# Correlation and fluctuation in chain of active particles

Prashant Singh and Anupam Kundu  International Centre for Theoretical
Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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.

##  1 Introduction

Complex systems with many interacting particles occur in different scientific
disciplines like physics [1, 2, 3], biology [4, 5, 6], chemistry [7, 8],
understanding which has brought these disciplines under the ambit of
statistical physics [9]. The motion of a tagged particle in the single file
diffusion where overtaking of particles is forbidden, is a classical problem
in non-equilibrium physics that has been extensively studied in the past
several decades [10, 11]. The mean squared diplacement (MSD) of the tagged
particle in single file diffusion in one dimension is known to scale sub-
diffusively at large times as $\sim\sqrt{t}$ contrary to the ordinary
diffusion where MSD grows linearly with time [11, 12, 13]. In addition, the
probability distribution associated to the position of a tagged particle is
also known [14, 15, 16]. Another interacting particle system where dynamics of
a tagged particle is studied in detail is the random average process [17]. In
this process, each particle can move by a random fraction of the space
available until the next nearest particle at some rate. For this process also,
various statistical features about the position of tagged particle like MSD,
correlation functions, mean squared auto fluctuations are well studied [18,
19, 20, 21]. Not only in the theoretical front, the advancements in the
experimental techniques have made it possible to probe the single particle
motion in the crowded medium by using various optical and magnetic techniques
[24, 25, 26]. While the motion of a tagged particle is quite extensively
studied for interacting passive particles over many years and a huge number of
results are well established theoretically and experimentally, a very handful
results exist for the interacting active particles in the context of single
file problems. Most of the studies for interacting active particles have
focussed on the hydrodynamic description of these systems [27, 28, 29, 30, 33,
34, 35, 36] (see [71] for a review).

Active particle systems has attracted a lot of interest in recent years as
both at the individual and collective levels they exhibit interesting
behaviours different from their passive counterparts. In active systems, the
particles are driven out of equilibrium through consumption of energy from the
environment and conversion of that energy to a systematic movement by some
internal mechanisms [27, 28, 29, 30, 31, 32]. Due to this, the dynamics of
these systems breaks the time reversal symmetry and hence breaks the detailed
balance. The “active” nature endows these systems with a variety of novel
phenomena like motility-induced phase separation [33, 34, 35, 36], flocking
[37, 38], clustering [39, 40], non-existence of equation of state in pressure
[41]. Run and Tumble Particle, active Ornstein Uhlenbeck Particle and active
Brownian Particle are three paradigmatic models for the dynamics of active
particles. At the level of single particle, these models are quite
substantially studied and a variety of its properties have been
characteried/understood. Examples include - probability distribution with and
without confining potential [42, 43, 44, 45, 46, 47, 48, 49, 50], non-trivial
persistent properties [51, 52, 53, 54], arcsine laws [55], convex hull [56],
local time statistics [57], escape problems [58], behaviour under resetting
[59, 60] and large deviations [61, 62, 63] to name a few. Generalizations of
these models in inhomogeneous medium have been recently studied in [64, 65].
There have also been reasonable amount of study with two interacting RTPs on
continuous as well as lattice space [66, 67, 68, 69, 70].

Recently the MSD and the correlations for active particles with repulsive
interaction were studied in different contexts [72, 73, 74, 75] where the MSD
in one dimension was shown to scale sub-diffusively at large times as
$\sim\sqrt{t}$ similar to single file problems of passive particles [11, 12,
13]. The prefactor depends on the particle denisty and the “activity”
parameters. Furthermore, it has been numerically observed in [75] that the MSD
satisfies a scaling function of the variable $t/\tau_{A}$ where $\tau_{A}$ is
a time scale associated to the correlation of the active noises. These studies
[72, 73, 74, 75] mostly rely on either hydrodynamic approximations or mean
field approximations and are largely numerical-based. A rigorous and
systematic analytical treatment of the microscopic dynamics of a tagged
particle in interacting active particle systems has still been lacking. In an
attempt towards this direction, we here study a simple and analytically
tractable system consisting of $N$ active particles interacting via nearest-
neighbour potential in one and two dimension. Using a harmonization technique
it has been shown in [77] that single file properties of a collections of
over-damped Brownian particles in an infinite line where two particles
interact via general two-body potential (with hardcore repulsion) can be well
approximated by a system where particles interact via nearest neighbour
quadratic potential. Following this paper we consider the harmonic interaction
between nearest neighbour particles in our system.

To this end, we study three models in this paper, namely, run and tumble
particle (RTP), active Ornstein-Uhlenbeck particle (AOUP) and active Brownian
particle (ABP). While RTP and AOUP are considered in one dimension, we
consider the ABP in two dimensions. For these three models, we analytically
compute the fluctuations and the correlations of the tagged particle’s
position. We emphasise that the MSD for the tagged particle in harmonic chain
of RTPs was considered recently in [76] where it was shown that an
intermediate regime exists between the ballistic (MSD $\sim t^{2}$) and sub-
diffusive regime (MSD $\sim\sqrt{t}$), where MSD grows super-diffusively as
$\sim t^{3/2}$. Also it was shown numerically that MSD exhibits interesting
scaling behaviours. In this paper, we go beyond MSD and study the equal and
unequal time correlation functions for two tagged particles for RTPs. In
particular we show that such intermediate regime of super-diffusive growth
also appears for AOUPs and ABPs with exponents different from $\frac{3}{2}$.
In addition, we also study crossover functions for MSD that connect various
scaling regimes. Such crossovers were not studied earlier.

Our aim in this paper is two-fold. Firstly, we aim to study the statistical
features of the dynamics of the tagged particle for active systems and compare
and contrast them with that of the passive systems. For this purpose, we will
focus on the MSD, covariance and the equal and unequal time correlations. The
second question that we address is - How do these quantities vary for
different active particle models? It is known that at times larger than the
“activity” time scale, all models converge to the Brownian motion with an
effective diffusion constant. Question is - What happens when the time is
comparable to (or smaller than) the “activity” time scale? We answer this
question in the context of the above mentioned three models, RTP, AOUP and
ABP.

The paper is organized as follows. We introduce our models in sec. 2 and also
provide the summary of our main results here. We devote sec. 3 for the
derivation of various correlation functions separately for the RTP chain. We
focus on the study of MSD in sec. 3.1, equal time correlation in sec. 3.2,
position auto correlations in sec. 3.3 and unequal time correlations in sec.
3.4. In sec. 3.5 we study these correlation for large but finite size $N$ of
the ring where we obtain interesting scaling behaviour for times of the order
of $\tau_{N}$. For AOUP, these correlation functions are studied in sec. 4 and
for ABP in sec. 5 which is followed by the conclusion in sec. 6.

Figure 1: Schematic illustration of the active particles interacting via
nearest-neighbour quadratic potential (shown by green springs) of spring
constant $K$ in one dimension. We study the dynamics of the tagged particle
(shown in red). The position of the $\alpha$-th particle is denoted by
$\vec{r}_{\alpha}(t)$.

##  2 Models and summary of the results

We consider a harmonic ring of $N$ active particles with spring constant $K$.
Denoting the position of the $\alpha$-th particle at time $t$ by
$\vec{r}_{\alpha}(t)=(x_{\alpha},y_{\alpha})$ where $\alpha=0,1,2,....N-1$. We
consider over-damped motion of the particles described by the following
evolution equations

$\displaystyle\frac{d\vec{r}_{\alpha}}{dt}=-K\left(2\vec{r}_{\alpha}-\vec{r}_{\alpha+1}-\vec{r}_{\alpha-1}\right)+\vec{F}_{\alpha}^{A}(t),~{}~{}~{}~{}~{}\text{
for }\alpha=1,...,N,$ (1)

with the periodic boundary condition
$\vec{r}_{\alpha+N}(t)=\vec{r}_{\alpha}(t)$. For simplicity, we have taken the
friction constant to be unity. In this paper, we consider RTP and AOUP in one
dimension and ABP in two dimension. Therefore, the position is denoted by
$\vec{r}_{\alpha}(t)=(x_{\alpha}(t),0)$ for RTP and AOUP and by
$\vec{r}_{\alpha}(t)=\left(x_{\alpha}(t),y_{\alpha}(t)\right)$ for ABP. Note
that the first term in the R.H.S. of Eqs. (1) arises due to the harmonic
interactions between successive particles with spring constant $K$. The second
term $\vec{F}_{\alpha}^{A}(t)$ represents the noises from some non-equilibrium
source which gives rise to the active nature of the particles. The form and
the properties of the noises are different for different models of active
particle dynamics. For RTP/AOUP, we denote the active noises by
$\vec{F}_{\alpha}^{A}(t)=(F_{\alpha}^{RTP/AOUP}(t)~{},~{}0)$ and for ABP by
$\vec{F}_{\alpha}^{A}(t)=\vec{F}_{\alpha}^{ABP}(t)=\left(\xi_{\alpha}(t),\psi_{\alpha}(t)\right)$.

For AOUP, the properties of the noise $F_{\alpha}^{AOUP}(t)$ are described by

$\displaystyle
F_{\alpha}^{AOUP}(t)=\lambda_{\alpha}(t),~{}~{}~{}\forall\alpha~{}~{}~{}~{}\text{
with},$
$\displaystyle\frac{d\lambda_{\alpha}}{dt}=-\gamma\lambda_{\alpha}+\sqrt{2D}\eta_{\alpha}(t),$
(2)

where $\eta_{\alpha}(t)$ is the Gaussian white noise of mean
$\langle\eta_{\alpha}(t)\rangle_{c}=0$ and variance
$\langle\eta_{\alpha}(t)\eta_{\beta}(t^{\prime})\rangle_{c}=\delta_{\alpha,\beta}~{}\delta(t-t^{\prime})$.
Here the parameters $\gamma>0$ and $D>0$ in Eq. (2). For the RTP model, the
noise $F_{\alpha}^{A}(t)$ (written as $F_{\alpha}^{RTP}(t)$) takes the form

$\displaystyle F_{\alpha}^{RTP}(t)$
$\displaystyle=v_{0}~{}\sigma_{\alpha}(t),~{}~{}~{}~{}\text{ for
}\alpha=0,1,..,N-1,$ (3)

where $v_{0}~{}(>0)$ is the speed of the particle and $\sigma_{\alpha}(t)$ is
the dichotomous noise that alternates between $\pm 1$ at a constant rate
$\zeta$. The noise has mean $\langle\sigma_{\alpha}(t)\rangle_{c}=0$ and
unequal time correlation given by

$\displaystyle\langle\sigma_{\alpha}(t)\sigma_{\beta}(t^{\prime})\rangle_{c}=\delta_{\alpha,\beta}~{}\text{exp}\left(-2\zeta|t-t^{\prime}|\right).$
(4)

Finally for ABP, the form of noise term $\vec{F}_{\alpha}^{A}(t)$ written as
$\vec{F}^{ABP}_{\alpha}(t)=\left(\xi_{\alpha}(t),\psi_{\alpha}(t)\right)$ is

$\displaystyle\xi_{\alpha}(t)=v_{A}\cos\phi_{\alpha}(t),~{}~{}~{}\psi_{\alpha}(t)=v_{A}\sin\phi_{\alpha}(t),$
(5)
$\displaystyle\frac{d\phi_{\alpha}}{dt}=\sqrt{2D_{rot}}~{}\eta_{\alpha}(t).$
(6)

Here $v_{A}~{}(>0)$ is the speed of the particle and $D_{rot}~{}(>0)$ is the
rotational diffusion constant which also sets the timescale for the actvity.
Also, here $\eta_{\alpha}(t)$ is the Gaussian white noise with
$\langle\eta_{\alpha}(t)\rangle=0$ and
$\langle\eta_{\alpha}(t)\eta_{\beta}(t^{\prime})\rangle=\delta_{\alpha,\beta}~{}\delta(t-t^{\prime})$.
To proceed further, we consider that for ABP, all particles are initially
oriented along $x$-axis, i.e. $\phi_{\alpha}(0)=0$ for all $\alpha$. For this
choice of the initial orientation, it was shown in [51] that the mean and
correlations of $\xi_{\alpha}(t)$ and $\psi_{\alpha}(t)$ are given by

$\displaystyle\langle\xi_{\alpha}(t)\rangle=v_{A}e^{-D_{rot}t},~{}~{}~{}\langle\psi_{\alpha}(t)\rangle=0,$
(7)
$\displaystyle\langle\xi_{\alpha}(t)\xi_{\beta}(t^{\prime})\rangle=\delta_{\alpha,\beta}\frac{v_{A}^{2}}{2}\left[e^{-D_{rot}|t-t^{\prime}|}+e^{-D_{rot}\\{t_{1}+t_{2}+2\text{min}(t_{1},t_{2})\\}}\right],$
(8)
$\displaystyle\langle\psi_{\alpha}(t)\psi_{\beta}(t^{\prime})\rangle=\delta_{\alpha,\beta}\frac{v_{A}^{2}}{2}\left[e^{-D_{rot}|t-t^{\prime}|}-e^{-D_{rot}\\{t_{1}+t_{2}+2\text{min}(t_{1},t_{2})\\}}\right],$
(9) $\displaystyle\langle\xi_{\alpha}(t)\psi_{\beta}(t^{\prime})\rangle=0.$
(10)

In this paper, we study the position statistics of two tagged particles for
the three models mentioned earlier. More specifically, we will study the equal
and unequal time correlation functions between these two particles and their
mean squared displacement and covariance.

Figure 2: Illustration of three time scales in the model of harmonic chain of
active particles - (i) $\tau_{K}=\frac{1}{K}$ due to the interaction among
particles, (ii)
$\tau_{A}=\frac{1}{\gamma}~{}/\frac{1}{\zeta}~{}/\frac{1}{D_{rot}}$ due to the
activity of AOUP / RTP /ABP and (iii) $\tau_{N}=\frac{N^{2}}{K}$ due to the
finiteness of the ring. As shown in the figure, we will consider
$\tau_{K}\ll\tau_{A}\ll\tau_{N}$ in our paper.

Let us briefly summarize our findings. We study two point correlation

$\displaystyle\begin{split}&C^{1}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle-\langle
x_{\alpha_{1}}(t_{1})\rangle\langle x_{\alpha_{2}}(t_{2})\rangle,\\\
&C^{2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
y_{\alpha_{1}}(t_{1})y_{\alpha_{2}}(t_{2})\rangle_{c}=\langle
y_{\alpha_{1}}(t_{1})y_{\alpha_{2}}(t_{2})\rangle-\langle
y_{\alpha_{1}}(t_{1})\rangle\langle y_{\alpha_{2}}(t_{2})\rangle.\end{split}$
(11)

Once again, we emphasise that $\vec{r}_{\alpha}(t)=(x_{\alpha}(t),0)$ for RTP
and AOUP, and $\vec{r}_{\alpha}(t)=\left(x_{\alpha}(t),y_{\alpha}(t)\right)$
for ABP. Consequently for RTP and AOUP, one has only the correlation
$C^{1}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})$, while for ABP one has both
$C^{1/2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})$. Note that, since the noises
$\xi_{\alpha}(t)$ and $\psi_{\beta}(t)$ are mutually uncorrelated [see Eq.
(10)], the cross correlations among the $x$ and $y$-coordinates for the
particles in the ABP chain are zero _i.e._ $\langle
x_{\alpha}(t_{1})y_{\beta}(t_{2})\rangle_{c}=0$. We study
$C^{1/2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})$ in different asymptotic regimes
of time for three models of active particle chains of length $N$. In all
models the particles are connected to each other by harmonic springs of
strength $K$ and each particle is driven by independent active noises
$\vec{F}^{A}(t)$ which have different statistical properties. There are three
time scales in this problem, namely, (i) the interaction time scale
$\tau_{K}=\frac{1}{K}$ (ii) the activity time scale $\tau_{A}$ and (iii) the
relaxation time scale $\tau_{N}$. The interaction time scale $\tau_{K}$ is
coming from interaction among the particles. The activity time scale
$\tau_{A}$ arises due to the activity present in the system. The activity time
scale $\tau_{A}=\frac{1}{\zeta}$ for RTP, $\tau_{A}=\frac{1}{\gamma}$ for AOUP
and $\tau_{A}=\frac{1}{D_{rot}}$ for ABP particles. Finally,
$\tau_{N}=\frac{N^{2}}{K}$ represents the relaxation time scale on a ring of
size $N$. Throughout this work, we will consider
$\tau_{K}\ll\tau_{A}\ll\tau_{N}$ (see Fig. 2). As illustrated in this work,
with this order of time scales we find various interesting scaling structures
and scaling functions for different position correlations. The other case of
$\tau_{A}\ll\tau_{K}\ll\tau_{N}$ does not give rise to these scaling
structures and only reproduces the results of the passive particles.

More precisely, in this paper we study the two point correlation
$C^{1/2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})$ in the following three regimes
(i) $t_{1},t_{2}\ll\tau_{K}$ (ii) $\tau_{K}\ll t_{1},t_{2}\ll\tau_{N}$ with
$N\to\infty$ (equivalently $\tau_{N}\to\infty$) (iii)
$t_{1},t_{2}\sim\tau_{N}$ for finite but large $N$. The MSD of a tagged
particle, say $\alpha$th, is obtained from
$\sigma^{2}_{1/2}(t)=C^{1/2}_{\alpha,\alpha}(t,t)$. Due to homogeneity and
periodic boundary of the chain, the MSD of a tagged particle is independent of
the tag of the particle. Covariance between the positions of two particles can
similarly be obtained from $C_{0,\beta}^{1}(t)=\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}$ and $C_{0,\beta}^{2}(t)=\langle
y_{0}(t)y_{\beta}(t)\rangle_{c}$ which are again same as $\langle
x_{\alpha}(t)x_{\alpha+\beta}(t)\rangle_{c}$ and $\langle
y_{\alpha}(t)y_{\alpha+\beta}(t)\rangle_{c}$ respectively due to translational
symmetry of the problem. We also study position auto correlation
$C_{\alpha,\alpha}^{1/2}(t_{1},t_{2})$ and finally un-equal time correlation
$C_{\alpha_{1},\alpha_{2}}^{1/2}(t_{1},t_{2})$. We consider two cases: (a)
$N\to\infty$ in which $\tau_{N}\to\infty$ and (b) finite but large $N$ in
which $\tau_{N}$ is finite. At times much smaller than $\tau_{K}$, the
statistical properties of one particle are independent of other particles as
in such short time they do not realise the presence of other particle. In this
time regime, the correlation behaves as
$C_{\alpha_{1},\alpha_{2}}^{1/2}(t_{1},t_{2})\sim
C_{\alpha_{1},\alpha_{1}}^{1/2}(t_{1},t_{2})\delta_{\alpha_{1},\alpha_{2}}$.
On the other hand, in case (a) we observe that the above four quantities
exhibit interesting scaling behaviours for times much larger than $\tau_{K}$.
For RTP and AOUP dynamics we get

$\displaystyle C_{\alpha,\alpha}^{1}(t,t)$ $\displaystyle\propto
t^{3/2}~{}\mathcal{T}^{RT/OU}\left(\frac{t}{\tau_{A}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{K}\ll
t\ll\tau_{N}$ (12) $\displaystyle C_{0,\beta}^{1}(t,t)$
$\displaystyle\propto\begin{cases}t^{\mu}~{}\Omega^{RT/OU}\left(\frac{\beta}{\sqrt{2Kt}}\right)~{};&~{}~{}~{}~{}~{}\tau_{K}\ll
t\ll\tau_{A}\\\
\sqrt{t}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right)~{};&~{}~{}~{}~{}~{}\tau_{A}\ll
t\ll\tau_{N},\end{cases}$ (13) $\displaystyle
C_{\alpha,\alpha}^{1}(t_{1},t_{2})$
$\displaystyle\propto\begin{cases}t_{1}^{\mu}~{}\mathcal{L}^{RT/OU}\left(\frac{t_{2}}{t_{1}}\right)~{};&~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{K}\ll
t_{1}\ll\tau_{A}\\\
\sqrt{t_{1}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right)~{};&~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{A}\ll
t_{1}\ll\tau_{N}\end{cases}$ (14) $\displaystyle
C_{\alpha_{1},\alpha_{2}}^{1}(t_{1},t_{2})$
$\displaystyle\propto\begin{cases}t_{1}^{\mu}~{}\mathcal{P}^{RT/OU}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)~{};&\tau_{K}\ll
t_{1}\ll\tau_{A}\\\
\sqrt{t_{1}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)~{};&\tau_{A}\ll
t\ll\tau_{N}\end{cases}$ (15)

where $\mu=\frac{3}{2}$ for RTP and $\mu=\frac{5}{2}$ for AOUP. Also, the
proportionality constants are given explicitly in the respective equations
while deriving these results. Similarly for ABP, in case (a), we get

$\displaystyle C_{\alpha,\alpha}^{p}(t,t)$ $\displaystyle\propto
t^{3/2}~{}\mathcal{T}^{AB}_{p}\left(\frac{t}{\tau_{A}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{K}\ll
t\ll\tau_{N}$ (16) $\displaystyle C_{0,\beta}^{p}(t,t)$
$\displaystyle\propto\begin{cases}t^{\nu}~{}\Omega^{AB}_{p}\left(\frac{\beta}{\sqrt{2Kt}}\right)~{};&~{}~{}~{}~{}~{}\tau_{K}\ll
t\ll\tau_{A}\\\
\sqrt{t}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right)~{};&~{}~{}~{}~{}~{}\tau_{A}\ll
t\ll\tau_{N},\end{cases}$ (17) $\displaystyle
C_{\alpha,\alpha}^{p}(t_{1},t_{2})$
$\displaystyle\propto\begin{cases}t_{1}^{\nu}~{}\mathcal{L}^{AB}_{p}\left(\frac{t_{2}}{t_{1}}\right)~{};&~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{K}\ll
t_{1}\ll\tau_{A}\\\
\sqrt{t_{1}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right)~{};&~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{A}\ll
t_{1}\ll\tau_{N}\end{cases}$ (18) $\displaystyle
C_{\alpha_{1},\alpha_{2}}^{p}(t_{1},t_{2})$
$\displaystyle\propto\begin{cases}t_{1}^{\nu}~{}\mathcal{P}^{AB}_{p}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)~{};&\tau_{K}\ll
t_{1}\ll\tau_{A}\\\
\sqrt{t_{1}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)~{};&\tau_{A}\ll
t\ll\tau_{N}\end{cases}$ (19)

where $p=1~{}\text{or}~{}2$ and $\nu=\frac{7}{2}$ for $p=1$ and
$\nu=\frac{5}{2}$ for $p=2$.

For large but finite $N$ in case (b), we find that the correlation function
$C_{0,\beta}^{p}(t_{1},t_{2})$ for all three models takes the scaling form

$\displaystyle
C_{0,\beta}^{p}(t_{1},t_{2})\simeq\frac{2D_{eff}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right),~{}~{}~{}t_{1},t_{2}\sim
O(\tau_{N})$ (20)

where $D_{eff}$ is the effective diffusion constants for the RTP
($D_{eff}=D_{R}=\frac{v_{0}^{2}}{2\zeta}$), AOUP
($D_{eff}=D_{A}=\frac{D}{\gamma^{2}}$) and ABP
($D_{eff}=D_{ABP}=\frac{v_{A}^{2}}{2D_{rot}}$) chains. A summary of these
results are given in Tables 1 and 2.

In what follows, we will study various statistical properties of the position
of the tagged particles. We will first consider the mean squared displacement
and covariance as functions of time which are followed by the studies of un-
equal time two-point position correlations. We first compute these quantities
for RTP and then for AOUP and ABP in the subsequent sections.

Figure 3: In Figure (a), we have plotted the scaling function
$\mathcal{T}^{RT}(y)$ in Eq. (37) for RTP and compared with the results of
numerical simulation. The Figure (b) shows the asymptotic forms of $\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ derived in Eqs. (41) for $\tau_{K}\ll
t\ll\tau_{A}$ (shown by black dotted line) and $\tau_{K}\ll\tau_{A}\ll t$
(shown by green dotted line) where $\tau_{A}=\frac{1}{\zeta}$. The solid pink
line corresponds to the theoretical result of $\langle
x_{\alpha}^{2}(t)\rangle_{c}$ obtained in Eq. (36). For both plots, we have
chosen $K=2,~{}\zeta=0.5$ and $v_{0}=1$. The simulation is done with $N=100$
particles on a ring.

##  3 TWO POINT CORRELATION FOR RTP CHAIN

We begin by computing the two point correlation function
$C_{\alpha,\beta}^{1}(t_{1},t_{2})$ in Eq. (11) when the particles perform RTP
dynamics. Recall that the RTP dynamics is considered in one dimension and
$\vec{r}_{\alpha}(t)=(x_{\alpha}(t),0)$ in Eq. (11). Let us first look at the
case when $t_{1}=t_{2}=t$ in $C_{\alpha,\beta}^{1}(t_{1},t_{2})$. To proceed,
we perform Fourier transform with respect to $\alpha$ on both sides of the set
of Langevin Eqs. (1). For a general function $g_{\alpha}(t)$, we define the
Fourier transformation as

$\displaystyle g_{\alpha}(t)=\sum_{s=0}^{N-1}e^{\frac{2\pi
is}{N}\alpha}~{}\bar{g}_{s}(t),$ (21)

where $i^{2}=-1$ and the inverse Fourier transformation as

$\displaystyle\bar{g}_{s}(t)=\frac{1}{N}\sum_{\alpha=0}^{N-1}e^{-\frac{2\pi
is}{N}\alpha}~{}g_{\alpha}(t).$ (22)

Writing the Langevin Eqs. (1) in terms of the Fourier variables

$\displaystyle\frac{d\bar{x}_{s}}{dt}=-a_{s}\bar{x}_{s}+\bar{F}_{s}^{A}(t)~{},$
(23)

one now gets rid of the cross terms. Also here

$\displaystyle\bar{x}_{s}(t)=\frac{1}{N}\sum_{\alpha=0}^{N-1}e^{-\frac{2\pi
is}{N}\alpha}~{}x_{\alpha}(t),$ (24)
$\displaystyle\bar{F}^{A}_{s}(t)=\frac{1}{N}\sum_{\alpha=0}^{N-1}e^{-\frac{2\pi
is}{N}\alpha}~{}F_{\alpha}^{A}(t),$ (25) $\displaystyle
a_{s}=4K\sin^{2}\left(\frac{\pi s}{N}\right).$ (26)

It is now straightforward to solve Eq. (23) for $\bar{x}_{s}(t)$ to get

$\displaystyle\bar{x}_{s}(t)=e^{-a_{s}t}~{}\bar{x}_{s}(0)+e^{-a_{s}t}\int_{0}^{t}d\tau~{}e^{a_{s}\tau}~{}\bar{F}_{s}^{A}(\tau),$
(27)

using which in Eq. (21) we get the displacement $x_{\alpha}(t)$ of the
$\alpha^{\text{th}}$ particle.

Remember we are interested to compute the two point correlation defined in Eq.
(11) which after some manipulations can be written as

$\displaystyle\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}=\sum_{s=0}^{N-1}\sum_{s=0}^{N-1}e^{\frac{2\pi
is}{N}\alpha_{1}}e^{\frac{2\pi
is^{\prime}}{N}\alpha_{2}}\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$
(28)

where

$\displaystyle\begin{split}\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}&={\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle-\langle\bar{x}_{s}(t)\rangle\langle\bar{x}_{s^{\prime}}^{*}(t)\rangle}\\\
&=e^{-(a_{s}+a_{s^{\prime}})t}\int_{0}^{t}d\tau_{1}~{}d\tau_{2}~{}e^{a_{s}\tau_{1}+a_{s^{\prime}}\tau_{2}}~{}\langle\bar{F}_{s}^{A}(\tau_{1})\bar{F}_{s^{\prime}}^{A*}(\tau_{2})\rangle_{c},\end{split}$
(29)

where $\bar{x}_{s}^{*}(t)$ is the complex conjugate of $\bar{x}_{s}(t)$ and
$\langle...\rangle$ denotes average over noise $\bar{\sigma}_{s}(t)$ for RTP.
We substitute $F^{A}_{\alpha}(t)$ from Eq. (3) for RTP in Eq. (25) to get
$\bar{F}^{A}_{s}(t)$. Inserting this expression in Eq. (29) and performing the
integration over $\tau_{1}$ and $\tau_{2}$ gives
$\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$. For continuity
of the presentation, we have relegated the details of this calculation to A
and present only the final results here. The final expressions for the
correlations read

$\displaystyle\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}^{RTP}$
$\displaystyle=\delta_{s,s^{\prime}}\frac{v_{0}^{2}}{N}~{}\mathcal{G}(2\zeta,a_{s},t),$
(30)

where $\delta_{s,s^{\prime}}$ is the Kronecker delta which takes value $1$ if
$s=s^{\prime}$ and $0$ otherwise. Also here, $\mathcal{G}(a,b,t)$ is defined
as

$\displaystyle\mathcal{G}(a,b,t)$
$\displaystyle=\frac{b\left[1+e^{-2bt}-2e^{-(a+b)t}\right]-a(1-e^{-2bt})}{b(b^{2}-a^{2})}.$
(31)

###  3.1 MSD for RTP chain

To compute the MSD, we use Eq. (24) to write $x_{\alpha}(t)$ in terms of
$\bar{x}_{s}(t)$ and then substitute the correlation in Eq. (30) to get

$\displaystyle\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$
$\displaystyle=\langle x_{\alpha}^{2}(t)\rangle^{RTP}-\left(\langle
x_{\alpha}(t)\rangle^{RTP}\right)^{2}$
$\displaystyle=\sum_{s=0}^{N-1}\langle|\bar{x}_{s}(t)|^{2}\rangle_{c}^{RTP},$
$\displaystyle=\frac{v_{0}^{2}}{N}\sum_{s=0}^{N-1}\mathcal{G}(2\zeta,a_{s},t),$
(32)

with $a_{s}$ and $\mathcal{G}(2\zeta,a_{s},t)$ given respectively in Eqs. (26)
and (31). Note that, due to the translational symmetry in $\alpha$, the R.H.S.
of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ is independent of $\alpha$. To
gain more understanding of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$, we
first analyze the expression in the limit $N\to\infty$. In this limit, the
summation in the R.H.S. of Eq. (32) can be changed to an integral as
$\frac{1}{N}\sum_{s}\to\frac{1}{2\pi}\int_{-\pi}^{\pi}dq$ where $q=\frac{2\pi
s}{N}$. Changing the summation to integration in Eq. (32), we get

$\displaystyle\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$
$\displaystyle\simeq\frac{v_{0}^{2}}{2\pi}\int_{-\pi}^{\pi}dq~{}\mathcal{G}\left(2\zeta,b_{q},t\right),$
(33) $\displaystyle\text{where}~{}~{}b_{q}$ $\displaystyle=4K\sin^{2}(q/2).$
(34)

As mentioned before (see discussion below Eq. (11)), we are interested in
$\tau_{K}\ll\tau_{A}$ in this work with $\tau_{N}\to\infty$ as $N\to\infty$.
Remember in this case $\tau_{A}=\frac{1}{\zeta}$. We first look at the mean
squared displacement for $t\ll\tau_{K}$ followed by a discussion for
$t\gg\tau_{K}$.

Since $\tau_{K}$ is the smallest time scale in the model, the observation time
$t$ is smaller than all the time scales (see Fig. 2). We can, then, consider
$\langle x_{\alpha}^{2}(t)\rangle_{c}$ in Eq. (33) upto leading order in $t$.
Expanding $\mathcal{G}(2\zeta,b_{q},t)$ in Eq. (31) in small $t$, we find that
$\mathcal{G}(2\zeta,b_{q},t)\simeq t^{2}$. Inserting this in the expression of
$\langle x_{\alpha}^{2}(t)\rangle_{c}$ and performing the integration over $q$
gives

$\displaystyle\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}\simeq
v_{0}^{2}t^{2},~{}~{}~{}~{}~{}~{}~{}\text{for }t\ll\tau_{K}.$ (35)

For small $t$, the particles do not feel the presence of the neighbouring
particles and move ballistically with speed $v_{0}$. This observation
essentially asserts the form of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$
in Eq. (35).

We now look at the other limit when $t\gg\tau_{K}$. Substituing the expression
of $\mathcal{G}\left(2\zeta,b_{q},t\right)$ and $b_{q}$ from Eqs. (31) and
(34) respectively in $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ in Eq. (33),
we find integrals of the form $\sim\int
dq~{}h(q)~{}\text{exp}\left[-Kt\sin^{2}(q/2)\right]$ where $h(q)$ is some
function of $q$. For $Kt\to\infty$, such integrals will be dominated by the
small values of $q$. Therefore, we approximate $b_{q}\simeq Kq^{2}$ in Eq.
(33). We have explicitly computed $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$
for $t\gg\tau_{K}$ in B. The final approximate expression obeys the scaling
form

$\displaystyle\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}t^{\frac{3}{2}}}{\pi}\sqrt{\frac{2}{K}}~{}~{}\mathcal{T}^{RT}\left(\zeta
t\right),$ (36)

where the scaling function $\mathcal{T}^{RT}(y)$ is given by,

$\displaystyle\mathcal{T}^{RT}(y)=\frac{1}{4}\int_{-\infty}^{\infty}dw~{}\mathcal{G}\left(2y,\frac{w^{2}}{2},1\right),~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\tau_{K}$ (37)

with $\mathcal{G}\left(2y,\frac{w^{2}}{2},1\right)$ given in Eq. (31). In Fig.
3(a), we have plotted the scaling function $\mathcal{T}^{RT}(y)$ and compared
it with the numerical simulation. We observe an excellent match between them.
Expanding $\mathcal{G}\left(2y,\frac{w^{2}}{2},1\right)$ in various limits of
$y$, it is easy to see that $\mathcal{T}^{RT}(y)$ has the following asymptotic
forms

$\displaystyle\mathcal{T}^{RT}(y)$
$\displaystyle\simeq\frac{2\sqrt{\pi}}{3}(2-\sqrt{2})-{\frac{4\sqrt{\pi}}{15}\left(8-5\sqrt{2}\right)y},~{}~{}~{}~{}\text{as
}y\to 0,$ (38)
$\displaystyle\simeq\frac{\sqrt{\pi}}{2y}+{O\left(\frac{1}{y^{2}}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (39)

Using these asymptotic forms in the expression of $\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ in Eq. (36), we observe that it crosses
from $t^{3/2}$ for $t\ll\frac{1}{\zeta}$ to $\sqrt{t}$ for
$t\gg\frac{1}{\zeta}$

$\displaystyle\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$
$\displaystyle\simeq\frac{4(\sqrt{2}-1)}{3\sqrt{\pi
K}}v_{0}^{2}t^{\frac{3}{2}}+{O\left(t^{5/2}\right)},~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\ll\frac{1}{\zeta},$ (40)
$\displaystyle\simeq\frac{v_{0}^{2}}{2\zeta}\sqrt{\frac{2t}{\pi
K}}+{O\left(t^{-1/2}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\frac{1}{\zeta}.$ (41)

For $t\gg\frac{1}{\zeta}$, the RTPs behave like Brownian particles with an
effective diffusion constant $D_{R}=\frac{v_{0}^{2}}{2\zeta}$. For
$N\to\infty$, the motion of the tagged particle is effectively like single
file diffusion which gives rise to the $\sim\sqrt{t}$ scaling for the MSD.
This behaviour of the MSD for persistent random walkers in one dimensional
lattice with hardcore interactions was obtained in [72, 75]. On the other
hand, for $\tau_{K}\ll t\ll\frac{1}{\zeta}$, we obtain that the MSD for RTPs
scales super-diffusively as $\sim t^{3/2}$ in contrast to the $t^{2}$ scaling
for non-interacting RTPs. We emphasise that the $t^{3/2}$ scaling of the MSD
arises due to the interplay of the activity of the particles and interactions
between them. Due to the interaction with other particles, the motion of the
tagged particle is hemmed and we consequently get $t^{3/2}$ scaling for the
MSD. The $t^{3/2}$ behaviour for the MSD was also obtained in [76] even though
the crossover function was not obtained.

In Fig. 3(b), we have plotted these asymptotic forms and compared with the
same obtained from the numerical simulations. We clearly observe two scaling
regimes of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ with respect to $t$
and an excellent match of the analytic results in Eqs. (40) and (41) with the
numerical simulations. Interestingly, the scaling form of the MSD in Eq. (36)
was also conjectured for RTPs with hardcore interactions in [75] however the
form of the scaling function was not provided. We have derived this form
explicitly in our simple setting of harmonic interaction.

Figure 4: (a) Comparision of the scaling function $\Omega^{RT}(y)$ in Eq. (47)
for run and tumble particles when $t\ll\frac{1}{\zeta}$. Solid orange line is
the result of theory in Eq. (47) while the symbols correspond to simulation at
different times. We have chosen $K=2,~{}\zeta=0.001$ and $v_{0}=1$. (b) Plot
of the scaling function $\mathcal{C}(y)$ in Eq. (51) (shown by red solid line)
obtained for RTPs when $t\gg\frac{1}{\zeta}$ and comparision with the
numerical simulations for three different values of $t$ (shown by symbols). We
have chosen $K=2,~{}\zeta=2$ and $v_{0}=1$. For both (a) and (b), simulation
is done with $N=200$ particles on a ring.

###  3.2 Covariance for RTP chain

We next look at the covariance for the chain of RTPs which is obtained by
putting $t_{1}=t_{2}$ in $C^{1}_{\alpha,\beta}(t_{1},t_{2})$ in Eq. (11). In
the previous section, we saw that the MSD possesses interesting scaling
behaviours with repect to $t$ depending on where the observation time $t$
lies. Question is - How does $C^{1}_{\alpha,\beta}(t,t)=\langle
x_{\alpha}(t)x_{\alpha+\beta}(t)\rangle_{c}$ behave for different regimes of
$t$? Before answering this question, we note that due to the translational
symmetry with respect to $\alpha$, $\langle
x_{\alpha}(t)x_{\alpha+\beta}(t)\rangle_{c}$ will be independent of $\alpha$.
Hence without loss of generality we take $\alpha=0$.

Proceeding further, we use the definition of Fourier transform in Eq. (21) to
write $x_{\alpha}(t)$ in terms of $\bar{x}_{s}(t)$ which finally gives the
correlation function as

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}=\sum_{s,s^{\prime}=0}^{N-1}e^{\frac{2\pi
is\beta}{N}}\langle\bar{x}_{s}(t)\bar{x}^{*}_{s^{\prime}}(t)\rangle_{c}^{RTP}.$
(42)

Inserting $\langle\bar{x}_{s}(t)\bar{x}^{*}_{s^{\prime}}(t)\rangle_{c}^{RTP}$
from Eq. (30) in Eq. (42), we find that the imaginary part of $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}$ vanishes and we get

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}=\frac{v_{0}^{2}}{N}\sum_{s=0}^{N-1}\cos\left(\frac{2\pi
s\beta}{N}\right)\mathcal{G}(2\zeta,a_{s},t),$ (43)

where $a_{s}$ and $\mathcal{G}(2\zeta,a_{s},t)$ are given by Eqs. (26) and
(31) respectively. We first analyse Eq. (43) in the limit $N\to\infty$. The
finite $N$ case will be discussed in sec. 3.5. For large $N$, we can change
the summation to integration on the R.H.S. of Eq. (43). Changing
$\frac{1}{N}\sum_{s}\to\frac{1}{2\pi}\int_{-\pi}^{\pi}dq$ where $q=\frac{2\pi
s}{N}$, the expression of $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ now
reads

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\int_{-\pi}^{\pi}dq~{}\cos\left(q\beta\right)~{}\mathcal{G}(2\zeta,b_{q},t),$
(44)

where $b_{q}=4K\sin^{2}\left(q/2\right)$.

As done for the MSD, we study correlations for different limits of $t$. For
$t\ll\tau_{K}$, as explained before, the motion of the two particles will be
uncorrelated, hence $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq
v_{0}^{2}t^{2}~{}\delta_{0,\beta}$. In the opposite limit $t\gg\tau_{K}$ they
get nontrivially correlated. As in the case of the MSD, here also, we get
terms like $\sim\int dq~{}h(q)~{}\text{exp}\left[-Kt\sin^{2}(q/2)\right]$ when
we insert $\mathcal{G}(2\zeta,b_{q},t)$ from Eq. (31) in Eq. (44). In the
limit $Kt\to\infty$, such integrals will be dominated by the small values of
$q$ where one can approximate $b_{q}\simeq Kq^{2}$. The details of the
calculation of $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ for
$t\gg\tau_{K}$ is relegated to C. The final expression reads

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{4\pi}\sqrt{\frac{2t^{3}}{K}}\int_{-\infty}^{\infty}dw~{}\cos\left(\frac{w\beta}{\sqrt{2Kt}}\right)~{}\mathcal{G}\left(2\zeta
t,\frac{w^{2}}{2},1\right),$ (45)

To gain more insights, we consider the expression in various limits of $\zeta
t$. Let us first consider $\zeta t\to 0$ limit for which we use Eq. (31) to
approximate $\mathcal{G}\left(2\zeta t\to 0,\frac{w^{2}}{2},1\right)\simeq
4\left(\frac{1-e^{-\frac{w^{2}}{2}}}{w^{2}}\right)^{2}$. Substituting this
form of $\mathcal{G}\left(2\zeta t,\frac{w^{2}}{2},1\right)$ in the R.H.S. of
Eq. (45) and performing the integration over $w$, we find that $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ obeys the scaling form

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{\pi}\sqrt{\frac{2t^{3}}{K}}~{}\Omega^{RT}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{K}\ll t\ll\frac{1}{\zeta},$ (46)

where the scaling function $\Omega^{RT}(y)$ is given as

$\displaystyle\begin{split}\Omega^{RT}(y)=&\frac{\sqrt{\pi}}{6}\left[2e^{-\frac{y^{2}}{4}}\left(4+y^{2}\right)-2\sqrt{2}e^{-\frac{y^{2}}{2}}\left(2+y^{2}\right)\right.\\\
&\left.-\sqrt{\pi}y(6+y^{2})~{}\text{Erfc}\left(\frac{y}{2}\right)+2\sqrt{\pi}y(3+y^{2})~{}\text{Erfc}\left(\frac{y}{\sqrt{2}}\right)\right].\end{split}$
(47)

In Figure 4(a) we have plotted $\Omega^{RT}(y)$ and compared with the results
of numerical simulations for three different times. For all values of $t$ in
range $\tau_{K}\ll t\ll\frac{1}{\zeta}$, the simulation data collapse over our
theoretical result in Eq. (47). The scaling function $\Omega^{RT}(y)$ has the
following asymptotic forms:

$\displaystyle\Omega^{RT}(y)$
$\displaystyle\simeq\frac{2\sqrt{\pi}}{3}(2-\sqrt{2}){-\sqrt{\pi}\left(\sqrt{2}-1\right)y^{2}+O(y^{3})},~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to 0,$ (48)
$\displaystyle\simeq\left(\frac{2\sqrt{\pi}}{y^{4}}+O\left(y^{-6}\right)\right)e^{-\frac{y^{2}}{4}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (49)

Inserting the asymptotic form of $\Omega^{RT}(y)$ for large $y$, one finds
that $\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ in Eq. (46) has faster
than exponential decay for large $\beta$ ($\gg\sqrt{Kt}$) with a decay length
$l_{d}=2\sqrt{2Kt}$. On the other hand, for $\beta\to 0$, $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ correctly reduces to the MSD obtained
in Eq. (40).

Let us now consider the opposite limit $\zeta t\to\infty$ for $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ in Eq. (45). In this limit, from Eq.
(31), we get $\mathcal{G}\left(2\zeta
t\to\infty,\frac{w^{2}}{2},1\right)\simeq\frac{1-e^{-w^{2}}}{\zeta tw^{2}}$
which can be used in Eq. (45) to obtain the scaling form

$\displaystyle\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq
D_{R}\sqrt{\frac{2t}{\pi
K}}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\frac{1}{\zeta},$ (50)

where $D_{R}=\frac{v_{0}^{2}}{2\zeta}$ and the scaling function
$\mathcal{C}(y)$ is given by

$\displaystyle\mathcal{C}(y)=e^{-\frac{y^{2}}{4}}-\frac{\sqrt{\pi}}{2}y~{}\text{Erfc}\left(\frac{y}{2}\right).$
(51)

Same scaling form for the correlation function has been observed in the
context of single file problems in random average process [20, 21, 18] and in
symmetric exclusion process [22, 23]. In Figure 4(b), we have plotted and
compared $\mathcal{C}(y)$ with the numerical simulations for three different
values of $t$ for $\zeta=2$. We observe an excellent agreement of the
theoretical result with the simulation results for all $t$. The scaling
function $\mathcal{C}(y)$ has the following asymptotic forms;

$\displaystyle\mathcal{C}(y)$ $\displaystyle\simeq
1-{\frac{\sqrt{\pi}}{2}y},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to 0,$ (52)
$\displaystyle\simeq{\left(\frac{2}{y^{2}}+O\left(y^{-4}\right)\right)e^{-\frac{y^{2}}{4}},}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (53)

Again, for $\beta\to 0$, the scaling form in Eq. (50) correctly reduces to the
MSD in Eq. (41) for $t\gg\frac{1}{\zeta}$. Also, $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ has faster than exponential decay for
large $\beta$ ($\gg\sqrt{Kt}$) with a decay length $l_{d}=2\sqrt{2Kt}$.

Figure 5: Illustration of the scaling forms obtained in (60) and (66) for
$\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ for
$t\gg\frac{1}{\zeta}$ and $t\ll\frac{1}{\zeta}$ respectively. In both (a) and
(b), the black solid line is the analytical result while symbols correspond to
the simulation data for different values of $t_{1}$. We have taken
$\zeta=0.001$ for plot (a) and $\zeta=2$ for plot (b). For both plots,
simulation is done for $N=200$ particles with $K=3$ and $v_{0}=1$.

###  3.3 Position auto correlation for RTP chain

We next study the position auto correlation $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$ between the positions of a
tagged particle (say $\alpha$th) at times $t_{1}$ and $t_{2}$. The
translational symmetry in $\alpha$ makes $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$ independent of $\alpha$. Also
without loss of generality, we assume $t_{1}\leq t_{2}$ throughout our
analysis. To proceed further, we insert the unequal time correlations of
$\bar{F}^{RTP}_{s}(t)$ derived in Eqs. (167) in $\bar{x}_{s}(t)$ in Eq. (27)
to yield

$\displaystyle\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}^{*}(t_{2})\rangle_{c}^{RTP}~{}=\delta_{s,s^{\prime}}~{}\frac{v_{0}^{2}}{N}\mathcal{H}\left(2\zeta,a_{s},t_{1},t_{2}\right),$
(54)

where $a_{s}$ is given in Eq. (26) and the function
$\mathcal{H}(a,b,t_{1},t_{2})$ is

$\displaystyle\begin{split}\mathcal{H}(a,b,t_{1},t_{2})=&-\frac{a}{b}\frac{e^{-b|t_{1}-t_{2}|}-e^{-b(t_{1}+t_{2})}}{(b^{2}-a^{2})}+\frac{e^{-a|t_{1}-t_{2}|}-e^{-(bt_{1}+at_{2})}}{(b^{2}-a^{2})}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{e^{-(at_{1}+bt_{2})}-e^{-b(t_{1}+t_{2})}}{(b^{2}-a^{2})}\end{split}$
(55)

The correlation function
$\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}(t_{2})\rangle_{c}$ can be
easily translated to $\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$
by writing $x_{\alpha}(t)$ in terms of $\bar{x}_{s}(t)$ from Eq. (21).
Finally, we get

$\displaystyle\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$
$\displaystyle=\sum_{s^{\prime},s=0}^{N-1}\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}^{*}(t_{2})\rangle_{c}^{RTP},$
(56)
$\displaystyle=\frac{v_{0}^{2}}{N}\sum_{s=0}^{N-1}\mathcal{H}\left(2\zeta,a_{s},t_{1},t_{2}\right),$
(57)

As we show later, for this case also, one can obtain various interesting
scaling forms and scaling functions in the limit $N\to\infty$. Changing the
summation in the R.H.S. of Eq. (57) to the integration as
$\frac{1}{N}\sum_{s}\to\frac{1}{2\pi}\int_{-\pi}^{\pi}dq$ in Eq. (57) with
$q=\frac{2\pi s}{N}$, we get

$\displaystyle\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\int_{-\pi}^{\pi}dq~{}\mathcal{H}\left(2\zeta,b_{q},t_{1},t_{2}\right).$
(58)

Here $b_{q}=4K\sin^{2}(q/2)$. We now look at this expression in various limits
of $t_{1}$ with respect to $\tau_{K}$ keeping the ratio $\frac{t_{2}}{t_{1}}$
fixed. We again emphasise that we consider $t_{1}\leq t_{2}$.

We begin with the case when $t_{1}\ll\tau_{K}$ and $t_{2}\ll\tau_{K}$ while
keeping the ratio $\frac{t_{2}}{t_{1}}$ is fixed. Recall that $\tau_{K}$ is
the smallest time scale in the model (see Fig. 2) which means that $t_{1}$ and
$t_{2}$ are smaller than all the time scales present in the model in this
case. At this small time scale, the particle moves ballistically and
independently. As a result, we expect $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ which can be easily shown
by putting $\mathcal{H}\left(2\zeta,b_{q},t_{1},t_{2}\right)\simeq t_{1}t_{2}$
in Eq. (58) and then performing the integration over $q$.

Figure 6: Non-monotonous behaviour of $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}$ for RTP (left panel) and AOUP (right
panel). In panel (a), the analytic result (shown by solid lines) of Eq. (74)
is compared against the results of simulations (shown by filled circles). We
have chosen $K=3,~{}\zeta=0.02,~{}v_{0}=1,~{}N=200,~{}t_{1}=10$. In panel (b),
same is done for AOUP (Eq. (103)) with
$D=0.1,~{}K=2,~{}\gamma=0.02,~{}N=100,~{}t_{1}=50$.

Let us now consider the other limit when $t_{1}\gg\tau_{K}$ and
$\frac{t_{2}}{t_{1}}$ is kept fixed. For this case also, when we insert the
explicit form of $\mathcal{H}(2\zeta,b_{q},t_{1},t_{2})$ from Eq. (55) in
$\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ in Eq. (58), we
get terms of the form $\sim\int dq~{}h(q)~{}e^{-kt\sin^{2}(q/2)}$ where $h(q)$
is some function of $q$. For $Kt_{1}$ very large, such integrals will be
dominated by the small values of $q$ which suggests us to approximate
$b_{q}\simeq Kq^{2}$ as before. Making this approximation, we find

$\displaystyle\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\sqrt{\frac{t_{1}^{3}}{K}}\int_{-\infty}^{\infty}dw~{}\mathcal{H}\left(2\zeta
t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}\text{for
}t_{1}\gg\tau_{K}.$ (59)

We emphasise that this equation is derived assuming the ratio
$\frac{t_{2}}{t_{1}}$ is fixed. To decipher interesting scaling forms, we
evaluate this equation in various limits of $\zeta t_{1}$. For $\zeta t_{1}\to
0$, we obtain, using the definition of $\mathcal{H}$ in Eq. (55), that
$\mathcal{H}\left(2\zeta t_{1}\to
0,w^{2},1,t_{r}\right)\simeq~{}e^{-w^{2}(1+t_{r})}\frac{(e^{t_{r}w^{2}}-1)(e^{w^{2}}-1)}{w^{4}}$
where $t_{r}=t_{2}/t_{1}$. Putting this expression in Eq. (59) and performing
the integration over $w$, we obtain that $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ has the scaling form

$\displaystyle\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$
$\displaystyle\simeq\frac{v_{0}^{2}}{2\pi}\sqrt{\frac{t_{1}^{3}}{K}}~{}\mathcal{L}^{RT}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}{\text{for
}\tau_{K}\ll t_{1}<t_{2}\ll\frac{1}{\zeta}},$ (60)

with the scaling function $\mathcal{L}^{RT}(t_{r})$ given by

$\displaystyle\mathcal{L}^{RT}(t_{r})=\frac{4\sqrt{\pi}}{3}\left[(1+t_{r})^{3/2}-1-t_{r}^{3/2}\right].$
(61)

In Fig. 5(a), we have presented the comparision of our analytic result with
the same obtained from the numerical simulation. We have considered three
different values of $t_{1}$ and for each $t_{1}$, we vary $t_{2}$. All
simulation results match with the analytic result. The asymptotic forms of
$\mathcal{L}^{RT}(t_{r})$ read

$\displaystyle\mathcal{L}^{RT}(t_{r})$
$\displaystyle\simeq\frac{8\sqrt{\pi}}{3}\left(\sqrt{2}-1\right){+2\sqrt{\pi}\left(\sqrt{2}-1\right)(t_{r}-1)},~{}~{}~{}~{}~{}~{}~{}\text{as
}t_{r}\to 1,$ (62) $\displaystyle\simeq 2\sqrt{\pi
t_{r}}-\frac{4\sqrt{\pi}}{3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}{t_{r}\gg 1},$ (63)

inserting which in Eq. (60), we find that the correlation $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ crosses over from
$t_{2}^{3/2}$ behaviour to the $\sqrt{t_{2}}$ behaviour as

$\displaystyle\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$
$\displaystyle\simeq\frac{4(\sqrt{2}-1)}{3\sqrt{\pi
K}}v_{0}^{2}~{}t_{1}^{\frac{3}{2}}+O\left(t_{2}-t_{1}\right),~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{2}\to t_{1},$ (64) $\displaystyle\simeq\frac{v_{0}^{2}t_{1}}{\sqrt{\pi
K}}\left(\sqrt{t_{2}}-\frac{2}{3}\sqrt{t_{1}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{2}\gg t_{1}.$ (65)

For $t_{1}=t_{2}$, $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ correctly reduces to the
expression of MSD in Eq. (40). Quite remarkably, for $t_{2}\gg t_{1}$,
$\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ increases with
increasing $t_{2}$ while both $t_{1}$ and $t_{2}$ being smaller than
$\tau_{A}$.

We next look at $\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$
in the other limit $\zeta t_{1}\to\infty$ while the ratio $t_{2}/t_{1}$ is
again held fixed. In this limit, we use the approximation
$\mathcal{H}\left(2\zeta
t_{1}\to\infty,w^{2},1,t_{r}\right)\simeq\frac{1}{2\zeta
t_{1}}\frac{e^{-w^{2}(t_{r}-1)}-e^{-w^{2}(t_{r}+1)}}{w^{2}}$ inserting which
in Eq. (59) and performing the integration over $w$ we obtain the scaling form

$\displaystyle\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}\simeq
D_{R}\sqrt{\frac{2t_{1}}{\pi
K}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}\text{for
}{t_{2}>t_{1}\gg\frac{1}{\zeta}},$ (66)

where $D_{R}=\frac{v_{0}^{2}}{2\gamma}$ and the scaling function
$\mathcal{M}(t_{r})$ is given by

$\displaystyle\mathcal{M}(t_{r})=\frac{1}{\sqrt{2}}\left[\sqrt{t_{r}+1}-\sqrt{t_{r}-1}\right].$
(67)

This form of the correlation function between the position of a tagged
particle at two different times has been obtained generally in the context of
single file diffusion of passive particles using macroscopic fluctuation
theory [78]. The scaling form in Eq. (66) and the corresponding scaling
function $\mathcal{M}(t_{r})$ are illustrated in Fig. 5(b) where we have also
compared our analytical results with the same obtained from the numerical
simulations for different values of times. The data for different times
collapse over the theoretical result. For further insights, it is instructive
to look at the asymptotic forms of the scaling function $\mathcal{M}(t_{r})$.
The asymptotic forms read

$\displaystyle\mathcal{M}(t_{r})$ $\displaystyle\simeq
1-{\sqrt{\frac{t_{r}-1}{2}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}t_{r}\to 1,$ (68)
$\displaystyle\simeq\frac{1}{\sqrt{2t_{r}}}+{\frac{1}{8\sqrt{2}~{}t_{r}^{\frac{5}{2}}}},~{}~{}~{}~{}~{}~{}~{}\text{as
}t_{r}>>1.$ (69)

Inserting these forms in Eq. (66) reveals that $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ crosses over from
$\sqrt{t_{2}}$ behaviour for $t_{2}\to t_{1}$ to $1/\sqrt{t_{2}}$ behaviour
for $t_{2}\gg t_{1}$ as

$\displaystyle\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$
$\displaystyle\simeq D_{R}\sqrt{\frac{2t_{1}}{\pi
K}}+{O\left(\sqrt{t_{2}-t_{1}}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{2}\to t_{1},$ (70) $\displaystyle\simeq\frac{D_{R}t_{1}}{\sqrt{\pi
Kt_{2}}}+{O\left(\frac{t_{1}^{3}}{t_{2}^{5/2}}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{2}\gg t_{1}.$ (71)

Note that $\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ for
$t_{1}=t_{2}$ reduces to the expression of MSD in Eq. (41) for $t\gg\tau_{A}$.
Moreover contrary to the $t_{1}\ll\tau_{A}$ case, we find that $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ for fixed
$t_{1}\gg\tau_{A}$ decays as $\sim t_{2}^{-\frac{1}{2}}$ for $t_{2}\gg t_{1}$.

Interestingly, for a given $t_{1}$, it turns out that the auto fluctuation has
a non-monotonic dependence on $t_{2}$. This is illustrated in Fig. 6(a)
$(\beta=0)$ where comparision with simulations is also done. From the figure,
we see that the auto fluctuation first increases with $t_{2}$, attains a
maximum value and then starts decreasing again. To understand this behaviour,
it is instructive to look at the asymptotic forms of $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ when $t_{2}\gg t_{1}$ for
$t_{2}\ll\frac{1}{\zeta}$ (Eq. (65)) and $t_{2}\gg\frac{1}{\zeta}$ (Eq. (71)).
While for $t_{2}\ll\frac{1}{\gamma}$, $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ increases with $t_{2}$,
it decreases with $t_{2}$ for $t_{2}\gg\frac{1}{\gamma}$. At times
$t_{2}\ll\tau_{K}$ the particles move ballistically and independently as seen
earlier. As $t_{2}$ increases they realise the presence of other particles
through harmonic springs and as a result their motion start getting
correlated. However at very very large time the particles behaves effectively
like Brownian particles and because of the finiteness of the ring they
eventually reach an equilibrium state with an effective temperature where
their motion again become less correlated. Hence for some intermediate
$t_{2}$, $\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}$ will
exhibit a maximum value.

Figure 7:  Comparison of the scaling forms obtained in Eqs. (76) and (79) for
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ for
$t_{1}\ll\frac{1}{\zeta}$ and $t_{2}\gg\frac{1}{\zeta}$ respectively with the
data obtained from the numerical simulations for different values of $t_{1}$
and $t_{2}$. In both plots, the soild lines correspond to the analytical
results while the filled circles are obtained from the simulation. For plot
(a), we have taken $t_{1}=10,~{}\zeta=0.001$ and simulation is done with
$N=200$ particles. On the other hand, for plot (b), we take
$t_{1}=50,~{}\zeta=2$ and simulation is conducted with $N=500$ particles.
Common parameters for both plots are $v_{0}=1,~{}K=2$.

###  3.4 Unequal Time Position Correlations For RTP chain

This section deals with the unequal time correlations $\langle
x_{\alpha}(t_{1})x_{\alpha+\beta}(t_{2})\rangle_{c}^{RTP}$ for the chain of
RTPs. We again consider $t_{1}\leq t_{2}$. Also, due to the translational
symmetry in $\alpha$, $\langle
x_{\alpha}(t_{1})x_{\alpha+\beta}(t_{2})\rangle_{c}^{RTP}$ will be independent
of $\alpha$ because of which we set $\alpha=0$ without loss of generality.
Starting from the Fourier transfrom in Eq. (21), one gets the correlation in
terms of $\bar{x}_{s}(t_{1})$ as

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$
$\displaystyle=\sum_{s^{\prime},s=0}^{N-1}e^{-\frac{2\pi
is\beta}{N}}\langle\bar{x}_{s^{\prime}}(t_{1})\bar{x}_{s}^{*}(t_{2})\rangle_{c}^{RTP},$
(72) $\displaystyle=\frac{v_{0}^{2}}{N}\sum_{s=0}^{N-1}\cos\left(\frac{2\pi
s\beta}{N}\right)\mathcal{H}\left(2\zeta,a_{s},t_{1},t_{2}\right).$ (73)

While going from first line to the second line, we have substituted
$\langle\bar{x}_{s^{\prime}}(t_{1})\bar{x}_{s}^{*}(t_{2})\rangle_{c}^{RTP}$
from Eq. (54) with $\mathcal{H}\left(2\zeta,a_{s},t_{1},t_{2}\right)$ given in
Eq. (55). Once again, we analyse this expression for $N\to\infty$ first and
then the finite $N$ case. Changing the summation in the R.H.S. to an
integration for $N\to\infty$ gives

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\int_{-\pi}^{\pi}dq~{}\cos(q\beta)~{}\mathcal{H}\left(2\zeta,b_{q},t_{1},t_{2}\right),$
(74)

where $b_{q}=4K\sin(q/2)$. Here also, one can compute correlations for
$t_{1}\ll\tau_{K}$ and $t_{1}\gg\tau_{K}$ limits of $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ keeping the ratio
$\frac{t_{2}}{t_{1}}$ fixed. It is easy to realise that for $t_{1}\ll\tau_{K}$
the correlation would be given by $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq
v_{0}^{2}t_{1}t_{2}\delta_{\beta,0}$. This can be easily proved in a similar
way as done for the previous cases.

In the opposite limit $t_{1}\gg\tau_{K}$, as observed earlier, it will be
convenient to approximate $b_{q}\simeq Kq^{2}$. Inserting
$\mathcal{H}\left(2\zeta,b_{q},t_{1},t_{2}\right)$ from Eq. (55) in Eq. (74)
along with $b_{q}\simeq Kq^{2}$ and finally taking $Kt_{1}\to\infty$ keeping
$\frac{t_{2}}{t_{1}}$ fixed gives (see discussion above Eq. (59))

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\sqrt{\frac{t_{1}^{3}}{K}}\int_{-\infty}^{\infty}dw~{}\cos\left(\frac{w\beta}{\sqrt{Kt_{1}}}\right)~{}\mathcal{H}\left(2\zeta
t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right),~{}~{}~{}\text{for
}t_{1}\gg\tau_{K}.$ (75)

Next we analyse this expression in various limits of $\zeta t_{1}$. In the
limit $\zeta t_{1}\to 0$, we get $\mathcal{H}\left(2\zeta
t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right)\simeq
e^{-w^{2}(1+t_{r})}\frac{(e^{t_{r}w^{2}}-1)(e^{w^{2}}-1)}{w^{4}}$ where
$t_{r}=t_{2}/t_{1}$. Putting this expression in Eq. (75) and performing the
integration over $w$ results in the following scaling form:

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{v_{0}^{2}}{2\pi}\sqrt{\frac{t_{1}^{3}}{K}}~{}\mathcal{P}^{RT}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}\text{for
}t_{1}<t_{2}\ll\frac{1}{\zeta},$ (76)

where the scaling function $\mathcal{P}^{RT}(y,t_{r})$ is given by,

$\displaystyle\mathcal{P}^{RT}(y,t_{r})=$
$\displaystyle\frac{\sqrt{\pi}}{3}\left[2\sqrt{1+t_{r}}(2+2t_{r}+y^{2})e^{-\frac{y^{2}}{2(1+t_{r})}}-2(2+y^{2})e^{-\frac{y^{2}}{2}}-2\sqrt{t_{r}}(2t_{r}+y^{2})e^{-\frac{y^{2}}{2t_{r}}}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\pi}(3+y^{2})y~{}\text{Erfc}\left(\frac{y}{\sqrt{2}}\right)+\sqrt{2\pi}(3t_{r}+y^{2})y~{}\text{Erfc}\left(\frac{y}{\sqrt{2t_{r}}}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\sqrt{2\pi}(3+3t_{r}+y^{2})y~{}\text{Erfc}\left(\frac{y}{\sqrt{2(1+t_{r})}}\right)\right].$
(77)

For $t_{r}\to 1~{}{i.e.}~{}\left(t_{1}\to t_{2}\right)$, it is easy to see
$P^{RT}(y,t_{r}\to 1)=2\sqrt{2}\Omega^{RT}(y)$ and correctly recovers the
scaling relation for $\langle x_{0}(t_{2})x_{\beta}(t_{2})\rangle_{c}$ in Eq.
(46). Also for $y\to 0$, we find $\mathcal{P}^{RT}(y\to
0,t_{r})\simeq\mathcal{L}^{RT}(t_{r})$ (illustrated later) and we recover the
scaling form of Eq. (60) for the mean squared auto fluctuations. In Fig. 7(a),
we have plotted the scaling function $\mathcal{P}^{RT}(y,t_{r})$ and compared
against the same obtained from the numerical simulation for three different
values of $t_{r}=\frac{t_{2}}{t_{1}}$. We observe an excellent match between
them. To see the decay of $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ with respect to $\beta$, we
look at the asymptotic forms of the scaling function
$\mathcal{P}^{RT}(y,t_{r})$ in various limits of $y$. The asymptotic forms
read

$\displaystyle\mathcal{P}^{RT}(y,t_{r})$
$\displaystyle\simeq\frac{4\sqrt{\pi}}{3}\left[(1+t_{r})^{3/2}-1-t_{r}^{3/2}\right]+{2\sqrt{\pi}\left(\sqrt{1+t_{r}}-1-\sqrt{t_{r}}\right)y^{2}},~{}~{}\text{as
}y\to 0,$
$\displaystyle\simeq\left(\frac{4\sqrt{\pi}\left(1+t_{r}\right)^{7/2}}{y^{4}}+{O\left(y^{-6}\right)}\right)\text{exp}\left(-\frac{y^{2}}{2(1+t_{r})}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (78)

Using these expressions, it is easy to show that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ for large $\beta$ has a faster
than exponential decay with a decay length $l_{d}=2\sqrt{K(t_{1}+t_{2})}$.

Let us now look at $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ of
Eq. (75) in the other limit $\zeta t_{1}\to\infty$. In this limit, we obtain
$\mathcal{H}\left(2\zeta
t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right)\simeq\frac{e^{-w^{2}(t_{r}-1)}-e^{-w^{2}(t_{r}+1)}}{w^{2}}$
which we insert in Eq. (75) to obtain that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ obeys the scaling form

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq
D_{R}\sqrt{\frac{2t_{1}}{\pi
K}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}\text{for
}t_{1}\gg\frac{1}{\zeta},$ (79)

where $D_{R}=\frac{v_{0}^{2}}{2\gamma}$ and the scaling function
$\mathcal{Q}(y,t_{r})$ is given by

$\displaystyle\mathcal{Q}(y,t_{r})$
$\displaystyle=\frac{1}{\sqrt{2}}e^{-\frac{y^{2}}{2(1+t_{r})}}\left[\sqrt{t_{r}+1}-\sqrt{t_{r}-1}e^{-\frac{y^{2}}{(t_{r}^{2}-1)}}\right]$
$\displaystyle~{}~{}~{}~{}+\frac{\sqrt{\pi}}{2}y\left[\text{Erf}\left(\frac{y}{\sqrt{2(t_{r}+1)}}\right)\right.\left.-\text{Erf}\left(\frac{y}{\sqrt{2(t_{r}-1)}}\right)\right].$
(80)

For $t_{r}\to 1~{}{i.e.}~{}\left(t_{1}\to t_{2}\right)$,
$\mathcal{Q}(y,t_{r}\to 1)=\mathcal{C}(y)$ in Eq. (51) and we correctly
recover the scaling form in Eq. (50). In Fig. 7, we have presented a
comparision of $\mathcal{Q}(y,t_{r})$ with the same obtained from numerical
simulation for different values of $t_{1}$ and $t_{2}$. We see an excellent
agreement. Next we study the asymptotic behaviour of $\mathcal{Q}(y,t_{r})$ in
various limits of $y$. The asymptotic forms are

$\displaystyle\mathcal{Q}(y,t_{r})$
$\displaystyle\simeq\frac{1}{\sqrt{2}}\left[\sqrt{t_{r}+1}-\sqrt{t_{r}-1}\right]-{\frac{y^{2}}{2\sqrt{2}}\left(\frac{1}{\sqrt{t_{r}-1}}-\frac{1}{\sqrt{t_{r}+1}}\right)},~{}~{}~{}\text{as
}y\to 0,$ (81)
$\displaystyle\simeq\left[\frac{(1+t_{r})^{\frac{3}{2}}}{\sqrt{2}y^{2}}+O\left(y^{-4}\right)\right]\text{exp}\left(-\frac{y^{2}}{2(1+t_{r})}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (82)

For $\beta=0$, $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ reduces
to the expression of the auto fluctuation in Eq. (66). Also for large $\beta$
($\gg\sqrt{Kt_{1}}$), using Eq. (82), we find that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ decays spatially over the
length scale $l_{d}=2\sqrt{K(t_{1}+t_{2})}$.

Interestingly, for a given $\beta$ and $t_{1}$, we find that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ shows non-monotonic behaviour
with $t_{2}$ where it rises initally with $t_{2}$, reaches its maximum value
and then decreases again. This behaviour is illustrated in Fig. 6(a). We have
observed this non-monotonic behaviour even in the auto-fluctuations
$(\beta=0)$ which gets extended for the general $\beta$.

Figure 8: Comparision of the scaling function $\Phi(y)$ in Eq. (88) with the
numerical simulation for RTPs and AOUPs. (a) We have plotted $\Phi(y)$ (shown
by solid line)and compared with the same obtained from the numerical
simulation of RTP dynamics for three different values of $N$ (shown by the
different symbols). We have chosen $v_{0}=\sqrt{2},~{}\zeta=1,~{}K=2$. (b)
Here we have conducted the same analysis with AOUP dynamics with parameters
$D=0.6,~{}\gamma=1,~{}K=2$

###  3.5 Two point correlation in RTP chain for large but finite $N$:

In this section we compute two point correlation between to tagged particle
positions for large but finite $N$ such that $\tau_{N}$ is not infinite. We
start with the exact expression of $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ in Eq. (73). Substituting
$\mathcal{H}\left(2\zeta,a_{s},t_{1},t_{2}\right)$ from Eq. (55) in Eq. (73)
and performing some manipulation, we finally obtain the following expression
upto leading order in $N$

$\displaystyle\begin{split}&\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{2D_{R}t_{1}}{N}\\\
&~{}~{}~{}~{}~{}~{}~{}+\frac{D_{R}N}{2\pi^{2}K}\sum_{s=1}^{\infty}\cos\left(\frac{2\pi
s\beta}{N}\right)\frac{e^{-\frac{4K\pi^{2}s^{2}}{N^{2}}(t_{2}-t_{1})}+e^{-\frac{4K\pi^{2}s^{2}}{N^{2}}(t_{2}+t_{1})}}{s^{2}}+O\left(\frac{1}{N}\right).\end{split}$
(83)

From this expression, we notice that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ possesses the scaling form

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{2D_{R}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right)$
(84)

where the scaling function $\mathcal{W}(t_{r},z,y)$ is

$\displaystyle\mathcal{W}(t_{r},z,y)=y+\frac{1}{2\pi^{2}}\sum_{s=1}^{\infty}\cos\left(2\pi
sz\right)\frac{\sinh\left(4\pi^{2}s^{2}y\right)}{s^{2}}e^{-4\pi^{2}s^{2}t_{r}y}$
(85)

For large $y$, we approximate $\mathcal{W}(t_{r},z,y)\simeq y$ which yields

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}\simeq\frac{2D_{R}t_{1}}{N},~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}Kt_{1}\gg N^{2}.$ (86)

On the other hand for $Kt_{1}\ll N^{2}$, $Kt_{2}\ll N^{2}$ keeping
$t_{2}/t_{1}$ fixed, the summation in Eq. (85) can be converted to an integral
and performing the integration we recover the scaling result obtained in Eq.
(79) for $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$. Taking
various limits of the arguments in the expression of the two-point correlation
in Eqs. (84) and (85) one can the MSD, covariance and auto correlations.

####  3.5.1 MSD:

To get MSD, we put $\beta=0$ and $t_{1}=t_{2}=t$ in Eqs. (84-85) and get

$\displaystyle\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}\simeq\frac{2D_{R}N}{K}~{}\Phi\left(\frac{Kt}{N^{2}}\right),$
(87)

where $D_{R}=\frac{v_{0}^{2}}{2\zeta}$ and the scaling function $\Phi(y)$ is
given by

$\displaystyle\Phi(y)=y+\frac{1}{4\pi^{2}}~{}\sum_{s=1}^{\infty}\frac{1-e^{-8\pi^{2}s^{2}y}}{s^{2}}.$
(88)

In Figure 8(a), we have compared the theoretical result for $\Phi(y)$ with the
results of the numerical simulations of RTP for three different values of $N$.
We observe an excellent agreement between them for all values of $N$. To
obtain the asymptotic forms of the MSD, we look at the various limits of
$\Phi(y)$. For $y\to\infty$, the first term in Eq. (88) dominates which gives
$\Phi(y\to\infty)\simeq y$. On the other hand, for $y\to 0$, the second term
dominates and gives $\Phi(y\to 0)\sim\sqrt{y}$. The approximate expressions
read

$\displaystyle\Phi(y)$
$\displaystyle\simeq\sqrt{\frac{y}{2\pi}}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to 0,$ (89) $\displaystyle\simeq
y+{\frac{1}{24}}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as }y\to\infty.$ (90)

Using these asymptotic expressions in Eq. (87) for $\langle
x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$, it is easy to show that there is a
crossover from $\sim\sqrt{t}$ behaviour to the $\sim t$ behaviour at
$t\sim\mathcal{O}\left(N^{2}\right)$ for the MSD.

$\displaystyle\langle x_{\alpha}^{2}(t)\rangle_{c}^{RTP}$ $\displaystyle\simeq
D_{R}\sqrt{\frac{2t}{\pi
K}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\ll\frac{N^{2}}{K},$ (91)
$\displaystyle\simeq\frac{2D_{R}}{N}t+O\left(t^{0}\right)~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\frac{N^{2}}{K}.$ (92)

Recall that for $t$ greater than the activity time scale $\tau_{A}$, the
dynamics of the $RTP$ is effectively described by that of the Brownian
particle with an effective diffusion constant
$D_{R}=\frac{v_{0}^{2}}{2\zeta}$. Therefore, the scaling form obtained in Eq.
(87) will remain valid for the harmonic chain of Brownian particles in an one
dimensional ring. We close this section by remarking that a similar scaling
form as in Eqs. (87, 88) was obtained for the mean position of the driven
tracer in the random average process [20].

####  3.5.2 Covariance:

To get the covariance, we put $t_{1}=t_{2}=t$ in Eqs. (84-85). We find

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{2D_{R}N}{K}~{}\Psi\left(\frac{\beta}{N},\frac{Kt}{N^{2}}\right),$
(93)

with $D_{R}=\frac{v_{0}^{2}}{2\zeta}$ and the scaling function $\Psi(z,y)$
defined as

$\displaystyle\Psi(z,y)=y+\frac{1}{4\pi^{2}}\sum_{s=1}^{\infty}\cos\left(2\pi
sz\right)\frac{1-e^{-8\pi^{2}s^{2}y}}{s^{2}}.$ (94)

For $y>>1$, we approximate $\Psi(z,y)\simeq y$ using which in Eq. (93) gives

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}\simeq\frac{2D_{R}t}{N},~{}~{}~{}~{}~{}~{}\text{for
}Kt\gg N^{2}.$ (95)

For $Kt\gg N^{2}$, the tagged particle has interacted with all the $N-1$
particles. This fact essentially asserts the $\beta$ independence of Eq. (95).
Let us now look at the opposite limit $Kt\ll N^{2}$ for which the second term
in the R.H.S. of Eq. (94) dominates and one recovers the scaling results for
$\langle x_{0}(t)x_{\beta}(t)\rangle_{c}^{RTP}$ in Eq. (50) for
$t\gg\frac{1}{\zeta}$. This can also be verified by changing the summation in
Eq. (94) to integration and explicitly performing the integration. We end this
section by re-emphasising that for $t\gg\frac{1}{\zeta}$, the run and tumble
particles behave effectively like the Brownian particles with effective
diffusion constant $D_{R}=\frac{v_{0}^{2}}{2\zeta}$. Therefore, the analytic
results obtained for the correlation function in Eqs. (50) and (93) will
remain valid also for the harmonic chain of the Brownian particles.

####  3.5.3 Auto-correlation:

Putting $\beta=0$ in Eqs. (84-85) one finds that the auto-correlation
satisfies the following scaling form

$\displaystyle\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}\simeq\frac{2D_{R}N}{K}~{}\mathcal{R}\left(\frac{t_{2}}{t_{1}},\frac{Kt_{1}}{N^{2}}\right),$
(96)

where the scaling function $\mathcal{R}(t_{r},z)$ is

$\displaystyle\mathcal{R}(t_{r},z)=z+\frac{1}{2\pi^{2}}\sum_{s=1}^{\infty}\frac{\sinh\left(4\pi^{2}s^{2}z\right)}{s^{2}}e^{-4\pi^{2}s^{2}t_{r}z}.$
(97)

For $t_{2}\to t_{1}$, the scaling form in Eq. (96) correctly reduces to that
of the MSD in Eq. (87). Also for $z>>1$ but fixed $t_{r}$,
$\mathcal{R}(t_{r},z)\simeq z$ which when inserted in Eq. (96) gives

$\displaystyle\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{RTP}\simeq\frac{2D_{R}t_{1}}{N},~{}~{}~{}~{}~{}~{}\text{for
}Kt_{1}\gg N^{2}.$ (98)

On the other hand, in the opposite limit $Kt_{1}\ll N^{2}$ and $t_{2}/t_{1}$
fixed, the second term in the R.H.S. of Eq. (97) dominates as observed before.
Analysing this sum by converting to an integral and carrying it out one
recovers the scaling result obtained in Eq. (66) and (67).

Figure 9: (a) Comparision of the scaling function $\mathcal{T}^{OU}(y)$
obtained in Eq. (107) for AOUPs with the results of numerical simulation. (b)
Plot of the asymptotic forms of $\langle x_{\alpha}^{2}(t)\rangle_{c}^{AOUP}$
obtained in Eqs. (110) for $\tau_{k}\ll t\ll\tau_{A}$ (shown by black dotted
line) and $\tau_{k}\ll\tau_{A}\ll t$ (shown by green dotted line) where
$\tau_{A}=\frac{1}{\gamma}$. The pink solid line corresponds to the
theoretical result in Eq. (106) while the blue filled circles correspond to
the numerical simulation. For both plots, we have chosen $K=2,~{}\gamma=0.5$
and $D=0.2$. The simulation is done with $N=100$ particles on a ring.

##  4 MSD, COVARIANCE AND TWO POINT CORRELATION FOR AOUP CHAIN

In this section, we investigate the correlations in the positions of the
tagged particles for the AOUP chain. To this end, we use the correlation
$\langle\bar{F}_{s}^{AOUP}(\tau_{1})\bar{F}_{s}^{AOUP*}(\tau_{2})\rangle_{c}$
in Eq. (170) to write the unequal time correlation for the Fourier variables
$\bar{x}_{s}(t)$ as

$\displaystyle\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}^{*}(t_{2})\rangle^{AOUP}_{c}=\delta_{s,s^{\prime}}\frac{D}{\gamma
N}~{}\mathcal{H}_{1}\left(\gamma,a_{s},t_{1},t_{2}\right),$ (99)

where $a_{s}$ is in Eq. (26) and $\mathcal{H}_{1}(a,b,t_{1},t_{2})$ is given
by

$\displaystyle\mathcal{H}_{1}(a,b,t_{1},t_{2})=\mathcal{H}(a,b,t_{1},t_{2})-\frac{\left(e^{-bt_{1}}-e^{-at_{1}}\right)\left(e^{-bt_{2}}-e^{-at_{2}}\right)}{(b-a)^{2}}$
(100)

with $\mathcal{H}(a,b,t_{1},t_{2})$ given in Eq. (55). Here also, we take
$t_{1}\leq t_{2}$ without loss of generality. Proceeding further, we use
$\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}^{*}(t_{2})\rangle^{AOUP}_{c}$
in Eq. (99) to write the unequal time correlation as

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$
$\displaystyle=\sum_{s,s^{\prime}=0}^{N-1}e^{-\frac{2i\pi\beta
s}{N}}\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}(t_{2})\rangle_{c}^{AOUP},$
(101) $\displaystyle=\frac{D}{\gamma N}\sum_{s=0}^{N-1}\cos\left(\frac{2\pi
s\beta}{N}\right)\mathcal{H}_{1}\left(\gamma,a_{s},t_{1},t_{2}\right).$ (102)

In going to the second line, we substitute
$\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}(t_{2})\rangle_{c}^{AOUP}$ from
Eq. (99). Note that for $\beta=0$ and $t_{1}=t_{2}$, Eq. (102) reduces to the
MSD of the tagged particle. On the other hand, for $\beta\neq 0$ and
$t_{1}=t_{2}$, it reduces to the covariance of the positions of the tagged
particles. Similarly, for $\beta=0$ and $t_{1}\neq t_{2}$, Eq. (102) becomes
the position auto correlation. In what follows, we look at $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ and use it to study the MSD,
covariance and position auto correlation by suitably changing $\beta,~{}t_{1}$
and $t_{2}$.

As seen for RTP, the correlation function $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ possesses various interesting
scaling behaviours depending on where the observation time $t$ lies. To
decipher these scaling forms, we analyse Eq. (102) in the limit $N\to\infty$
for which we change the summation in the R.H.S. to the integration and rewrite

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq\frac{D}{2\pi\gamma}\int_{-\pi}^{\pi}dq\cos\left(q\beta\right)~{}\mathcal{H}_{1}\left(\gamma,b_{q},t_{1},t_{2}\right),$
(103)

where $b_{q}=4K\sin^{2}\left(q/2\right)$. In conjunction with the previous
case of RTP, we analyse this expression for (i)
$t_{1}\ll\tau_{k},~{}t_{2}\ll\tau_{K}$ and (ii)
$t_{1}\gg\tau_{K},~{}t_{2}\gg\tau_{K}$ keeping $\frac{t_{2}}{t_{1}}$ fixed.

Figure 10: (a) Illustration of the scaling function
$\mathcal{P}^{OU}(y,t_{r})$ in Eq. (112) associated with $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$. We have shown comparision
with the results of numerical simulation obtained for three sets of $t_{1}$
and $t_{2}$ keeping the ratio $t_{r}=\frac{t_{2}}{t_{1}}$ fixed which for our
case is $t_{r}=2$. We have chosen $D=0.2$, $K=2$, $\gamma=0.001$ and $N=100$.
(b) Comparision of $\mathcal{Q}(y,t_{r})$ in Eq. (117) with the same obtained
from the simulation of AOUP dynamics. The comparision is shown for three sets
of $t_{1}$ and $t_{2}$ with $t_{r}=\frac{t_{2}}{t_{1}}$ fixed which again is
chosen as $t_{r}=2$. The parameters taken are $D=0.2$, $K=2$, $\gamma=1$ and
$N=200$.

###  4.1 Case I: $t_{1}\ll\tau_{K},~{}t_{2}\ll\tau_{K}$:

Let us consider $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ when
$t_{1}\ll\tau_{K}$, $t_{2}\ll\tau_{K}$ while the ratio $\frac{t_{2}}{t_{1}}$
is kept fixed. Since $\tau_{K}$ is the smallest time scale, $t_{1}$ and
$t_{2}$ are smaller than all the time scales present in the model. Therefore
we expand the R.H.S. of Eq. (103) in $t_{1}$ and $t_{2}$. By direct expansion
of Eq. (100), we find that $\mathcal{H}_{1}(m,n,t_{1},t_{2})\simeq
mt_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)$ inserting which in Eq. (103), we
find

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq
Dt_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)\delta_{\beta,0},~{}~{}~{}~{}~{}~{}\text{as
}t_{1}\leq t_{2}\ll\tau_{K}.$ (104)

Since the particles do not feel the presence of other particles when
$t_{1}\ll\tau_{K}$ and $t_{2}\ll\tau_{K}$, we get zero correlation for
$\beta\neq 0$. Also, for $\beta=0$ and $t_{1}=t_{2}=t$, we find $\langle
x_{0}^{2}(t)\rangle_{c}^{AOUP}\simeq\frac{2D}{3}t^{3}$ which is just the MSD
for a single AOUP.

###  4.2 Case II: $t_{1}\gg\tau_{K},~{}t_{2}\gg\tau_{K}$:

We next look at $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in
Eq. (103) in the limit when $t_{1}\gg\tau_{K}$ and $t_{2}\gg\tau_{K}$ keeping
$t_{2}/t_{1}$ fixed. Proceeding as for the RTP chain (see discussion above Eq.
(75)), it is easy to show that Eq. (103) yields

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq$
$\displaystyle\frac{D}{2\pi\gamma}\sqrt{\frac{t_{1}^{3}}{K}}\int_{-\infty}^{\infty}dw\cos\left(\frac{w\beta}{\sqrt{Kt_{1}}}\right)\mathcal{H}_{1}\left(\gamma
t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right).$ (105)

####  4.2.1 MSD:

We first look at the MSD $\langle x_{0}^{2}(t)\rangle_{c}^{ABP}$ which is
otained by putting $\beta=0$ and $t_{1}=t_{2}=t$ in Eq. (105). Inserting these
values, we find that the MSD possesses the scaling form

$\displaystyle\langle x_{0}^{2}(t)\rangle_{c}^{AOUP}$
$\displaystyle\simeq\frac{Dt^{\frac{3}{2}}}{2\pi\gamma\sqrt{K}}~{}\mathcal{T}^{OU}\left(\gamma
t\right),~{}~{}~{}~{}~{}~{}~{}~{}\text{for }t\gg\tau_{K},$ (106)

where the scaling function $\mathcal{T}^{OU}(y)$ is given by

$\displaystyle\mathcal{T}^{OU}(y)=\int_{-\infty}^{\infty}~{}dw~{}\mathcal{H}_{1}\left(y,w^{2},1,1\right).$
(107)

In Figure 9(a), we have plotted $\mathcal{T}^{OU}(y)$ and compared against the
same obtained from numerical simulation. We observe an excellent agreement.
Proceeding further, we analyse the asymptotic forms of $\mathcal{T}^{OU}(y)$
for $y\to 0$ and $y\to\infty$. To get the these forms, we expand the term
inside integral on the R.H.S. of Eq. (107) in these limits. It is
straightforward to show that the asymptotic forms read

$\displaystyle\mathcal{T}^{OU}(y)$
$\displaystyle\simeq\frac{32\sqrt{\pi}}{15}\left(\sqrt{2}-1\right)y-{\frac{32\sqrt{\pi}}{105}\left(8\sqrt{2}-9\right)y^{2}},~{}~{}~{}~{}~{}\text{as
}y\to 0,$
$\displaystyle\simeq\frac{2\sqrt{2\pi}}{y}+O(y^{-2}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (108)

Using these asymptotic forms, we find that there is crossover from
$t^{\frac{5}{2}}$ to $\sqrt{t}$ at $t\sim\frac{1}{\gamma}$ as

$\displaystyle\langle x_{0}^{2}(t)\rangle_{c}^{AOUP}$
$\displaystyle\simeq\frac{16(\sqrt{2}-1)D}{15\sqrt{\pi
K}}t^{\frac{5}{2}}+O\left(t^{\frac{7}{2}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\ll\frac{1}{\gamma},$ (109)
$\displaystyle\simeq\frac{D}{\gamma^{2}}\sqrt{\frac{2t}{\pi
K}}+O\left(t^{-\frac{1}{2}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\frac{1}{\gamma}.$ (110)

Quite remarkably for $t\ll\tau_{A}$, we find that the MSD scales as $\sim
t^{\frac{5}{2}}$ for AOUP which is different than $\sim t^{\frac{3}{2}}$
scaling for RTP as seen in Eq. (40). Again, this scaling arises due to the
interplay of the activity and interactions between the particles. The caging
effect due to the interactions causes the scaling of the MSD to change from
$t^{3}$ to $t^{\frac{5}{2}}$. On the other hand, for $t\gg\tau_{A}$, both
models give $\sim\sqrt{t}$ scaling for the MSD. In Figure 9(b), we have
illustrated this cross-over behaviour from $t^{5/2}$ to $t^{1/2}$ by comparing
them against the same obtained from numerical simulation. As seen in Figure
9(b), the asymptotic results of Eqs. (109) and (110) are consistent with the
numerical simulation. Finally, we allude that a similar scaling form as in Eq.
(107) was conjectured in [75] for the AOUPs with hadrcore interaction even
though exact form of the scaling function was not obtained. Our consideration
of the simple model of harmonic chain has given both the scaling form and the
scaling function.

Coming back to $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in Eq.
(105), we study this expression in various limits of $\gamma t_{1}$ to get the
effect of activity in the correlation function. In what follows, we look at
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ for $t_{1}\leq
t_{2}\ll\tau_{A}$ and $\tau_{A}\ll t_{1}\leq t_{2}$ separately.

Figure 11: Comparision of the scaling forms obtained for $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}^{AOUP}$ in Eqs. (114) and (118)
for $t\ll\frac{1}{\gamma}$ and $t\gg\frac{1}{\gamma}$ respectively. In both
(a) and (b), the black solid line is the analytical results in in Eqs. (114)
and (118) respectively, while the symbols correspond to the simulation data
for different values of $t_{1}$. We have taken $\gamma=0.001$ for plot (a) and
$\gamma=2$ for plot (b). For both plots, simulation is done for $N=100$
particles with $K=3$ and $D=0.2$.

####  4.2.2 Covariance and unequal time correlations for
${t_{1}\ll\tau_{A},~{}t_{2}\ll\tau_{A}}$:

Let us first look at $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$
in Eq. (105) when both $t_{1}$ and $t_{2}$ are smaller than the activity time
scale $\tau_{A}=\frac{1}{\gamma}$. For $\gamma t_{1}\to 0$, we can expand
$\mathcal{H}_{1}(\gamma t_{1},w^{2},1,t_{r})$ in Eq. (105) in $\gamma t_{1}$
which gives $\mathcal{H}_{1}(\gamma t_{1},w^{2},1,t_{r})\simeq\frac{\gamma
t_{1}}{w^{6}}\left[-2+2e^{-w^{2}}-e^{-(t_{r}+1)w^{2}}+2e^{-t_{r}w^{2}}-e^{-(t_{r}-1)w^{2}}+2w^{2}\right]$
up to leading order in $\gamma t_{1}$. Substituting this expression in Eq.
(105) and performing the integration over $w$, we find that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ has the scaling form

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq\frac{D}{2\pi}\sqrt{\frac{t_{1}^{5}}{K}}~{}\mathcal{P}^{OU}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\frac{1}{\gamma},$ (111)

where the scaling function $\mathcal{P}^{OU}\left(y,t_{r}\right)$ is given by

$\displaystyle\mathcal{P}^{OU}$
$\displaystyle\left(y,\right.\left.t_{r}\right)=\frac{\sqrt{\pi}}{30}\left[2\sqrt{1+t_{r}}\left(8+8t_{r}^{2}+9y^{2}+y^{4}+t_{r}(16+9y^{2})\right)e^{-\frac{y^{2}}{2(t_{r}+1)}}-30\sqrt{2\pi}y\text{Erf}\left(\frac{y}{\sqrt{2}}\right)\right.$
$\displaystyle-2(t_{r}-1)^{\frac{3}{2}}\left(7t_{r}-7+y^{2}\right)e^{-\frac{y^{2}}{2(t_{r}-1)}}-4\left(8+9y^{2}+y^{4}\right)e^{-\frac{y^{2}}{2}}-4\sqrt{t_{r}}\left(8t_{r}^{2}+9t_{r}y^{2}+y^{4}\right)e^{-\frac{y^{2}}{2t_{r}}}$
$\displaystyle\left.+\left(15+15t_{r}^{2}-10y^{2}+y^{4}+10t_{r}(-3+y^{2})\right)\Bigg{\\{}2\sqrt{t_{r}-1}e^{-\frac{y^{2}}{2(t_{r}-1)}}+\sqrt{2\pi}y\text{Erf}\left(\frac{y}{\sqrt{2(t_{r}-1)}}\right)\Bigg{\\}}\right.$
$\displaystyle+2\sqrt{2\pi}y^{3}\left(10+y^{2}\right)\text{Erfc}\left(\frac{y}{\sqrt{2}}\right)-2\sqrt{2\pi}y\left(15t_{r}^{2}+10t_{r}y^{2}+y^{4}\right)\text{Erf}\left(\frac{y}{\sqrt{2t_{r}}}\right)$
$\displaystyle\left.+\sqrt{2\pi}y\left(15+15t_{r}^{2}+10y^{2}+y^{4}+10t_{r}(3+y^{2})\right)\text{Erf}\left(\frac{y}{\sqrt{2(t_{r}+1)}}\right)\right].$
(112)

In Fig. 10(a), we have illustrated the scaling function
$\mathcal{P}^{OU}(y,t_{r})$ and also compared with the same obtained from
numerical simulations. We have conducted the comparision for three sets of
$t_{1}$ and $t_{2}$ keeping the ratio $t_{r}=\frac{t_{2}}{t_{1}}$ constant for
all. All simuation data collapse over the analytical result. For large $y$, we
find $\mathcal{P}^{OU}(y,t_{r})$ decays as $\sim
y^{-6}~{}\text{exp}\left(-\frac{y^{2}}{2(t_{r}+1)}\right)$ inserting which in
Eq. (111), it is seen that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ has faster than exponential
decay with decay length $l_{d}=2\sqrt{K(t_{1}+t_{2})}$. On the other hand, for
$y=0$, we find that $\mathcal{P}^{OU}(y=0,t_{r})=\mathcal{L}^{OU}(t_{r})$
where $\mathcal{L}^{OU}(t_{r})$ is given by

$\displaystyle\mathcal{L}^{OU}(t_{r})=\frac{8\sqrt{\pi}}{15}\left[(t_{r}+1)^{5/2}+(t_{r}-1)^{5/2}-2(1+t_{r}^{5/2})\right].$
(113)

Inserting this expression in $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in Eq. (111) for $\beta=0$, we
find that the position auto fluctuation satisfies the scaling form

$\displaystyle\langle
x_{0}(t_{1})x_{0}(t_{2})\rangle_{c}^{AOUP}\simeq\frac{D}{2\pi}\sqrt{\frac{t_{1}^{5}}{K}}~{}\mathcal{L}^{OU}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\frac{1}{\gamma}.$ (114)

We have presented the comparision of the scaling function
$\mathcal{L}^{OU}(t_{r})$ with the same obtained from the numerical simulation
in Fig. 11(a). We have performed the comparision for three different $t_{1}$
and for each $t_{1}$, we vary $t_{2}$. All simulation results match with our
theoretical result.

We next look at $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in
Eq. (111) for general $\beta$ but $t_{1}=t_{2}=t$ for which it reduces to the
covariance. As evident from Eq. (111), the covariance satifies the scaling

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{AOUP}\simeq\frac{D~{}t^{\frac{5}{2}}}{2\pi\sqrt{K}}~{}\Omega^{OU}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}\text{for
}t\ll\frac{1}{\gamma},$ (115)

with the scaling function $\Omega^{OU}(y)$ given by

$\displaystyle\Omega^{OU}(y)=\mathcal{P}^{OU}\left(y,t_{r}=1\right),$ (116)

where $\mathcal{P}^{OU}\left(y,t_{r}=1\right)$ is given in Eq. (112). In
Figure 12(a), we have plotted the scaling function $\Omega^{OU}(y)$ and
compared with the results of the numerical simulations for three different
times. We observe an excellent agreement of the theoretical result with that
of the simulations. Quite expectedly, this scaling function is different than
the same obtained for RTP in Eq. (47). To see this more clearly, we look at
the large $y$ behaviour of $\Omega^{OU}(y)$. By direct expansion, we find that
$\Omega^{OU}(y)\sim y^{-6}~{}e^{-\frac{y^{2}}{4}}$ which is different than the
same obtained in Eq. (49) for RTP. Even though both decay as $\sim
e^{-\frac{y^{2}}{4}}$, but the corresponding prefactors are very different.

Figure 12: (a) Comparision of the scaling function $\Omega^{OU}(y)$ in Eq.
(116) for AOUPs (shown by solid black line) with the results obtained from
numerical simulation (shown by symbols) for three different times. We have
chosen $K=2,~{}\gamma=0.001$ and $D=0.5$. (b) Plot of the scaling form and
scaling function $\mathcal{C}(y)$ obtained in Eqs. (119) and (51) respectively
(shown by solid black line) and comparision with the same obtained from
numerical simulation for three different values of $t$ (shown by different
symbols). The parameters chosen are $K=2,~{}\gamma=2$ and $D=1.7$. For both
(a) and (b), the simulation is done with $N=150$ particles.

####  4.2.3 Covariance and unequal time correlations for
$t_{1}\gg\tau_{A},~{}t_{2}\gg\tau_{A}$:

We now look at $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in Eq.
(105) when both $t_{1}$ and $t_{2}$ are larger than the activity time scale
$\tau_{A}$. For $\gamma t_{1}\to\infty$, we use Eq. (100) to approximate
$\mathcal{H}_{1}(\gamma t_{1},w^{2},1,t_{r})\simeq\frac{1}{\gamma
t_{1}w^{2}}\left(e^{-(t_{r}-1)w^{2}}-e^{-(t_{r}+1)w^{2}}\right)$ and insert it
in Eq. (105) and performing integration over $w$ gives the scaling relation

$\displaystyle\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq
D_{A}\sqrt{\frac{2t_{1}}{\pi
K}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}\text{for
}\frac{1}{\gamma}\ll t_{1}\leq t_{2},$ (117)

where the scaling function $\mathcal{Q}(y,t_{r})$ is given in Eq. (80) and
$D_{A}=\frac{D}{\gamma^{2}}$. The result is identical to that of the RTP in
Eq. (79) as both models converge to the Brownian motion when $t\gg\tau_{A}$.
In Fig. 10(b), we have presented a comparision of the scaling function
$\mathcal{Q}(y,t_{r})$ with the same obtained from the numerical simulations
for three sets $t_{1}$ and $t_{2}$. We observe an excellent match of
simulation results with the analytical form of $\mathcal{Q}(y,t_{r})$.

By appropriately manipulating $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in Eq. (117), it is easy to
show that the auto correlation and covariance have the scaling form

$\displaystyle\langle x_{0}(t_{1})x_{0}(t_{2})\rangle_{c}^{AOUP}\simeq
D_{A}\sqrt{\frac{2t_{1}}{\pi
K}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{A}\ll t_{1}\leq t_{2},$ (118) $\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{AOUP}\simeq D_{A}\sqrt{\frac{2t}{\pi
K}}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}t\gg\tau_{A},$ (119)

where the scaling functions $\mathcal{C}(y)$ and $\mathcal{M}(t_{r})$ are
given in Eqs. (51) and (67) respectively. In Figure 11(b) and 12(b), we have
numerically illustrated the scaling relations in Eqs. (118) and (119)
respectively. We observe excellent agreement with the numerical simulations.
Once again, we emphasise that in this time regime $(\gg\tau_{A})$, the scaling
functions for RTP and AOUP models are same because they essentially become
harmonically coupled chain of Brownian particles.

Before closing this section, we accentuate that for AOUP also, the correlation
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ exhibits non-
monotonic behaviour with respect to $t_{2}$. This is shown in Fig. 6(b) for
$\beta=0$ and $\beta\neq 0$ along with the comparision with the simulations.
We see that it increases initially with rise in $t_{2}$, attains its maximum
value and then starts decreasing thereafter. As discussed for RTP, the non-
monotonic behaviour can be understood by looking at the asymptotic expressions
of $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ for $t_{2}\gg
t_{1}$ when $t_{2}\ll\frac{1}{\gamma}$ and $t_{2}\gg\frac{1}{\gamma}$. When
both $t_{1}\ll\frac{1}{\gamma}$ and $t_{2}\ll\frac{1}{\gamma}$, it can be
shown using Eq. (111) that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ increases as
$\sim\sqrt{t_{2}}$ for $t_{2}\gg t_{1}$. On the other hand when
$t_{1}\gg\tau_{A}$ and $t_{2}\gg\tau_{A}$, using Eq. (117), we find that
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ decays as $\sim
1/\sqrt{t_{2}}$. This means that at some intermediate $t_{2}$, it will exhibit
the maximum value.

###  4.3 Large but finite $N$:

We now look at the correlation $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ when $N$ is large but has
finite value. As done for RTP, one can, in principle, start from the the exact
expression of $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}$ in Eq.
(102) and analyse it for the large but finite $N$ case. However, we follow a
different route by noting that the dynamics of RTP and AOUP are identical to
that of the Brownian motion when the obervation time is larger than the
activity time scale. Hence the scaling results obtained in Eq. (84) and (85)
for $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{RTP}$ will also be valid
for AOUP with $D_{R}$ replaced by $D_{A}=\frac{D}{\gamma^{2}}$. Accordingly we
have

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}^{AOUP}\simeq\frac{2D_{A}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right),$
(120)

where the scaling function $\mathcal{W}(t_{r},z,y)$ is given in Eq. (85). Once
again, we can use Eq.(120) to obtain various scaling relations for the
variance, covariance and auto correlation as

$\displaystyle\langle
x_{0}(t_{1})x_{0}(t_{2})\rangle_{c}^{AOUP}\simeq\frac{2D_{A}N}{K}~{}\mathcal{R}\left(\frac{t_{2}}{t_{1}},\frac{Kt_{1}}{N^{2}}\right),$
(121) $\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{AOUP}\simeq\frac{2D_{A}N}{K}~{}\Psi\left(\frac{\beta}{N},\frac{Kt}{N^{2}}\right),$
(122) $\displaystyle\langle
x_{0}^{2}(t)\rangle_{c}^{AOUP}\simeq\frac{2D_{A}N}{K}\Phi\left(\frac{Kt}{N^{2}}\right),$
(123)

where the scaling functions $\mathcal{R}(t_{r},z),~{}\Psi(z,y)$ and $\Phi(y)$
are given in Eqs. (97), (94) and (88) respectively. In Figure 8(b), we have
compared the scaling form in Eq. (123) with numerical simulations for three
different values of $N$. We observe excellent agreement for all values of $N$.

Figure 13: Comparision of the scaling functions $\mathcal{T}_{1}^{AB}(y)$
(left panel) and $\mathcal{T}_{2}^{AB}(y)$ (right panel) in Eqs. (139) and
(140) with the numerical simulations. For both plots, we have chosen
$K=2,~{}D_{rot}=0.5$ and $v_{A}=1$. Simulation is done with $N=100$.

##  5 MSD, COVARIANCE AND TWO POINT CORRELATION FOR ABP CHAIN

We now study the two point correlation function for $N$ active Brownian
particles with nearest neighbour harmonic interactions in a chain. As
discussed before, for ABP, we have
$\vec{r}_{\alpha}(t)=\left(x_{\alpha}(t),y_{\alpha}(t)\right)$. To solve the
Langevin equations (1), once again, we take the Fourier transformation with
respect to $\alpha$ (see Eq. (21)) and rewrite Eqs. (1) in terms of the
Foruier variables as

$\displaystyle\frac{d\vec{\bar{r}}_{s}}{dt}=-a_{s}\vec{\bar{r}}_{s}+\vec{\bar{F}}_{s}^{ABP}(t),$
(124)

where $\vec{\bar{r}}_{s}(t)=\left(\bar{x}_{s}(t),\bar{y}_{s}(t)\right)$ and
$\vec{\bar{F}}_{s}^{ABP}(t)=\left(\bar{\xi}_{s}(t),\bar{\psi}_{s}(t)\right)$.
As done for RTP and AOUP in A, we solve Eq. (124) and use the correlations in
Eqs. (7), (8) and (9) to write the connected correlations for $\bar{x}_{s}(t)$
and $\bar{z}_{s}(t)$ as

$\displaystyle\langle\bar{x}_{s}(t_{1})\bar{x}_{s^{\prime}}^{*}(t_{2})\rangle_{c}^{ABP}=\frac{v_{A}^{2}}{2N}~{}\mathbb{H}_{1}\left(D_{rot},a_{s},t_{1},t_{2}\right)\delta_{s,s^{\prime}},$
(125)
$\displaystyle\langle\bar{y}_{s}(t_{1})\bar{y}_{s^{\prime}}^{*}(t_{2})\rangle_{c}^{ABP}=\frac{v_{A}^{2}}{2N}~{}\mathbb{H}_{2}\left(D_{rot},a_{s},t_{1},t_{2}\right)\delta_{s,s^{\prime}}.$
(126)

Note that here also, we take $t_{1}\leq t_{2}$ without loss of generality.
Also, $a_{s}$ is given in Eq. (26) and the functions
$\mathbb{H}_{1/2}(m,n,t_{1},t_{2})$ are given by

$\displaystyle\mathbb{H}_{p}(a,b,t_{1},t_{2})$
$\displaystyle=\mathcal{H}(a,b,t_{1},t_{2})+(-1)^{p-1}\mathcal{V}(a,b,t_{1},t_{2})$
$\displaystyle-2\delta_{p,1}\frac{\left(e^{-at_{1}}-e^{-bt_{1}}\right)\left(e^{-at_{2}}-e^{-bt_{2}}\right)}{(b-a)^{2}},$
(127)

with $p=1,2$ and $\mathcal{H}(a,b,t_{1},t_{2})$ is given in Eq. (55) and
$\mathcal{V}(m,n,t_{1},t_{2})$ is defined as

$\displaystyle\begin{split}\mathcal{V}(a,b,t_{1},t_{2})=&\frac{ae^{-b|t_{1}-t_{2}|}\left(e^{-4at_{1}}-e^{-2bt_{1}}\right)}{(b-a)(b-3a)(b-2a)}\\\
&-\frac{e^{-bt_{2}}\left(e^{-at_{1}}-e^{-bt_{1}}\right)-e^{-at_{2}}\left(e^{-3at_{1}}-e^{-bt_{1}}\right)}{(b-a)(b-3a)}.\end{split}$
(128)

Next we use correlations of the Fourier variables $\bar{x}_{s}(t)$ and
$\bar{y}_{s}(t)$ in Eqs. (125) and (126) to write the unequal time correlation
in positions as

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}=\frac{v_{A}^{2}}{2N}\sum_{s=0}^{N-1}\cos\left(\frac{2\pi
s\beta}{N}\right)\mathbb{H}_{1}\left(D_{rot},a_{s},t_{1},t_{2}\right),$ (129)
$\displaystyle\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}=\frac{v_{A}^{2}}{2N}\sum_{s=0}^{N-1}\cos\left(\frac{2\pi
s\beta}{N}\right)\mathbb{H}_{2}\left(D_{rot},a_{s},t_{1},t_{2}\right).$ (130)

As done for RTP and AOUP, we will analyse the correlations in the $N\to\infty$
limit to extract various scaling forms and scaling functions. In this limit,
one can change the summation in the R.H.S. of Eqs. (129) and (130) to
integrals and rewrite

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}}{4\pi}\int_{-\pi}^{\pi}dq~{}\cos(q\beta)~{}\mathbb{H}_{1}\left(D_{rot},b_{q},t_{1},t_{2}\right),$
(131) $\displaystyle\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}}{4\pi}\int_{-\pi}^{\pi}dq~{}\cos(q\beta)~{}\mathbb{H}_{2}\left(D_{rot},b_{q},t_{1},t_{2}\right),$
(132)

where $b_{q}=4K\sin^{2}(q/2)$. In what follows, we will analyse $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}$ and $\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}$ for (i)
$t_{1}\ll\tau_{K},~{}t_{2}\ll\tau_{K}$ and (ii)
$t_{1}\gg\tau_{K},~{}t_{2}\gg\tau_{K}$ keeping the ratio $\frac{t_{2}}{t_{1}}$
fixed. Finally, we look at these correlations for finite but large $N$.

Figure 14: Illustration of the asymptotic behaviours of $\langle
x_{\alpha}^{2}(t)\rangle_{c}^{ABP}$ (left panel) and $\langle
y_{\alpha}^{2}(t)\rangle_{c}^{ABP}$ (right panel) for $\tau_{k}<<t<<\tau_{A}$
(shown by green line in both plots) and $\tau_{k}<<\tau_{A}<<t$ (shown by
black line in both plots) where $\tau_{A}=\frac{1}{D_{rot}}$. The
corresponding analytic expressions are given in Eqs .(145) - (148). For both
plots, we have chosen $K=2,~{}D_{rot}=0.5$ and $v_{A}=1$. Simulation is done
with $N=100$

###  5.1 Case I: $t_{1}\ll\tau_{K},~{}t_{2}\ll\tau_{K}$:

We first look at the correlations in Eqs. (131) and (132) when both $t_{1}$
and $t_{2}$ are smaller than $\tau_{K}$. Since $\tau_{K}$ is the smallest time
scale in the problem, we expand the correlations up to leading order in
$t_{1}$ and $t_{2}$. Using Eq. (127), we approximate
$\mathbb{H}_{1}(D_{rot},a_{s},t_{1},t_{2})\simeq\frac{2}{3}D_{rot}^{2}t_{1}^{3}(2t_{2}-t_{1})$
and $\mathbb{H}_{2}(D_{rot},a_{s},t_{1},t_{2})\simeq
2D_{rot}t_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)$ and insert these
expressions in Eqs. (131) and (132) to get

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{1}{3}v_{A}^{2}D_{rot}^{2}t_{1}^{3}(2t_{2}-t_{1})\delta_{\beta,0},~{}~{}~{}~{}~{}t_{1}\leq
t_{2}\ll\tau_{K},$ (133) $\displaystyle\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq
v_{A}^{2}D_{rot}t_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)\delta_{\beta,0},~{}~{}~{}~{}~{}t_{1}\leq
t_{2}\ll\tau_{K}.$ (134)

Once again, the $\delta_{\beta,0}$ arises due to the fact that the particles
do not interact with each other during this time scale. Also, note that
putting $\beta=0$ and $t_{1}=t_{2}=t$, we recover the expression for variance
as $\langle
x_{0}^{2}(t)\rangle_{c}^{ABP}\simeq\frac{1}{3}v_{A}^{2}D_{rot}^{2}t^{4}$ and
$\langle y_{0}^{2}(t)\rangle_{c}^{ABP}\simeq\frac{2}{3}v_{A}^{2}D_{rot}t^{3}$
. These expressions agree with the results obtained in [51] for a single ABP.

###  5.2 Case II: $t_{1}\gg\tau_{K},~{}t_{2}\gg\tau_{K}$:

We next consider $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}$ and
$\langle y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}$ in Eqs. (131) and
(132) when both $t_{1}$ and $t_{2}$ are larger than $\tau_{K}$. Inserting
$\mathbb{H}_{1/2}\left(D_{rot},b_{q},t_{1},t_{2}\right)$ from Eq. (127) in
Eqs. (131) and (132), we find integrals of the form $\sim\int
dq~{}h(q)~{}\text{exp}\left[-Kt_{1}\sin^{2}(q/2)\right]$ where $h(q)$ is some
function of $q$. For $Kt_{1}\to\infty$ with $N\to\infty$ already taken, such
integrals will be dominated by small values of $q$ which means we can
approximate $b_{q}\simeq Kq^{2}$ as done before. With this approximation, the
correlations become

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}}{4\pi}\sqrt{\frac{t_{1}^{3}}{K}}\int_{-\infty}^{\infty}dw~{}\cos\left(\frac{w\beta}{\sqrt{Kt_{1}}}\right)~{}\mathbb{H}_{1}\left(D_{rot}t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right),$
(135) $\displaystyle\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}}{4\pi}\sqrt{\frac{t_{1}^{3}}{K}}\int_{-\infty}^{\infty}dw~{}\cos\left(\frac{w\beta}{\sqrt{Kt_{1}}}\right)~{}\mathbb{H}_{2}\left(D_{rot}t_{1},w^{2},1,\frac{t_{2}}{t_{1}}\right).$
(136)

Figure 15:  Comparision of the scaling functions
$\mathcal{P}_{1}^{AB}(y,t_{r})$ (left panel) and
$\mathcal{P}_{2}^{AB}(y,t_{r})$ (right panel) given in Eqs. (150) and (152)
with the numerical simulations. We have conducted the comparision for three
sets of values of $t_{1}$ and $t_{2}$ keeping the ratio
$t_{r}=\frac{t_{2}}{t_{1}}$ fixed. For both plots, we have taken
$K=2,~{}D_{rot}=0.001,~{}v_{A}=1.5$ and $N=100$ for the simulation.

####  5.2.1 MSD:

Using these expressions, we first study the MSD of the position of tagged
particle and consider other correlations subsequently. To get the MSD, we put
$\beta=0$ and $t_{1}=t_{2}=t$ in Eqs. (135) and (136) and rewrite

$\displaystyle\langle
x_{0}^{2}(t)\rangle_{c}^{ABP}\simeq\frac{v_{A}^{2}t^{3/2}}{4\pi\sqrt{K}}~{}\mathcal{T}_{1}^{AB}\left(D_{rot}t\right),~{}~{}~{}~{}~{}t\gg\tau_{K},$
(137) $\displaystyle\langle
y_{0}^{2}(t)\rangle_{c}^{ABP}\simeq\frac{v_{A}^{2}t^{3/2}}{4\pi\sqrt{K}}~{}\mathcal{T}_{2}^{AB}\left(D_{rot}t\right),~{}~{}~{}~{}~{}t\gg\tau_{K},$
(138)

where the scaling functions $\mathcal{T}_{1}^{AB}(y)$ and
$\mathcal{T}_{2}^{AB}(y)$ are given by

$\displaystyle\mathcal{T}_{1}^{AB}(y)=\int_{-\infty}^{\infty}dw~{}\mathbb{H}_{1}(y,w^{2},1,1),$
(139)
$\displaystyle\mathcal{T}_{2}^{AB}(y)=\int_{-\infty}^{\infty}dw~{}\mathbb{H}_{2}(y,w^{2},1,1).$
(140)

In Fig. 13, we have compared the scaling functions with the results of the
numerical simulation. We get an excellent match. To get interesting scaling of
the MSD with respect to $t$, we look at the asymptotics of these scaling
functions. We find that $\mathcal{T}_{1}^{AB}(y)$ and
$\mathcal{T}_{2}^{AB}(y)$ has the following asymptotics:

$\displaystyle\mathcal{T}_{1}^{AB}(y)$
$\displaystyle\simeq\frac{256(\sqrt{2}-1)\sqrt{\pi}}{105}y^{2}-{\frac{256\sqrt{\pi}\left(16\sqrt{2}-17\right)}{945}y^{3}},~{}~{}~{}~{}~{}\text{as
}y\to 0,$ (141)
$\displaystyle\simeq\frac{2\sqrt{2\pi}}{y}+{O\left(y^{-2}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty,$ (142) $\displaystyle\mathcal{T}_{2}^{AB}(y)$
$\displaystyle\simeq\frac{64(\sqrt{2}-1)\sqrt{\pi}}{15}y-{\frac{64\sqrt{\pi}}{105}\left(8\sqrt{2}-9\right)y^{2}},~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to 0,$ (143)
$\displaystyle\simeq\frac{2\sqrt{2\pi}}{y}+{O\left(y^{-2}\right)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{as
}y\to\infty.$ (144)

Inserting these asymptotics in the expression of $\langle
x_{0}^{2}(t)\rangle_{c}^{ABP}$ and $\langle y_{0}^{2}(t)\rangle_{c}^{ABP}$ in
Eqs. (137) and (138), we get

$\displaystyle\langle x_{0}^{2}(t)\rangle_{c}^{ABP}$
$\displaystyle\simeq\frac{64(\sqrt{2}-1)}{105\sqrt{\pi}}\frac{v_{A}^{2}D_{rot}^{2}}{\sqrt{K}}~{}t^{7/2}+O\left(t^{9/2}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{K}\ll t\ll\tau_{A},$ (145)
$\displaystyle\simeq\frac{v_{A}^{2}}{2D_{rot}}\sqrt{\frac{2t}{\pi
K}}+O\left(t^{-1/2}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{K}\ll\tau_{A}\ll t,$ (146) $\displaystyle\langle
y_{0}^{2}(t)\rangle_{c}^{ABP}$
$\displaystyle\simeq\frac{16(\sqrt{2}-1)}{15\sqrt{\pi}}\frac{v_{A}^{2}D_{rot}}{\sqrt{K}}~{}t^{5/2}+O\left(t^{7/2}\right),~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{K}\ll t\ll\tau_{A},$ (147)
$\displaystyle\simeq\frac{v_{A}^{2}}{2D_{rot}}\sqrt{\frac{2t}{\pi
K}}+O\left(t^{-1/2}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{K}\ll\tau_{A}\ll t.$ (148)

We have illustrated these asymptotic behaviours in Fig. 14 for both $\langle
x_{0}^{2}(t)\rangle_{c}^{ABP}$ (left panel) and $\langle
y_{0}^{2}(t)\rangle_{c}^{ABP}$ (right panel) where we have also compared with
the numerical simulations. We observe an excellent match. Note that for
$t\ll\tau_{A}$, we see that $\langle x_{0}^{2}(t)\rangle_{c}^{ABP}$ scales as
$t^{7/2}$ which is different than the $t^{3/2}$ and $t^{5/2}$ scaling for RTP
and AOUP respectively. This non-trivial scaling arises due to the interplay of
activity and caging effects of the neighbouring particles. On the other hand,
we find that $\langle y_{0}^{2}(t)\rangle_{c}^{ABP}$ has $t^{5/2}$ scaling for
$t\ll\tau_{A}$ which is same as that for the AOUP (see Eq. (109)). This
results from the fact that for $t\ll\tau_{A}$, the dynamics for the
$y$-coordinate of ABP and AOUP is exactly same. For $t\gg\tau_{A}$, we recover
the result of the Brownian motion with an effective diffusion constant
$D_{ABP}=\frac{v_{A}^{2}}{2D_{rot}}$.

We now revert to the unequal time position correlations in Eqs. (135) and
(136) and analyse it for (i) $\tau_{K}\ll t_{1}\leq t_{2}\ll\tau_{A}$ and (ii)
$\tau_{K}\ll\tau_{A}\ll t_{1}\leq t_{2}$.

Figure 16:  Comparision of the scaling functions $\mathcal{L}_{1}^{AB}(t_{r})$
(left panel) and $\mathcal{L}_{2}^{AB}(y,t_{r})$ (right panel) given in Eqs.
(155) and (156) with the numerical simulations (shown by symbols) for three
different values of $t_{1}$. For both plots, wehave taken
$K=2,~{}D_{rot}=0.0001,~{}v_{A}=1$ and $N=100$ for the simulation.

####  5.2.2 Covariance and unequal time correlations for
${t_{1}\ll\tau_{A},~{}t_{2}\ll\tau_{A}}$:

We first consider $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}$ and
$\langle y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}$ when both $t_{1}$ and
$t_{2}$ are smaller than the activity time scale $\tau_{A}=\frac{1}{D_{rot}}$.
For this case, we use Eq. (127) to approximate

$\displaystyle\mathbb{H}_{1}\left(D_{rot}t_{1}\to
0,w^{2},1,t_{r}\right)\simeq\frac{2D_{rot}^{2}t_{1}^{2}}{w^{8}}$
$\displaystyle\left[e^{-w^{2}(1+t_{r})}-4e^{-w^{2}t_{r}}-4e^{-w^{2}}\right.$
$\displaystyle\left.+(3-2w^{2})e^{-w^{2}(t_{r}-1)}+4(2-2w^{2}+w^{4})\right],$

inserting which in Eq. (135), we find that $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}$ has the scaling form

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}D_{rot}^{2}t_{1}^{7/2}}{2\pi\sqrt{K}}~{}\mathcal{P}_{1}^{AB}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\tau_{A},$ (149)

where the scaling function $\mathcal{P}_{1}^{AB}(y,t_{r})$ is given by

$\displaystyle\begin{split}\mathcal{P}_{1}^{AB}(y,t_{r})&=\int_{-\infty}^{\infty}dw~{}\frac{\cos\left(\sqrt{2}wy\right)}{w^{8}}\left[e^{-w^{2}(1+t_{r})}-4e^{-w^{2}t_{r}}-4e^{-w^{2}}\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(3-2w^{2})e^{-w^{2}(t_{r}-1)}\Big{.}+4(2-2w^{2}+w^{4})\Big{]}.\end{split}$
(150)

Similarly for the $y$-coordinate, we get

$\displaystyle\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}D_{rot}t_{1}^{5/2}}{2\pi\sqrt{K}}~{}\mathcal{P}_{2}^{AB}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\tau_{A},$ (151)

where the scaling function $\mathcal{P}_{2}^{AB}(y,t_{r})$ is given in terms
of $\mathcal{P}^{OU}(y,t_{r})$ in Eq. (112) as

$\displaystyle\mathcal{P}_{2}^{AB}(y,t_{r})=\mathcal{P}^{OU}(y,t_{r}).$ (152)

In Fig. 15, we have compared the scaling functions
$\mathcal{P}_{1}^{AB}(y,t_{r})$ (left panel) and
$\mathcal{P}_{2}^{AB}(y,t_{r})$ (right panel) with the numerical simulations.
The comparision is performed for three sets of $t_{1}$ and $t_{2}$ keeping the
ratio $t_{r}=\frac{t_{2}}{t_{1}}$ fixed. For all values, the numerical data
match with the analytic results. Note that the scaling form in Eq. (149) for
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}$ is different than the
same for the RTP and AOUP in Eqs. (76) and (111) repsectively. To illustrate
this difference more clearly, it is instructive to look at the large $y$
behaviour of $\mathcal{P}_{1}^{AB}(y,t_{r})$. For large $y$, we find
$\mathcal{P}_{1}^{AB}(y\to\infty,t_{r})\sim
y^{-10}~{}\text{exp}\left(-\frac{y^{2}}{2(t_{r}+1)}\right)$. Although for all
models, the scaling function decays as $\sim
f(y)\text{exp}\left(-\frac{y^{2}}{2(t_{r}+1)}\right)$, the form of $f(y)$ is
very different, namely (i) $f(y)=y^{-4}$ (RTP), (ii) $f(y)=y^{-6}$ (AOUP) and
(iii) $f(y)=y^{-8}$ (ABP $x$\- coordinate). On the other hand, the scaling
form for $\langle y_{0}(t_{1})y_{\beta}(t_{2})\rangle_{c}^{ABP}$ in Eq. (151)
is exactly same as for AOUP in Eq. (111). This arises due to the fact,
mentioned earlier, that for observation time smaller than the activity time
scale, the dynamics of AOUP and that of the $y$-coordinate of the ABP are
essentially the same.

Note that for $\beta=0$, Eqs. (149) and (151) reduce to the auto correlation
of the position $\langle x_{0}(t_{1})x_{0}(t_{2})\rangle^{ABP}_{c}$ and
$\langle y_{0}(t_{1})y_{0}(t_{2})\rangle^{ABP}_{c}$ which follow the scaling
form

$\displaystyle\langle
x_{0}(t_{1})x_{0}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}D_{rot}^{2}t_{1}^{7/2}}{2\pi\sqrt{K}}~{}\mathcal{L}_{1}^{AB}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\tau_{A},$ (153) $\displaystyle\langle
y_{0}(t_{1})y_{0}(t_{2})\rangle^{ABP}_{c}\simeq\frac{v_{A}^{2}D_{rot}t_{1}^{5/2}}{2\pi\sqrt{K}}~{}\mathcal{L}_{2}^{AB}\left(\frac{t_{2}}{t_{1}}\right),~{}~{}~{}~{}~{}\text{for
}t_{1}\leq t_{2}\ll\tau_{A},$ (154)

where the scaling functions $\mathcal{L}_{1,2}^{AB}(t_{r})$ are given by

$\displaystyle\mathcal{L}_{1}^{AB}(t_{r})=\frac{16\sqrt{\pi}}{105}\left[(1+t_{r})^{7/2}-4\left(1+t_{r}^{7/2}\right)+\sqrt{t_{r}-1}\left(4-5t_{r}-2t_{r}^{2}+3t_{r}^{3}\right)\right],$
(155)
$\displaystyle\mathcal{L}_{2}^{AB}(t_{r})=\frac{8\sqrt{\pi}}{15}\left[(t_{r}+1)^{5/2}+(t_{r}-1)^{5/2}-2(1+t_{r}^{5/2})\right].$
(156)

Figure 17: Comparision of $\Omega^{AB}_{1}(y)$ and $\Omega^{AB}_{2}(y)$ in
Eqs. (159) and (160) with the numerical simulations for three different values
of $t$ (shown by different symbols). For both plots, we have chosen
$K=2,~{}D_{rot}=0.001$ and $v_{A}=1.5$. For simulations, we have taken
$N=200$.

Expectedly for $y$-coordinate, the scaling relation in Eq. (154) is same as
for the AOUP in Eq. (114). In Fig. 16, we have compared the scaling functions
$\mathcal{L}_{1,2}^{AB}(t_{r})$ with the same obtained from the numerical
simulations for three different values of $t_{1}$. For all values, the
numerical data match with our analytic expressions.

Finally, we look at the equal time correlations $\langle
x_{0}(t)x_{\beta}(t)\rangle^{ABP}_{c}$ and $\langle
y_{0}(t)y_{\beta}(t)\rangle^{ABP}_{c}$ which is obtained by putting
$t_{1}=t_{2}=t$ in Eqs. Eqs. (149) and (151) respectively. The correlations
possess the scaling form

$\displaystyle\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}^{ABP}\simeq\frac{v_{A}^{2}D_{rot}^{2}t^{7/2}}{2\pi\sqrt{K}}~{}\Omega_{1}^{AB}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{k}\ll t\ll\tau_{A},$ (157) $\displaystyle\langle
y_{0}(t)y_{\beta}(t)\rangle_{c}^{ABP}\simeq\frac{v_{A}^{2}D_{rot}t^{5/2}}{2\pi\sqrt{K}}~{}\Omega_{2}^{AB}\left(\frac{\beta}{\sqrt{2Kt}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
}\tau_{k}\ll t\ll\tau_{A},$ (158)

where the scaling functions $\Omega_{1/2}^{AB}(y)$ are given by

$\displaystyle\begin{split}\Omega_{1}^{AB}(y)=\frac{\sqrt{\pi}}{630}&\left[2\sqrt{2}e^{-\frac{y^{2}}{4}}\left(384+348y^{2}+40y^{4}+y^{6}\right)\right.\\\
&-16e^{-\frac{y^{2}}{2}}\left(48+87y^{2}+20y^{4}+y^{6}\right)\\\
&-\sqrt{2\pi}y~{}\text{Erfc}\left(\frac{y}{2}\right)\left(840+420y^{2}+42y^{4}+y^{6}\right)\\\
&\Big{.}+8\sqrt{2\pi}y~{}\text{Erfc}\left(\frac{y}{\sqrt{2}}\right)\left(105+105y^{2}+21y^{4}+y^{6}\right)\Big{]},~{}~{}~{}\text{and,}\end{split}$
(159) $\displaystyle\Omega_{2}^{AB}(y)=$ $\displaystyle\Omega^{OU}(y),$ (160)

where $\Omega^{OU}(y)$ is given in Eq.(116). We have illustrated these scaling
behaviours in Fig.17 where we have also performed comparision with the
numerical simulations for three different values of $t$. For all $t$, we
observe convergence of the numerical data with our analytic results.

####  5.2.3 Covariance and unequal time correlations for
$t_{1}\gg\tau_{A},~{}t_{2}\gg\tau_{A}$:

When the observation time is greater than the activity time scale, all three
models, namely ABP, RTP and AOUP converge to the Brownian motion with an
effective diffusion constant. This means that the scaling results obtained for
$\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{RTP}_{c}$ in Eq. (79) for RTP
will also remain valid for the ABP with $D_{R}$ replaced by
$D_{ABP}=\frac{v_{A}^{2}}{2D_{rot}}$ . Consequently, we get

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq
D_{ABP}\sqrt{\frac{2t_{1}}{\pi
K}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right),$
(161)

for $\tau_{A}\ll t_{1}\leq t_{2}$ where $\mathcal{Q}(y,t_{r})$ is given in Eq.
(80). One can also get Eq. (161) starting from Eqs. (135) and (136) and
performing same simplifications as done for RTP and AOUP. By suitably changing
$\beta,~{}t_{1}$ and $t_{2}$ in this equation, one can easily obtain the
scaling relations for the MSD, covariance and position auto correlations as
seen for RTP and AOUP as given in Eqs. (118, 119) with $D_{A}$ replaced by
$D_{ABP}=v_{A}^{2}/(2D_{rot})$. In Figure 18, we have verified the scaling
relation in Eq. (161) with the results of the numerical simulations. We
observe an excellent match.

###  5.3 Large but finite $N$

Since for this case also the observation time is larger than the activity time
scale, the scaling results obtained for $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{RTP}_{c}$ in Eq. (84) for RTP will also
remain valid for the ABP as a result of which we have

$\displaystyle\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\langle
y_{0}(t_{1})y_{\beta}(t_{2})\rangle^{ABP}_{c}\simeq\frac{2D_{ABP}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right)$
(162)

where $\mathcal{W}(t_{r},y,z)$ is given in Eq. (85) and
$D_{ABP}=\frac{v_{A}^{2}}{2D_{rot}}$. Once again, we can use Eq. (162) to
obtain the scaling relations for the MSD, covariance and position auto
fluctuations as done for RTP and AOUP.

Figure 18: Comparision of the scaling relation in Eq. (161) for $x$-coordinate
(left panel) and $y$-coordinate (right panel) of the ABP (shown by black line)
with the same obtained from the numerical simulation (shown by symbols). For
both plots, we have chosen $K=2,~{}v_{A}=1.5,~{}D_{rot}=1$ and $N=100$ for
simulation.

##  6 Conclusions

To summarize, we have studied a chain of $N$ active particles with nearest
neighbour quadratic interaction. We considered three models of active
particles, namely, run and tumble particle, active Ornstein-Uhlenbeck particle
and active Brownian particle. While the first two models are considered in one
dimension, the ABP has been studied in two dimension. For all models, there
exists three time scales - (i) $\tau_{K}=\frac{1}{K}$ due to the interaction
among particles, (ii)
$\tau_{A}=\frac{1}{\zeta}~{}/\frac{1}{\gamma}~{}/\frac{1}{D_{rot}}$ due to the
activity of RTP / AOUP / ABP. and (iii) $\tau_{N}=\frac{N^{2}}{K}$ due to the
finiteness of the ring (note that $\tau_{K}\ll\tau_{A}\ll\tau_{N}$ as shown in
Fig. 2 ). We investigated various statistical quantities for the position of
the tagged particles which were shown to exhibit interesting scaling
structures depending on where the observation time $t$ lies. We emphasise that
using our simple and analytically tractable model, we were able to re-derive
many features which are seen in particles with non-trivial interactions.

We first looked at the MSD of the tagged particle’s position. For all models,
there is a crossover of the MSD from super-diffusive scaling $(\sim t^{\mu})$
for $t\ll\tau_{A}$ to the sub-diffusive scaling $(\sim\sqrt{t})$ for
$t\gg\tau_{A}$ with $\mu=\frac{3}{2}$ for RTP, $\mu=\frac{5}{2}$ for AOUP,
$\mu=\frac{7}{2}~{}\left(=\frac{5}{2}\right)$ for $x~{}(y)$-coordinate of ABP.
We analytically computed the crossover function that connects these two
distinct scaling regimes in Eqs. (37) and (107) for RTP and AOUP respectively
and in Eqs. (139) and (140) for ABP. Note that $t^{\mu}$ scaling of the MSD
arises due to the interplay of the activity and caging effect due to the
surrounding particles. Furthermore, we also studied the MSD for
$t\sim\tau_{N}$ and showed that it exhibits crossover from $\sqrt{t}$ scaling
for $t\ll\tau_{N}$ to the diffusive scaling for $t\gg\tau_{N}$ for all models.
Here also, we explicitly computed the corresponding crossover function in Eq.
(88). Such crossover behaviour for the MSD of active particles with hardcore
repulsive interactions was predicted in [75] which we have rigorously derived
in the simple setting of harmonic chain.

Next we studied the equal time two-point correlation function $\langle
x_{0}(t)x_{\beta}(t)\rangle_{c}$ for the two models. It was shown that
$\langle x_{0}(t)x_{\beta}(t)\rangle_{c}$ exhibits scaling forms with respect
to $y=\frac{\beta}{\sqrt{2Kt}}$ with distinct scaling functions
$\Omega^{RT}(y)$, $\Omega^{OU}(y)$ and $\Omega^{AB}_{1/2}(y)$ which were
derived in Eqs. (47), (116), (159) and (160) respectively. On the other hand,
all three models give rise to the same scaling form (as expected) for
$t\gg\tau_{A}$ with the corresponding scaling function $\mathcal{C}(y)$ given
in Eq. (51). In addition to these extreme limits, we also computed the
correlation function when $t\sim\tau_{A}$.

We have also investigated the mean squared auto fluctuations $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$ with $t_{1}\leq t_{2}$. Keeping
the ratio $t_{r}=\frac{t_{2}}{t_{1}}$ fixed, we analysed the auto fluctuation
for two different limits for $t_{1}$ and $t_{2}$ \- (A) $t_{1}\ll\tau_{A}$,
$t_{2}\ll\tau_{A}$ and (B) $t_{1}\gg\tau_{A}$, $t_{2}\gg\tau_{A}$. For case
(A), the three models exhibit distinct scaling functions
$\mathcal{L}^{RT}(t_{r})$, $\mathcal{L}^{OU}(t_{r})$ and
$\mathcal{L}^{AB}_{1/2}(t_{r})$ which we have computed in Eqs. (61), (113),
(155) and (156) respectively. However for case (B), all models converge to the
same scaling function $\mathcal{M}(t_{r})$ in Eq. (67) for the auto
correlation. We also showed that the auto correlation has a non-monotonic
dependence on $t_{2}$ in which $\langle
x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$ first increases with $t_{2}$,
reaches a maximum value and then starts decreasing again. This non-monotonic
nature of $\langle x_{\alpha}(t_{1})x_{\alpha}(t_{2})\rangle_{c}$ is again the
signature of the activity of the particles.

Finally, we studied the unequal time correlation function $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}$ ($t_{1}\leq t_{2}$) which also shows
various scaling structures depending on where $t_{1}$ and $t_{2}$ lie. For
case (A), $\langle x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}$ displays scaling
form with respect to $t_{r}=\frac{t_{2}}{t_{1}}$ and
$y=\frac{\beta}{\sqrt{2Kt_{1}}}$ with the corresponding scaling functions
$\mathcal{P}^{RT}(y,t_{r})$, $\mathcal{P}^{OU}(y,t_{r})$ and
$\mathcal{P}^{AB}_{1/2}(y,t_{r})$ computed in Eqs. (77), (112), (150) and
(152). For case (B), in conjunction to the previous cases, here also, the two
models converge to the same scaling function $\mathcal{Q}(y,t_{r})$ given in
Eq. (80). Interestingly, for a given $\beta$, $\langle
x_{0}(t_{1})x_{\beta}(t_{2})\rangle_{c}$ also exhibits non-monotonic behaviour
with respect to $t_{2}$ in which it increases with $t_{2}$, attains a maximum
and then starts decreasing again. Once again, this non-monotonic nature is the
signature of the activity of the particles. A summary of results for all three
models are given in Tables 1 and 2.

##  7 Acknowledgement

We thank Julien Cividini for fruitful discussions and important comments on
the draft. AK and PS acknowledge support of the Department of Atomic Energy,
Government of India, under project no.12-R&D-TFR-5.10-1100. AK acknowledges
support from DST, Government of India grant under project No. ECR/2017/000634.

$C_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}$
---
Quantities | Model - RTP /AOUP, $\tau_{K}=\frac{1}{K},~{}\tau_{A}=\frac{1}{\zeta}~{}/\frac{1}{\gamma}$ and $\tau_{N}=\frac{N^{2}}{K}$.
MSD $C_{\alpha,\alpha}(t,t)$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\simeq\begin{cases}v_{0}^{2}t^{2},\text{ (RTP)}\\\ \frac{2D}{3}t^{3},\text{ (AOUP)}\end{cases}$ | $\sim t^{3/2}~{}\mathcal{T}^{RT/OU}\left(\frac{t}{\tau_{A}}\right)$
| Eqs. (35) and (104) | with $\tau_{A}=\frac{1}{\zeta}~{}/\frac{1}{\gamma}$ for RTP /AOUP,
|  |  | see Eqs. (36) and (106)
[.5]() |  Large but finite $N$ | $\simeq\frac{2D_{eff}N}{K}\Phi\left(\frac{Kt}{N^{2}}\right)$
| where
$\Phi(y)=y+\sum_{s=1}^{\infty}\frac{1-\text{exp}({-8\pi^{2}s^{2}y})}{4\pi^{2}s^{2}}$
| and $D_{eff}=D_{R}~{}/D_{A}$ for RTP /AOUP
Covariance $C_{0,\beta}(t,t)$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\simeq\delta_{0,\beta}\begin{cases}v_{0}^{2}t^{2},~{}\text{(RTP)}\\\ \frac{2D}{3}t^{3}~{}\text{(AOUP)}\end{cases}$ | $\sim\begin{cases}t^{\mu}~{}\Omega^{RT/OU}\left(\frac{\beta}{\sqrt{2Kt}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t\ll\tau_{A}\\\ \sqrt{t}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t\gg\tau_{A}\end{cases}$
|  $\mu=\frac{3}{2}~{}/\frac{5}{2}$ for RTP/AOUP, see
| Eqs. (47) and (115) for $\Omega^{RT/OU}(y)$
| and Eq. (51) for $\mathcal{C}(y)$.
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{eff}N}{K}\Psi\left(\frac{\beta}{N},\frac{Kt}{N^{2}}\right)$ where,
| where $\Psi(z,y)=y+\sum_{s=1}^{\infty}\cos(2\pi
sz)\frac{1-\text{exp}({-8\pi^{2}s^{2}y})}{4\pi^{2}s^{2}}$
| and $D_{eff}=D_{R}~{}/D_{A}$ for RTP /AOUP
Position Auto Correlation $C_{\alpha,\alpha}(t_{1},t_{2})$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\simeq\begin{cases}v_{0}^{2}t_{1}t_{2}~{}\text{(RTP)}\\\ Dt_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)\\\ ~{}\text{(AOUP)}\end{cases}$ | $\sim\begin{cases}t_{1}^{\mu}~{}\mathcal{L}^{RT/OU}\left(\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t_{1}\leq t_{2}\ll\tau_{A}\\\ \sqrt{t_{1}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}\tau_{A}\ll t_{1}\leq t_{2}\end{cases}$
|  $\mu=\frac{3}{2}~{}/\frac{5}{2}$ for RTP/AOUP, see
| Eqs. (61) and (113) for $\mathcal{L}^{RT/OU}(t_{r})$
|  | and Eq. (67) for $\mathcal{M}(t_{r})$
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{eff}N}{K}\mathcal{R}\left(\frac{t_{2}}{t_{1}},\frac{Kt}{N^{2}}\right)$ where,
| where
$\mathcal{R}(t_{r},y)=y+\sum_{s=1}^{\infty}\text{e}^{-4\pi^{2}s^{2}yt_{r}}\frac{\text{Sinh}\left({4\pi^{2}s^{2}y}\right)}{2\pi^{2}s^{2}}$
| and $D_{eff}=D_{R}~{}/D_{A}$ for RTP /AOUP.
$C_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}$
---
Quantities | Model - RTP /AOUP, $\tau_{K}=\frac{1}{K},~{}\tau_{A}=\frac{1}{\zeta}~{}/\frac{1}{\gamma}$ and $\tau_{N}=\frac{N^{2}}{K}$.
Unequal time correlation $C_{0,\beta}(t_{1},t_{2})$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
$\simeq\delta_{\beta,0}\begin{cases}v_{0}^{2}t_{1}t_{2}~{}\text{(RTP)}\\\ Dt_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)\\\ ~{}\text{(AOUP)}\end{cases}$ | $\sim\begin{cases}t_{1}^{\mu}~{}\mathcal{P}^{RT/OU}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t_{1}\leq t_{2}\ll\tau_{A}\\\ \sqrt{t_{1}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}\tau_{A}\ll t_{1}\leq t_{2}\end{cases}$
|  $\mu=\frac{3}{2}~{}/\frac{5}{2}$ for RTP/AOUP, see
| Eqs. (77) and (112) for $\mathcal{P}^{RT/OU}(y,t_{r})$
|  | and Eq. (80) for $\mathcal{Q}(y,t_{r})$.
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{eff}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right)$ where,
| where $\mathcal{W}(t_{r},z,y)=y+\sum_{s=1}^{\infty}\cos\left(2\pi
sz\right)\text{e}^{-4\pi^{2}s^{2}yt_{r}}\frac{\text{Sinh}\left({4\pi^{2}s^{2}y}\right)}{2\pi^{2}s^{2}}$
| and $D_{eff}=D_{R}~{}/D_{A}$ for RTP /AOUP.
Table 1: Summary of the results on two-point position correlations in the
harmonic chain of RTP and AOUP.
$C^{1}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}$
---
$C^{2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
y_{\alpha_{1}}(t_{1})y_{\alpha_{2}}(t_{2})\rangle_{c}$
Quantities | Model - ABP, $\tau_{K}=\frac{1}{K},~{}\tau_{A}=\frac{1}{D_{rot}}$ and $\tau_{N}=\frac{N^{2}}{K}$.
MSD $C_{\alpha,\alpha}^{p}(t,t)$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\sim\begin{cases}t^{4},~{}~{}~{}p=1\\\ t^{3},~{}~{}~{}p=2\end{cases}$ | $\sim t^{3/2}~{}\mathcal{T}_{p}^{AB}\left(D_{rot}t\right)$
| see Eqs. (133) and (134) | see Eqs. (137) and (138)
[.5]() |  Large but finite $N$ | $\simeq\frac{2D_{ABP}N}{K}\Phi\left(\frac{Kt}{N^{2}}\right)$
|
$\Phi(z,y)=y+\sum_{s=1}^{\infty}\frac{1-\text{exp}({-8\pi^{2}s^{2}y})}{4\pi^{2}s^{2}}$
Covariance $C_{0,\beta}^{p}(t,t)$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\sim\begin{cases}t^{4}\delta_{0,\beta},~{}~{}p=1\\\ t^{3}\delta_{0,\beta},~{}~{}p=2\end{cases}$ | $\sim\begin{cases}t^{\nu}~{}\Omega^{AB}_{p}\left(\frac{\beta}{\sqrt{2Kt}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t\ll\tau_{A}\\\ {\sqrt{t}~{}\mathcal{C}\left(\frac{\beta}{\sqrt{2Kt}}\right)}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t\gg\tau_{A}\end{cases}$
| see Eqs. (133) and (134) | where $\nu=\frac{7}{2}~{}/\frac{5}{2}$ for $p=1/2$,
|  |  $\Omega_{1/2}^{AB}(y)$ are given in
|  | Eqs. (159) and (160)
|  | and $\mathcal{C}(y)$ in Eq. (51).
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{ABP}N}{K}\Psi\left(\frac{\beta}{N},\frac{Kt}{N^{2}}\right)$ where,
| $\Psi(z,y)=y+\sum_{s=1}^{\infty}\cos(2\pi
sz)\frac{1-\text{exp}({-8\pi^{2}s^{2}y})}{4\pi^{2}s^{2}}$
Position Auto Correlation $C^{p}_{\alpha,\alpha}(t_{1},t_{2})$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
[.5]() | $\sim\begin{cases}t_{1}^{3}(2t_{2}-t_{1}),~{}~{}p=1\\\ t_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right),~{}~{}p=2\end{cases}$ | $\sim\begin{cases}t_{1}^{\nu}~{}\mathcal{L}^{AB}_{p}\left(\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t_{1}\leq t_{2}\ll\tau_{A}\\\ {\sqrt{t_{1}}~{}\mathcal{M}\left(\frac{t_{2}}{t_{1}}\right)}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}\tau_{A}\ll t_{1}\leq t_{2}\end{cases}$
| see Eqs. (133) and (134) |  $\nu=\frac{7}{2}~{}/\frac{5}{2}$ for $p=1/2$, $\mathcal{M}(t_{r})$
|  | is in Eq. (67) and $\mathcal{L}^{AB}_{p}(t_{r})$
|  | are in Eqs. (155) and (156).
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{ABP}N}{K}\mathcal{R}\left(\frac{t_{2}}{t_{1}},\frac{Kt}{N^{2}}\right)$ where,
|
$\mathcal{R}(t_{r},y)=y+\sum_{s=1}^{\infty}\text{e}^{-4\pi^{2}s^{2}yt_{r}}\frac{\text{Sinh}\left({4\pi^{2}s^{2}y}\right)}{2\pi^{2}s^{2}}$
$C^{1}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
x_{\alpha_{1}}(t_{1})x_{\alpha_{2}}(t_{2})\rangle_{c}$
---
$C^{2}_{\alpha_{1},\alpha_{2}}(t_{1},t_{2})=\langle
y_{\alpha_{1}}(t_{1})y_{\alpha_{2}}(t_{2})\rangle_{c}$
Quantities | Model - ABP, $\tau_{K}=\frac{1}{K},~{}\tau_{A}=\frac{1}{D_{rot}}$ and $\tau_{N}=\frac{N^{2}}{K}$.
Unequal time correlation $C_{0,\beta}^{p}(t_{1},t_{2})$ | $N\to\infty$ | Time $\ll\tau_{K}$ | Time $\gg\tau_{K}$
$\begin{cases}\sim t_{1}^{3}(2t_{2}-t_{1})\delta_{\beta,0},\\\ ~{}~{}~{}~{}~{}\text{for }p=1\\\ \sim t_{1}^{2}\left(t_{2}-\frac{t_{1}}{3}\right)\delta_{\beta,0},\\\ ~{}~{}~{}~{}~{}\text{for }p=2\end{cases}$ | $\sim\begin{cases}t_{1}^{\nu}~{}\mathcal{P}^{AB}_{p}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}t_{1}\leq t_{2}\ll\tau_{A}\\\ \sqrt{t_{1}}~{}\mathcal{Q}\left(\frac{\beta}{\sqrt{2Kt_{1}}},\frac{t_{2}}{t_{1}}\right)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};~{}\tau_{A}\ll t_{1}\leq t_{2}\end{cases}$
| where $\nu=\frac{7}{2}~{}/\frac{5}{2}$ for $p=1/2$,
|  $\mathcal{Q}(y,t_{r})$ is in Eq. (79) and
| see Eqs. (133) and (134) |  $\mathcal{P}^{AB}_{p}(y,t_{r})$ are in Eqs. (150)
|  | and (152).
[.5]() |  Large but finite $N$ |  $\simeq\frac{2D_{ABP}N}{K}~{}\mathcal{W}\left(\frac{t_{2}}{t_{1}},\frac{\beta}{N},\frac{Kt_{1}}{N^{2}}\right)$ where,
| $\mathcal{W}(t_{r},z,y)=y+\sum_{s=1}^{\infty}\cos\left(2\pi
sz\right)\text{e}^{-4\pi^{2}s^{2}yt_{r}}\frac{\text{Sinh}\left({4\pi^{2}s^{2}y}\right)}{2\pi^{2}s^{2}}$
Table 2: Summary of the results on two-point position correlations in the
harmonic chain of ABP.

##  Appendix A Computation of
$\langle\bar{x}_{s}(t)\bar{x}_{s^{\prime}}^{*}(t)\rangle_{c}$ in Eqs. (30) and
(99)

In this appendix, we will compute