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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

This paper proposes an effective diffusion equation method to analyze nuclear magnetic resonance (NMR) relaxation. NMR relaxation is a spin system recovery process, where the evolution of the spin system is affected by the random field due to Hamiltonians, such as dipolar couplings. The evolution of magnetization can be treated as a random walk in phase space described either by a normal or fractional phase diffusion equation. Based on these phase diffusion equations, the NMR relaxation rates and equations can be obtained, exemplified in the analysis of relaxations affected by an arbitrary random field, and by dipolar coupling for both like and unlike spins. The obtained theoretical results are consistent with the reported results in the literature. Additionally, the anomalous relaxation expression obtained from the Mittag-Leffler function based time correlation function can successfully fit the previously reported 13C T1 NMR experimental data of polyisobutylene (PIB) in the blend of PIB and head-to-head poly(propylene) (hhPP). Furthermore, the proposed phase diffusion approach provides an intuitive way to interpret NMR relaxation, particularly for the fractional NMR relaxation, which is still a challenge to explain by the available theoretical methods. The paper provides additional insights into NMR and magnetic resonance imaging (MRI) relaxation experiments., Comment: 17 pages, 3 figures

This paper employs the general time-space fractional diffusion equation to derive correlation time function for analyzing nuclear magnetic resonance (NMR) relaxation. Both the anomalous rotational and translational diffusion are treated. NMR relaxation time affected by various Hamilton interactions such as dipolar or quadrupolar couplings can be calculated from the Mittag-Leffler type time correlation and their corresponded spectral density functions obtained. Additionally, to verify the results, the theoretical expressions are applied to fit reported experimental data of NMR quadrupolar coupling relaxation of head-to-head poly(propylene) (hhPP) in a polymer blend. The fitting is excellent and more convenient than the fitting utilizing the traditional modified Kohlrausch-Williams-Watts (KWW) formalism. Further, it is found that the temperature dependence behavior of the segmental dynamics in anomalous diffusion may obey a different Vogel-Tamman-Fulcher (VTF) expression. The paper proposes new, general formalisms for analyzing various NMR relaxation experiments in macromolecular systems., Comment: 10 pages, 1 figure

A quantum-dot thermal transistor consisting of three Coulomb-coupled quantum dots coupled to respective electronic reservoirs by tunnel contacts is established. The heat flows through the collector and emitter can be controlled by the temperature of the base. It is found that a small change in the base heat flow can induce a large heat flow change in the collector and emitter. The huge amplification factor can be obtained by optimizing the Coulomb interaction between the collector and the emitter or by decreasing the energy-dependent tunneling rate at the base. The proposed quantum-dot thermal transistor may open up potential applications in low-temperature solid-state thermal circuits at the nanoscale., Comment: 14 pages, 6 figures

A modified-Bloch equation based on the fractal derivative is proposed to analyze pulsed field gradient (PFG) anomalous diffusion. Anomalous diffusion exists in many systems such as in polymer or biological systems. PFG anomalous diffusion could be analyzed based on the fractal derivative or the fractional derivative. Compared to the fractional derivative, the fractal derivative is simpler, and it is faster in numerical evaluations. In this paper, the fractal derivative is employed to build the modified-Bloch equation that is a fundamental method to describe the spin magnetization evolution affected by fractional diffusion, Larmor precession, and relaxation. An equivalent form of the fractal derivative is proposed to convert the fractional diffusion equation, which can then be combined with the precession and relaxation equations to get the modified-Bloch equation. This modified-Bloch equation yields a general PFG signal attenuation expression that includes the finite gradient pulse width (FGPW) effect, namely, the signal attenuation during field gradient pulse. The FGPW effect needs to be considered in most clinical MRI applications, and including FGPW effect allows the detecting of slower diffusion that is often encountered in polymer systems. Additionally, the spin-spin relaxation effect can be analyzed, which provides a broad view of the dynamic process in materials. The modified-Bloch equation based on the fractal derivative could provide a fundamental theoretical model for PFG anomalous diffusion., Comment: 12 pages, 4 figures, this article draws heavily from arXiv:1702.07116

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. An general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion based on the fractional derivative, has not been obtained, where {\alpha} and \b{eta} are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. There is good agreement between the theory and the CTRW simulation. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI., Comment: 25 pages,6 figures

Anomalous diffusion has been investigated in many polymer and biological systems. The analysis of PFG anomalous diffusion relies on the ability to obtain the signal attenuation expression. However, the general analytical PFG signal attenuation expression based on the fractional derivative has not been previously reported. Additionally, the reported modified-Bloch equations for PFG anomalous diffusion in the literature yielded different results due to their different forms. Here, a new integral type modified-Bloch equation based on the fractional derivative for PFG anomalous diffusion is proposed, which is significantly different from the conventional differential type modified-Bloch equation. The merit of the integral type modified-Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. From the modified-Bloch equation, the general solutions are derived, which includes the finite gradient pulse width (FGPW) effect. The numerical evaluation of these PFG signal attenuation expressions can be obtained either by the Adomian decomposition, or a direct integration method that is fast and practicable. The theoretical results agree with the continuous-time random walk (CTRW) simulations performed in this paper. Additionally, the relaxation effect in PFG anomalous diffusion is found to be different from that in PFG normal diffusion. The new modified-Bloch equations and their solutions provide a fundamental tool to analyze PFG anomalous diffusion in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI)., Comment: 24 pages, 5 figures. arXiv admin note: text overlap with arXiv:1702.07116

The studying of anomalous diffusion by pulsed field gradient (PFG) diffusion technique still faces challenges. Two different research groups have proposed modified Bloch equation for anomalous diffusion. However, these equations have different forms and, therefore, yield inconsistent results. The discrepancy in these reported modified Bloch equations may arise from different ways of combining the fractional diffusion equation with the precession equation where the time derivatives have different derivative orders and forms. Moreover, to the best of my knowledge, the general PFG signal attenuation expression including finite gradient pulse width (FGPW) effect for time-space fractional diffusion based on the fractional derivative has yet to be reported by other methods. Here, based on different combination strategy, two new modified Bloch equations are proposed, which belong to two significantly different types: a differential type based on the fractal derivative and an integral type based on the fractional derivative. The merit of the integral type modified Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. The general solutions including the FGPW effect were derived from these two equations as well as from two other methods: a method observing the signal intensity at the origin and the recently reported effective phase shift diffusion equation method. The relaxation effect was also considered. It is found that the relaxation behavior influenced by fractional diffusion based on the fractional derivative deviates from that of normal diffusion. The general solution agrees perfectly with continuous-time random walk (CTRW) simulations as well as reported literature results. The new modified Bloch equations is a valuable tool to describe PFG anomalous diffusion in NMR and MRI., Comment: Submitted to Physical Review E

Anomalous diffusion has been investigated in many systems. Pulsed field gradient (PFG) anomalous diffusion is much more complicated than PFG normal diffusion. There have been many theoretical and experimental studies for PFG isotropic anomalous diffusion, but there are very few theoretical treatments reported for anisotropic anomalous diffusion. Currently, there is not a general PFG signal attenuation expression, which includes the finite gradient pulse effect and can treat all three types of anisotropic fractional diffusions: general fractional diffusion, time fractional diffusion, and space-fractional diffusion. In this paper, the recently developed instantaneous signal attenuation (ISA) method was applied to obtain PFG signal attenuation expression for free and restricted anisotropic anomalous diffusion with two models: fractal derivative and fractional derivative models. The obtained PFG signal attenuation expression for anisotropic anomalous diffusion can reduce to the reported result for PFG anisotropic normal diffusion. The results can also reduce to reported PFG isotropic anomalous diffusion results obtained by effective phase shift diffusion equation method and instantaneous signal attenuation method. For anisotropic space-fractional diffusion, the obtained result agrees with that obtained by the modified Bloch equation method. Additionally, The PFG signal attenuation expressions for free and restricted anisotropic curvilinear diffusions were derived by the traditional method, the results of which agree with the PFG anisotropic fractional diffusion results based on the fractional derivative model. The powder pattern of PFG anisotropic diffusion was also discussed. The results here improve our understanding of PFG anomalous diffusion, and provide new formalisms for PFG anisotropic anomalous diffusion in NMR and MRI., Comment: PFG diffusion NMR and MRI, anomalous diffusion, diffusion tensor

Pulsed field gradient (PFG) has been increasingly employed to study anomalous diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). However, the analysis of PFG anomalous diffusion is complicated. In this paper, a fractal derivative model based modified Gaussian phase distribution method is proposed to describe PFG anomalous diffusion. By using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives, and employing some of the traditional Gaussian phase distribution approximation techniques, a general signal attenuation expression for free fractional diffusion is derived. This expression describes a stretched exponential function based attenuation, which is distinct from both the exponential attenuation for normal diffusion obtained from conventional Gaussian phase distribution approximation, and the Mittag-Leffler function based attenuation for anomalous diffusion obtained from fractional derivative. The obtained signal attenuation expression can analyze the finite gradient pulse width (FGPW) effect. Additionally, it can generally be applied to all three types of PFG fractional diffusions classified based on time derivative order alpha and space derivative order beta. These three types of fractional diffusions include time-fractional diffusion, space-fractional diffusion, and general fractional diffusion. The results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method. This method provides a new, convenient approximation formalism for analyzing PFG anomalous diffusion experiments., Comment: 19 pages, 3 figures, PFG signal attenuation of anomalous diffusion based on fractal derivative model, modified Gaussian phase distribution approximation method, submitted to Physica A and under review

Anomalous diffusion exists widely in polymer and biological systems. Pulsed field gradient (PFG) techniques have been increasingly used to study anomalous diffusion in NMR and MRI. However, the interpretation of PFG anomalous diffusion is complicated. Moreover, there is not an exact signal attenuation expression based on fractional derivatives for PFG anomalous diffusion, which includes the finite gradient pulse width effect. In this paper, a new method, a Mainardi-Luchko-Pagnini (MLP) phase distribution approximation, is proposed to describe PFG fractional diffusion. MLP phase distribution is a non-Gaussian phase distribution. From the fractional diffusion equation based on fractional derivatives in both real space and phase space, the obtained probability distribution function is a MLP distribution. The MLP distribution leads to a Mittag-Leffler function based PFG signal attenuation rather than the exponential or stretched exponential attenuation that is obtained from a Gaussian phase distribution (GPD) under a short gradient pulse approximation. The MLP phase distribution approximation is employed to get a complete signal attenuation expression E{\alpha}(-Dfb*{\alpha},\b{eta}) that includes the finite gradient pulse width effect for all three types of PFG fractional diffusion. The results obtained in this study are in agreement with the results from the literature. These results provide a new, convenient approximation formalism to interpret complex PFG fractional diffusion experiments., Comment: 16 pages, 3 figures, PFG signal attenuation for general fractional diffusion,submitted to J. Chem. Phys. and under review