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Although our existing one-dimensional (1D) model provides a successful quantitative description of rupture events, a 1D description is somewhat limited. We therefore derive a two-dimensional (2D) model which allows us to investigate characteristics of earthquakes (EQs) and slow slip events (SSEs) that are only apparent in a second dimension. We find that the leading edge of an EQ rupture in the direction of the global shear stress (x-direction) is wider in the plane of the crustal fault (y-direction) than the trailing edge. The direction of the slip velocity is primarily in the x-direction. EQ ruptures expand in both the x- and y-directions. In SSEs, the rupture also expands in both directions for a short period of time, then, after pulses are formed, there is no further expandsion, i.e., the pulse shape remains practically unchanged. The 2D simulations show the seismic moment (M) versus time (T) scaling law may be expressed by the relation , where . For EQs, is approximately equal to 3, while for SSEs, tends to 1 as the pulses develop and become fully established. The 2D model quantitatively describes the parameters of a seismic pulse in episodic tremor and slip (ETS) phenomena and also explains important features of non-volcanic tremor associated with this pulse, such as reverse tremor migration. The 2D model confirms the basic findings obtained previously by the 1D model: (i) The type of a seismic event, EQ or SSE, and the type of a rupture, crack-like or pulse-like, are determined solely by the fault strength, the ratio of the shear to normal stress, and the gradient in this ratio; (ii) The shape of a rupture and position of the maximum slip is determined by the spatial distribution of the initial stress., Comment: 24 pages, 8 figures

We have developed a model that describes the major characteristics of a rupture, ranging from regular earthquakes (EQs) to slow slip events (SSEs), including episodic tremor and slip (ETS). Previous model predictions, while accurate, are based on a highly idealized initial stress distribution and a simple velocity-dependent expression for friction. The full scope of the model has, therefore, not been fully demonstrated. Further developments, presented here, include more physically realistic treatments of both the initial conditions and friction. Model predictions are: (1) The type of a seismic event, i.e. regular EQ or SSE, is determined by the fault strength, the shear to normal stress ratio, and the gradient in the ratio. Quantitative values for these crucial parameters are also obtained here; (2) Rupture velocities for regular EQs range from a fraction of the shear wave velocity up to the supershear velocity. The maximum slip velocity for regular EQs is typically on the order of 1 m/s. For SSEs, the slip velocity ranges widely from a few cm/year up to 0.1 m/s. The magnitude of the stress drop may vary in the range from 1% to 10% of the initial shear stress for regular EQs and from 0.1% to 10% for the SSEs; (3) The rupture can expend as a crack-like mode or as a self-healing pulse mode. The type of rupture mode is determined by the stress ratio and its gradient. If stress heterogeneity has a step-like shape, the rupture is always crack-like. If the heterogeneity is localized, the rupture is crack-like for sufficiently large values of the stress and its gradient and is pulse-like for smaller values; (4) After an instability develops, rupture dynamics do not depend on the relative values of the rate-state a and b parameters, i.e., it does not matter if frictional sliding is velocity-weakening or velocity-strengthening. Models based on the rate-state are not consistent with this result., Comment: 45 pages, 13 figures

The Bloch equation is the fundamental dynamical model applicable to arbitrary two-level systems. Analytical solutions to date are incomplete for a number of reasons that motivate further investigation. The solution obtained here for the propagator, which generates the time evolution of the system and embodies all the system dynamics, is compact and completely general. The parameter space that results in division by zero in previous treatments is explicitly defined and accommodated in the solution. Polynomial roots required for the solution are expressed in terms of a single real root obtained using simple functional forms. A simple graphical rendition of this root is developed that clarifies and characterizes its dependence on the physical parameters of the problem. As a result, the explicit time dependence of the system as a function of its physical parameters is immediately evident. Several intuitive models of system dynamics are also developed. In particular, the Bloch equation is separable in the proper coordinate system, written as the sum of a relaxation operator and either a null operator or a commuting rotation. The propagator thus drives either pure relaxation or relaxation followed by a rotation. The paper provides a basis for increased physical insight into the Bloch equation and its widespread applications., Comment: 20 pages, 4 figures

A binary mixture of super-paramagnetic colloidal particles is confined between glass plates such that the large particles become fixed and provide a two-dimensional disordered matrix for the still mobile small particles, which form a fluid. By varying fluid and matrix area fractions and tuning the interactions between the super-paramagnetic particles via an external magnetic field, different regions of the state diagram are explored. The mobile particles exhibit delocalized dynamics at small matrix area fractions and localised motion at high matrix area fractions, and the localization transition is rounded by the soft interactions [T. O. E. Skinner et al, Phys. Rev. Lett. {\bf 111}, 128301 (2013)]. Expanding on previous work, we find the dynamics of the tracers to be strongly heterogeneous and show that molecular dynamics simulations of an ideal gas confined in a fixed matrix exhibit similar behavior. The simulations show how these soft interactions make the dynamics more heterogenous compared to the disordered Lorentz gas and lead to strong non-Gaussian fluctuations.

Description of the transitional process from a static to a dynamic frictional regime is a fundamental problem of modern physics. Previously we developed a model based on the well-known Frenkel-Kontorova model to describe dry macroscopic friction. Here this model has been modified to include the effect of dissipation in derived relations between the kinematic and dynamic parameters of a transition process. The main (somewhat counterintuitive) result is a demonstration that the rupture (i.e. detachment front) velocity of the slip pulse which arises during the transition does not depend on friction. The only parameter (besides the elastic and plastic properties of the medium) controlling the rupture velocity is the shear to normal stress ratio. In contrast to the rupture velocity, the slip velocity does depend on friction. The model we have developed describes these processes over a wide range of rupture and slip velocities (up to 7 orders of magnitude) allowing, in particular, the consideration of seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities of a few km/day., Comment: 21 pages, 12 figures

An exact general theory of heteronuclear decoupling is presented for spin-1/2 IS systems. RF irradiation applied to the I spins decouples the S spins by reversing the sign of antiphase magnetization. A one-to-one correspondence is demonstrated between the sign of $2S_yI_z$ and the sense of the S-spin coupling evolution, with magnetization $S_x$ refocused when $\int 2S_yI_z\,dt = 0$. The exact instantaneous coupling at any time during the decoupling sequence is readily obtained in terms of the system states, showing that the creation of two-spin coherence is crucial for reducing the coupling, as required during the refocusing process. Representative examples from standard cyclic, periodic composite-pulse and adiabatic decoupling sequences illustrate the decoupling mechanism. More general aperiodic sequences, obtained using optimal control, realize the potential inherent in the theory for significantly improved decoupling. The recently derived equivalence between the dynamics of any N-level quantum system and a system of classical coupled harmonic oscillators provides additional insight into the mechanics of decoupling., Comment: 9 pages, 4 figures

We apply our theory of conservative solute transport, based on concepts from percolation theory, directly and without modification to reactive solute transport. This theory has previously been shown to predict the observed range of dispersivity values for conservative solute transport over ten orders of magnitude of length scale. We now show that the temporal dependence derived for the solute velocity accurately predicts the time-dependence for the weathering of silicate minerals over nine orders of magnitude of time scale, while its predicted length dependence agrees with data obtained for reaction rates over five orders of magnitude of length scale. In both cases, it is possible to unify lab and field results. Thus, net reaction rates appear to be limited by solute transport velocities. We suggest the possible relevance of our results to landscape evolution of the earth's terrestrial surface.

The dynamics of two-dimensional fluids confined within a random matrix of obstacles is investigated using both colloidal model experiments and molecular dynamics simulations. By varying fluid and matrix area fractions in the experiment, we find delocalized tracer particle dynamics at small matrix area fractions and localized motion of the tracers at high matrix area fractions. In the delocalized region, the dynamics is subdiffusive at intermediate times, and diffusive at long times, while in the localized regime, trapping in finite pockets of the matrix is observed. These observations are found to agree with the simulation of an ideal gas confined in a weakly correlated matrix. Our results show that Lorentz gas systems with soft interactions are exhibiting a smoothening of the critical dynamics and consequently a rounded delocalization-to-localization transition., Comment: 5 pages, 3 figures

The dynamics of states representing arbitrary N-level quantum systems, including dissipative systems, can be modelled exactly by the dynamics of classical coupled oscillators. There is a direct one-to-one correspondence between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. The time evolution of an N-level system is represented as the rotation of a real state vector in hyperspace, as previously known for density matrix states but generalized here to Schrodinger states. A single rotor in n dimensions is then mapped directly to n oscillators in one physical dimension. The number of oscillators needed to represent N-level systems scales linearly with N for Schrodinger states, in contrast to N^2 for the density matrix formalism. Although the well-known equivalence (SU(2), SO(3) homomorphism) of 2-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary N-level systems, representing quantum dynamics by a system of coupled harmonic oscillators in one physical dimension is general for any N. Values for the classical coupling constants are readily obtained from the system Hamiltonian, allowing construction of classical mechanical systems that can provide visual insight into the dynamics of abstract quantum systems as well as a metric for characterizing the interface between quantum and classical mechanics., Comment: 6 pages, 1 figure; added illustrative examples, discussed and clarified implications, corrected typo in figure

In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions.

We present highly robust, optimal control-based shaped pulses designed to replace all 90{\deg} and 180{\deg} hard pulses in a given pulse sequence for improved performance. Special attention was devoted to ensuring that the pulses can be simply substituted in a one-to-one fashion for the original hard pulses without any additional modification of the existing sequence. The set of four pulses for each nucleus therefore consists of 90{\deg} and 180{\deg} point-to-point (PP) and universal rotation (UR) pulses of identical duration. These 1 ms pulses provide uniform performance over resonance offsets of 20 kHz (1H) and 35 kHz (13C) and tolerate reasonably large radio frequency (RF) inhomogeneity/miscalibration of (+/-)15% (1H) and (+/-)10% (13C), making them especially suitable for NMR of small-to-medium-sized molecules (for which relaxation effects during the pulse are negligible) at an accessible and widely utilized spectrometer field strength of 600 MHz. The experimental performance of conventional hard-pulse sequences is shown to be greatly improved by incorporating the new pulses, each set referred to as the Fantastic Four (Fanta4)., Comment: 28 pages, 19 figures

Existing optimal control protocols for mitigating the effects of relaxation and/or RF inhomogeneity on broadband pulse performance are extended to the more difficult problem of designing robust, refocused, frequency selective excitation pulses. For the demanding case of T1 and T2 equal to the pulse length, anticipated signal losses can be significantly reduced while achieving nearly ideal frequency selectivity. Improvements in performance are the result of allowing residual unrefocused magnetization after applying relaxation-compensated selective excitation by optimized pulses (RC-SEBOP). We demonstrate simple pulse sequence elements for eliminating this unwanted residual signal.

Broadband inversion pulses that rotate all magnetization components 180 degrees about a given fixed axis are necessary for refocusing and mixing in high-resolution NMR spectroscopy. The relative merits of various methodologies for generating pulses suitable for broadband refocusing are considered. The de novo design of 180 degree universal rotation pulses using optimal control can provide improved performance compared to schemes which construct refocusing pulses as composites of existing pulses. The advantages of broadband universal rotation by optimized pulses (BURBOP) are most evident for pulse design that includes tolerance to RF inhomogeneity or miscalibration. We present new modifications of the optimal control algorithm that incorporate symmetry principles and relax conservative limits on peak RF pulse amplitude for short time periods that pose no threat to the probe. We apply them to generate a set of pulses suitable for widespread use in Carbon-13 spectroscopy on the majority of available probes.

We present robust radio frequency (rf) pulses that tolerate a factor of six inhomogeneity in the B1 field, significantly enhancing the potential of toroid cavity resonators for NMR spectroscopic applications. Both point-to-point (PP) and unitary rotation (UR) pulses were optimized for excitation, inversion, and refocusing using the gradient ascent pulse engineering (GRAPE) algorithm based on optimal control theory. In addition, the optimized parameterization (OP) algorithm applied to the adiabatic BIR-4 UR pulse scheme enabled ultra-short (50 microsec) pulses with acceptable performance compared to standard implementations. OP also discovered a new class of non-adiabatic pulse shapes with improved performance within the BIR-4 framework. However, none of the OP-BIR4 pulses are competitive with the more generally optimized UR pulses. The advantages of the new pulses are demonstrated in simulations and experiments. In particular, the DQF COSY result presented here represents the first implementation of 2D NMR spectroscopy using a toroid probe., Comment: 32 pages, 8 figures pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4) submitted to Journal of Magnetic Resonance