Transport properties of the continuous-time random walk with a long-tailed waiting-time density

Journal Article · Sun Oct 01 04:00:00 EDT 1989 · Journal of Statistical Physics; (USA)

DOI:https://doi.org/10.1007/BF01023645· OSTI ID:6140000

Weissman, H; Havlin, S ^[1]; Weiss, G H ^[2]

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1. BarIlan Univ., Ramat Gan (Israel)
2. National Institutes of Health, Bethesda, MD (USA)

The authors derive asymptotic properties of the propagator p(r, t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic form {psi}(t) {approximately} T{sup {alpha}}/t{sup {alpha}+1} when t >> T and 0 < {alpha} < 1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption they consider both random walks with and without a bias. The principle results of their analysis is that one needs two forms to characterize p(r, t), depending on whether r is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for large r. Several results can be demonstrated that contrast with the case in which = {integral}{sub 0}{sup {infinity}} {tau}{psi}({tau})d{tau} is finite. One is that the asymptotic behavior of p(0, t) is demonstrated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a field p(r, t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs at r = 0, and the effect of the field is to break the symmetry that occurs when < {infinity}. Finally, they calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean.

• OSTI ID: 6140000
• Journal Information: Journal of Statistical Physics; (USA), Journal Name: Journal of Statistical Physics; (USA) Vol. 57:1-2; ISSN 0022-4715; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

656002* -- Condensed Matter Physics-- General Techniques in Condensed Matter-- (1987-)
657002 -- Theoretical &amp; Mathematical Physics-- Classical &amp; Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
BOUNDARY CONDITIONS
CRYSTAL MODELS
INTEGRAL TRANSFORMATIONS
LAPLACE TRANSFORMATION
MATHEMATICAL MODELS
MECHANICS
ONE-DIMENSIONAL CALCULATIONS
ORDER-DISORDER TRANSFORMATIONS
PHASE TRANSFORMATIONS
PROBABILITY
PROPAGATOR
RANDOMNESS
SCALING LAWS
SERIES EXPANSION
STATISTICAL MECHANICS
SYMMETRY BREAKING
TIME DEPENDENCE
TRANSFORMATIONS
TRANSPORT THEORY