Moongyu Park

Scaling Laws and Fokker--Planck Equations for 3-Dimensional Porous Media with Fractal Mesoscale

Transport is studied in three-scale porous media with fractal mesoscale. On the microscale the dispersive mixing is governed by a stochastic ordinary differential equation with stationary, ergodic, Markovian drift velocity and an

THE COMPLEXITY OF BROWNIAN PROCESSES RUN WITH NONLINEAR CLOCKS

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On upscaling operator-stable Lévy motions in fractal porous media

-stable Levy diffusion. The inclusion of Levy diffusion allows one to study self-motile particles such as flagellated microbes. On the mesoscale it is assumed that a fractal Eulerian flow field with spatially homogeneous increments gives rise to a fractal drift velocity with temporally stationary increments. The drift velocity is assumed to be Levy. These processes have stationary increments and fractal graphs. The diffusive structure on the mesoscale is determined by the asymptotic microscale process. On the macroscale there is no additional drift; i.e., the physics is controlled solely by the asymptotic behavior of the mesoscale process. Scaling laws from the micro- to meso- and meso- to macroscales are obtained as well as the Fokker--Planck equations for the transition...

Operator-stable Lévy motions and renormalizing the chaotic dynamics of microbes in anisotropic porous media

Anomalous diffusion occurs in many branches of physics. Examples include diffusion in confined nanofilms, Richardson turbulence in the atmosphere, near-surface ocean currents, fracture flow in porous formations and vortex arrays in rotating flows. Classically, anomalous diffusion is characterized by a power law exponent related to the mean-square displacement of a particle or squared separation of pairs of particles: 〈|X(t)|2〉 ~tγ. The exponent γ is often thought to relate to the fractal dimension of the underlying process. If γ > 1 the flow is super-diffusive, if it equals 1 it is classical, otherwise it is sub-diffusive. In this work we illustrate how time-changed Brownian position processes can be employed to model sub-, super-, and classical diffusion, while time-changed Brownian velocity processes can be used to model super-diffusion alone. Specific examples presented include transport in turbulent fluids and renormalized transport in porous media. Intuitively, a time-changed Brownian process is a classical Brownian motion running with a nonlinear clock (Bm-nlc). The major difference between classical and Bm-nlc is that the time-changed case has nonstationary increments. An important novelty of this approach is that, unlike fractional Brownian motion, the fractal dimension of the process (space filling character) driving anomalous diffusion as modeled by Bm-nlc positions or velocities does not change with the scaling exponent, γ.

Fractional Brownian motion run with a multi-scaling clock mimics diffusion of spherical colloids in microstructural fluids.

The dynamics of motile particles, such as microbes, in random porous media are modeled with a hierarchical set of stochastic differential equations which correspond to micro, meso and macro scales. On the microscale the motile particle is modeled as an operator stable Levy process with stationary, ergodic, Markov drift. The micro to meso and meso to macro scale homogenization is handled with generalized central limit theorems. On the mesoscale the Lagrangian drift (or the Lagrangian acceleration) is assumed Levy to account for the fractal character of many natural porous systems. Diffusion on the mesoscale is a result of the microscale asymptotics while diffusion on the macroscale results from the mesoscale asymptotics. Renormalized Fokker-Planck equations with time dependent dispersion tensors and fractional derivatives are presented at the macro scale.

Upscaling Lévy motions in porous media with long range correlations

Self-motile particles, such as microbes, that can preferentially orient their microscale movement are embedded in a multiscale anisotropic porous medium. Such an environment is consistent with most natural geological systems. The preferential directional orientation and position of the self-motile particles at the microscale is accounted for with an operator-stable Levy process. The anisotropy of the porous medium on the mesoscale is accounted for with an operator-stable velocity with either finite or infinite second moments. Upscaling is accomplished with generalized central limit theorems which can be shown to be equivalent to a renormalized group approach. If the mesoscale drift is operator-stable Levy (fractal), then the macroscale Fokker–Planck equation has time-dependent diffusion tensor and fractional derivatives which are directionally dependent.

Upscaling Interpretation of Nonlocal Fields, Gradients, and Divergences

The collective molecular reorientations within a nematic liquid crystal fluid bathing a spherical colloid cause the colloid to diffuse anomalously on a short time scale (i.e., as a non-Brownian particle). The deformations and fluctuations of long-range orientational order in the liquid crystal profoundly influence the transient diffusive regimes. Here we show that an anisotropic fractional Brownian process run with a nonlinear multiscaling clock effectively mimics this collective and transient phenomenon. This novel process has memory, Gaussian increments, and a multiscale mean square displacement that can be chosen independently from the fractal dimension of a particle trajectory. The process is capable of modeling multiscale sub-, super-, or classical diffusion. The finite-size Lyapunov exponents for this multiscaling process are defined for future analysis of related mixing processes.

Stability analysis and error estimate of flowfield-dependent variation (FDV) method for first order linear hyperbolic equations

Bacterial motility has been modeled by Levy motions which were upscaled in porous media, with drift processes such as Levy processes and time-changed Brownian motion, via central limit theorems. It was possible to prove the limit theorems and upscale the processes because of the independence of their increments. Therefore, it has not been applied to processes with correlated increments such as fractional Brownian motions. In this paper, the upscaling approach is generalized to porous media with long-range correlated processes. The processes are modeled by a fractional Brownian velocity process and p-diffusive position processes that were defined and used to classify diffusion processes by O’Malley and Cushman [“A renormalization group classification of nonstationary and/or infinite second moment diffusive processes,” J. Stat. Phys. 146, 989–1000 (2012)]10.1007/s10955-012-0448-3. A few examples of p-diffusive processes are discussed by computing the values of the parameter p.

Stability analysis of VEISV propagation modeling for network worm attack

Recently, nonlocal generalizations of the classical gradient, divergence, and curl have been introduced to examine nonlocal field problems, and develop a nonlocal vector calculus. Here we introduce, by definition, the concept of a nonlocal field variable and relate it to its classical local counterpart. Subsequently, we relate the concept of measurement via a convolution (an upscaling) to the nonlocal field variables and the nonlocal operators. It is shown via Fourier transform that the nonlocal gradient and divergence can be thought of as a very special convolution averaging of their classical counterparts. A nonlocal self-diffusion equation is upscaled and written in terms of nonlocal operators.

Anomalous diffusion as modeled by a nonstationary extension of Brownian motion.

Flowfield-dependent variation (FDV) method has been used in fluid mechanics and astrophysics. This method has been developed to solve many flow problems such as the interactions of shock waves with turbulent boundary layers in transonic flow and hypersonic flow, and chemically reacting flows. However, stability analysis and error estimate are missing in the numerical method. In this paper we analyze FDV method for a first-order linear hyperbolic equation, and apply finite difference method to discretize the space variable. Stability conditions and optimal error estimates are proved.