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A Simple Stochastic Kinetic Transport Model

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2016

Michel Dekking  and
Derong Kong
Michel Dekking*
Affiliation:
Delft University of Technology
Derong Kong*
Affiliation:
Delft University of Technology
*
∗ Postal address: 3TU Applied Mathematics Institute, Delft University of Technology, Faculty EWI, PO Box 5031, 2600 GA Delft, The Netherlands.
∗ Postal address: 3TU Applied Mathematics Institute, Delft University of Technology, Faculty EWI, PO Box 5031, 2600 GA Delft, The Netherlands.
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Abstract

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We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

Keywords

Markov binomial distributionreactive transportkinetic adsorptionsolute transportmultimodalitydouble peak

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 874 - 885
Copyright
© Applied Probability Trust 

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