We study the motion of a particle governed by a generalized Langevin equation. We show that, when no fluctuation-dissipation relation holds, the long-time behavior of the particle may be from stationary to superdiffusive, along with subdiffusive and diffusive. When the random force is Gaussian, we derive the exact equations for the joint and marginal probability density functions for the position and velocity of the particle and find their solutions. © 1996 The American Physical Society.

This paper investigates the motion of a Brownian particle experiencing both a friction (biased) force and a randomly fluctuating force with a long-time-correlation function ${\mathit{C}}_{\mathit{f}}$(t)∼${\mathit{t}}^{\mathrm{-}\mathrm{\beta }}$, 0<β<1, 1<β<2, and β=1, instead of a Dirac δ function. The generalized Langevin equation and Fokker-Planck equation and corresponding solution are presented. It is shown that when 0<β<1 or 1<β<2, the diffusion motion of the Brownian particle is the anomalous diffusion that is related to fractal Brownian motion (FBM). But when β=1 the diffusion motion is anomalous diffusion with no connection to FBM. The effects of friction retardation result in a probability density function for finding the particle at displacement X at time t that depends on the initial value of velocity of the particle. The approach in this paper may provide a systematic method for the study of particles diffusing in fractal media.

The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using the Laplace analysis we derive exact expressions for the relaxation functions of the particle in terms of generalized Mittag-Leffler functions and its derivatives from a generalized Langevin equation. Our results show that the oscillator displays an anomalous diffusive behavior. In the strictly asymptotic limit, the dynamics of the harmonic oscillator corresponds to an oscillator driven by a noise with a pure power-law autocorrelation function. However, at short and intermediate times the dynamics has qualitative difference due to the presence of the characteristic time of the noise.