Simulation of continuous-time random walks with anharmonicity


Anomalous diffusion processes are best described and analyzed by a continuous-time random walk (CTRW) formalism. The latter is a generalization of a transport model using random walk processes that are not limited to Markovian systems. Using a robust numerical generation of continuous trajectories based on Langevin equations, sample paths from a CTRW system encountering a nonlinear net force were generated and analyzed. The resultant force comprised of a linear repelling term and an anharmonic term, that is, f(x) = -k₁ x + k₂ x³. The effect of anharmonicity through varying the k₂ values on the trajectories was manifested on the waiting time. This is the time at which the particle is relatively stationary. At a fixed stability index α, the results show that generally, waiting times would increase as k₂ values increased and that there is an optimum value of k₂ where the waiting time is longest. These results are compared to the case when only the linear force is present.