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Normal and Anomalous Diffusion: An Analytical Study Based on Quantum Collision Dynamics and Boltzmann Transport TheoryPublished in American Chemical Society

2016

Volume: 120

Issue: 36

Pages: 9608 - 9620

Diffusion, 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 = ηc, 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. © 2016 American Chemical Society.

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About the journal

Journal | Data powered by TypesetJournal of Physical Chemistry B |
---|---|

Publisher | Data powered by TypesetAmerican Chemical Society |

ISSN | 15206106 |

Open Access | No |

Concepts (14)

- Boltzmann equation
- Diffusion
- Molecules
- Quantum theory
- Statistical mechanics
- BIOLOGICAL ENVIRONMENTS
- BOLTZMANN TRANSPORT THEORY
- CONTINUOUS PHASE TRANSITIONS
- DYNAMICAL CHARACTERISTICS
- Dynamical environment
- EFFECTIVE INTERACTIONS
- Mean square displacement
- NONEQUILIBRIUM TRANSPORT
- Quantum chemistry