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 a method based on observing the signal intensity at the origin. 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 the 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 a field gradient, and their results will help the analysis of PFG anomalous diffusion.

中文翻译：

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PFG异常扩散的实空间和相空间方法分析

脉冲场梯度（PFG）扩散实验可用于测量许多聚合物或生物系统中的异常扩散。但是，分析PFG异常扩散仍然很复杂，尤其是有限梯度脉冲宽度（FGPW）效应。在实际应用中，例如在临床扩散磁共振成像（MRI）中，可能不会忽略FGPW效应。在此，提出了两种截然不同的方法来分析PFG异常扩散：有效相移扩散方程（EPSDE）方法和基于观测原点信号强度的方法。EPSDE方法描述了虚拟相空间中的相位演化，而观察原点信号强度的方法描述了实际空间中的磁化演化。然而，这两种方法给出了相同的一般PFG信号衰减，包括FGPW效应，可以通过直接积分法对其进行数值评估。直接集成方法快速且无溢出。这是用于Mittag-Leffler函数型PFG信号衰减的便捷数值评估方法。这里的方法提供了在场梯度下自旋演化的清晰视图，其结果将有助于分析PFG异常扩散。