Power law diffusion coefficient and anomalous diffusion: Analysis of solutions and first passage time

Abstract

We investigate one-dimensional equations for the diffusion with a nonconstant diffusion coefficient inside the second derivative and between the derivatives. In particular, we employ the diffusion coefficient D(x) proportional to /x/(-theta)(theta in R) and a quartic potential. These diffusion equations present a rich variety of behaviors associated with different regimes. Results of two approaches are analyzed and compared. We also investigate the mean first passage time of these systems. We show that the system with the coefficient D(x) between the derivatives can produce different behaviors for the mean first passage time in comparison with those obtained by the system with the coefficient inside the derivatives.

The Importance of Mean Free Path in Determining Gas Micro Flow Behaviour

“…The MD data [12] highlights the limitations of the exponential form of probability distribution function for gases in the transition regime, as it can only provide an accurate description of a gas under thermodynamic equilibrium [9]. Anomalous diffusive transport is often better described using Lévy and power-law (PL) probability distributions [13,14]. Anomalous transport can be characterised by incorporating a spatial dependence in the diffusion coefficient, D; for example, D(x) ∝ | x | −n [13].…”

“…Anomalous diffusive transport is often better described using Lévy and power-law (PL) probability distributions [13,14]. Anomalous transport can be characterised by incorporating a spatial dependence in the diffusion coefficient, D; for example, D(x) ∝ | x | −n [13]. For ideal gases, D =(1/3)vλ and ρD = µ, with v being the molecular mean velocity, µ the dynamic viscosity of the gas and ρ the density of the gas [15].…”

Influence of external potentials on heterogeneous diffusion processes

“…The same conclusion that the exponent of the anomalous diffusion does not depend on the prescription has been obtained in Refs. [58,73,74] for equations describing diffusion without the presence of an external force. …”

Random walks with shrinking steps: First-passage characteristics

“…So we have to consider another approach. To study how much discrepancy arises, we have simulated the random walk for different extremity points and computed the mean first passage time minus the values that is given by (16) , that is for each λ we computed…”

“…For example the hunt will be over when the prey and predator meets for the first time, or a neuron will fire when the electric potential first reaches a threshold value, etc. For further information on the study of first passage time for various systems we refer the reader to [11] (and references therein) for a review and introductory book and to [12]- [16] for recent literature.…”