Abstract
The problem of nonlinear natural convection in a fluid saturated porous layer heated from below is reviewed focusing on the specific result of a collapse of the wave function. When the conditions for the onset of convection are met, a wave function is obtained as the solution of the linearized equations expressed in terms of a Fourier expansion. Only one mode of this expansion survives at the onset of convection, a result that can be seen as the “collapse of the wave function” in a very similar fashion as in quantum mechanics, although the explanations of the latter are very distinct from the ones in quantum mechanics. The reasons behind the “collapse of the wave function” result in natural convection are discussed and the analysis is extended into the nonlinear domain of convection, by using a weak nonlinear analysis.
Highlights
Following Bohm [1], Griffiths [2] and Bowman [3] the kernel of quantum mechanics is the Schrödinger wave equation (Schrödinger [4,5,6,7,8,9])
The latter was challenged, by Schrödinger, and by a large group of physicists led by Albert Einstein who claimed that the quantum mechanical description of physical reality cannot be considered complete, as shown in their famous EPR paper (Einstein, Podolsky and Rosen [11])
Hooft [17] shows that while “it is often claimed that the collapse of the wave function and Born’s rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation” it is not true in certain models “where quantum behavior can be attributed to underlying deterministic equations
Summary
Following Bohm [1], Griffiths [2] and Bowman [3] the kernel of quantum mechanics is the Schrödinger wave equation (Schrödinger [4,5,6,7,8,9]). The statistical approach was entrenched in quantum mechanics as a technical means of providing answers and solutions to sub-atomic phenomena but as a “complete” interpretation of the physical “reality” following Bohr’s and Heisenberg’s “Copenhagen interpretation” that became mainstream physics The latter was challenged, by Schrödinger, and by a large group of physicists led by Albert Einstein who claimed that the quantum mechanical description of physical reality cannot be considered complete, as shown in their famous EPR paper (Einstein, Podolsky and Rosen [11]). Hooft [17] shows that while “it is often claimed that the collapse of the wave function and Born’s rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation” it is not true in certain models “where quantum behavior can be attributed to underlying deterministic equations. Selecting natural convection in porous media to demonstrate such a wave function collapse is just an example choice, and many more such examples are available from fluid mechanics, such as natural convection in pure fluids, Taylor-Couette instability, and others
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