The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.     

Hierarchical modeling of diffusive transport through nanochannels by coupling molecular dynamics with finite element method

We present a successful hierarchical modeling approach which accounts for interface effects on diffusivity, ignored in classical continuum theories. A molecular dynamics derived diffusivity scaling scheme is incorporated into a finite element method to model transport through a nanochannel. In a 5 nm nanochannel, the approach predicts 2.2 times slower mass release than predicted by Fick’s law by comparing time spent to release 90% of mass. The scheme was validated by predicting experimental glucose diffusion through a nanofluidic membrane with a correlation coefficient of 0.999. Comparison with experiments through a nanofluidic membrane showed interface effects to be crucial. We show robustness of our discrete continuum model in addressing complex diffusion phenomena in biomedical and engineering applications by providing flexible hierarchical coupling of molecular scale effects and preserving computational finite element method speed.     

Estimation of diffusion coefficient by photoemission electron microscopy in ion-implanted nanostructures

We have fabricated parallel stripes of nanostructures in an n-type Si substrate by implanting 30 keV Ga⁺ ions from a focused ion beam (FIB) source. Two sets of implantation were carried out. In one case, during implantation the substrate was held at room temperature and in the other case at 400 °C. Photoemission electron microscopy (PEEM) was carried out on these samples. The implanted parallel stripes, each with a nominal dimension of 4000 nm × 100 nm, appear as bright regions in the PEEM image. Line scans of the intensities from the PEEM image were recorded along and across these stripes. The intensity profile at the edges of a line scan is broader for the implantation carried out at 400 °C compared to room temperature. From the analysis of this intensity profile, the lateral diffusion coefficient of Ga in silicon was estimated assuming that the PEEM intensity is proportional to Ga concentration. The diffusion coefficient at 400 °C has been estimated to be ∼1.3 × 10⁻¹⁵ m²/s. Across the stripes an asymmetric diffusion profile has been observed, which has been related to the sequence of implantation of these stripes and the associated defect distribution due to lateral straggling of the implanted ions.     

Hopping on a zig-zag course

We study the diffusion coefficient of Active Brownian particles in two dimensions. In addition to usual attributes of active motion we let the particles turn in preferred directions over random times. This angular motion is modeled by an effective Lorentz force with time dependent frequency switching between two values at exponentially distributed random times. The diffusion coefficient is calculated by the Taylor-Kubo formula where distributions found from a Fokker-Planck equation or from a continuous time random walk approach have been inserted for averaging. Eventually properties of the diffusion coefficient will be discussed.     

Finite size effects on transport coefficients for models of atomic wires coupled to phonons

We consider models of quasi-1-d, planar atomic wires consisting of several, laterally coupled rows of atoms, with mutually non-interacting electrons. This electronic wire system is coupled to phonons, corresponding, e.g., to some substrate. We aim at computing diffusion coefficients in dependence on the wire widths and the lateral coupling. To this end we firstly construct a numerically manageable linear collision term for the dynamics of the electronic occupation numbers by following a certain projection operator approach. By means of this collision term we set up a linear Boltzmann equation. A formula for extracting diffusion coefficients from such Boltzmann equations is given. We find in the regime of a few atomic rows and intermediate lateral coupling a significant and non-trivial dependence of the diffusion coefficient on both, the width and the lateral coupling. These results, in principle, suggest the possible applicability of such atomic wires as electronic devices, such as, e.g., switches.     

The super long-time decay of velocity fluctuations in a two-dimensional fluid

We have calculated the velocity autocorrelation function for a tracer particle in a model two-dimensional fluid. The fluid was represented by a lattice Boltzmann equation with imposed fluctuations. By choosing a low Boltzmann diffusion coefficient for the tracer, the diverging contribution to the diffusion coefficient can be made to exceed the Boltzmann value even at short times. We were thus able to find evidence for the renormalized, or ‘super long-time’, decay of the VACF in a two-dimensional fluid. We find quantitative evidence for the 1/t√ln(t) decay predicted by theory.     

Adsorption and diffusion of argon confined in ordered and disordered microporous carbons

We use a combination of grand canonical Monte Carlo and microcanonical molecular dynamics simulations to study the adsorption and diffusion of argon at 77 K and 120 K confined in previously generated models of a disordered bituminous coal-based carbon, BPL, and an ordered carbon replica of Faujasite zeolite (C-FAU). Both materials exhibit a maximum in the diffusion coefficient as well as anomalous (sub-diffusive) behavior in the mean-squared displacements at short times at some relative pressures. In BPL, the anomalous diffusion occurs at low relative pressures, due to the trapping of argon atoms in small pores. In C-FAU, the anomalous diffusion occurs at high relative pressures, due to competitive diffusion of atoms traveling through windows and constrictions which interconnect the pores. All diffusion eventually tends to Fickian diffusion at longer times.     

Simulation study of lateral diffusion in lipid-sterol bilayer mixtures

We employ off-lattice Monte Carlo simulations to study lateral diffusion in lipid-sterol bilayers using a two-dimensional model system which has been designed to simulate the experimental phase diagrams of both lipid-cholesterol and lipid-lanosterol systems. We focus on the effects of varying sterol concentration and temperature on the tracer diffusion coefficient, D, which characterizes the lateral motion of single tagged lipids in a bilayer. Generally, we find that increasing the cholesterol concentration suppresses D due to an increased conformational ordering of lipid chains. We argue that this effect competes with an increase in the average free area per lipid, which favours an increase in D. At temperatures close to the main transition temperature, the competition between the two effects leads to intriguing behavior of D. Overall, the model results are in excellent qualitative agreement with available experimental results for lipid-cholesterol mixtures. Additional studies of a model lipid-lanosterol system, for which experimental diffusion results are not available, predict that the presence of lanosterol has a smaller effect than cholesterol on reducing D relative to the pure lipid system. We conclude that the molecular model employed contains the essential features required to describe many of the qualitative features of the lateral diffusion behavior in lipid-sterol systems. Received 24 November 2000 and Received in final form 30 April 2001     

Flexoelectricity of lyotropics and biomembranes

Summary Flexoelectric properties of the following lyotropic systems are described: monolayers, bilayers, lamellar lipid-water phases, black lipid membranes and biomembranes. Lipid layers (one-component and mixed) and lipid-protein layers are considered. Different molecular mechanisms (dipolar and quadrupolar) at free and blocked flip-flop and at free and blocked lateral diffusion are discussed in details. Surface potential measurements in monolayers and diamagnetic anisotropy of bilayers are used to evaluate the contribution of the different mechanisms. The area flexoelectric coefficient is typically −5·10⁻¹¹ statC. Biomembranes with free lateral diffusion of the integral proteins (conical and dipolar ones) would exhibit dipolar flexoelectricity, while those with blocked lateral diffusion (high protein content) would exhibit quadrupolar flexoelectricity. The flexoelectric coefficient of biomembranes seems to be one order of magnitude higher than that of the protein-free bilayers and both positive and negative signs are possible. Some mechano-electrical phenomena in membrane systems are discussed in connection to the flexoelectricity. Paper presented at the ?Meeting on Lyotropics and Related Fields?, held in Rende, Cosenza, September 13–18, 1982.     

Dynamical perturbation for classical fluids: A solvable model

We investigate on a one-dimensional model the perturbation to the timedependent correlations in a classical fluid when a small interaction is added to a hard core. Various formulas have already been proposed for this correction. We verify on this model, for which everything can be calculated explicitly, that the expressions proposed by Frisch and Berne yield strongly divergent time integrals for the diffusion coefficient. On the contrary, when all corrections are accounted for, the correction to the velocity time correlation is shown to decay like (Int)/t ₂ at large times, yielding a finite first-order correction to the diffusion coefficient. The extension of this calculation to a gas of hard rods in the case of a perturbation with an infinite range is discussed.     

Uphill anomalous transport in a deterministic system with speed-dependent friction coefficient

We investigate the transport of a deterministic Brownian particle theoretically, which moves in simple onedimensional, symmetric periodic potentials under the influence of both a time periodic and a static biasing force. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed.This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusion in terms of localization(diffusion coefficient with the form D(t) ~t~(-1) at long times). We also have observed the non-trivially anomalous subdiffusion which is significantly deviated from the localization; whereas most of the downhill motion evolves chaotically, with the normal diffusion.     

Decay of scalar turbulence revisited

We demonstrate that at long times the rate of passive scalar decay in a turbulent, or simply chaotic, flow is dominated by regions where mixing is less efficient. We examine two situations. The first is of a spatially homogeneous stationary turbulent flow with both viscous and inertial scales present. It is shown that at large times scalar fluctuations decay algebraically in time at all spatial scales. The second example explains chaotic stationary flow in a disk/pipe. The boundary region of the flow controls the long-time decay, which is algebraic at some transient times, but becomes exponential, with the decay rate dependent on the scalar diffusion coefficient, at longer times.     

Membrane gas diffusion measurements with MRI

Gas transport across polymeric membranes is fundamental to many filtering and separation technologies. To elucidate transport mechanisms, and understand the behaviors of membrane materials, accurate measurement of transport properties is required. We report a new magnetic resonance imaging (MRI) methodology to measure membrane gas phase diffusion coefficients. The MRI challenges of low spin density and short gas phase relaxation times, especially for hydrogen gas, have been successfully overcome with a modified one-dimensional, single-point ramped imaging with T(1) enhancement, measurement. We have measured the diffusion coefficients of both hydrogen gas and sulfur-hexafluoride in a model polymeric membrane of potential interest as a gas separator in metal hydride batteries. The experimental apparatus is a modified one-dimensional diaphragm cell which permits measurement of the diffusion coefficient in experimental times of less than 1 min. The H(2) gas diffusion coefficient in the membrane was 0.54 +/- 0.01 mm(2)/s, while that of sulfur-hexafluoride was 0.14 +/- 0.01 mm(2)/s, at ambient conditions.     

Skew Disperson and Continuity of Local Time

Results are provided that highlight the effect of interfacial discontinuities in the diffusion coefficient on the behavior of certain basic functionals of the diffusion, such as local times and occupation times, extending previous results in (Appuhamillage et al., Ann Appl Probab 21:183–214, 2011; Water Resour Res 46:W07511, 2009) on the behavior of first passage times. The main goal is to obtain a characterization of large scale parameters and behavior by an analysis at the fine scale of stochastic particle motions. In particular, considering particle concentration modeled by a diffusion equation with piecewise constant diffusion coefficient, it is shown that the continuity of a natural modification of local time is the individual (stochastic) particle scale equivalent to continuity of flux at the scale of the (macroscopic) particle concentrations. Consequences of this involve the determination of a skewness transmission probability in the presence of an interface, as well as corollaries concerning interfacial effects on occupation time of the associated stochastic particles.     

Temperature effects on the diffusion of water and monovalent counterions in the hydrated montmorillonite

We investigate here the effect of temperature on the diffusion of water and cations in the Wyoming-type montmorillonite clay. The considered cations are monovalent compensating ions, such as Li⁺, Na⁺, K⁺, Rb⁺ and Cs⁺ in one-, two- and three-hydration states. For this purpose, molecular dynamics simulations have been performed to obtain the dynamic behaviour regarding the interlayer ions and water molecules under a temperature range between 260 and 400 K. The diffusion coefficient of water and cations in different hydrated clays increases with temperature. The influence of temperature on the diffusion of water is much greater than that of cations in one-, two- or three-hydrated clay. The degree of hydration plays an important role on the diffusion behaviour of water and counterions. We found that the effect of temperature is negligible in weakly hydrated clay, whereas it became significant in highly hydrated one. Besides, the size and mass of cations’ hydrate also affect the diffusion behaviour of water and cations in the interlayer space of hydrated clay.     

Diffusive motion in a model for particle hopping

The diffusive motion of adsorbates on crystal planes is studied by means of a lattice gas model with stochastic dynamics, in the disordered phase and at half coverage. The diffusion coefficient and the time-correlation functions measured in field-emission experiments are calculated. These correlation functions are shown to have the proper hydrodynamic power law decay at long times. It is pointed out that if experiments are done at times before the onset of the hydrodynamic regime the value of the diffusion coefficient obtained will be too small. Our results show also that correlations among the adsorbed particles persist for times longer than predicted by a hydrodynamical approximation.     

Nonergodicity mimics inhomogeneity in single particle tracking

Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with a power-law distribution of waiting times psi(t) proportional to t{-1-alpha}, with 0相似文献   

Measurement of the nuclear spin diffusion coefficient in organic glasses doped with paramagnetic centers: Orthoterphenyl and polymeric glasses

We have recently developed a method of studying spin diffusion coefficients by doping the materials with paramagnetic centers and measuring the nuclear relaxation in a tilted rotating frame. Using this method, we measure here the spin diffusion coefficient of orthoterphenyl, a molecular organic glass, and of three polymer glasses: poly(4-vinylpyridine), poly(vinylacetate) and poly(methyl methacrylate). We explore a possible dependence of the measured orthoterphenyl spin diffusion coefficient on the electronic relaxation time and concentration of the paramagnetic centers. We conclude that the experiments can be performed at higher concentrations than previously thought. We also show that our method applies to polymers in the glassy state if one works at sufficiently small tilt angle, in spite of a short value ofT _1ρ. We had anticipated that the distribution of proton pairs in these materials precludes the standard dependence of the spin diffusion coefficient on the proton density and free induction decay characteristic decay time. Our results fully confirm such expectation.