Stochastic dispersive transport: An excursion from statistical physics to automated production line design
✍ Scribed by Hongler, M.-O.
- Book ID
- 102754695
- Publisher
- John Wiley and Sons
- Year
- 1993
- Tongue
- English
- Weight
- 778 KB
- Volume
- 9
- Category
- Article
- ISSN
- 8755-0024
No coin nor oath required. For personal study only.
✦ Synopsis
Both the sediment transport dynamics and the population level of a buffer in automated production line systems can be described by the same class of stochastic differential equations. The ubiquitous noise is generated by continuous time Markov chains. The probability densities which describe the dynamics are governed by high-order hyperbolic systems of partial differential equations. While this hyperbolic nature clearly exhibits a non-diffusive character of the processes (diffusion would imply a parabolic evolution of the probability densities), we nevertheless can use a central limit theorem which holds for large-time regimes. This enables analytical estimations of the time evolution of the moments of these processes. Particular emphasis is devoted to non-Markovian, dichotomous alternating renewal processes, which enter directly into the description ,of the applications presented.
KEY WORDS Stochastic buffered flows Transport processes Stochastic differential equations
Alternating renewal processes Markov chains First passage times Variability of sojourn times
1. Introduction
The purely deterministic description of numerous transport phenomena is very often incomplete in obtaining a reliable prediction of the expected flows and other macro-and mesoscopic quantities of interest. Indeed, stochastic perturbations always corrupt the mean values of the external control parameters governing the system and hence can very often strongly affect the expected behaviour of the relevant observables. Therefore, one should definitely include the influence of noise sources in the modelling. The estimation of the statistical properties of stochastic flows of matter and/or information, is a common need in both science and technology. One of the pioneers in this class of mathematical modelling problems is Sydney Goldstein, who is reported' to have said that 'to be a good applied mathematician, you have to be one third physicist, one third engineer and one third mathematician'. The exact cocktail of expertise consists of those requirements needed to bridge the gaps existing between certain aspects of physics and technology. In the spirit of Goldstein's guidelines, let us, more specifically, introduce the class of problems to be discussed by considering separately three different points of view.