Crossover behaviours exhibited by fluctuations and correlations in a chain of active particles

We study motion of tagged particles in a harmonic chain of active particles. We consider three models of active particle dynamics - run and tumble particle, active Ornstein-Uhlenbeck particle and active Brownian particle. We investigate the variance, auto correlation, covariance and unequal time cross correlation of two tagged particles. For all three models, we observe that the mean squared displacement undergoes a crossover from the super-diffusive $\sim t^{\mu}$ scaling for $t \ll \tau_A$ ($\tau_A$ being the time scale arising due to the activity) to the sub-diffusive $\sim \sqrt{t}$ scaling for $t \gg \tau_A$, where $\mu = \frac{3}{2}$ for RTP, $\mu = \frac{5}{2}$ for AOUP. For the $x$ and $y$-coordinates of ABPs we get $\mu=\frac{7}{2}$ and $\mu=\frac{5}{2}$ respectively. We show that these crossover behaviours in each case can be described by appropriate crossover function that connects these two scaling regimes. We compute these crossover functions explicitly. In addition, we also find that the equal and unequal time auto and cross correlations obey interesting scaling forms in the appropriate limits of the observation time $t$. The associated scaling functions are rigorously derived in all cases. All our analytical results are well supported by the numerical simulations.