By Sergio Chibbaro, Jean-Pierre Minier
Since their first creation in usual sciences in the course of the paintings of Einstein on Brownian movement in 1905 and extra works, specifically by way of Langevin, Smoluchowski and others, stochastic procedures were utilized in numerous parts of technological know-how and expertise. for instance, they've been utilized in chemical stories, or in fluid turbulence and for combustion and reactive flows. The articles during this publication supply a common and unified framework within which stochastic procedures are provided as modeling instruments for varied matters in engineering, physics and chemistry, with specific specialize in fluid mechanics and significantly dispersed two-phase flows. the purpose is to increase what can often called stochastic modeling for an entire variety of applications.
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Additional info for Stochastic Methods in Fluid Mechanics
Sample text
3 State-vector In the Langevin treatment of the Brownian motion (Langevin, 1908), the relevant variables to describe the whole physics are the velocity and the position: x, v. These two variables define therefore the state-vector of the system. Of course other choices can be possible, for instance the sole position as it was done by Einstein, or adding other variables like acceleration. It appears that a hierarchy between state-vectors naturally arises with regard to the information content desired in the physical approach (Minier and Peirano, 2001): the more information is detailed, the larger the number of variables possibly contained in the state vector.
Such a step not only allows researchers to be free from any misconceptions (whereas even distinguished physicists fell into the poor belief that Ito definition was ‘a mathematical vagary’. . ) but, more importantly, will avoid them to induce spurious drifts by flawed mathematical calculus and also potentially inconsistent numerical schemes. Bibliography L. Arnold, Stochastic Differential Equations, theory and applications, John Wiley & Sons, New-York, 1974. I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Springer Verlag, Berlin, 2cd edition, 1991.
The existence of the extra term entering Ito calculus compared to classical calculus is actually simple to understand: it stems from a Taylorseries development made to the second-order in dt since it must be remembered that (dWt )2 = dt in a mean-square sense and, therefore (contrary to the rules of classical calculus with differentiable functions), second-order terms of the development can give contributions to the first order √ in dt. As a practical rule-of-the-thumb, it is often written that dWt ∼ dt.