Self-similar Gaussian processes for modeling anomalous diffusion.
ABSTRACT We study some Gaussian models for anomalous diffusion, which include the time-rescaled Brownian motion, two types of fractional Brownian motion, and models associated with fractional Brownian motion based on the generalized Langevin equation. Gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean-square displacement to vary with t(alpha), 0<alpha<2. However, these processes have different properties, thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism. Although the two versions of fractional Brownian motion and time-rescaled Brownian motion all have the same probability distribution function, the Slepian theorem can be used to compare their first passage time distributions, which are different. Finally, in order to model anomalous diffusion with a variable exponent alpha(t) it is necessary to consider the multifractional extensions of these Gaussian processes.
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ABSTRACT: We study the effect of an arbitrary stationary random force on the motion of damped particles. Using a Langevin description, we derive exact expressions for the dispersion of the particle position, of the particle velocity, and their cross dispersion. The particles can exhibit anomalous diffusion, and the connection between this behavior and the functional form of the noise correlations is investigated in detail. We also study anomalous diffusion for the special cases of overdamped and undamped particles.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 01/2001; 62(6 Pt A):7729-34.
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ABSTRACT: 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 Cf(t)~t-beta, 0Physical Review A 02/1992; 45(2):833-837. · 3.04 Impact Factor
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ABSTRACT: Fractional Brownian motion (FBM) is widely used in the modeling of phenomena with power spectral density of power-law type. However, FBM has its limitation since it can only describe phenomena with monofractal structure or a uniform degree of irregularity characterized by the constant Holder exponent. For more realistic modeling, it is necessary to take into consideration the local variation of irregularity, with the Holder exponent allowed to vary with time (or space). One way to achieve such a generalization is to extend the standard FBM to multifractional Brownian motion (MBM) indexed by a Holder exponent that is a function of time. This paper proposes an alternative generalization to MBM based on the FBM defined by the Riemann-Liouville type of fractional integral. The local properties of the Riemann-Liouville MBM (RLMBM) are studied and they are found to be similar to that of the standard MBM. A numerical scheme to simulate the locally self-similar sample paths of the RLMBM for various types of time-varying Holder exponents is given. The local scaling exponents are estimated based on the local growth of the variance and the wavelet scalogram methods. Finally, an example of the possible applications of RLMBM in the modeling of multifractal time series is illustrated.Physical Review E 05/2001; 63(4 Pt 2):046104. · 2.31 Impact Factor