. Czopnik and P. Garbaczewky, “Brownian motion in a magnetic field”, Phys. Rev. E63,021105 (2001).

T. P. Sim ̃oes and R. E. Lagos, “Kramers equation for a charged Brownian particle: The exactsolution”, Physica A355, 274 (2005)

F. N. C. Paraan, M. P. Solon, and J. P. Esguerra, “Brownian motion of a charged particledriven internally by correlated noise”, Phys. Rev. E77, 022101 (2008)

L. J. Hou, Z. L. Miskovic, A. Piel, and P. K. Shukla, “Brownian dynamics of charged particlesin a constant magnetic field”, Phys. Plasmas16, 053705 (2009)

E. Bringuier, “From mechanical motion to Brownian motion, thermodynamics and particletransport theory”, Eur. J. Phys.29, 1243 (2008).

J. T ́othov ́a and G. Vasziov ́a and L. Glod and V. Lis ́y, “Langevin theory of anomalous Brownianmotion made simple”, Eur. J. Phys.32, 645 (2011).

D. S. Lemons and D. L. Kaufman, “Brownian Motion of a Charged Particle in a MagneticField”, IEEE Transactions of Plasma Science,27, 1288 (1999).

E. Bringuier, “On the Langevin approach to particle transport”, Eur. J. Phys.27, 373 (2006).

J. I. Jim ́enez-Aquino and M. R. Romero-Bastida, “Brownian motion of a charged particle ina magnetic field”, Rev. Mex. Fis. E52(2) 182 (2006).

Jim ́enez-Aquino, J. I. and Velasco, R. M. and Uribe, F. J., “Brownian motion of a classicalharmonic oscillator in a magnetic field”, Phys. Rev. E77, 051105 (2008).

M.Yaghoubi, M. E. Foulaadvand, A. B ́erut and J. Luczka, “Energetics of a driven Brownianharmonic oscillator”, J. of Stat. Mech. Theo. and Exp.2017, 113206 (2017).

M. Mandrysz and B. Dybiec, “Energetics of the undamped stochastic harmonic oscillator”,A. Phy. Pol. B49, 871-882 (2018).

S. F.Nørrelykke and H. Flyvbjerg, “Harmonic oscillator in heat bath: Exact simulation oftime-lapse-recorded data and exact analytical benchmark statistics”, Phys. Rev. E83, 041103(2011).

M. J. Madsen and A. D. Skowronski, “Brownian motion of a trapped microsphere ions”, Am.J. Phys.82, 934 (2014).

G. Volpe, “Simulation of a Brownian particle in an optical trap”, Am. J. Phys.81, 224 (2013).

T. Li and M. G. Raizen, “Brownian motion at short time scales”, Ann. Der Phys.525, 281(2013).

A. Darras, J. Fiscina, N. Vandewalle, and G. Lumay, “Relating Brownian motion to diffusionwith superparamagnetic colloids”, Am. J. Phys.85, 265 (2017).

Y. Pomeaua and J. Piasecki, “The Langevin equation”, C. R. PHYS.18, 570 (2017).

R. Newburgh, J. Peidle, and W. Rueckner “Einstein, Perrin, and the reality of atoms: 1905revisited”, Am. J. Phys.74, 478 (2006).

N. Lucero-Azuara, N. S ́anchez-Salas, and J.I. Jim ́enez-Aquino “Brownian motion across amagnetic field: Langevin approach revisited”. Eur. J. Phys.41, 3 (2020).

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


  • There are currently no refbacks.

Revista Mexicana de Física E

ISSN: 2683-2216 (on line), 1870-3542 (print)

Semiannual publication of Sociedad Mexicana de Física, A.C.
Departamento de Física, 2o. Piso, Facultad de Ciencias, UNAM.
Circuito Exterior s/n, Ciudad Universitaria. C. P. 04510 Ciudad de México.
Apartado Postal 70-348, Coyoacán, 04511 Ciudad de México.
Tel/Fax: (52) 55-5622-4946, (52) 55-5622-4840. rmf@ciencias.unam.mx