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The general theoretical description of spin self-diffusion under nonlinear gradient is proposed, which extends the effective phase diffusion method for linear gradient field. Based on the phase diffusion, the proposed method reveals the general features of phase evolutions in non-nonlinear gradient fields. There are three types of phase evolutions: phase diffusion, float phase evolution, and shift based on the starting position. For spin diffusion near the origin of the nonlinear field, these three phase evolutions significantly affect the NMR signal. The traditional methods have difficulties in handling these three-phase evolutions. Notably, the phase from float phase evolution is missed or misplaced in traditional methods, which leads to incorrect NMR signal attenuation or phase shift. The method here shows that the diffusing and float phase evolutions come from the first and second derivatives of the gradient field. Based on these three phase evolutions, the phase variance and corresponding NMR signal attenuation are obtained, demonstrated by calculating the phase diffusions under the parabolic and cubic fields. The results indicate that signal attenuation obeys Gaussian attenuation for a short time, then changes to Lorentzian or Mittag Leffler function attenuations when time increases, significantly different from Gaussian attenuation. For spins starting diffusion far away from the origin, the signal attenuation is Gaussian, but the float phase still has an important effect on the total phase shift of even-order gradient fields, which could be used to measure the diffusion coefficient directly. Random walk simulations are performed, which support the obtained theoretical results. The obtained general theoretical expressions can handle random order nonlinear gradient field. The results could help develop advanced experimental techniques for NMR and MRI., Comment: 20 pages, 3 figures

Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation. The discrete asymmetric random walk provides a basis for understanding biased drift-diffusion. However, the current reported theoretical results for discrete random walks do not give a general theoretical treatment for the asymmetry due to unequal jump times. In this paper, the theoretical expressions for asymmetric random walks with unequal jump probabilities, times, and lengths are derived. The obtained theoretical results can be reduced to reported results when jump times are equal. Additionally, discrete random walk simulations are performed to verify the obtained theoretical results. There are good agreements between the theoretical predictions and simulation results., Comment: 15 pages, 2 figures

The phase time coupling effect on NMR relaxation is investigated based on coupled and uncoupled phase diffusion. The results indicate that phase and time coupling could significantly impact the NMR relaxation time. The spectral density term in the relaxation time expression is modified with an apparent angular frequency, which could be twice the conventional angular frequency when the phase-time coupling is strong. A phase-time coupling constant is proposed to modify the conventional angular frequency to the apparent angular frequency. The modified NMR relaxation time can successfully fit the experimental data taken from Ref. [6]. The results could improve our understanding and analysis of NMR relaxation., Comment: 21 pages, 3 figures

This paper proposes an effective phase diffusion method to analyze chemical exchange in nuclear magnetic resonance (NMR). The chemical exchange involves spin jumps around different sites where the spin angular frequencies vary, which leads to a random phase walk viewed from the rotating frame reference. Therefore, the random walk in phase space can be treated by the effective phase diffusion method. Both the coupled and uncoupled phase diffusions are considered; additionally, it includes normal diffusion as well as fractional diffusion. Based on these phase diffusion equations, the line shape of NMR exchange spectrum can be analyzed. By comparing these theoretical results with the conventional theory, this phase diffusion approach works for fast exchange, ranging from slightly faster than intermediate exchange to very fast exchange. For normal diffusion models, the theoretically predicted curves agree with those predicted from traditional models in the literature, and the characteristic exchange time obtained from phase diffusion with a fixed jump time is the same as that obtained from the conventional model. However, the phase diffusion with a monoexponential time distribution gives a characteristic exchange time constant which is half of that obtained from the traditional model. Additionally, the fractional diffusion obtains a significantly different line shape than that predicted based on normal diffusion., Comment: 20 pages, 4 figures