This paper studies the diffusive behaviors of the two-state model generalized Lévy walk with jumps (GLWJ) in a constant biased field. Assuming that the motion time density of the GLW process follows a power-law distribution, and the corresponding velocity is defined via a power-law coupling with its motion time, while the random jump density follows a Gaussian distribution. The diffusive behaviors are analyzed and explored through calculating the mean-squared displacement (MSD) and the probability density function (PDF). The results indicate that the MSD exhibits a crossover phenomenon driven by three distinct mechanisms and the PDF also shows a directional deviation. Notably, although the biased field endows the walker with the capability of directional transport, it does not always change the diffusion result, which contradicts our intuition. Meanwhile, since the fluctuations originate from different mechanisms, even if a diffusion term has a smaller diffusion exponent, as long as its fluctuation intensity is much stronger than the others, it can still dominate the result on a large observation timescale. This study reveals the rich diffusion phenomena of the biased GLWJ model, and can provide theoretical guidance for particle transport in the related biological and physical experimental observations.

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• Classical statistical mechanics
• Classical transport
• Diffusion
• Random walks
• Stochastic processes