CPMG relaxation by diffusion with constant magnetic field gradient in a restricted geometry: numerical simulation and application

Carr-Purcell-Meiboom-Gill (CPMG) measurements are the primary nuclear magnetic resonance (NMR) technique used for evaluating formation properties and reservoir fluid properties in the well logging industry and laboratory sample analysis. The estimation of bulk volume irreducible (BVI), permeability, and fluid type relies on the accurate interpretation of the spin-spin relaxation time (T(2)) distribution. The interpretation is complicated when spin's self-diffusion in an inhomogeneous field and restricted geometry becomes dominant. The combined effects of field gradient, diffusion, and a restricted geometry are not easily evaluated analytically. We used a numerical method to evaluate the dependence of the free and restricted diffusion on the system parameters in the absence of surface relaxation, which usually can be neglected for the non-wetting fluids (e.g., oil or gas). The parameter space that defines the relaxation process is reduced to two dimensionless groups: D* and tau*. Three relaxation regimes: free diffusion, localization, and motionally averaging regimes are identified in the (log(10)D*, log(10)tau*) domain. The hypothesis that the normalized magnetization, M*, relaxes as a single exponential with a constant dimensionless relaxation time T*(2) is justified for most regions of the parameter space. The numerical simulation results are compared with the analytical solutions from the contour plots of T*(2). The locations of the boundaries between different relaxation regimes, derived from equalizing length scales, are challenged by observed discrepancies between numerical and analytical solutions. After adjustment of boundaries by equalizing T*(2), numerical simulation result and analytical solution match each other for every relaxation regime. The parameters, fluid diffusivity and pore length, can be estimated from analytical solutions in the free diffusion and motionally averaging regimes, respectively. Estimation of the parameters near the boundaries of the regimes may require numerical simulation.     

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.     

Long-term fluctuations in globally coupled phase oscillators with general coupling: finite size effects

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D～O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D～O(1/N(a)) with a certain constant a>0 in the coherent regime and D～O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.     

Phase diagram of the Holstein polaron in one dimension

The behavior of the 1D Holstein polaron is described, with emphasis on lattice coarsening effects, by distinguishing between adiabatic and nonadiabatic contributions to the local correlations and dispersion properties. The original and unifying systematization of the crossovers between the different polaron behaviors, usually considered in the literature, is obtained in terms of quantum to classical, weak coupling to strong coupling, adiabatic to nonadiabatic, itinerant to self-trapped polarons and large to small polarons. It is argued that the relationship between various aspects of polaron states can be specified by five regimes: the weak-coupling regime, the regime of large adiabatic polarons, the regime of small adiabatic polarons, the regime of small nonadiabatic (Lang-Firsov) polarons, and the transitory regime of small pinned polarons for which the adiabatic and nonadiabatic contributions are inextricably mixed in the polaron dispersion properties. The crossovers between these five regimes are positioned in the parameter space of the Holstein Hamiltonian.     

Study of the Nonlinear Dynamics of a Traveling-Wave-Tube Oscillator with Delayed Feedback

We present the results of numerical simulations of the nonlinear dynamics of a traveling-wave-tube (TWT) oscillator with delayed feedback. Basic properties of stationary single-frequency oscillation regimes are considered, and the onset of self-modulation is studied in detail. Various route-to-chaos scenarios corresponding to successively increasing values of the beam current are simulated numerically. It is shown that the basic scenario is a quasi-periodic route to chaos, while the beam deceleration in strongly nonlinear regimes causes transitions via intermittency to regimes based on modes with higher frequencies. Competition between these two scenarios leads to a complex picture of regular and chaotic self-modulation regimes in the parameter space. Such a behavior is typical of distributed electron–wave self-excited oscillators with delayed feedback.     

Universality in lattice models of dynamic arrest: introduction of an order parameter

We introduce an order parameter for dynamical arrest. Dynamically available volume (unoccupied space that is available to the motion of particles) is expressed as holes for the simple lattice models we study. Near the arrest transition the system is dilute in holes, so we expand dynamical quantities in a series of hole density. Unlike the situation when presented in particle density, all cases of simple models that we examine have a quadratic dependence of the diffusion constant on hole density. This observation implies that in certain regimes ideal dynamical arrest transitions may possess a hitherto unnoticed degree of universality.     

Coarsening and frozen faceted structures in the supercritical complex Swift-Hohenberg equation

The supercritical complex Swift-Hohenberg equation models pattern formation in lasers, optical parametric oscillators and photorefractive oscillators. Simulations of this equation in one spatial dimension reveal that much of the observed dynamics can be understood in terms of the properties of exact solutions of phase-winding type. With real coefficients these states take the form of time-independent spatial oscillations with a constant phase difference between the real and imaginary parts of the order parameter and may be unstable to a longwave instability. Depending on parameters the evolution of this instability may or may not conserve phase. In the former case the system undergoes slow coarsening described by a Cahn-Hilliard equation; in the latter it undergoes repeated phase-slips leading either to a stable phase-winding state or to a faceted state consisting of an array of frozen defects connecting phase-winding states with equal and opposite phase. The transitions between these regimes are studied and their location in parameter space is determined.     

Dynamics of Overlap in Pair-Correlated Hopfield Model

The first step evolution of main overlaps in the pair-correlated Hopfield network is calculated by an analytical approach with probability theory, the results are in good agreemen t with computer simulations. On the basis of some assumptions, we obtain the final value of main overlaps at t = ∞, from which three regimes in parameter space, pattern-retrieval, pairretrieval and paramagnetic regimes, are discovered.     

Parameter space for collisionless RF sheaths

As plasma processing reactors approach higher density, the sheath models which neglect the radio frequency (RF) response of the ions become invalid. This work show that the nature of the collisionless RE ion sheath can be described in a number of different regimes of parameter space. These regimes can all be visualized on a single two-dimensional (2-D) plot where the horizontal axis is the ion plasma frequency divided by the frequency, and the vertical axis is the electron oscillating velocity divided by the ion sound speed     

Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting

In this work we demonstrate that the unstable trigger waves, connecting stable and unstable spatially uniform steady states, can create intricate dynamic regimes in one-dimensional three-component reaction-diffusion model describing blood clotting. Among the most interesting regimes are the composite and replicating waves running at a constant velocity. The front part of the running composite wave remains constant, while its rear part oscillates in a complex manner. The rear part of the running replicating wave periodically gives rise to new daughter waves, which propagate in the direction opposite the parent wave. The domain of these intricate regimes in parameter space lies in the region of monostability near the region of bistability.     

Crossover from one to three dimensions for a gas of hard-core bosons

We develop a variational theory of the crossover from the one-dimensional (1D) regime to the 3D regime for ultracold Bose gases in thin waveguides. Within the 1D regime we map out the parameter space for fermionization, which may span the full 1D regime for suitable transverse confinement.     

Map-based model of the cardiac action potential

A simple computationally efficient model which is capable of replicating the basic features of cardiac cell action potential is proposed. The model is a four-dimensional map and demonstrates good correspondence with real cardiac cells. Various regimes of cardiac activity, which can be reproduced by the proposed model, are shown. Bifurcation mechanisms of these regimes transitions are explained using phase space analysis. The dynamics of 1D and 2D lattices of coupled maps which model the behavior of electrically connected cells is discussed in the context of synchronization theory.     

On-Off Intermittency in Continuous Systems

On-off intermittency is investigated for the model χ= (a + Г(t))χ-χ³ with Г(t) being a stochastic force. The laminar phase distribution ω(T) is studied in the parameter space of bifurcation parameter a, noise intensity D and noise correlation time τ. It is found that increasing D may stabilize the fixed point χ= 0 and reduce the exponential tail in ω(T) for a>0. An analytical solution of the laminar phase distributions is obtained for white noise and colored noise cases, respectively, which agrees with numerical simulations well.     

Antiphase dynamics of a multimode quantum well semiconductor laser

A novel multilongitudinal mode model of a semiconductor laser is presented. The model takes into account four-wave mixing of the longitudinal modes and is based on the correct procedure of simultaneous expansion of the population inversion in time and space series. It is shown that the model has antiphase regimes similar to those observed in experiment. Such behavior exists in a narrow range of the carrier diffusion coefficient, which allows us to estimate the value of this parameter.     

Scaling at the chaos threshold for interacting electrons in a quantum Dot

The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated. Two regimes are distinguished in the dependence of the typical number of Fock states that are mixed into an eigenstate on the interaction strength V, the excitation energy varepsilon, and the level spacing Delta. In both regimes the number is large (indicating delocalization in Fock space). However, only the large- V regime is described by the golden rule (indicating chaotic mixing). The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is (varepsilonV/Delta(2))ln(Delta/V).     

Regions of stability of high-power gyrotron oscillators

The problem of mode competition in a high-power gyrotron oscillator is considered. The regions of parameter space in which a preexisting large-amplitude mode is able to suppress competing satellites are determined for cases in which the coupling coefficients and cavity quality factors for the operating and satellite modes are different. In addition, the effect of beam quality on the regimes of stable single-mode operation is investigated. Generally speaking, the requirement of stable operation favors devices whose interaction length measured by the parameter μ is not too large. It is found that for μ near 10 the operation is relatively stable and μ near 17 is not     

Nonstationary processes in a diffraction-output orotron

The nonlinear dynamics of an orotron with diffraction output is studied. The evolution of the longitudinal field distribution is described by means of a parabolic equation subject to the condition of zero reflection at the collector end of the interaction space. For stationary regimes, the regions of high efficiency are delineated on the parameter plane, with the parameters being the departure from the critical frequency and the reduced length of the interaction space. From numerical results, the parameter plane is divided into the regions of stationary operation, periodic self-modulation, and stochastic self-modulation. It is demonstrated that self-modulation oscillating modes can be obtained most easily if the effective bounce frequency slightly exceeds the cutoff frequency, with the former being the blinking frequency of the dipole comprising an electron and its reflection in the regularly corrugated slow-wave guide.     

光合作用体系捕光复合物线性和非线性光谱的理论研究：修正的Redfield方法的验证

使用数值精确的级联方程方法对其计算可靠性进行了验证．目前已知,修正的Redfield方法所用的微扰和马尔科夫近似,在中间耦合区对激发态能量转移会给出不正确的描述．因此研究其在计算各 种光谱信号时是否仍然有效有重要意义．利用不同参数下的二聚体模型和Fenna-Matthews-Olson复合物模型,使用级联方程方法和修正的Redfield方法,计算并比较了它们的吸收谱、发射谱和二维电子相干光谱．研究发现在较大的参数范围内两种方法给出一致的结果,这两种方法的对比也增进了对光合作用捕光复合物的2D光谱中量子拍频信号来源的认识．     

Characterizing localization properties of two spinless electrons in a one-dimensional Harper model with concurrence

By mapping the Fock space of many local fermionic modes isomorphically onto a many-qubit space and using the measure of concurrence, this paper studies numerically the mode entanglement of two spinless electrons with on-site interaction U moving in the one-dimensional Harper model. Generally speaking, for electrons in extended regimes （potential parameter λ 〈 2）, the spectrum-averaged concurrence N（C） first decreases slowly as A increases until its local minimum, then increases with λ until its peak at λ = 2, while for electrons in localized regimes （λ 〉 2）, N（C） decreases drastically as λ increases. The functions of N（C） versus λ are different for electrons in extended and localized regimes. The maximum of N（C） occurs at the point λ= 2, which is the critical value in the one-dimensional singleparticle Harper model. From these studies it can distinguish extended, localized and critical regimes for the two-particle system. It is also found for the same λ that the interaction U always induce the decreases of concurrence, i.e., the concurrence can reflect the localization effect due to the interaction. All these provide us a new quantity to understand the localization properties of eigenstates of two interacting particles.     

Influence of the temperature of positive ions on the sheath formation and parameter space region in magnetized electronegative plasmas

The parameter space regions and the sheath formation in an electronegative discharge in the presence of thermal positive ions and oblique magnetic field are investigated. It is assumed that the negative species are in thermal equilibrium and the positive ions have a finite temperature. Three regimes of uniform, multilayer stratified and pure stratified are found as functions of positive and negative ion temperature, electronegativity and the magnetic field. The influence of positive ion temperature in the presence of magnetic field on the profiles of the positive ion density, positive ion velocity and electric potential are investigated. The positive ion flux at the sheath edge as a function of magnetic field is obtained for different collisionality and positive and negative ion temperatures. Finally, the influence of the magnetic field, collision frequency and the positive ion temperature on the parameter space regions are discussed.