SELFTRAPPING DYNAMICS IN TWO-DIMENSIONAL NONLINEAR LATTICES

1999 ◽  
Vol 13 (24) ◽  
pp. 837-847 ◽  
Author(s):  
M. I. MOLINA
Keyword(s):  
Mean Square Displacement ◽  
Critical Value ◽  
Mean Square ◽  
Two Dimensional ◽  
Nonlinear Lattices ◽  
Long Time ◽  
Average Probability ◽  
The Mean ◽  
Time Average ◽  
Dnls Equation

We compute numerically the selftrapping dynamics for an electron or excitation initially located on a single site of a two-dimensional nonlinear lattice of arbitrary nonlinear exponent. The time evolution is given by the Discrete Nonlinear Schrödinger (DNLS) equation and we focus on the long-time average probability at the initial site and the mean square displacement in terms of both the exponent and strength of the nonlinearity. For the square and triangular nonlinear lattices, we find selftrapping for nonlinearity parameters greater than an exponent-dependent critical value, whose magnitude increases (decreases) with the nonlinear exponent when this is larger (smaller) than one, approximately.

RANDOM WALKS ON KEBAB LATTICES: LOGARITHMIC DIFFUSION ON ORDERED STRUCTURES

1995 ◽  
Vol 09 (10) ◽  
pp. 601-606 ◽  
Author(s):  
D. CASSI ◽  
S. REGINA
Keyword(s):  
Random Walks ◽  
Linear Chain ◽  
Mean Square Displacement ◽  
Analytical Techniques ◽  
Mean Square ◽  
Ordered Structure ◽  
Ordered Structures ◽  
Asymptotic Expressions ◽  
Long Time ◽  
The Mean

Kebab lattices are ordered lattices obtained matching an infinite two-dimensional lattice to each point of a linear chain. Discrete time random walks on these structures are studied by analytical techniques. The exact asymptotic expressions of the mean square displacement and of the RW Green functions show an unexpected logarithmic behavior that is the first example of such kind of law on an ordered structure. Moreover the probability of returning to the origin shows the fastest long time decay ever found for recursive random walks.

Velocity and displacement correlation functions for fractional generalized Langevin equations

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Trifce Sandev ◽  
Ralf Metzler ◽  
Živorad Tomovski
Keyword(s):  
Initial Conditions ◽  
Mean Square Displacement ◽  
Langevin Equations ◽  
Average Particle ◽  
Mean Square ◽  
Relaxation Functions ◽  
Long Time ◽  
The Mean ◽  
Diffusive Processes ◽  
Frictional Memory Kernel

AbstractWe study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

scholarly journals VIRIAL THEOREM FOR ROTATING SELF-GRAVITATING BROWNIAN PARTICLES AND TWO-DIMENSIONAL POINT VORTICES

2012 ◽  
Vol 26 (12) ◽  
pp. 1241002 ◽  
Author(s):  
PIERRE-HENRI CHAVANIS
Keyword(s):  
Virial Theorem ◽  
Mean Square Displacement ◽  
Point Vortices ◽  
Mean Square ◽  
Two Dimensional ◽  
Exact Analytical Expression ◽  
Brownian Particles ◽  
The Mean ◽  
Gravitating Systems ◽  
Interacting Brownian Particles

We derive the virial theorem for an overdamped system of rotating self-gravitating Brownian particles. We show that, in the two-dimensional case, it takes a closed form that can be used to obtain general results about the dynamics without being required to solve the Smoluchowski–Poisson system explicitly. In particular, we obtain the exact analytical expression of the mean square displacement 〈r2〉(t) of the interacting Brownian particles. We exhibit a critical temperature below which the system collapses, and above which it evaporates, and we determine how this temperature is affected by a solid rotation. We also develop an analogy between self-gravitating systems and two-dimensional point vortices. We derive a virial-like relation for point vortices at statistical equilibrium relating the angular velocity to the angular momentum and the temperature.

Intracrystalline Transport Barriers Affecting the Self-Diffusion of CH4 in Zeolites |Na12|-A and |Na12-xKx|-A

2018 ◽  
Author(s):  
Niklas Hedin ◽  
Przemyslaw Rzepka ◽  
Alma Jasso-Salcedo ◽  
Tamara L. Church ◽  
Diana Bernin
Keyword(s):  
Close Contact ◽  
Mean Square Displacement ◽  
Diffusion Time ◽  
Mean Square ◽  
Zeolite A ◽  
Self Diffusion ◽  
Transport Barriers ◽  
Long Time ◽  
The Mean ◽  
Short Time

<p>Removing carbon dioxide is important for the upgrading of biogas or natural gas into compressed or liquefied methane, and adsorption-driven separation of CO<sub>2</sub> could be further advanced by developing for example new adsorbents. Zeolite adsorbents can select CO<sub>2</sub> over CH<sub>4</sub>, and we here confirmed that the adsorption of CH<sub>4</sub> on zeolite |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A was significantly lower for samples with a high K<sup>+</sup> content, i.e. <i>x</i> > 2. Nevertheless, these samples adsorb CH<sub>4</sub> after long equilibration times as determined with <sup>1</sup>H NMR experiments. To assess further the intracrystalline diffusion of CH<sub>4</sub> in these zeolites, pulsed-field gradient NMR experiments were performed. In large crystals of zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A, the long-time diffusion coefficients of CH<sub>4</sub> did not vary with <i>x</i>, and the mean square displacement was about 1.5 mm irrespective of the diffusion time. Also for zeolite |Na<sub>12</sub>|-A samples of three different particle sizes (~0.44, ~2.9, ~10.6 mm), the mean-square displacement of CH<sub>4</sub> was 1.5 mm and largely independent of the diffusion time. This similarity provided further evidence for an intracrystalline diffusion restriction for CH<sub>4</sub> within the medium- and large-sized zeolite A crystals, and possibly of clustering and close contact among the small zeolite A crystals. The long-time diffusion coefficient of adsorbed CH<sub>4</sub> was (at 1 atm and 298 K) about 1 ´ 10<sup>–10</sup> m<sup>2</sup>/s irrespective of the size of the zeolite particle or the studied content of K<sup>+</sup> in zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A and |Na<sub>12</sub>|-A. The <i>T</i><sub>1</sub><i> </i>relaxation time for adsorbed CH<sub>4</sub> on zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A with <i>x</i> > 2 was smaller than for those with <i>x</i> < 2, indicating that the short-time diffusion of CH<sub>4</sub> was hindered.</p>

Intracrystalline Transport Barriers Affecting the Self-Diffusion of CH4 in Zeolites |Na12|-A and |Na12-xKx|-A

2018 ◽  
Author(s):  
Niklas Hedin ◽  
Przemyslaw Rzepka ◽  
Alma Jasso-Salcedo ◽  
Tamara L. Church ◽  
Diana Bernin
Keyword(s):  
Close Contact ◽  
Mean Square Displacement ◽  
Diffusion Time ◽  
Mean Square ◽  
Zeolite A ◽  
Self Diffusion ◽  
Transport Barriers ◽  
Long Time ◽  
The Mean ◽  
Short Time

<p>Removing carbon dioxide is important for the upgrading of biogas or natural gas into compressed or liquefied methane, and adsorption-driven separation of CO<sub>2</sub> could be further advanced by developing for example new adsorbents. Zeolite adsorbents can select CO<sub>2</sub> over CH<sub>4</sub>, and we here confirmed that the adsorption of CH<sub>4</sub> on zeolite |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A was significantly lower for samples with a high K<sup>+</sup> content, i.e. <i>x</i> > 2. Nevertheless, these samples adsorb CH<sub>4</sub> after long equilibration times as determined with <sup>1</sup>H NMR experiments. To assess further the intracrystalline diffusion of CH<sub>4</sub> in these zeolites, pulsed-field gradient NMR experiments were performed. In large crystals of zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A, the long-time diffusion coefficients of CH<sub>4</sub> did not vary with <i>x</i>, and the mean square displacement was about 1.5 mm irrespective of the diffusion time. Also for zeolite |Na<sub>12</sub>|-A samples of three different particle sizes (~0.44, ~2.9, ~10.6 mm), the mean-square displacement of CH<sub>4</sub> was 1.5 mm and largely independent of the diffusion time. This similarity provided further evidence for an intracrystalline diffusion restriction for CH<sub>4</sub> within the medium- and large-sized zeolite A crystals, and possibly of clustering and close contact among the small zeolite A crystals. The long-time diffusion coefficient of adsorbed CH<sub>4</sub> was (at 1 atm and 298 K) about 1 ´ 10<sup>–10</sup> m<sup>2</sup>/s irrespective of the size of the zeolite particle or the studied content of K<sup>+</sup> in zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A and |Na<sub>12</sub>|-A. The <i>T</i><sub>1</sub><i> </i>relaxation time for adsorbed CH<sub>4</sub> on zeolites |Na<sub>12-<i>x</i></sub>K<i><sub>x</sub></i>|-A with <i>x</i> > 2 was smaller than for those with <i>x</i> < 2, indicating that the short-time diffusion of CH<sub>4</sub> was hindered.</p>

scholarly journals Identifying short- and long-time modes of the mean-square displacement: An improved nonlinear fitting approach

AIP Advances ◽  
2019 ◽  
Vol 9 (5) ◽  
pp. 055112
Author(s):  
M. K. Riahi ◽  
I. A. Qattan ◽  
J. Hassan ◽  
D. Homouz
Keyword(s):  
Mean Square Displacement ◽  
Mean Square ◽  
Nonlinear Fitting ◽  
Long Time ◽  
The Mean

Equilibrium structure and diffusion in concentrated hydrodynamically interacting suspensions confined by a spherical cavity

2017 ◽  
Vol 836 ◽  
pp. 413-450 ◽  
Author(s):  
Christian Aponte-Rivera ◽  
Yu Su ◽  
Roseanna N. Zia
Keyword(s):  
Spherical Cavity ◽  
Volume Fraction ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Cavity Size ◽  
Cavity Radius ◽  
Particle Volume ◽  
Wide Range ◽  
Long Time ◽  
The Mean

The short- and long-time equilibrium transport properties of a hydrodynamically interacting suspension confined by a spherical cavity are studied via Stokesian dynamics simulations for a wide range of particle-to-cavity size ratios and particle concentrations. Many-body hydrodynamic and lubrication interactions between particles and with the cavity are accounted for utilizing recently developed mobility and resistance tensors for spherically confined suspensions (Aponte-Rivera & Zia, Phys. Rev. Fluids, vol. 1(2), 2016, 023301). Study of particle volume fractions in the range $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.40$ reveals that confinement exerts a qualitative influence on particle diffusion. First, the mean-square displacement over all time scales depends on the position in the cavity. Additionally, at short times, the diffusivity is anisotropic, with diffusion along the cavity radius slower than diffusion tangential to the cavity wall, due to the anisotropy of hydrodynamic coupling and to confinement-induced spatial heterogeneity in particle concentration. The mean-square displacement is anisotropic at intermediate times as well and, surprisingly, exhibits superdiffusive and subdiffusive behaviours for motion along and perpendicular to the cavity radius respectively, depending on the suspension volume fraction and the particle-to-cavity size ratio. No long-time self-diffusive regime exists; instead, the mean-square displacement reaches a long-time plateau, a result of entropic restriction to a finite volume. In this long-time limit, the higher the volume fraction is, the longer the particles take to reach the long-time plateau, as cooperative rearrangements are required as the cavity becomes crowded. The ordered dynamical heterogeneity seen here promotes self-organization of particles based on their size and self-mobility, which may be of particular relevance in biophysical systems.

LONG-TIME ASYMPTOTICS FOR DIFFUSING CLUSTERS WITH POISSON GROWTH STATISTICS

Fractals ◽  
1996 ◽  
Vol 04 (04) ◽  
pp. 543-546 ◽  
Author(s):  
JERZY ŁUCZKA ◽  
RYSZARD RUDNICKI
Keyword(s):  
Fractal Dimension ◽  
Power Function ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Poisson Statistics ◽  
Fractal Clusters ◽  
Time Asymptotics ◽  
Long Time ◽  
The Mean ◽  
Long Time Asymptotics

Long-time asymptotics for the mean square displacement of diffusing fractal clusters, masses of which randomly grow according to the Poisson statistics, is proven to be a power function of time with the prefactor and exponent depending explicitly upon the fractal dimension of clusters.

scholarly journals Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 111
Author(s):  
Leonid M. Ivanov ◽  
Collins A. Collins ◽  
Tetyana Margolina
Keyword(s):  
Black Sea ◽  
Diffusion Coefficients ◽  
Current System ◽  
Mean Square Displacement ◽  
California Current ◽  
The Black Sea ◽  
Mean Square ◽  
California Current System ◽  
The Mean ◽  
Non Gaussian

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

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