Velocity distribution and diffusion of an athermal inertial run-and-tumble particle in a shear-thickening medium

We study the dynamics of an athermal inertial run-and-tumble particle moving in a shear-thickening medium in d=1. The viscosity of the medium is represented by a nonlinear function f(v)∼tan(v), while a symmetric dichotomous noise of strength Σ and flipping rate λ models the activity of the particle. Starting from the Fokker-Planck (FP) equation for the time-dependent probability distribution W_{±Σ}(v,t) of the particle's velocity v at time t and the active force is ±Σ, we analytically derive the steady-state velocity distribution function W_{s}(v) and a quadrature expression for the effective diffusion coefficient D_{eff}. For a fixed Σ, W_{s}(v) undergoes multiple transitions with varying λ, and we have identified the corresponding transition points. We then numerically compute W_{s}(v), the mean-squared velocity 〈v^{2}〉(t), and the diffusion coefficient D_{eff}, all of which show excellent agreement with the analytical results in the steady state. Finally, we test the robustness of the transitions in W_{s}(v) by considering an alternative f(v) function that also captures the shear-thickening behavior of the medium.