Limit Of Slow Diffusion Research Articles

First-encounter time of two diffusing particles in two- and three-dimensional confinement.

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability S(t) and the associated first-encounter time probability density H(t) over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time 〈T〉, as well as for the decay time T characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound t_{B} for the time at which S(t) starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to T depends only on the total diffusivity D=D_{1}+D_{2}, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity D. In two dimensions, the first subleading contribution to T is found to depend weakly on the ratio D_{1}/D_{2}. We also investigate the slow-diffusion limit when D_{2}≪D_{1}, and we discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when T can be expected to be a good approximation for 〈T〉.

Fixation time of the rock-paper-scissors model: rigorous results in the well-mixed setting

The rock-paper-scissors model simulates the effect of cyclic dominance in a finite population of size N and has received considerable attention in applied literature. In the well-mixed version of the model, population densities fluctuate around periodic orbits of a deterministic ODE approximation, and for large N the time to fixation (complete dominance by one species) has been observed by simulation to be approximately τN where τ is a positive, finite random variable. We give a rigorous proof of this observation by establishing a slow diffusion limit for a conserved quantity of the deterministic approximation, together with a careful analysis of the behaviour at the boundary, both in the limit as N→∞, and for large but finite N.

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.

The larmor frequency shift in magnetically heterogeneous media depends on their mesoscopic structure.

Recent studies have addressed the determination of the NMR precession frequency in biological tissues containing magnetic susceptibility differences between cell types. The purpose of this study is to investigate the dependence of the precession frequency on medium microstructure using a simple physical model. This dependence is governed by diffusion of NMR-visible molecules in magnetically heterogeneous microenvironments. In the limit of fast diffusion, the precession frequency is determined by the average susceptibility-induced magnetic field, whereas in the limit of slow diffusion it is determined by the average local phase factor of precessing spins. The main method used is Monte Carlo simulation of isotropic suspensions of impermeable magnetized spheres. In addition, NMR spectroscopy was performed in aqueous suspensions of polystyrene microbeads. The precession frequency depends on the structural organization of magnetized objects in the medium. Monte Carlo simulations demonstrated a nonmonotonic transition between the regimes of fast and slow diffusion. NMR experiments confirmed the transition, but were unable to confirm its precise form. Results for a given pattern of structural organization obey a scaling law. The NMR precession frequency exhibits a complex dependence on medium structure. Our results suggest that the commonly assumed limit of fast water diffusion holds for biological tissues with small cells. Magn Reson Med 79:1101-1110, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

A discrete dislocation analysis of hydrogen-assisted mode-I fracture

Fracture of engineering alloys in the presence of hydrogen commonly occurs by decohesion along grain boundaries via a mechanism known as hydrogen induced decohesion (HID). This mechanism is investigated here by analysing the mode-I fracture of a single crystal with plastic flow in the crystal described by discrete dislocation plasticity (DDP) and material separation (decohesion) modelled using a cohesive zone formulation. The motion of dislocations is assumed to be unaffected by hydrogen diffusion. While the cohesive strength is assumed to be reduced proportional to the local hydrogen concentration. Two limiting cases are analysed: (i) the fast diffusion limit where the hydrogen within the material is assumed to be at chemical equilibrium throughout the loading so that there is a high hydrogen concentration in the regions of high hydrostatic stress around dislocations and near the crack tip and (ii) the slow diffusion limit where we assume that there is no appreciable hydrogen diffusion over the duration of loading and thus the hydrogen concentration remains spatially uniform as in a stress-free material. The lower cohesive strength at high hydrogen concentrations results in reduced dislocation activity around the crack tip and a reduction in the material toughness. In fact, at the highest hydrogen concentrations analysed here, crack growth primarily occurs in an elastic manner. However, surprisingly the calculations predicted that the toughness in the fast diffusion case was no more than 12% lower compared to the slow diffusion case suggesting that the stress concentrations due to the dislocation structures and the crack tip fields have only a minor effect on the toughness reduction in the presence of hydrogen. The DDP calculations are finally used to investigate the sensitivity of the material toughness to the grain boundary cohesive strength. The calculations show that the toughness of materials with a small cohesive opening at the peak cohesive traction are more sensitive to hydrogen loading. We speculate that this result might be used as a guide in grain boundary engineering to design alloys that are less sensitive to hydrogen embrittlement by the HID mechanism.

Microfluidic enzyme-linked immunosorbent assay in a region of finite length

The current article presents a mathematical description of the distribution of the enzyme reaction product around an enzyme-linked immunosorbent assay (ELISA) region of finite length created within a microfluidic channel. An analytical expression has been derived for this diffusion-reaction system relating the measured signal to the various operating parameters in the limit of slow diffusion across the assay segments. The predictions of this model have been shown to agree well with the recent experimental reports by Yanagisawa et al.[18] on such a device. The current analysis also shows that quantitating ELISAs based on measurements made in the interfacial region between two assay segments is prone to error. However, such errors can be practically eliminated if the assay signal is collected from a region where this quantity deviates from its asymptotic limit far away from the interface by less than 5%. Moreover, the mathematical analysis suggests that the axial extent of an assay region (L) in these devices may be reduced to about 3 mm before signal cross-talk with its neighboring segments begins to affect the quantitation process. The reported value for L corresponds to a sample volume requirement of 3 nL per assayed analyte in a 10 μm deep and 100 μm wide microfluidic channel, which is nearly 3 orders of magnitude smaller than that required on microarray based platforms.

General solution to gradient-induced transverse and longitudinal relaxation of spins undergoing restricted diffusion

We develop an approach, by calculating the autocorrelation function of spins, to derive the magnetic field gradient-induced transverse (${T}_{2}$) relaxation of spins undergoing restricted diffusion. This approach is an extension to the method adopted by McGregor. McGregor's approach solves the problem only in the fast diffusion limit; however, our approach yields a single analytical solution suitable in all diffusion regimes, including the intermediate regime. This establishes a direct connection between the well-known slow diffusion result of Torrey and the fast diffusion result. We also perform free induction decay measurements on spin-exchange optically polarized ${}^{3}$He gas with different diffusion constants. The measured transverse relaxation profiles are compared with the theory and satisfactory agreement has been found throughout all diffusion regimes. In addition to the transverse relaxation, this approach is also applicable to solving the longitudinal relaxation (${T}_{1}$) regardless of the diffusion limits. It turns out that the longitudinal relaxation in the slow diffusion limit differs by a factor of 2 from that in the fast diffusion limit.

One‐gas models with height‐dependent mean molecular weight: Effects on gravity wave propagation

Many models of the thermosphere employ the one‐gas approximation where the governing equations apply only to the total gas and the physical properties of the gas that depend on composition (mean molecular weight and specific heats) are height‐dependent. It is further assumed that the physical properties of the gas are locally constant; thus motion‐induced perturbations are nil. However, motion in a diffusively separated atmosphere perturbs local values of mean molecular weight and specific heats. These motion‐induced changes are opposed by mutual diffusion of the constituent gases, which attempts to restore diffusive equilibrium. Assuming that composition is locally constant is equivalent to assuming that diffusion instantaneously damps the changes that winds attempt to produce. This is the limit of fast diffusion. In the limit of slow diffusion, gas properties are constant (conserved) following the motion but are perturbed locally by advection. An analysis of the static stability shows that composition effects significantly change the static stability, with greater changes for the slow‐diffusion limit than for the fast‐diffusion limit. We have used a one‐gas full‐wave model to examine the effects of wave‐perturbed composition on gravity waves propagating through the lower thermosphere. We have augmented the conventional system (fixed gas properties) with predictive equations for composition‐dependent gas properties. These equations include vertical advection and mutual diffusion. The latter is included in parameterized form as second‐order scale‐dependent diffusion. We have found that the fast diffusion implied by locally fixed properties has a significant effect on the dynamics. Predicted temperatures are larger for locally fixed composition than for conserved composition. The simulations with parameterized mutual diffusion gave results that are much closer to the results for conserved gas properties than for fixed properties. We found that the divergence between the fast and slow limits was greatest for fast waves and for colder thermospheres. This is because the propagation characteristics of fast waves are sensitive to changes in the static stability and because compositional gradients are stronger for colder thermospheres. We conclude that future models that use the one‐gas approximation for fast waves in the lower thermosphere should include, at minimum, the simplification of conserved rather than fixed properties, especially for colder thermospheres.

Fractional power dependence of the mean lifetime of a first order reaction on the time scale of environment relaxation in the slow diffusion limit

We investigated the fractional power dependence of the mean lifetime on a time scale of environment relaxation for first order reactions described in terms of the reaction–diffusion equation with bounded diffusion. Modifying the static (nondiffusing) approximation so that the decay rate at long times becomes correct, we calculated the values of the powers in the slow diffusion limit. As a result, they were found to be fractional, depending on the sink term and the initial condition. The values of the powers obtained by the theory are in agreement with those found by direct numerical calculations for slow diffusion. In the case of the harmonic sink, we proved that the theory gives the correct values of the powers in the slow diffusion limit.

Viscosity and Free Energy Dependence of Photochemical Charge Separation

Contrary to the conventional (“exponential”) model of contact charge creation and recombination, we consider here a recombination of a geminate ion pair occurring in a remote rectangular spherical layer. This simple model available for analytical solution enables one to discriminate between inner and outer generation of charges with respect to a thin recombination layer. In the slow diffusion limit, the qualitatively different viscosity and free energy dependencies of effective recombination rate are predicted for these alternative cases. The experimental verification of these predictions can provide clear evidence of noncontact electron transfer in liquid solutions.

Landau-Ginzburg theory of self-organized criticality.

Self-organized criticality occurs through a nonlinear feedback mechanism triggering transitions between different metastable states. These transitions take the form of intermittent avalanchelike events distributed according to a power law. We present the first and simplest fully continuous partial differential formalism of this phenomenon, based on the introduction of a subcritical dynamics. SOC is identified as the regime where diffusive relaxation is faster than the instability growth rate. In the other limit of slow diffusion, avalanches comparable to the system size become dominant. This provides a general correspondence between SOC and synchronization of threshold oscillators.

1H spin‐lattice relaxation and observation of interfacial material in solid polyethylenes

AbstractThe 1H NMR laboratory‐ and rotating‐frame (LF,RF) spin‐lattice relaxation (SLR) behaviour in solid polyethylenes is reviewed in light of the two region spin diffusion coupled model. Highly crystalline high‐density materials show single T1 values indicative of the fast diffusion limit whilst RFSLR is close to the slow diffusion limit, allowing direct determination of the dimensions of the crystalline and disordered regions. The LFSLR and RFSLR of the HDPE's suggest the existence of an interfacial material which the SLR behaviour includes as part of the crystalline region in contrast to XRD and DSC. 13C CPMAS measurements of 1H RFSLR in a LLDPE provide an additional insight into this problem. The 1H RFSLR is best represented by a superposition of two exponentials, measured by either 1H or 13C NMR. The 1H RFSLR, measured at different chemical shifts in the 13C NMR spectrum, provides direct evidence of the interfacial material. The possible use of spin‐locking at angles other than 90° in the rotating frame to scale the spin diffusion rates is reviewed and its possible utility for LLDPE materials explored.

Populations of the excited states and the radical transients produced in the electron transfer reactions in liquid solutions

The response functions of the decay of the excited singlet state of the donor (acceptor) in the presence of the acceptor (donor) and the population of the radical pairs formed due to the electron transfer in liquid solutions are derived. The velocity averaged response functions were obtained assuming that the molecular transport in a liquid system can be described in terms of the stochastic linear Langevin equations. The results derived are valid in the slow diffusion limit. Model calculations for the time evolution of the radical transients in rigid and liquid solutions are presented.

1H nuclear magnetic resonance and spin–lattice relaxation in solid, high-density polyethylene

1 H spin–lattice relaxation, in both laboratory and on-resonance rotating frames, are reported for five samples of a high-density, linear polyethylene prepared so as to produce a range of lamellar crystalline widths. The samples are also characterised by measurements of density, Raman longitudinal acoustic modes (LAM) crystallinity (DSC) and small-angle X-ray scattering (SAXS). The laboratory-frame spin–lattice relaxation (LFSLR) is well represented by a single process, and the relaxation rates are linearly related to the crystalline fraction (DSC), which is consistent with the prediction of the fast-diffusion limit of the two-region, spin-diffusion-coupled model of relaxation in these systems. The rotating-frame relaxation (RFSLR) is, as found in previous studies, a multi-exponential process. It is shown that a fit to four components is optimum in terms of quality of fit and signal-to-noise ratio and that this leads to accurate values for the amplitude and time constant of the dominant, longest relaxation-time component. It is also shown that the use of a solid-echo refocussing pulse, following the spin-locking substantially increases the total signal observed but has a much smaller effect on the amplitudes and time constants. The values of the relaxation times and amplitudes for the two longest relaxation time components for each sample are compared with the predictions of the slow-diffusion limit of the spin-diffusion model. The longest relaxation-time component is found to be proportional to the square of the lamellar thickness determined from Raman LAM, and this is used to deduce a value for the spin-diffusion coefficient in the crystalline region, D2≈ 2 × 1016 m2 s–1. Assuming the slow-diffusion limit, the amplitudes of the long-time components in the RFSLR are used to derive values for the size of the disordered region, l1, using both the Raman and NMR derived values of l2, the size of the crystalline region. The l1 and l2 values obtained from the RFSLR are used to derive densities for regions 1 and 2 from the measured sample densities. The values of (l1+l2) from NMR are compared with those obtained from SAXS, and good agreement is found. The validity of the spin-diffusion model is further supported by the data reported here. The relaxation processes (both LF and RF) seem to imply a higher degree of ‘crystallinity’ than revealed by DSC or X-ray diffraction. The LFSLR and RFSLR are internally self-consistent, and this suggests that for SLR the crystalline component includes a fraction of the material characterised as non-crystalline by other methods (XRD, DSC).

Theory of conduction-electron spin resonance

The influence of the conduction-electron magnetization density, induced in a metallic slab by an incident microwave electromagnetic field, on the power absorbed from the microwave field, and on the field transmitted through the slab, is studied theoretically. The motion of the magnetization density is described by a Bloch equation, modified to include diffusion; this approach is equivalent to, but simpler than, that of Dyson. For classical skin effect conditions, exact formulas for the amplitudes of the transmitted fields and for the surface impedance are obtained for symmetric excitation, for antisymmetric excitation, and for one-sided excitation. In performing the calculations, it is found to be convenient to separate the magnetization density into two parts, a forced part and a free part; the forced part is the more important in the slow-diffusion limit, while the free part is the more important in the rapid-diffusion limit. A calculation of the amplitude of the transmitted field, valid under anomalous skin effect conditions, is given for the limiting cases where the diffusion is either very rapid or very slow. Also, the dependence of the polarization of the transmitted fields on the angle which the static applied magnetic field makes with the normal to the slab is discussed.