A comparison study of diffusion properties in Brownian motion models: From the stochastic to discrete and continuous chaotic-based models

The Brownian motion has been studied from different perspectives. Einstein proposed the first mathematical description of the Brownian motion of a free particle in one dimension; later, Langevin proposed another model using stochastic differential equations based on Newton's second law, and recently, deterministic models based on Langevin equations have been proposed, where the fluctuating acceleration is replaced, by the jerk equation and by a discrete system “booster” capable of generating chaos. In this work, we compare the statistical properties of the Brownian motion generated by a chaotic map, the Jerk equation, and the classical Langevin Model under parameter variations. We analyze their properties, such as the mean square displacement, the probability distribution for average displacement, and the type of diffusion generated by these models through the detrended fluctuation analysis and the diffusion coefficient calculation. Our results reveal that deterministic models can serve as viable alternatives to classical stochastic models, offering comparable statistical properties for particle diffusion.