Lévy walk with asymmetric walking times in complex environments

Abstract

The Lévy walk serves as an important model for superdiffusion and holds significant importance in biological motion research. This study addresses the anisotropic deviations observed in practical biological movements by proposing an asymmetric walking-time Lévy walk model based on diffusion direction, while systematically analyzing its dynamical behavior under linear potential fields. The spatiotemporal coupling challenge is resolved through the Hermite polynomial approximation method, enabling precise computation of key statistical quantities. Research findings demonstrate that in the absence of potential fields, both exponential and power-law distributed walking times exhibit ballistic diffusion, with the latter showing variance scaling affected by asymmetry $\text{Var}\left[x\left(t\right)\right]\propto t^{2-\left|\alpha \right|}$ (where $\alpha ={\alpha }_r-{\alpha }_l$ represents the difference in power-law exponents of asymmetric walking-time distributions), where the dynamics are governed collectively by initial velocity $v_0$, jumping probability $\gamma$, and asymmetric time parameters. When subjected to a linear potential field, exponential walking times maintain the $t^2$ scaling of ballistic diffusion, while power-law distributions enhance to superdiffusive behavior with $〈x^2\left(t\right)〉\propto t^4$, though the dominant variance term depends on $min({\alpha }_r,{\alpha }_l)$, with the dynamics jointly controlled by acceleration and asymmetric time parameters. This research provides theoretical foundations for designing random walk models with specific transport properties, offering substantial value for both biological motion mechanism studies and engineering applications.