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 $f(v)\sim \tan (v)$, while a symmetric dichotomous noise of strength $\mathrm{\Sigma }$ and flipping rate $\lambda$ models the activity of the particle. Starting from the Fokker-Planck~(FP) equation for the time-dependent probability distribution $W_{\pm \mathrm{\Sigma }}(v,t)$ of the particle’s velocity $v$ at time $t$ and the active force is $\pm \mathrm{\Sigma }$, we analytically derive the steady-state velocity distribution function $W_s(v)$ and a quadrature expression for the effective diffusion coefficient $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. For a fixed $\mathrm{\Sigma }$, $W_s(v)$ undergoes multiple transitions with varying $\lambda$, 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 $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, 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 capture the shear-thickening behavior of the medium.