Velocity Resetting of Inertial Run-and-Tumble Particles in Non-Newtonian Media: Velocity Distribution, Diffusion and First-Passage Time
We study the dynamics of an athermal inertial run-and-tumble particle moving through a non-Newtonian medium in $d=1$, where the particle's velocity $v$ is reset to zero at a constant rate $r$. The drag force from the non-Newtonian medium is represented by a nonlinear velocity-dependent function $g(v)$. The run-and-tumble dynamics is modeled by a symmetric dichotomous noise with strength $\Sigma$ and flipping rate $\lambda$. We begin with the Fokker-Planck (FP) equation for the velocity distribution $P(v,t)$ of the particle. In the presence of resetting, however, the FP equation does not yield a closed-form solution even in the steady state. We therefore compute the steady-state velocity distribution $P_s(v)$ directly from particle trajectories and compare it with the numerical solution of the FP equation, finding good agreement between the two approaches. For sufficiently large $r$, $P_s(v)$ shows a cusp-like singularity at $v=0$ and the particles display diffusive motion at long times. The effective diffusion coefficient $D_{\mathrm{eff}}$ decays as $r^{-2}$ in the large-$r$ regime. These results hold irrespective of the specific form of $g(v)$ and the values of $\lambda$ and $\Sigma$. However, the mean first-passage time exhibits a strong dependence on the nature of the medium as the resetting rate $r$ is varied. In shear-thickening media, there exists an optimal resetting rate that minimizes the time required to reach the target velocity $v_t$. In contrast, no such optimal resetting rate is observed in shear-thinning media.