Run-and-tumble dynamics with non-reciprocal transitions between three velocity states

We investigate the transport properties of active particles undergoing a three-state run-and-tumble dynamics in one dimension, induced by non-reciprocal transition rates between self-propelling velocity states $\{-v, 0, +v\}$ that explicitly break microscopic reversibility. Departing from conventional reciprocal models, our formulation introduces a minimal yet rich framework for studying non-equilibrium transport driven by internal state asymmetries. Using kinetic Monte Carlo simulations and analytical methods, we characterize the particle's transport properties across the transition-rates space. The model exhibits a variety of non-equilibrium behaviors, including ballistic transport, giant diffusion, and Gaussian or non-Gaussian transients, depending on the degree of asymmetry in the transition rates. We identify a manifold in transition-rate space where long-time diffusive behavior emerges despite the absence of microscopic reversibility. Exact expressions are obtained for the drift, effective diffusion coefficient, and moments of the position distribution. Our results establish how internal-state irreversibility governs macroscopic transport, providing a tractable framework to study non-equilibrium active motion beyond reciprocal dynamics.