Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude $1/L$ in an interval of length $L$ found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement $langle X^2(t)ranglesimeq t^{alpha}$ with $1<alpha<2$, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion ($0<alpha<1$) this behaviour is reversed and the particle density is depleted close to the boundaries. The mean squared displacement in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent $alpha$. Our a priori surprising results may have interesting consequences for the application of fractional Brownian motion for processes such as molecule or tracer diffusion in the confined of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

A note on the Voigt profile function

A Voigt profile function emerges in several physical investigations (e.g. atmospheric radiative transfer, astrophysical spectroscopy, plasma waves and acoustics) and it turns out to be the convolution of the Gaussian and the Lorentzian densities. Its relation with a number of special functions has been widely derived in literature starting from its Fourier type integral representation. The main aim of the present paper is to introduce the Mellin-Barnes integral representation as a useful tool to obtain new analytical results. Here, starting from the Mellin-Barnes integral representation, the Voigt function is expressed in terms of the Fox H-function which includes representations in terms of the Meijer G-function and previously well-known representations with other special functions.

On the Well-Mixed Quasi-One-Dimensional Formulation of Lagrangian Stochastic Models for Turbulent Relative Dispersion

This paper has been withdrawn by the author due to a crucial error in the formulation.

The dependence of relative dispersion on turbulence scales in Lagrangian Stochastic Models

The aim of the article is to investigate the relative dispersion properties of the Well Mixed class of Lagrangian Stochastic Models. Dimensional analysis shows that given a model in the class, its properties depend solely on a non-dimensional parameter, which measures the relative weight of Lagrangian-to-Eulerian scales. This parameter is formulated in terms of Kolmogorov constants, and model properties are then studied by modifying its value in a range that contains the experimental variability. Large variations are found for the quantity $g^*=2gC_0^{-1}$, where $g$ is the Richardson constant, and for the duration of the $t^3$ regime. Asymptotic analysis of model behaviour clarifies some inconsistencies in the literature and excludes the Ornstein-Uhlenbeck process from being considered a reliable model for relative dispersion.