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 $f\left(v\right)\sim tan\left(v\right)$, 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 $W_s\left(v\right)$ and a quadrature expression for the effective diffusion coefficient $D_{\mathrm{eff}}$. For a fixed $\mathrm{\Sigma }$, $W_s\left(v\right)$ undergoes multiple transitions with varying $\lambda$, and we have identified the corresponding transition points. We then numerically compute $W_s\left(v\right)$, the mean-squared velocity $\langle v^2\rangle \left(t\right)$, and the diffusion coefficient $D_{\mathrm{eff}}$, all of which show excellent agreement with the analytical results in the steady state. Finally, we test the robustness of the transitions in $W_s\left(v\right)$ by considering an alternative $f\left(v\right)$ function that also captures the shear-thickening behavior of the medium.

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• Diffusion
• Shear thickening
• Langevin equation
• Probability theory