Anomalous diffusion resulting from strongly asymmetric random walks

We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights (steps of constant velocity) and sticking (pauses). The sticking time probability distribution function (PDF) decays as P(t)similar to t(-nu). Previous work considered the case of a flight PDF decaying as P(t)similar to t(-mu) [Weeks ct al., Physica D 97, 291 (1996)]; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmetries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: P(L)(t)similar to t(-mu) and P(R)(t)similar to t(-eta) With mu not equal eta . We calculate the dependence of the variance exponent gamma (sigma(2) similar to t(gamma)) On the PDF exponents nu, mu, and eta. We find that gamma is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a Levy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive (1 < y < 2). When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive (y < 1); this is in contrast with the case with mu=eta.