APPLICATION OF FRACTAL THEORY IN MOLECULAR DYNAMICS SIMULATION
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Fractional Brownian motion
Brownian dynamics
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Hurst exponent
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Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the one-parameter fractional Brownian motion is constructed using a two-parameter Poisson process. The proof involves the tightness and identification of finite-dimensional distributions.
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In the paper we present several numerical algorithms to simulate the fractional Brownian motion (the generalized Wiener process). The algorithms have been developed on the basis of the integral representation of the fractional Brownian motion and the spectral decomposition for its increments. The convergence of the numerical methods has been studied.
Fractional Brownian motion
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Fractional Brownian motions (fBm) provide important models for a wide range of physical phenomena whose empirical spectra obey power laws of fractional order. Extensions of fBm to higher dimension has become increasingly important. In this paper we study isotropic d-dimensional fBm in the framework of inhomogeneous random fields, and we derive exact expressions for the dual-wavenumber spectrum of fractional Brownian fields (fBf). Based on the spectral correlation structure of fBf we develop an algorithm for synthesizing fBf. The proposed algorithm is accurate and allow us to generate fractional Brownian motions of arbitrary dimension.
Fractional Brownian motion
Wavenumber
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Single-file diffusion behaves as normal diffusion at small time and as anomalous subdiffusion at large time. These properties can be described by fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann-Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as solution of fractional Langevin equation with zero damping. Various types of fractional Langevin equations and their generalizations are then considered to decide whether their solutions provide the correct description of the long and short time behaviors of single-file diffusion. The cases where dissipative memory kernel is a Dirac delta function, a power-law function, and a combination of both of these functions, are studied in detail. In addition to the case where the short time behavior of single-file diffusion behaves as normal diffusion, we also consider the possibility of the process that begins as ballistic motion.
Fractional Brownian motion
Hurst exponent
Anomalous Diffusion
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In some applications, for instance, finance, biomechanics, turbulence or internet traffic, it is relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is dependent on the frequency. In this contribution, we describe the multiscale fractional Brownian motions which present a parameter H as a piecewise constant function of the frequency. We provide the main properties of these processes: long-memory and smoothness of the paths. Then we propose a statistical method based on wavelet analysis to estimate the different parameters and prove a functional Central Limit Theorem satisfied by the empirical variance of the wavelet coefficients.
Fractional Brownian motion
Hurst exponent
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An approach to develop stochastic models for studying anomalous diffusion is proposed. In particular, in this approach the stochastic particle trajectory is based on the fractional Brownian motion but, for any realization, it is multiplied by an independent random variable properly distributed. The resulting probability density function for particle displacement can be represented by an integral formula of subordination type and, in the single-point case, it emerges to be equal to the solution of the spatially symmetric space-time fractional diffusion equation. Due to the fractional Brownian motion, this class of stochastic processes is self-similar with stationary increments in nature and uniquely defined by the mean and the auto-covariance structure analogously to the Gaussian processes. Special cases are the time-fractional diffusion, the space-fractional diffusion and the classical Gaussian diffusion.
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Anomalous Diffusion
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Derivative (finance)
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