Distribution Functions and Moments of Harmonically Trapped Inertial Run‐and‐Tumble Particles in a Shear‐Thickening Medium

We study the dynamics of an athermal inertial active particle confined in a harmonic potential and moving through shear‐thickening media in one dimension. The activity is modeled using a symmetric dichotomous noise of strength and flipping rate , which mimics a run‐and‐tumble particle. The shear‐thickening response of the medium is incorporated through a velocity‐dependent viscous force of the form . Starting from the Fokker–Planck equations corresponding to the particle's dynamics, we numerically compute the time‐dependent joint probability density function using a finite‐volume upwind scheme . Next, we numerically solve the particle's equations of motion to obtain the steady‐state position and velocity distributions, and compare them with the marginal distributions obtained from at sufficiently large . The two approaches show excellent agreement. Both and exhibit several intriguing transitions as is varied, and they eventually approach a Gaussian‐like profile at sufficiently large . Our study of the time‐dependent position‐velocity correlation, , shows that larger values of lead to faster relaxation to the steady‐state when all other parameters are constant.