A new efficient numerical method for contact mechanics of rough surfaces

[1] Since aquifer parameters may have statistical dependence structures that are present across a huge range of scales, the concepts of fractional Brownian motion (fBm) have been used in both analytic and numerical stochastic settings. Most previous models have used isotropic scaling characterized by a single scalar Hurst coefficient. Any real-world anisotropy has been handled by an elliptical stretching random K field. We define a d-dimensional extension of fBm in which the fractional-order integration may take on different orders in the d primary (possibly nonorthogonal) scaling directions, and the degree of connectivity and long-range dependence is freely assigned via a probability measure on the unit sphere. This approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of K in certain directions. It also allows for the representation of fracture networks in a continuum setting: The eigenvectors of the scaling matrix describe the primary fracture scaling directions, and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions. In a numerical experiment, the motion of solutes through 2-D “operator-fractional” Gaussian fields depends very much on transverse Hurst coefficients. Transverse scaling in the range of fractional Gaussian noise engenders greater plume mixing and a transition to a Fickian regime. Higher orders of integration, in the range of fractional Brownian motion, are associated with thicker layering of aquifer sediments and more preferential, unmixed transport. Therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport.

/pdf/aquifer-operator-scaling-and-the-effect-on-solute-mixing-and-3w3gv8g5qp.pdf

Aquifer operator scaling and the effect on solute mixing and dispersion

A proposed new technique extracts frequency domain information from the observed trajectory of a single microscopic particle in a complex fluid.

Spectral Content of a Single Non-Brownian Trajectory

Time-dependent processes are often analysed using the power spectral density (PSD), calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble-average. Frequently, the available experimental data sets are too small for such ensemble averages, and hence it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from $S(f,T)$, the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable, parametrized by frequency $f$ and observation-time $T$, for a broad family of anomalous diffusions---fractional Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability density function. We show that $S(f,T)$ is proportional---up to a random numerical factor whose universal distribution we determine---to the ensemble-averaged PSD. For subdiffusion ($H 1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for $H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on $T$. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations.

Spectral content of a single non-Brownian trajectory

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

/pdf/fractional-laplace-motion-4r8coki40w.pdf

Fractional Laplace motion

This paper focuses on simulating fractional Brownian motion (fBm). Despite the availability of several exact simulation methods, attention has been paid to approximate simulation (i.e., the output is approximately fBm), particularly because of possible time savings. In this paper, we study the class of approximate methods that are based on the spectral properties of fBm's stationary incremental process, usually called fractional Gaussian noise (fGn). The main contribution is a proof of asymptotical exactness (in a sense that is made precise) of these spectral methods. Moreover, we establish the connection between the spectral simulation approach and a widely used method, originally proposed by Paxson, that lacked a formal mathematical justification. The insights enable us to evaluate the Paxson method in more detail. It is also shown that spectral simulation is related to the fastest known exact method.

On spectral simulation of fractional Brownian motion

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On spectral simulation of fractional Brownian motion