Turbulence deterministic theory

If we solve this equation with Uj = uj and compare the solution with observations, we would find in reality that the material spreads out more than predicted. This extra spreading is, in fact, what is referred to as turbulent diffusion and results from the influence of the random component wj, which we have ignored. Now let us solve this equation with the precise velocity field Uj. We should then find that the solution agrees exactly with the observations (assuming, of course, that molecular diffusion is negligible), implying that if we knew the velocity field precisely at all locations and times, there would be no such phenomenon as turbulent diffusion. Thus turbulent diffusion is an artifact of our lack of complete knowledge of the true velocity field. Consequently, one of the fundamental tasks in turbulent diffusion theory is to define the deterministic and stochastic components of the velocity field. 

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

Besides these stochastic interpretations, deterministic interpretations are presently developed GRAY [12], KUMPINSKY and EPSTEIN [13], propose systemic approaches, commonly used in chemical engineering several ideal reactors are coupled by conservative flows with expandable coefficients, so that by-passes or dead zones may be taken into account. NICOLIS and FRISCH 14] use a quasi-Semenov equation in the limit of large diffusion coefficients and obtain a renormalization of k , DEWEL et al. [15] use a phenomenological theory of turbulent mixing to study surface effects produced by the feed of the reactor. 

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