Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Analyze PFG Anomalous Diffusion via Real Space and Phase Space Approaches

Version 1 : Received: 13 December 2017 / Approved: 14 December 2017 / Online: 14 December 2017 (11:27:15 CET)

A peer-reviewed article of this Preprint also exists.

Lin, G. Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches. Mathematics 2018, 6, 17. Lin, G. Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches. Mathematics 2018, 6, 17.


Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase shift diffusion equation (EPSDE) method and an observing the signal intensity at the origin method. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function type PFG signal attenuation. The methods here provide a clear view of spin evolution under field gradient, and their results will help the analysis of PFG anomalous diffusion.


PFG anomalous diffusion; fractional derivative; NMR; MRI


Computer Science and Mathematics, Probability and Statistics

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