The exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket

1992 ◽  
Vol 93 (3) ◽  
pp. 273-284 ◽  
Author(s):  
Tetsuya Hattori ◽  
Shigeo Kusuoka
Keyword(s):  
Random Walk ◽  
Sierpinski Gasket ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Sierpiński Gasket ◽  
The Mean

scholarly journals Generalized Diffusion Equation Associated with a Power-Law Correlated Continuous Time Random Walk

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Long Shi
Keyword(s):  
Random Walk ◽  
Diffusion Equation ◽  
Power Law ◽  
Continuous Time ◽  
Continuum Limit ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Continuous Time Random Walk ◽  
Mean Square ◽  
The Mean

In this work, a generalization of continuous time random walk is considered, where the waiting times among the subsequent jumps are power-law correlated with kernel function M(t)=tρ(ρ>-1). In a continuum limit, the correlated continuous time random walk converges in distribution a subordinated process. The mean square displacement of the proposed process is computed, which is of the form 〈x2(t)〉∝tH=t1/(1+ρ+1/α). The anomy exponent H varies from α to α/(1+α) when -1<ρ<0 and from α/(1+α) to 0 when ρ>0. The generalized diffusion equation of the process is also derived, which has a unified form for the above two cases.

EXACT FORMULA FOR THE MEAN LENGTH OF A RANDOM WALK ON THE SIERPINSKI TOWER

2002 ◽  
Vol 12 (11) ◽  
pp. 2379-2385 ◽  
Author(s):  
JOHN J. KOZAK ◽  
V. BALAKRISHNAN
Keyword(s):  
Random Walk ◽  
Arbitrary Number ◽  
Sierpinski Gasket ◽  
Exact Formula ◽  
Spectral Dimension ◽  
Sierpiński Gasket ◽  
Large N ◽  
The Mean

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula [Formula: see text] The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by [Formula: see text] This also establishes the leading large-n behavior [Formula: see text] that may be expected on general grounds, where Nn is the number of sites on the nth generation tower and [Formula: see text] is the spectral dimension of the fractal.

scholarly journals On the mean square displacement of a random walk on a graph

2016 ◽  
Vol 51 ◽  
pp. 227-235 ◽  
Author(s):  
Seonghyuk Im ◽  
Hwidong Kim ◽  
Jiho Maeng ◽  
Jihwan Yu ◽  
Yongwook Cha ◽  
...  
Keyword(s):  
Random Walk ◽  
Mean Square Displacement ◽  
Mean Square ◽  
The Mean

A description of random walks with collision anisotropy and with a nonconstant mean free path

1990 ◽  
Vol 68 (9) ◽  
pp. 912-917 ◽  
Author(s):  
Thomas Goulet ◽  
Isabelle Mattél ◽  
Jean-Paul Jay-Gerin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Path ◽  
Radial Distribution ◽  
Mean Square Displacement ◽  
Mean Free Path ◽  
Mean Square ◽  
Satisfactory Description ◽  
Average Value ◽  
The Mean

We study various characteristics of a particle's random walk both analytically and with the help of Monte-Carlo simulation techniques. With the analytical approach, we derive the expression[Formula: see text]which relates the mean-square displacement [Formula: see text] to (i) the number of steps N in the walk, (ii) the mean-square displacement [Formula: see text] on each of the steps, and (iii) a coefficient of collision anisotropy A defined as the average value of the cosine of the scattering angle θ. This expression is general in the sense that it holds for any value of N and A. It is, however, restricted to cases where the mean free path is constant throughout the random walk. The results of the simulations allow a further generalization to random walks with a nonconstant mean free path. They also allow the study of the radial distribution f(r) of particles after the walk. We find that a set of six functions fi(r) is necessary to give a satisfactory description of the particles' radial distribution for arbitrary values of N and A.

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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