Stochastic Dynamics Crossover From 1/ƒ3 to Flicker Noise

Abstract

AbstractFinite time processes within the limits of the Newton equation and zero inertia motion (i.e., road to chaos) are studied by numerically solving the ordinary, stochastic Langevin equation in 1D for a free particle with inertia moving in a medium with viscosity γ. In these simulations, the scaling behaviour of particle trajectories x(t) and velocities v(t) with time are derived and the inclusion of non‐zero particle masses is shown to define the asymptotic time limit τc at which — independently of γ — the system evolves into the well‐known statistically stationary state characterized by x2(t)> æ t and flicker noise. The time τc is further analysed from the correlation length given by the two‐point autocorrelation function of the particle velocity at each value of γ. It is found that the noise power spectrum of v(t) is characterized by flicker noise for frequencies ƒ ≦ ƒc ≈ 1/τc, whereas for ƒ > ƒc, the noise power spectra behaves as 1/ƒv, where v varies between the limits of Newton's equation (i.e., v = 3) and road to chaos (i. e., v = 1). Furthermore, at times τ < τc and 0 < γ < ∞, an ad‐hoc exponent for the scaling of the variance of particle velocities with time is shown to characterize a subset of multifractal dimensions df(γ) while the single particle trajectories are shown to display a rather different subset of exponents on increasing γ. Generic features of this transition are nicely given by Poincaré maps in the velocity space.