1 Analyze PFG Anomalous Diffusion via Real Space 2 and Phase Space Approaches 3

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous 7 diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG 8 anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical 9 applications, the FGPW effect may not be neglected such as in clinical diffusion magnetic resonance 10 imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous 11 diffusion: the effective phase shift diffusion equation (EPSDE) method and an observing the signal 12 intensity at the origin method. The EPSDE method describes the phase evolution in virtual phase 13 space, while the method to observe the signal intensity at the origin describes the magnetization 14 evolution in real space. However, these two approaches give the same general PFG signal 15 attenuation including FGPW effect, which can be numerically evaluated by a direct integration 16 method. The direct integration method is fast and without overflow. It is a convenient numerical 17 evaluation method for Mittag-Leffler function type PFG signal attenuation. The methods here 18 provide a clear view of spin evolution under field gradient, and their results will help the analysis 19 of PFG anomalous diffusion. 20