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# Normal and Anomalous Diffusion: An Analytical Study Based on Quantum Collision Dynamics and Boltzmann Transport Theory

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posted on 2016-08-23, 00:00 authored by Sathiya Mahakrishnan, Subrata Chakraborty, Amrendra VijayDiffusion, an emergent nonequilibrium
transport phenomenon, is
a nontrivial manifestation of the correlation between the microscopic
dynamics of individual molecules and their statistical behavior observed
in experiments. We present a thorough investigation of this viewpoint
using the mathematical tools of quantum scattering, within the framework
of Boltzmann transport theory. In particular, we ask: (a) How and
when does a normal diffusive transport become anomalous? (b) What
physical attribute of the system is conceptually useful to faithfully
rationalize large variations in the coefficient of normal diffusion,
observed particularly within the dynamical environment of biological
cells? To characterize the diffusive transport, we introduce, analogous
to continuous phase transitions, the curvature of the mean square
displacement as an order parameter and use the notion of quantum scattering
length, which measures the effective interactions between the diffusing
molecules and the surrounding, to define a tuning variable, η.
We show that the curvature signature conveniently differentiates the
normal diffusion regime from the superdiffusion and subdiffusion regimes
and the critical point, η = η

_{c}, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example) as a dynamical cause of the perpetual molecular movement, which eventually manifests as an ordered motion, called the diffusion.