A Brief Survey of Lévy Walks : with applications to probe diffusion

Abstract: Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion(i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions p(|r|) yielding infinite mean square displacements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are dominated by a few long steps, but most steps are much, much smaller. The semi-experimental part ofthis work dealt with how fluorescent probes moved in systems of cationic starch and latex/solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks couldbe detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Microsoft Excel could be used to generate normal diffusion as well as anomalous diffusion.