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Self-similar Gaussian processes for modeling anomalous diffusion.

S C Lim1, S V Muniandy

  • 1School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. sclim@pkrisc.cc.ukm.my

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2002
PubMed
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Gaussian models for anomalous diffusion, including fractional Brownian motion, exhibit different properties despite similar probability distributions. A single anomalous diffusion parameter is insufficient for mechanism characterization.

Area of Science:

  • Physics
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • Anomalous diffusion describes particle transport deviating from standard Brownian motion.
  • Gaussian processes are frequently used to model anomalous diffusion phenomena.
  • The mean-square displacement scaling with t(alpha) is a key indicator of anomalous diffusion.

Purpose of the Study:

  • To investigate various Gaussian models for anomalous diffusion.
  • To determine if a single anomalous diffusion parameter sufficiently characterizes underlying mechanisms.
  • To explore the necessity of multifractional models for variable anomalous diffusion exponents.

Main Methods:

  • Analysis of time-rescaled Brownian motion.
  • Examination of two types of fractional Brownian motion.

Related Experiment Videos

  • Application of the generalized Langevin equation.
  • Utilizing the Slepian theorem to compare first passage time distributions.

    • Main Results:

    • Multiple Gaussian models satisfy the anomalous diffusion relation (mean-square displacement ~ t(alpha)).
    • Despite identical probability distributions, fractional Brownian motion variants and time-rescaled Brownian motion show distinct first passage time distributions.
    • A single anomalous diffusion parameter (alpha) is insufficient to differentiate these models.

    Conclusions:

    • The anomalous diffusion relation with a single parameter is inadequate for characterizing diverse underlying mechanisms.
    • Multifractional extensions of Gaussian processes are required to model anomalous diffusion with a variable exponent alpha(t).