If the mean-square displacement of a stochastic process is proportional to t;{beta} , beta not equal1 , then it is said to be anomalous. We construct a family of Markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean-square displacement. We label the family as an extended Brownian motion and show that they satisfy a Langevin equation with time-dependent diffusion coefficient. If the time derivative of the variance of the process is homogeneous, then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the Brownian motion. For two particles initially separated by a distance x , the finite-size Lyapunov exponent (FSLE) measures the average rate of exponential separation to a distance ax . An analytical expression is developed for the FSLEs of the extended Brownian processes and numerical examples presented. The explicit construction of these processes illustrates that contrary to what has been stated in the literature, a power-law mean-square displacement is not necessarily related to a breakdown in the classical central limit theorem (CLT) caused by, for example, correlation (fractional Brownian motion or correlated continuous-time random-walk schemes) or infinite variance (Levy motion). The classical CLT, coupled with nonstationary increments, can and often does give rise to power-law moments such as the mean-square displacement.
[1] Super-diffusive mixing in geophysics occurs in atmospheric turbulence, near surface currents in oceans, and macro-pore flow in the subsurface to name three of many areas. Models of super-diffusion have been around for almost a century, yet here we put forth a new perspective on the topic which clarifies and substantially expands on classical approaches. Eighty years after it was introduced, we trivially derive Richardson's scaling law for atmospheric super-diffusion, where the mean square separation of two tracer particles goes as t 3 , by assuming the Lagrangian velocity can be represented as a Brownian process. Next we generalize in the spirit of Mandelbrot's intermittency to other types of flows by employing Lagrangian velocities represented as a-stable Lévy processes. For a specific flow field, we show how to obtain the stability parameter, a, from tracer experiments and the finite-size Lyapunov exponent.
Over some range of scales natural porous media often display a fractal Eulerian velocity or conductivity field. If one assumes the fractal conductivity gives rise to fractal drift velocities, then particle paths may be studied in the framework of stochastic differential equations (SODEs). On the microscale, trajectories are modeled as solutions to a SODE with Markovian, stationary, ergodic drift subject to a fluctuating Lévy force. The Lévy force allows for self-motile particles such as flagellated microbes. On the mesoscale the trajectories are modeled as solutions to a SODE with Lévy (fractal) drift and diffusion arising from the microscale asymptotics. On the macroscale the process is driven by the asymptotics of the mesoscale drift without diffusion. Asymptotic scaling laws and dispersion equations are presented.
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