Tagged particle behavior in a harmonic chain of direction reversing active Brownian particles

We study the tagged particle dynamics in a harmonic chain of direction reversing active Brownian particles, with spring constant $k$, rotation diffusion coefficient $D_{\text{r}}$, and directional reversal rate $\gamma$. We exactly compute the tagged particle position variance for quenched and annealed initial orientations of the particles. For well-separated time scales, $k^{-1}$, $D_{\text{r}}^{-1}$ and $\gamma^{-1}$, the strength of spring constant $k$ relative to $D_{\text{r}}$ and $\gamma$ gives rise to different coupling limits and for each coupling limit there are short, intermediate, and long time regimes. In the thermodynamic limit, we show that, to the leading order, the tagged particle variance exhibits an algebraic growth $t^{\nu}$, where the value of the exponent $\nu$ depends on the specific regime. For a quenched initial orientation, the exponent $\nu$ crosses over from $3$ to $1/2$, via intermediate values $5/2$ or $1$, depending on the specific coupling limits. On the other hand, for the annealed initial orientation, $\nu$ crosses over from $2$ to $1/2$ via an intermediate value $3/2$ or $1$ for strong coupling limit and weak coupling limit respectively. An additional time scale $t_N=N^2/k$ emerges for a system with a finite number of oscillators $N$. We show that the behavior of the tagged particle variance across $t_N$ can be expressed in terms of a crossover scaling function, which we find exactly. Finally, we characterize the stationary state behavior of the separation between two consecutive particles by calculating the corresponding spatio-temporal correlation function.