Anomalous diffusion resulting from strongly asymmetric random walks

Abstract

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 $P\left(t\right)\mathrm{}\sim t^{-\nu }$. Previous work considered the case of a flight PDF decaying as $P\left(t\right)\mathrm{}\sim t^{-\mu }$ [Weeks et 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: $P_L\mathrm{}\left(t\right)\mathrm{}\sim t^{-\mu }$ and $P_R\mathrm{}\left(t\right)\mathrm{}\sim t^{-\eta }$, with $\mu \ne \eta$. We calculate the dependence of the variance exponent $\gamma \left({\sigma }^2\sim t^{\gamma }\right)$ 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 Lévy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive $(1<\gamma <2)$. When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive $(\gamma <1)$; this is in contrast with the case with μ=η.