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A diffusion model of surface soil pollution based on planar finite-velocity stochastic motion with random lifetime

Alexander D. Kolesnik

TL;DR

The paper develops a diffusion model for surface soil pollution from a stationary source by modeling pollutant transport as planar finite-velocity stochastic motion with a random lifetime. The stationary pollution density on the plane is obtained by averaging the planar random flight density against the lifetime distribution $q(t)$, yielding explicit results for different lifetime laws. For heavy particles with an exponential lifetime, the stationary density is given by a McDonald-function series; for lighter particles with a gamma lifetime, it admits a definite-integral form and a hypergeometric-series representation, both amenable to numerical computation. The work also discusses asymmetric wind directions via a von Mises model and emphasizes numerical approaches for practical application to soil-percolation problems into deeper layers.

Abstract

We present a diffusion model of surface soil pollution from a stationary source based on the symmetric stochastic motion at finite speed in the plane $\Bbb R^2$, also called the planar Markov random flight, whose lifetime is a random variable with given distribution. We consider a heavy-particle model, in which the lifetime is supposed to be an exponentially-distributed random variable, and obtain an explicit formula for the stationary probability density of the pollution process expressed in terms of McDonald functions with variable indices. We also study a light-particle model, in which the lifetime is a gamma-distributed random variable. In this case, the stationary probability density of the pollution process is given in the form of a definite integral calculated numerically, as well as in the form of a functional series composed of the hypergeometric functions with variable coefficients. These stationary densities are plotted in a figure and numerically calculated tables that demonstrate the behaviour of the pollution process on long time intervals. Some remarks on the pollution model based on asymmetric finite-velocity planar stochastic motion are also given.

A diffusion model of surface soil pollution based on planar finite-velocity stochastic motion with random lifetime

TL;DR

The paper develops a diffusion model for surface soil pollution from a stationary source by modeling pollutant transport as planar finite-velocity stochastic motion with a random lifetime. The stationary pollution density on the plane is obtained by averaging the planar random flight density against the lifetime distribution , yielding explicit results for different lifetime laws. For heavy particles with an exponential lifetime, the stationary density is given by a McDonald-function series; for lighter particles with a gamma lifetime, it admits a definite-integral form and a hypergeometric-series representation, both amenable to numerical computation. The work also discusses asymmetric wind directions via a von Mises model and emphasizes numerical approaches for practical application to soil-percolation problems into deeper layers.

Abstract

We present a diffusion model of surface soil pollution from a stationary source based on the symmetric stochastic motion at finite speed in the plane , also called the planar Markov random flight, whose lifetime is a random variable with given distribution. We consider a heavy-particle model, in which the lifetime is supposed to be an exponentially-distributed random variable, and obtain an explicit formula for the stationary probability density of the pollution process expressed in terms of McDonald functions with variable indices. We also study a light-particle model, in which the lifetime is a gamma-distributed random variable. In this case, the stationary probability density of the pollution process is given in the form of a definite integral calculated numerically, as well as in the form of a functional series composed of the hypergeometric functions with variable coefficients. These stationary densities are plotted in a figure and numerically calculated tables that demonstrate the behaviour of the pollution process on long time intervals. Some remarks on the pollution model based on asymmetric finite-velocity planar stochastic motion are also given.

Paper Structure

This paper contains 7 sections, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Planar symmetric finite-velocity stochastic motion
  3. General diffusion model of surface soil pollution
  4. Heavy-particle diffusion model
  5. Light-particle diffusion model
  6. Some remarks on the asymmetric case
  7. Conclusions

Figures (1)

  • Figure 1: The shape of density $p_h(\bold x)$ on the interval $\Vert\bold x\Vert\in (0, 5]$ (for $\lambda = 1, \mu=2, c=3$)

Theorems & Definitions (2)

  • proof
  • proof