An approach to generate deterministic Brownian motion G. Huerta-Cuellar a,b,⇑ , E. Jiménez-López b , E. Campos-Cantón b , A.N. Pisarchik c,d a Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique Díaz de León 1144, Paseos de la Montaña, 47460 Lagos de Moreno, Jalisco, Mexico b Instituto Potosino de Investigación Científica y Tecnológica, División de Matemáticas Aplicadas, Camino a la Presa San José 2055, col. Lomas 4 sección, 78216 San Luis Potosí (SLP), Mexico c Centro de Investigaciones en Óptica, Loma del Bosque 115, Lomas del Campestre, Leon, Guanajuato 37150, Mexico d Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain article info Article history: Received 15 October 2013 Received in revised form 22 December 2013 Accepted 12 January 2014 Available online xxxx Keywords: Brownian motion Deterministic Brownian motion Unstable dissipative systems DFA analysis abstract We propose an approach for generation of deterministic Brownian motion. By adding an additional degree of freedom to the Langevin equation and transforming it into a system of three linear differential equations, we determine the position of switching surfaces, which act as a multi-well potential with a short fluctuation escape time. Although the model is based on the Langevin equation, the final system does not contain a stochastic term, and therefore the obtained motion is deterministic. Nevertheless, the system behav- ior exhibits important characteristic properties of Brownian motion, namely, a linear growth in time of the mean square displacement, a Gaussian distribution, and a 2 power law of the frequency spectrum. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Brownian motion has been extensively studied since the findings of the biologist Brown in 1828 [1] and first described by the mathematician Thiele [2] in his paper on the least squares method published in 1880. At that time, Brownian motion was defined as a continuous-time stochastic (or probabilistic) process characterized by normal distribution. The nature of the Brownian motion is uncertain and many questions still remain open of how it could depend on particle interactions with the environment, is this process stochastic or deterministic? After the Thiele’s paper, the study of Brownian motion has been followed independently by Bachelier [3] and Albert Ein- stein [4], who gave the first mathematical description of a free particle Brownian motion. Later, Smoluchowski [5] brought the solution of the problem to the attention of physicists. In 1908, Langevin [6] obtained the same result as Einstein, using a macroscopically description based on the Newton’s second law. He referred his approach to as ‘‘infinitely simplest’’ because it was much simpler than the one proposed by Einstein. Since the pioneering work of Langevin, many papers have been de- voted to the description of Brownian motion [7–16], where characteristic features of this behavior have been defined. The dynamical model of Brownian motion provided by Langevin [6], who used a second-order differential equation with a stochastic term, seems apparently from the nature of randomness. On the other hand, it is widely believed that Brownian motion can be rigorously derived from totally deterministic Hamiltonian models of classical mechanics. One of the reasons for this conviction is related to the widely used Van Hove’s method [17–19]. In one way or another, many attempts to establish a unified view of mechanics and thermodynamics [20] traced back to the Van Hove’s approach. The result of their method depended on whether one adopted the Heisenberg perspective corresponding to the time evolution of observables, 1007-5704/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2014.01.010 ⇑ Corresponding author at: Universidad de Guadalajara, Centro Universitario de los Lagos, Enrique Díaz de León 1144, Paseos de la Montaña, 47460 Lagos de Moreno, Jalisco, Mexico. Tel.: +52 4747424314. E-mail address: g.huerta@lagos.udg.mx (G. Huerta-Cuellar). Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Please cite this article in press as: Huerta-Cuellar G et al. An approach to generate deterministic Brownian motion. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.010