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01.Evolution Equations for Grain Growth and Coarsening (pp.5-60)
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Exact Solution to Einstein’s Evolution Equation for Fluid with Exponential Free Paths Distribution (pp.165-180) $100.00
Authors:  (G.L. Aranovich, M.D. Donohue, Department of Chemical & Biomolecular Engineering, The Johns Hopkins University, Baltimore, Maryland, USA)
The evolution of fluid density in time and space is considered in the framework of
Einstein’s theory involving expansions of the density distribution in powers of
displacement and time. By truncating these expansions in the evolution equation, Einstein
derived the classical model of diffusion. Here, we consider Einstein’s model without
truncating the density expansions. This demonstrates limits to Einstein’s truncations and
demonstrates non-classical behavior for diffusion phenomenology in fluids with large
mean-free paths, i.e. for fluids between the Knudsen and Fickian limits.
An exact solution to Einstein’s evolution equation for a fluid is presented. This
solution shows that the classical diffusion model is valid for fluids only in the limit of
small mean-free paths, λ.
Here we demonstrate that, at large λ, Einstein’s evolution equation predicts not only
significant quantitative deviations from classical diffusion profiles, but also qualitative
metamorphoses in diffusion mechanisms, such as co-existence of “laminar” and
“turbulent” fluxes and “cooperative” transitions between molecular chaos and
predominantly ballistic motion of molecules.
This has implications for fundamentals of fluids between the Knudsen and Fickian
limits, and for a variety of fields where evolution of a system includes random, multiscale
displacement of particles, such as nanotechnology, vacuum techniques, biological
systems, turbulence, and astrophysics. 

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Exact Solution to Einstein’s Evolution Equation for Fluid with Exponential Free Paths Distribution (pp.165-180)