Crossover from ballistic transport to normal diffusion: a kinetic view

TL;DR

This paper introduces a kinetic model that captures the transition from ballistic transport to normal diffusion in systems, supported by analytical and numerical evidence, enhancing understanding of dispersion pattern crossovers.

Contribution

It presents a novel kinetic framework that models the crossover from ballistic to diffusive behavior, integrating intracellular processes and noise effects.

Findings

The model demonstrates a clear crossover between transport regimes.

Analytical and numerical methods confirm the transition.

Two distinct macroscopic limits are identified.

Abstract

The crossover between dispersion patterns has been frequently observed in various systems. Inspired by the pathway-based kinetic model for E. coli chemotaxis that accounts for the intracellular adaptation process and noise, we propose a kinetic model that can exhibit a crossover from ballistic transport to normal diffusion at the population level. At the particle level, this framework aligns with a stochastic individual-based model. Using numerical simulations and rigorous asymptotic analysis, we demonstrate this crossover both analytically and computationally. Notably, under suitable scaling, the model reveals two distinct limits in which the macroscopic densities exhibit either ballistic transport or normal diffusion.