THE EFFECTS OF QUENCHED DISORDER ON THE CHAOTIC DIFFUSION FOR SIMPLE MAPS

1999 ◽  
Vol 13 (01) ◽  
pp. 83-95 ◽  
Author(s):  
HSEN-CHE TSENG ◽  
HUNG-JUNG CHEN
Keyword(s):  
Initial Conditions ◽  
Mean Square Displacement ◽  
Quenched Disorder ◽  
Mean Square ◽  
Numerical Errors ◽  
Limited Sampling ◽  
Chaotic Diffusion ◽  
Normal Diffusion ◽  
The Mean ◽  
Diffusive Processes

That both normal and anomalous chaotic diffusions are suppressed by the presence of quenched disorder for a large class of maps was established by G. Radons.1 In this paper, we consider simple maps (which exhibit normal diffusion) modified by discrete disorder. By decomposing the mean square displacement (MSD) σ2(t) of the system into three terms, namely, [Formula: see text], we find that the MSD of the random walk which corresponds to disorder, [Formula: see text], enhances that of the original unmodified map, [Formula: see text] and that the term 2σ01(t), which describes the correlation between the diffusion fronts of the previous two diffusive processes, just essentially cancels the sum of [Formula: see text] and [Formula: see text]. In consequence, the trajectories of the system are effectively localized. In this formalism, exact numerical calculations without any round-off error can be achieved, the numerical errors coming only from the limited sampling of the initial conditions.

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 Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Long Shi ◽  
Aiguo Xiao
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Weak Ergodicity ◽  
Normal Diffusion ◽  
Weak Ergodicity Breaking ◽  
The Mean ◽  
Ergodicity Breaking

We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0<α<1/3, normal diffusion when α=1/3, and superdiffusion when 1/3<α<1. The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

scholarly journals Diffusion and Superdiffusion in Lattice Models for Colliding Particles with Stored Momentum

2019 ◽  
Vol 177 (6) ◽  
pp. 1240-1262
Author(s):  
Edward Crane ◽  
Sean Ledger ◽  
Bálint Tóth
Keyword(s):  
Initial Conditions ◽  
Mean Square Displacement ◽  
Lattice Models ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Discrete Models ◽  
One Dimension ◽  
Logarithmic Correction ◽  
Mean Square ◽  
The Mean

Abstract We introduce two discrete models of a collection of colliding particles with stored momentum and study the asymptotic growth of the mean-square displacement of an active particle. We prove that the models are superdiffusive in one dimension (with power law correction) and diffusive in three and higher dimensions. In two dimensions we demonstrate superdiffusivity (with logarithmic correction) for certain anisotropic initial conditions.

GROWTH KINETICS OF A NONEQUILIBRIUM ISING MODEL

2000 ◽  
Vol 14 (27n28) ◽  
pp. 983-993
Author(s):  
C. M. SOUKOULIS ◽  
M. C. TRINGIDES ◽  
M. J. VELGAKIS
Keyword(s):  
Time Dependence ◽  
Growth Kinetics ◽  
Disordered Systems ◽  
Spin Exchange ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Domain Boundaries ◽  
The Mean ◽  
Diffusive Processes ◽  
Kinetics Of

The growth kinetics of a system with spin-exchange dynamics has been investigated by means of computer simulations. These systems are used to simulate diffusive processes and kinetic phenomena far away from equilibrium. As a measure of growth we have studied the mean square displacement <R2> of tagged particles. It is found that <R2>t1-n follows a sublinear time dependence, which is explained in terms of the changes of the distribution of atoms between sites within the ordered region and sites of the domain boundaries. The time-dependence of the diffusion coefficient has been derived. The analogies to the relaxation in disordered systems, such as in a:Si–H and in O/W(110), is discussed.

Numerics for the fractional Langevin equation driven by the fractional Brownian motion

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peng Guo ◽  
Caibin Zeng ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Order ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Hurst Parameter ◽  
Mean Square ◽  
Fractional Langevin Equation ◽  
Normal Diffusion ◽  
The Mean

AbstractWe study analytically and numerically the fractional Langevin equation driven by the fractional Brownian motion. The fractional derivative is in Caputo’s sense and the fractional order in this paper is α = 2 − 2H, where H ∈ ($\tfrac{1} {2} $, 1) is the Hurst parameter (or, index). We give numerical schemes for the fractional Langevin equation with or without an external force. From the figures we can find that the mean square displacement of the fractional Langevin equation has the property of the anomalous diffusion. When the fractional order tends to an integer, the diffusion reduces to the normal diffusion.

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.

Molecular Dynamics Study of Microscopic Mechanism of Diffusion in Li2SiO3 System

1991 ◽  
Vol 46 (7) ◽  
pp. 616-620 ◽  
Author(s):  
Junko Habasaki
Keyword(s):  
Molecular Dynamics ◽  
Coordination Number ◽  
Md Simulation ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Lithium Ions ◽  
Microscopic Mechanism ◽  
Trapping Model ◽  
The Mean

MD simulation has been performed to learn the microscopic mechanism of diffusion of ions in the Li2SiO3 system. The motion of lithium ions can be explained by the trapping model, where lithium is trapped in the polyhedron and moves with fluctuation of the coordination number. The mean square displacement of lithium was found to correlate well with the net changes in coordination number.

THE EFFECT OF PREMELTING: STUDY FOR SEMI-INFINITE CRYSTALS AND NANO-SYSTEMS

1994 ◽  
Vol 08 (24) ◽  
pp. 3411-3422 ◽  
Author(s):  
W. SCHOMMERS
Keyword(s):  
Pair Correlation ◽  
Mean Square Displacement ◽  
Phonon Density ◽  
Mean Square ◽  
Phonon Density Of States ◽  
Molecular Dynamics Calculations ◽  
Theory Of Melting ◽  
The Mean ◽  
Dynamical Properties ◽  
Reliable Interaction

The effect of premelting is of particular interest in connection with the theory of melting. In this paper, we discuss the structural and dynamical properties of the surfaces of semi-infinite crystals as well as of nano-clusters, which show the effect of premelting. The investigations are based on molecular-dynamics calculations: different models are used for the systematic study of the effect of premelting. In particular, the behaviour of the following functions have been studied: pair correlation function, generalized phonon density of states, and the mean-square displacement as a function of time. The calculations have been done for krypton since for this substance a reliable interaction potential is available.

Non-Markovian effect of the fractional damping environment and Newton’s second law of motion

2018 ◽  
Vol 32 (19) ◽  
pp. 1850210
Author(s):  
Chun-Yang Wang ◽  
Zhao-Peng Sun ◽  
Ming Qin ◽  
Yu-Qing Xu ◽  
Shu-Qin Lv ◽  
...  
Keyword(s):  
Diffusion Process ◽  
Mean Square Displacement ◽  
Dynamical Analysis ◽  
Second Law ◽  
Mean Square ◽  
Fractional Damping ◽  
Dynamical Mechanism ◽  
Brownian Particles ◽  
The Mean ◽  
Dissipative Environments

We report, in this paper, a recent study on the dynamical mechanism of Brownian particles diffusing in the fractional damping environment, where several important quantities such as the mean square displacement (MSD) and mean square velocity are calculated for dynamical analysis. A particular type of backward motion is found in the diffusion process. The reason of it is analyzed intrinsically by comparing with the diffusion in various dissipative environments. Results show that the diffusion in the fractional damping environment obeys the Langevin dynamics which is quite different form what is expected.

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