Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which a regime of superdiffusion may lurk underneath one of normal diffusion.
Giampaolo Cristadoro, Thomas Gilbert, Marco Lenci, David P Sanders (2015). Lévy walks on lattices as multi-state processes. JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT, 2015, P05012-01-P05012-25 [10.1088/1742-5468/2015/05/P05012].
Lévy walks on lattices as multi-state processes
CRISTADORO, GIAMPAOLO;LENCI, MARCO;
2015
Abstract
Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which a regime of superdiffusion may lurk underneath one of normal diffusion.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.