# Diffusion of an active particle bound to a generalized elastic model:
fractional Langevin equation

Alessandro Taloni
Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, via dei
Taurini 19, 00185 Rome, Italy
<EMAIL_ADDRESS>

###### Abstract

We investigate the influence of a self-propelling, out-of-equilibrium active
particle on generalized elastic systems, including flexible and semiflexible
polymers, fluid membranes, and fluctuating interfaces, while accounting for
long-ranged hydrodynamic effects. We derive the fractional Langevin equation
governing the dynamics of the active particle, as well as that of any other
passive particle (or probe) bound to the elastic system. This equation
demonstrates analytically how the active particle dynamics is influenced by
the interplay of both the non-equilibrium force and of the viscoelastic
environment. Our study explores the diffusional behavior emerging for both the
active particle and a distant probe.The active particle undergoes three
different surprising and counterintuitive regimes identified by the distinct
dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease of
the mobility and an asymptotic subdiffusive regime.

_K_ eywords active Ornstein-Uhlenbeck; generalized elastic model; fractional
Langevin equation

##  1 Introduction

Active matter refers to a class of materials or systems whose individual
components are active, meaning they can convert athermal energy from the
environment or internal sources into directed motion or mechanical forces.
These systems are characterized by the ability of their constituents to
exhibit self-propelled motion, leading to collective behaviors and dynamic
patterns that distinguish them from traditional equilibrium systems [1, 2, 3,
4, 5]. Thus, by definition, active systems violate the fluctuation-dissipation
theorem (FDT) and encompasses a wide range of phenomena observed in biological
systems, such as swarming of birds [6, 7, 8, 9, 10, 11] or schooling of fish
[12, 13, 14], the run-and-tumble dynamics of microswimmers [15, 16, 17, 18,
19] and the molecular motors-driven transport phenomena inside the cell [20,
21, 22, 23]. Active matter can also originate synthetically, as seen in
systems composed of Janus particles that become active due to chemical
reactions [24, 25], magnetic [26, 27, 28] or electrodynamical forces [29, 30,
31, 32].

In the past decade, stochastic models have been devised to capture and
reproduce observed out-of-equilibrium scenarios and their properties [33, 34,
35], as well as the resulting collective dynamics [36, 29, 37, 38, 39, 36, 40,
41]. These models encompass run-and-tumble particle models [16, 42, 43],
active Brownian particle (ABP) models [44, 45, 25], and active Ornstein-
Uhlenbeck particle (AOUP) models [46, 47, 48, 49, 18, 1, 50].

Particular attention has been dedicated to the investigation of active polymer
systems, specifically polymeric chains composed of active particles [51, 52,
53, 54] or immersed in an active bath [55, 56, 53, 50, 57, 58]. This interest
has been partially motivated by the observed out-of-equilibrium intracellular
transport and collective phenomena, where biopolymers and active elements
coexist [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75,
76, 77, 78, 79, 80, 81, 82, 83, 84, 21, 85, 86, 87, 88, 37, 89, 90, 91, 92,
93], and partially by the design of new complex active materials. Usually,
simulations of active polymers are enforced by either a self-propelling force
tangential to the elastic chain [94, 95, 96, 97, 98], or by considering the
monomers as active particles, thus resorting to the aforementioned models [52,
53, 99, 100]. The phenomenology exhibited by active polymers is heterogeneous,
ranging from translational [101, 102] to reptation motion [95, 96], swelling
[52, 53, 103], looping [55], swirling [104] or shrinkage [105, 106], according
to the different parameters characterizing the active polymer model
scrutinized.

As emphasized, the numerous numerical studies mentioned above primarily focus
on active polymers or polymers embedded in a thermal bath. Nevertheless, three
notable studies, both numerical and analytical, specifically investigate the
impact of an active particle (or force) confined to a local portion of
polymeric chains. In the works by Natali et al. [107] and Joo et al. [108],
the dynamics of flexible (Rouse) polymers were examined with the addition of
an active monomer, specifically an AOUP. In the first study [107], the head of
the polymer was permitted to exhibit active non-equilibrium motion. In the
second [108], the AOUP was positioned as the middle monomer, and the study
extended to a flexible polymer network with the central cross-linker being the
AOUP. The third study [109], conceptualized and conducted by part of the team
from the second study, concentrated on investigating the impact of an AOUP on
a semiflexible polymer network. These works exhibit several common features:
$i)$ they are primarily numerical studies, $ii)$ their analysis is rooted in
discrete polymer models, and $iii)$ they reveal intriguing and non-trivial
physical scenarios, both in the dynamics of the AOUP and in that of the other
monomers. Particular interesting are the transition from globule to elongated
conformational dynamics observed in [107], the intermediate slowing down of
the AOUP diffusional motion, in spite of the active self-propelling force, in
[108, 109] and the fact that the AOUP dynamics can be described by a
fractional Langevin equation [110].

The analysis presented in this paper expands the scope of the applicability of
the reported models. Specifically, we explore how an AOUP influences the
thermal dynamics of a general viscoelastic system, whether it be a polymer, a
membrane, a fluctuating interface, or any other system falling under the
category of a generalized elastic model (GEM) [110, 111]. The GEM is defined
by its stochastic evolution equation:

$\frac{\partial}{\partial t}\mathbf{h}\left(\vec{x},t\right)=\int
d^{d}x^{\prime}\Lambda\left(\vec{x}-\vec{x}^{\prime}\right)C\frac{\partial^{z}}{\partial\left|\vec{x}^{\prime}\right|^{z}}\mathbf{h}(\vec{x}^{\prime},t)+\boldsymbol{\eta}\left(\vec{x},t\right).$
(1)

It is formally defined for $D$-dimensional stochastic field $\mathbf{h}$
defined in the $d$-dimensional infinite space $\vec{x}$. The GEM encompasses
also the presence of hydrodynamic effects, as in the case of a Zimm polymer
model by instance [112]. The hydrodynamic friction kernel is

$\Lambda\left(\vec{r}\right)=\frac{B}{\gamma\left|\vec{r}\right|^{\alpha}}$
(2)

where $\frac{d-1}{2}<\alpha<d$, $B$ is a constant with the dimensions of
$L^{\alpha-d}$ and $\gamma$ is the friction constant. The Fourier transform of
(2) is
$\Lambda\left(\vec{q}\right)=\frac{(4\pi)^{d/2}}{2^{\alpha}}\frac{\Gamma\left((d-\alpha)/2\right)}{\Gamma\left(\alpha/2\right)}\frac{B}{\gamma}\left|\vec{q}\right|^{\alpha-d}=A\left|\vec{q}\right|^{\alpha-d}$.
In case of local hydrodynamic interactions, (2) simplifies to
$\Lambda\left(\vec{r}\right)=\frac{\delta(\vec{r})}{\gamma}$, with $\delta$
the Dirac’s delta function. The fractional derivative appearing in the right
hand side of (1) is commonly defined as fractional Laplacian
($\frac{\partial^{z}}{\partial\left|\vec{x}\right|^{z}}:=-\left(-\nabla^{2}\right)^{z/2}$
[113]), and is expressed in terms of its Fourier transform ${\cal
F}_{\vec{q}}\left\\{\frac{\partial^{z}}{\partial\left|\vec{x}\right|^{z}}\right\\}\equiv-\left|\vec{q}\right|^{z}$
[114]. Finally the Gaussian random noise source satisfies the fluctuation-
dissipation (FD) relation

$\langle\eta_{j}\left(\vec{x},t\right)\eta_{k}\left(\vec{x}^{\prime},t^{\prime}\right)\rangle=2k_{B}T\,l^{d}\Lambda\left(\vec{x}-\vec{x}^{\prime}\right)\delta_{j\,k}\delta(t-t^{\prime}),$
(3)

where $j,k\in[1,D]$, $l$ is the microscopical length scale of the model,
$k_{B}$ is the Boltzmann constant and $T$ is the temperature.

Although the GEM equation (1) is continuous, it simplifies to the Rouse chain
model for $z=2$, $d=1$, the semiflexible polymer model for $z=4$, $d=1$, and
the Zimm polymer model for $z=2$, $d=1$, and $\alpha=1/2$. The influence of an
active non-equilibrium particle on the elastic system described by (1) can be
examined using the framework developed in [115, 116, 117], where the effect of
a local external perturbation was investigated within the context of the Kubo
fluctuation relations. It’s worth noting that this analysis can be formally
extended to the case of out-of-equilibrium stochastic forces.

###  1.1 Generalized elastic model with active Brownian particle

According with the theory outlined in [115, 116, 117], it is possible to
include in the GEM stochastic evolution (1) the action of an ABP at a given
position $\vec{x}^{\star}$:

$\frac{\partial}{\partial t}\mathbf{h}\left(\vec{x},t\right)=\int
d^{d}x^{\prime}\Lambda\left(\vec{x}-\vec{x}^{\prime}\right)\left[C\frac{\partial^{z}}{\partial\left|\vec{x}^{\prime}\right|^{z}}\mathbf{h}(\vec{x}^{\prime},t)+\frac{\gamma\sigma^{d}}{\gamma_{A}}\boldsymbol{\xi}(t)\delta(\vec{x}^{\prime}-\vec{x}^{\star})\right]+\boldsymbol{\eta}\left(\vec{x},t\right).$
(4)

Here the active noise $\boldsymbol{\xi}(t)$ is due to an athermal energy
source, leading to the breakdown of the FDT [18, 33, 118]. It is governed by
the Langevin equation [118, 52]

$\frac{\partial\boldsymbol{\xi}}{\boldsymbol{\partial}t}=-\frac{\boldsymbol{\xi}(t)}{\tau_{A}}+\sqrt{\frac{2\gamma_{A}\nu_{p}}{3\tau_{A}}}\boldsymbol{\lambda}(t)$
(5)

where $\sigma$, $\gamma_{A}$, $\nu_{p}$ and $\tau_{A}$ are the characteristic
length, the frictional coefficient, the propulsion velocity and the
correlation times of the Brownian active particle. The zero-mean Gaussian
noise in Eq.(5) satisfies the fluctuation-dissipation relation
$\langle\lambda_{\mu}(t)\lambda_{\nu}(t^{\prime})\rangle=\delta_{\mu,\nu}\delta(t-t^{\prime})$
($\mu,\nu\in[1,D]$). The Langevin Eq.(5) rules the dynamics of the active
Ornstein-Ulhenbeck (OU) noise [118, 33, 18] with $\langle\xi(t)\rangle=0$ and
exponential decaying autocorrelation function

$\langle\xi_{\mu}(t)\xi_{\nu}(t^{\prime})\rangle=\frac{\gamma^{2}\nu_{p}^{2}}{3}\delta_{\mu,\nu}e^{-\frac{|t-t^{\prime}|}{\tau_{A}}}.$
(6)

Hence, hereafter, we will refer to the self-propelling particle at
$\vec{x}^{\star}$ as the AOUP. The Eq.(4) jibes with that furnished in [108],
provided that $C=kl^{2}$, where $k$ is the flexible polymer’s elastic
constant. On the other side, the corresponding GEM equation for AOUP in a
semiflexible chain is obtained by setting
$\Lambda\left(\vec{r}\right)=\frac{\delta(\vec{r})}{\gamma},z=4,d=1$ [111]. It
is possible to reconcile the evolution equation (4) with that introduced in
[109], assuming that $C=k_{B}Tl_{p}l$, where $l_{p}$ is the persistent length
of the semiflexible polymer.

As anticipated, we demonstrate that Eq.(4) constitutes the more general and
suitable framework to study the effect of a self-propelling AOUP on a systems
whose interactions are non-local and linear, and possibly mediated by long-
ranged hydrodynamics. In Sec.2 we derive the fractional Langevin equation for
the AOUP and for any particle belonging to the elastic system at a generic
position $\vec{x}$, hereafter called probe. In Sec.3 we derive the position
autocorrelation function within the FLE framework. In Sec.4 we furnish the
analytical derivation of the mean squared displacement of the AOUP and of the
different regimes attained. Moreover we describe qualitatively the effect of
the OU noise on the other regions (or probes) belonging to the elastic system
and far from the AOUP. In Sec.5 we end up with concluding remarks.

##  2 Fractional Langevin equation

We firstly furnish the solution of the Eq.(4) in the Fourier space. We first
define the Fourier transform in space and time as

$\mathbf{h}\left(\vec{q},\omega\right)=\int_{-\infty}^{+\infty}d^{d}x\int_{-\infty}^{+\infty}dt\,\mathbf{h}\left(\vec{x},t\right)\,e^{-i\left(\vec{q}\cdot\vec{x}-\omega
t\right)}$ (7)

The solution of Eq.(4) reads

$\mathbf{h}\left(\vec{q},\omega\right)=\frac{A\sigma^{d}\gamma}{\gamma_{A}}\frac{\boldsymbol{\xi}(\omega)e^{-i\vec{q}\cdot\vec{x}^{\star}}}{\left|\vec{q}\right|^{d-\alpha}\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)}+\frac{\boldsymbol{\eta}\left(\vec{q},\omega\right)}{-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}}.$
(8)

We then multiply both sides of the former equation by
$K^{+}(-i\omega)^{\beta}$ where

$\beta=\frac{z-d}{z+\alpha-d}$ (9)

and

$K^{+}=\pi^{d/2-1}\sin\left(\pi\beta\right)\frac{\Gamma(d/2)}{2^{2-d}}\frac{(z+\alpha-d)}{A^{\beta}C^{\beta-1}},$
(10)

achieving

$K^{+}(-i\omega)^{\beta}\mathbf{h}\left(\vec{q},\omega\right)=\frac{\sigma^{d}\gamma}{\gamma_{A}}\frac{\boldsymbol{\xi}(\omega)\,K^{+}A(-i\omega)^{\beta}e^{-i\vec{q}\cdot\vec{x}^{\star}}}{\left|\vec{q}\right|^{d-\alpha}\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)}+\frac{\boldsymbol{\eta}\left(\vec{q},\omega\right)K^{+}(-i\omega)^{\beta}}{-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}}.$
(11)

In analogy to [115, 116, 117] we define the force-propagator in the Fourier
space as

$\Theta\left(\left|\vec{q}\right|,\omega\right)=K^{+}A\frac{(-i\omega)^{\beta}e^{-i\vec{q}\cdot\vec{x}^{\star}}}{\left|\vec{q}\right|^{d-\alpha}\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)}.$
(12)

Inverting in space and time, the force-propagator reads [116, 117]

$\Theta\left(\left|\vec{x}-\vec{x}^{\star}\right|,t\right)=\frac{AK^{+}\left|\vec{x}\right|^{1-d/2}}{(2\pi)^{d/2}}\int_{0}^{+\infty}d\left|\vec{q}\right|\left|\vec{q}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}-\vec{x}^{\star}\right|\right)_{-\infty}D_{t}^{\beta}\left(e^{-AC\left|\vec{q}\right|^{\gamma/2}t}\theta(t)\right),$
(13)

where $\theta(t)$ is the Heaviside step function, $J_{d/2-1}$ is the Bessel
function of fractional order $d/2-1$ and the pseudo-differential operator

${}_{a}D_{t}^{\beta}\phi(t)=\frac{1}{\Gamma\left(1-\beta\right)}\frac{d}{dt}\int_{a}^{t}dt^{\prime}\frac{1}{\left(t-t^{\prime}\right)^{\beta}}\phi\left(t^{\prime}\right),\
\ 0<\beta<1,$ (14)

represents the left side Riemann-Liouville derivative with lower bound $a<t$
[119, 120].

We also introduce the noise-propagator as

$\Phi\left(\left|\vec{q}\right|,\omega\right)=K^{+}\frac{(-i\omega)^{\beta}}{-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}},$
(15)

which, once inverted, attains the following form in space and time

$\Phi\left(\left|\vec{x}\right|,t\right)=\frac{AK^{+}\left|\vec{x}\right|^{1-d/2}}{(2\pi)^{d/2}}\int_{0}^{+\infty}d\left|\vec{q}\right|\left|\vec{q}\right|^{d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}\right|\right)_{-\infty}D_{t}^{\beta}\left(e^{-AC\left|\vec{q}\right|^{\gamma/2}t}\theta(t)\right).$
(16)

Thanks to the definitions (12) and (15), we can formally invert the Eq.(11):

$K^{+}\,_{-\infty}D_{t}^{\beta}\mathbf{h}\left(\vec{x},t\right)=\frac{\sigma^{d}\gamma}{\gamma_{A}}\int_{-\infty}^{t}dt^{\prime}\boldsymbol{\xi}(t^{\prime})\Theta\left(\left|\vec{x}-\vec{x}^{\star}\right|,t-t^{\prime}\right)+\boldsymbol{\zeta}\left(\vec{x},t\right),$
(17)

where the non-Markovian noise, defined as

$\boldsymbol{\zeta}\left(\vec{x},t\right)=\int_{-\infty}^{+\infty}d\vec{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}\boldsymbol{\eta}\left(\vec{x}^{\prime},t^{\prime}\right)\Phi\left(\left|\vec{x}^{\prime}-\vec{x}\right|,t-t^{\prime}\right),$
(18)

fulfills the generalized fluctuation-dissipation relation

$\langle\zeta_{\mu}\left(\vec{x},t\right)\zeta_{\nu}\left(\vec{x},t^{\prime}\right)\rangle=k_{B}T\delta_{\mu,\nu}\frac{K^{+}}{\Gamma\left(1-\beta\right)\left|t-t^{\prime}\right|^{\beta}}.$
(19)

The Eq.(17) is the fractional equation governing the dynamics of any probe
placed at the generic position $\vec{x}$. By a close inspection, its structure
unveils how the action of the AOUP in the position $\vec{x}^{*}$ affects the
stochastic behavior of a distant point $\vec{x}$, through the propagator (13).
This is mirrored by the fGN definition (18), resulting from the convolution of
any stochastic Gaussian force
$\boldsymbol{\eta}\left(\vec{x}^{\prime},t^{\prime}\right)$, performing on the
entire GEM, with the noise-propagator (16). Notice that the Riemann-Liouville
fractional derivative in (14) could be safely replaced by the Caputo
fractional derivative in the FLE (17), as both definitions coincide when the
lower bound tends to $-\infty$. [119, 120]

We now turn to the expression of the effective stochastic equation ruling out
the motion of the AOUP. For such a purpose we first perform the Fourier
inverse transform in space of the expression (12):

$\Theta\left(\left|\vec{x}-\vec{x}^{*}\right|,\omega\right)=\frac{AK^{+}(-i\omega)^{\beta}\left|\vec{x}\right|^{1-d/2}}{(2\pi)^{d/2}}\int_{0}^{+\infty}d\left|\vec{q}\right|\frac{\left|\vec{q}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)}{-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}},$
(20)

hence we use the expansion of the Bessel function for small arguments as [121]

$J_{d/2-1}(r)\sim\frac{1}{\Gamma(d/2)}\left(\frac{2}{r}\right)^{1-d/2},$ (21)

getting

$\Theta\left(0,\omega\right)\sim\frac{A^{\beta}K^{+}2^{1-d/2}}{(2\pi)^{d/2}\Gamma\left(\frac{d}{2}\right)C^{1-\beta}}\int_{0}^{+\infty}dy\frac{y^{\alpha-1}}{1+y^{z+\alpha-d}}.$
(22)

Solving the integral [122], we finally obtain that

$\Theta\left(0,\omega\right)\sim\frac{1}{2l^{d}}\delta(t-t^{\prime}).$ (23)

Therefore the Eq.(17) reduces to the following fractional equation for the
AOUP:

$K^{+}\,_{-\infty}D_{t}^{\beta}\mathbf{h}\left(\vec{x}^{*},t\right)=\frac{\sigma^{d}\gamma}{2l^{d}\gamma_{A}}\boldsymbol{\xi}(t)+\boldsymbol{\zeta}\left(\vec{x}^{*},t\right).$
(24)

Thus, it is immediately seen how the active noise applies to the AOUP as a
_bare_ stochastic force. Similar equations were conjectured in Ref.s[108] and
[109], while analyzing the case of flexible and semiflexible polymers under
the action of an ABP. We hereby offered the rigorous derivation in the more
general case represented by the GEM (1). Moreover we established the formal
validity of the FLE framework for any particle, not only for the AOUP.

Most importantly, Eqs.(17) and (24) highlight how the motion of any probe in
the elastic system, being the AOUP at $\vec{x}^{*}$ or another at a position
$\vec{x}$, corresponds to a fractional Brownian motion [123].

##  3 $h$-autocorrelation function

The MSD of any particle belonging to the GEM is affected by the presence of
the active noise $\boldsymbol{\xi}(t)$. As a matter of fact the general
expression for the MSD reads

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle\equiv\langle\left[h\left(\vec{x},t\right)-h\left(\vec{x},0\right)\right]^{2}\rangle=2\left[\langle
h^{2}\left(\vec{x},t\right)\rangle-\langle
h\left(\vec{x},t\right)h\left(\vec{x},0\right)\rangle\right]$ (25)

where $h\left(\vec{x},t\right)$ represents one of the components of the
stochastic field $\mathbf{h}\left(\vec{x},t\right)$. From the definition (25)
it turns out that the calculation of the MSD, as of any other physical
observable, needs the explicit expression of the two times autocorrelation
function $\langle
h\left(\vec{x},t\right)h\left(\vec{x},t^{\prime}\right)\rangle$.

Starting from the single component solution of the Eq.(17) in the Fourier
space

$h\left(\vec{x},\omega\right)=\frac{\sigma^{d}\gamma}{\gamma_{A}}\frac{\xi(\omega)\Theta\left(\left|\vec{x}-\vec{x}^{\star}\right|,\omega\right)}{K^{+}(-i\omega)^{\beta}}+\frac{\zeta\left(\vec{x},\omega\right)}{K^{+}(-i\omega)^{\beta}},$
(26)

we can write down

$\langle
h\left(\vec{x},\omega\right)h\left(\vec{x},\omega^{\prime}\right)\rangle=\left(\frac{\sigma^{d}\gamma}{\gamma_{A}}\right)^{2}\left|\vec{x}-\vec{x}^{\star}\right|^{2-d}\int_{0}^{+\infty}d\left|\vec{q}\right|\left|\vec{q}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\times\\\
\int_{0}^{+\infty}d\left|\vec{q}^{\prime}\right|\left|\vec{q}^{\prime}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}^{\prime}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\frac{\langle\xi(\omega)\xi(\omega^{\prime})\rangle}{\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)\left(-i\omega^{\prime}+AC\left|\vec{q}^{\prime}\right|^{z+\alpha-d}\right)}+\\\
\frac{\langle\zeta(\vec{x},\omega)\zeta(\vec{x},\omega^{\prime})\rangle}{K^{+2}(-i\omega)^{\beta}(-i\omega^{\prime})^{\beta}},$
(27)

where we made use of the force-propagator Fourier transform (20). The first
addendum on the RHS of the previous equation can be further simplified by
resorting to the active noise correlation properties in Fourier space, i.e.

$\langle\xi(\omega)\xi(\omega^{\prime})=\frac{4\pi\nu_{p}^{2}\gamma_{A}^{2}}{3\tau_{A}}\frac{\delta(\omega+\omega^{\prime})}{\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}}$

.

The second addendum is the usual term accounting for the correlations inherent
to the GEM [111]. Therefore the $h-$autocorrelation function gets the final
form

$\langle
h\left(\vec{x},t\right)h\left(\vec{x},t^{\prime}\right)\rangle=\frac{(\sigma^{d}\gamma
A\nu_{p})^{2}\left|\vec{x}-\vec{x}^{\star}\right|^{2-d}}{2^{d-1}\pi^{d+1}3\tau_{A}}\times\\\
\int_{0}^{+\infty}d\left|\vec{q}\right|\left|\vec{q}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\int_{0}^{+\infty}d\left|\vec{q}^{\prime}\right|\left|\vec{q}^{\prime}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}^{\prime}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\times\\\
\int_{-\infty}^{\infty}d\omega\frac{e^{-i\omega(t-t^{\prime})}}{\left[\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}\right]\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)\left(-i\omega+AC\left|\vec{q}^{\prime}\right|^{z+\alpha-d}\right)}+\\\
\frac{2^{2-d}k_{B}Tl^{d}A^{\beta}C^{\beta-1}}{(z+\alpha-d)\pi^{d/2}\Gamma\left(\frac{d}{2}\right)\cos\left(\frac{\pi\beta}{2}\right)}\int_{0}^{\infty}d\omega\frac{\cos[\omega(t-t^{\prime})]}{\omega^{1+\beta}},$
(28)

The two terms appearing on the RHS of the autocorrelation function have a
different origin and different behaviors. While in the first one it appears
clear the dependence on the internal coordinate $\vec{x}$ as well as the
absence of any divergence in the $\omega$ space, the second does not depend on
the specific internal position but diverges as $|\omega|^{-(1+\beta)}$ in the
limit of $\omega\to 0$ [111]. Therefore, any physically measurable quantity
must be organized in a manner that ensures the cancellation of this
divergence.

The $h-$autocorrelation function of the AOUP is obtained from Eq.(28) using
the property (21)

$\langle
h\left(\vec{x}^{*},t\right)h\left(\vec{x}^{*},t^{\prime}\right)\rangle=\left(\frac{\sigma^{d}\gamma\nu_{p}}{l^{d}K^{+}}\right)^{2}\frac{1}{6\pi\tau_{A}}\int_{0}^{\infty}d\omega\frac{\cos[\omega(t-t^{\prime})]}{\omega^{2\beta}\left[\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}\right]}+\\\
\frac{2^{2-d}k_{B}Tl^{d}A^{\beta}C^{\beta-1}}{(z+\alpha-d)\pi^{d/2}\Gamma\left(\frac{d}{2}\right)\cos\left(\frac{\pi\beta}{2}\right)}\int_{0}^{\infty}d\omega\frac{\cos[\omega(t-t^{\prime})]}{\omega^{1+\beta}}$
(29)

The same expression is achievable from the solution of the Eq.(24) in Fourier
space:

$h\left(\vec{x}^{*},\omega\right)=\frac{\sigma^{d}\gamma}{2l^{d}\gamma_{A}}\frac{\xi(\omega)}{K^{+}(-i\omega)^{\beta}}+\frac{\zeta\left(\vec{x}^{*},\omega\right)}{K^{+}(-i\omega)^{\beta}},$
(30)

##  4 Mean Square Displacement

By substituting the expression (28) into the MSD definition (25), it becomes
evident that the MSD is composed of the sum of two contributions:

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle=\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{OUN}+\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{fGN}.$
(31)

As noticed in Ref.[109], this superposition makes non-trivial the possible
diffusive scenario arising when we are consdering both the AOUP and a generic
probe. We consider the case of high Péclet number where the active dynamics is
definitely larger that the thermal counterpart. Indeed, the first in Eq.(31)
arises from the action of the non-equilibrium Ornstein-Uhlenbeck noise ruled
by the Langevin equation (5), whereas the second stems from the fractional
Gaussian noise (18) and represents the typical subdiffusive thermal dynamics
exhibited by any element belonging to the GEM (1) [111, 110]:

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{fGN}=\frac{4k_{B}T\pi^{d/2}(AC)^{\beta}\Gamma(1-\beta)}{(2\pi)^{d}\Gamma(d/2)(z-d)C}t^{\beta}$
(32)

The first term on the RHS of (31) is formally defined as

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{OUN}=\frac{(\sigma^{d}\gamma
A\nu_{p})^{2}\left|\vec{x}-\vec{x}^{\star}\right|^{2-d}}{2^{d-1}\pi^{d+1}3\tau_{A}}\times\\\
\int_{0}^{+\infty}d\left|\vec{q}\right|\left|\vec{q}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\int_{0}^{+\infty}d\left|\vec{q}^{\prime}\right|\left|\vec{q}^{\prime}\right|^{\alpha-d/2}J_{d/2-1}\left(\left|\vec{q}^{\prime}\right|\left|\vec{x}-\vec{x}^{*}\right|\right)\times\\\
\int_{-\infty}^{\infty}d\omega\frac{1-e^{-i\omega
t}}{\left[\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}\right]\left(-i\omega+AC\left|\vec{q}\right|^{z+\alpha-d}\right)\left(-i\omega+AC\left|\vec{q}^{\prime}\right|^{z+\alpha-d}\right)}.$
(33)

The equivalent term in case of the AOUP can be derived from (29)

$\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{OUN}=\left(\frac{\sigma^{d}\gamma\nu_{p}}{l^{d}K^{+}}\right)^{2}\frac{1}{3\pi\tau_{A}}\int_{0}^{\infty}d\omega\frac{1-\cos(\omega
t)}{\omega^{2\beta}\left[\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}\right]}$
(34)

and it is definitely easier to treat. Hence, in the following we develop our
analysis starting from the expression (34).

###  4.1 AOUP’s MSD

Firstly we notice how in the integral

${\cal I}=\int_{0}^{\infty}d\omega\frac{1-\cos(\omega
t)}{\omega^{2\beta}\left[\left(\frac{1}{\tau_{A}}\right)^{2}+\omega^{2}\right]}$

there is no issue of divergence in the limit as $\omega\to 0$. In fact, the
integrand function behaves as $\sim\omega^{2(1-\beta)}$ for small frequencies.
We then introduce a change of variable, $y=\omega\tau_{A}$, so that

${\cal
I}=\tau_{A}^{1+2\beta}\int_{0}^{\infty}dy\frac{1-\cos\left(\frac{yt}{\tau_{A}}\right)}{y^{2\beta}\left(1+y^{2}\right)}.$
(35)

Its time behavior differs whether $t\ll\tau_{A}$ or $t\gg\tau_{A}$ and we will
treat them separately.

####  4.1.1 $t\ll\tau_{A}$

The integral (35) must be handled differently in the three cases
$0<\beta<1/2$, $\beta=1/2$ and $1/2<\beta<1$.

  * •

$0<\beta<1/2$.

We can split the integral and solve the first one [122]:

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]-\int_{0}^{\infty}dy\frac{\cos\left(\frac{yt}{\tau_{A}}\right)}{y^{2\beta}\left(1+y^{2}\right)}\right\\}.$
(36)

Hence we integrate the second by parts and we expand the resulting
trigonometric functions for small arguments

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]\right.-\\\
\left.\frac{1}{1-2\beta}\left[\left(\frac{t}{\tau_{A}}\right)^{2}\left(\int_{0}^{\infty}dy\frac{y^{2-2\beta}}{1+y^{2}}-\int_{0}^{\infty}dy\frac{y^{3-2\beta}}{(1+y^{2})^{2}}\right)+2\int_{0}^{\infty}dy\frac{y^{2-2\beta}}{(1+y^{2})^{2}}\right]\right\\}.$
(37)

By evaluating the remaining integrals we obtain the final result

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}\left(\frac{t}{\tau_{A}}\right)^{2}\left[\frac{cosec\left[\frac{(2\beta-3)\pi}{2}\right]}{1-2\beta}+\frac{cosec\left[\frac{(1-2\beta)\pi}{2}\right]}{2}\right]+\right.\\\
\left.\pi\left[\frac{cosec\left[\frac{(1-2\beta)\pi}{2}\right]}{2}-\frac{\beta\,cosec\left[\pi\beta\right]}{1-2\beta}\right]\right\\}.$
(38)

  * •

$\beta=1/2$.

The integral (35) is in this case

${\cal
I}=\tau_{A}^{2}\int_{0}^{\infty}dy\frac{1-\cos\left(\frac{yt}{\tau_{A}}\right)}{y\left(1+y^{2}\right)}.$
(39)

Integrating by parts we have

${\cal
I}=\tau_{A}^{2}\left\\{-\frac{t}{\tau_{A}}\int_{0}^{\infty}\frac{\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)}{1+y^{2}}+2\int_{0}^{\infty}dy\frac{\ln(y)\left[1-\cos\left(\frac{yt}{\tau_{A}}\right)\right]}{(1+y^{2})^{2}}\right\\}.$
(40)

We can neglect the second and split the first into two contributions

${\cal
I}=\tau_{A}^{2}\left\\{-\frac{t}{\tau_{A}}\int_{0}^{1}\frac{\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)}{1+y^{2}}+-\frac{t}{\tau_{A}}\int_{1}^{\infty}\frac{\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)}{1+y^{2}}\right\\}.$
(41)

Then we can retain only the first one, as the second is nearly zero, and
expand the sine for small arguments, getting [122]

${\cal
I}=\tau_{A}^{2}\left\\{-\left(\frac{t}{\tau_{A}}\right)^{2}\int_{0}^{1}\frac{y\ln(y)}{1+y^{2}}\right\\}=t^{2}\frac{\pi^{2}}{48}.$
(42)

  * •

$1/2<\beta<1$.

This case is the easier to be handled. Expanding the cosine for small
arguments in (35) yields

${\cal
I}=\frac{t^{2}}{\tau_{A}^{1-2\beta}}\frac{1}{2}\int_{0}^{\infty}dy\frac{y^{2(1-\beta)}}{1+y^{2}}=\frac{t^{2}}{\tau_{A}^{1-2\beta}}\frac{\pi}{4}cosec\left[\frac{(3-2\beta)\pi}{2}\right]$
(43)

####  4.1.2 $t\gg\tau_{A}$

This time limit presents the same symmetry of the previous one, therefore we
study the behaviour of ${\cal I}$ in the three cases $0<\beta<1/2$,
$\beta=1/2$ and $1/2<\beta<1$.

  * •

$0<\beta<1/2$.

From (36), after integrating by parts, it is obtained

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]-\frac{t}{\tau_{A}}\frac{1}{1-2\beta}\int_{0}^{\infty}dy\frac{\sin\left(\frac{yt}{\tau_{A}}\right)}{y^{2\beta-1}\left(1+y^{2}\right)}-\right.\\\
\left.\frac{2}{1-2\beta}\int_{0}^{\infty}dy\frac{\cos\left(\frac{yt}{\tau_{A}}\right)}{y^{2\beta-2}\left(1+y^{2}\right)}\right\\}.$
(44)

The major contributions to the integrals appearing in (44) come from $y\sim
0$, hence ${\cal I}$ may be properly approximated to

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]-\frac{t}{\tau_{A}}\frac{1}{1-2\beta}\int_{0}^{\infty}dy\,y^{1-2\beta}\sin\left(\frac{yt}{\tau_{A}}\right)-\right.\\\
\left.\frac{2}{1-2\beta}\int_{0}^{\infty}dy\,y^{2-2\beta}\cos\left(\frac{yt}{\tau_{A}}\right)\right\\}.$
(45)

We can thus use the method of summation of improper integrals [124, 115] to
finalize the calculation

${\cal
I}=\tau_{A}^{1+2\beta}\left\\{\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]-\left(\frac{t}{\tau_{A}}\right)^{2\beta-1}\frac{\Gamma(2-2\beta)\sin[\pi(1-\beta)]}{1-2\beta}-\right.\\\
\left.\left(\frac{t}{\tau_{A}}\right)^{2\beta-3}\frac{\Gamma(3-2\beta)\cos\left[\frac{\pi(3-2\beta)}{2}\right]}{1-2\beta}\right\\}.$
(46)

  * •

$\beta=1/2$.

We recap from the expression (40), neglecting the second integral on the RHS
and retaining only the contributions coming from $y\sim 0$ in the first:

${\cal
I}=\tau_{A}^{2}\left\\{-\frac{t}{\tau_{A}}\int_{0}^{\infty}\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)\right\\}.$
(47)

Then we can split the resulting integral into two terms

${\cal
I}=\tau_{A}^{2}\left\\{-\frac{t}{\tau_{A}}\int_{0}^{1}\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)+-\frac{t}{\tau_{A}}\int_{1}^{\infty}\ln(y)\sin\left(\frac{yt}{\tau_{A}}\right)\right\\}.$
(48)

and consider only the first one as the second yields almost no contribution.
Finally, solving the integral [122]:

${\cal
I}=\tau_{A}^{2}\left\\{\gamma+\ln\left(\frac{t}{\tau_{A}}\right)-Ci\left(\frac{t}{\tau_{A}}\right)\right\\},$
(49)

where $\gamma$ is the Euler-Mascheroni constant and $Ci$ is the cosine
integral [121].

  * •

$1/2<\beta<1$.

As in the previous situations the main contributions to the integral in (35)
will arise from $y\sim 0$, hence

${\cal
I}=\tau_{A}^{1+2\beta}\int_{0}^{\infty}dy\frac{1-\cos\left(\frac{yt}{\tau_{A}}\right)}{y^{2\beta}}.$
(50)

By integration by parts it becomes

${\cal
I}=\frac{\tau_{A}^{1+2\beta}}{2\beta-1}\left(\frac{t}{\tau_{A}}\right)\int_{0}^{\infty}dy\,y^{1-2\beta}\sin\left(\frac{yt}{\tau_{A}}\right)$
(51)

and, using the methods of improper integrals [124], the final result is

${\cal
I}=\tau_{A}^{1+2\beta}\left(\frac{t}{\tau_{A}}\right)^{2\beta-1}\frac{\Gamma(2-2\beta)}{2\beta-1}\sin\left[\pi(1-\beta)\right]$
(52)

Now, collecting the expressions achieved in these subsections, we can wrap
them up in a unique compact formula:

when $t\ll\tau_{A}$:

$\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{OUN}\simeq\left(\frac{\sigma^{d}\gamma\nu_{p}}{l^{d}K^{+}}\right)^{2}\frac{t^{2}}{3\pi\tau_{A}}\left\\{\begin{array}[]{cc}\tau_{A}^{2\beta-1}\frac{\pi}{2}\left[\frac{cosec\left[\frac{(2\beta-3)\pi}{2}\right]}{1-2\beta}+\frac{cosec\left[\frac{(1-2\beta)\pi}{2}\right]}{2}\right]&0<\beta<1/2\\\
\frac{\pi^{2}}{48}&\beta=1/2\\\
\tau_{A}^{2\beta-1}\frac{\pi}{4}cosec\left[\frac{(3-2\beta)\pi}{2}\right]&1/2<\beta<1;\end{array}\right.$
(53)

when $t\gg\tau_{A}$:

$\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{OUN}\simeq\left(\frac{\sigma^{d}\gamma\nu_{p}}{l^{d}K^{+}}\right)^{2}\frac{1}{3\pi\tau_{A}}\left\\{\begin{array}[]{cc}\tau_{A}^{1+2\beta}\frac{\pi}{2}cosec\left[\frac{(1-2\beta)\pi}{2}\right]&0<\beta<1/2\\\
\tau_{A}^{2}\ln\left(\frac{t}{\tau_{A}}\right)&\beta=1/2\\\
t^{2\beta-1}\,\tau_{A}^{2}\frac{\Gamma(2-2\beta)}{2\beta-1}\sin\left[\pi(1-\beta)\right]&1/2<\beta<1;\end{array}\right.$
(54)

Hence, we can infer that the impact of the OUN on the AOUP diffusive dynamics
exhibits pseudo-ballistic behavior for time intervals shorter than the active
decorrelation time ($\tau_{A}$). However, this impact varies depending on
whether $\beta$ is less than, equal to, or greater than 1/2. This result must
be summed to the contribution arising from the fGn according to the formula
(31). It is clear that, in the long time limit, the term
$\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{fGn}$ will dominate.
However, the transition from the the dynamics dictated by
$\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{OUN}$ to the asymptotic
one will depend on the values of the parameters appearing in (4). Defining
this transition time as $\tau_{sub}$ and assuming $\tau_{sub}\gg\tau_{A}$ a
rough estimate of $\tau_{sub}$ could be given by equating the contributions of
the expression (54) with (32):
$\langle\delta^{2}h\left(\vec{x}^{*},\tau_{sub}\right)\rangle_{OUN}=\langle\delta^{2}h\left(\vec{x}^{*},\tau_{sub}\right)\rangle_{fGn}$.
This is schematically shown in Fig.1.

However, it’s essential to emphasize that when we relax the assumption of
$\tau_{\text{sub}}\gg\tau_{A}$, the dynamics become less straightforward.
Although the asymptotic behavior remains unchanged, the intermediate ultraslow
dynamics can be significantly compressed or reduced.

Figure 1: MSD of the AOUP. The three situations described in the text,
$0<\beta<1/2$ (black curve), $\beta=0$ (red curve) and $1/2<\beta<1$ (green
curve) are qualitatively shown. Assuming $\tau_{sub}\gg\tau_{A}$ the three
regimes appear well distinct. After a pseudo- ballistic initial phase, the
behaviors in (54) represent a considerable slowing down of the diffusive
dynamics, which is followed by the asymptotic subdiffusive GEM usual behaviour
(32).

###  4.2 MSD at a generic position $\vec{x}$

The expression (33) is not straightforward to manipulate, making it
challenging to deduce the impact of the AOUP dynamics on a tracer positioned
at a distance of $|\vec{x}-\vec{x}^{*}|$. The reason is the appearance of a
correlation time

$\tau(|\vec{x}-\vec{x}^{*}|)=\frac{|\vec{x}-\vec{x}^{*}|^{z+\alpha-d}}{CA},$
(55)

which can be seen as the time up to which the dynamics of two distinct probes
in $\vec{x}$ and $\vec{x}^{*}$ is uncorrelated [115, 125, 116, 117]. As a
matter of fact, both the force- and the noise-correlators are expressed as
$\Theta\left(\frac{t}{\tau(|\vec{x}|)}\right)$ and
$\Phi\left(\frac{t}{\tau(|\vec{x}|)}\right)$. The dependence of (31) and, in
particular, the dependence of the $\langle\delta^{2}h(\vec{x},t)\rangle$ on
the correlation time (55) can be achieved by the following changes of variable
in the integrals of (33):
$y=\left(\frac{CA}{-i\omega}\right)^{1/(z+\alpha-d)}|\vec{q}|$,
$y^{\prime}=\left(\frac{CA}{-i\omega}\right)^{1/(z+\alpha-d)}|\vec{q}^{\prime}|$
and $\lambda=\omega t$:

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{OUN}=\frac{(\sigma^{d}\gamma
A\nu_{p})^{2}}{2^{d-1}\pi^{d+1}3\tau_{A}}\frac{\left|\vec{x}-\vec{x}^{\star}\right|^{-\alpha}}{(CA)^{\frac{\alpha}{z+\alpha-d}}}\left(\frac{t}{\tau(\left|\vec{x}-\vec{x}^{\star}\right|)}\right)^{\frac{2+\alpha-d}{z+\alpha-d}}\int_{-\infty}^{\infty}d\lambda\frac{1-e^{-i\lambda}}{\left(\frac{1}{\tau_{A}}\right)^{2}+\left(\frac{\lambda}{t}\right)^{2}}\times\\\
\int_{0}^{+\infty}dy\,\frac{y^{\alpha-d/2}}{1+y^{z+\alpha-d}}J_{d/2-1}\left(\left(\frac{-i\lambda\tau(\left|\vec{x}-\vec{x}^{\star}\right|)}{t}\right)^{\frac{1}{z+\alpha-d}}y\right)\times\\\
\int_{0}^{+\infty}dy^{\prime}\,\frac{y^{\prime\alpha-d/2}}{1+y^{\prime
z+\alpha-d}}J_{d/2-1}\left(\left(\frac{-i\lambda\tau(\left|\vec{x}-\vec{x}^{\star}\right|)}{t}\right)^{\frac{1}{z+\alpha-d}}y^{\prime}\right).$
(56)

As previously mentioned, the formal analysis of (33) is not straightforward,
and it will be the focus of an upcoming investigation. Nonetheless, we can
qualitatively examine the limiting behaviors of the equivalent expression
(56).

In the regime where $t\ll\tau(|\vec{x}-\vec{x}^{*}|)$, the Bessel functions
exhibit high oscillations. As a result, the major contributions to the
integrals come from values where $y\simeq 0$, $y^{\prime}\simeq 0$, and
$\lambda\simeq 0$. Surprisingly, these contributions are almost negligible,
leading to:

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle\simeq\langle\delta^{2}h\left(\vec{x},t\right)\rangle_{fGn}.$
(57)

In the opposite limit, where $t\gg\tau(|\vec{x}-\vec{x}^{*}|)$, we can employ
the expansion of the Bessel functions for small arguments (21). This results
in the same expression (34) as that valid for the AOUP. Consequently, we
obtain:

$\langle\delta^{2}h\left(\vec{x},t\right)\rangle\simeq\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{OUN}+\langle\delta^{2}h\left(\vec{x}^{*},t\right)\rangle_{fGn}.$
(58)

Given these limiting situations, we can provide an approximate study of the
possible scenarios:

  * •

$\tau(|\vec{x}-\vec{x}^{*}|)\ll\tau_{A}$.

Probes very close to the AOUP exhibit an initial thermal subdiffusive behavior
$\propto t^{\beta}$. Subsequently, the probe at $\vec{x}$ behaves identically
to the AOUP.

  * •

$\tau_{A}\ll\tau(|\vec{x}-\vec{x}^{*}|)\ll\tau_{sub}$.

In this case, the probe’s diffusion is primarily governed by thermal noise.
The non-equilibrium dynamics becomes significant only for
$\tau(|\vec{x}-\vec{x}^{*}|)\ll t\ll\tau_{sub}$, leading to the results in
(54). For longer times, i.e., $t\gg\tau_{sub}$, the thermal MSD (32) is
recovered.

  * •

$\tau_{sub}\ll\tau(|\vec{x}-\vec{x}^{*}|)$.

Probes that satisfy this condition, i.e., probes far away from the AOUP, are
not influenced by the action of the active force.

##  5 Concluding remarks

In this paper we have examined the diffusional dynamics of an AOUP connected
to an elastic system such a (semi)flexible polymer, a membrane, or a
fluctuating interface, including hydrodynamic fluid-mediated interactions.
Moreover we have investigated the action of the non-equilibrium OU force on
the other element belonging to the elastic system. We have demonstrated that
the FLE constitutes the correct description of the AOUP dynamics, where the
thermal contributions deriving from the rest of the elastic system are
incorporated into the fGn, in addition to a renormalized OUN (24). We also
have derived the FLE for any other probe placed at an arbitrary distance from
the AOUP $|\vec{x}-\vec{x}^{*}|$. Here the effect of the nonequilibrium force
is delayed in time, propagating through the medium thanks to the force-
propagator $\Theta(|\vec{x}-\vec{x}^{*}|,t)$ (17).

Our analytical theory constitutes a significant improvement over the arguments
presented in previous works on this subject. In Ref. [108], the FLE for the
AOUP was inferred from the numerical evidence of the velocity autocorrelation
function’s time behavior and from that of the MSD. In other words, it was
proposed as an effective equation supported by analytical calculations drawn
from a normal mode expansion [94, 126, 127, 128]. However, a formal derivation
of the FLE, based on the analysis by Panja [129, 130], was not attempted.

In Ref. [109], the mesoscopic FLE for the AOUP attached to a (semi)flexible
chain was introduced to reproduce its stochastic non-equilibrium dynamics. The
fractional equation, namely the damping kernel, was derived by resorting to
the tension propagation theory in the absence of active noise. However, even
in this case, a formal derivation from the semiflexible evolution equation was
not provided.

In this article, we have offered such an analytical derivation, significantly
expanding the domain of its applicability to any elastic system, including
hydrodynamics. This became possible because the framework of the FLE with a
localized applied potential [115] could be entirely shifted to the case of the
AOUP particle bound to an elastic system. Moreover, this framework entails the
derivation of the FLE for any other probe belonging to the GEM, different from
the AOUP.

The FLE framework provides a crucial additional value through the formally
easy calculation of any observable composed as a function of the elementary
correlation function (28). In particular, we examined the MSD of AOUP,
uncovering some unexpected diffusional scenarios, which include, as special
cases, those found in the simulations and theoretical analyses of Refs. [109,
108].

Firstly, we observed how three different types of diffusion arise depending on
the value of $\beta$. This is not immediately apparent in the early time
regime, i.e., $t\ll\tau_{A}$, when the action of the non-equilibrium OU noise
drives the directional motion of the AOUP. In this regime, a superdiffusive
pseudo-ballistic dynamics emerges, analogous to the result in [108] where the
Rouse model yields $\beta=1/2$ [111, 110]. However, although the $\sim t^{2}$
behavior is maintained for any value of $\beta$, the prefactors change
significantly as shown in Eq. (53). Moreover, we observe that the ballistic
regime may not distinctly emerge from simulations. Specifically, at low Péclet
numbers, the contribution to the MSD from the thermal part, attributed to the
action of the fractional Gaussian noise in (31), cannot be entirely neglected.
This observation might underlie the $\sim t^{3/2}$ behavior exhibited by the
AOUP connected to semiflexible polymers [109]. Simultaneously, this
discrepancy could be attributed to the fact that, in the presented simulation
results, the AOUP was attached to a network of four semiflexible arms.
Regardless, as highlighted by the same authors, in the simulations, ’the
superdiffusion for $t\ll\tau_{A}$ occurs with an anomalous exponent slightly
greater than 3/2’.

For times $t\gg\tau_{A}$ and for high Péclet number, the diffusion is still
dictated by the non-equilibrium active force, and the diffusional scenario are
different according to the values of $\beta$, see Eq.(54). For example, we
both recover the logarithmic dependence of the Rouse chains, obtained
analytically in [108], and the $t^{1/2}$ behaviour observed in the numerical
simulations of Ref.[109]. In case of the Zimm polymers model instead, we
predict $\sim t^{1/3}$ in this regime, being $\beta=\frac{2}{3}$. Importantly,
we anticipate that for $\beta$ in the range $0<\beta<1/2$, the diffusion tends
to slow down till to a constant value, see Eq.(54).

The asymptotic diffusion requires special attention. The FLEs (17) and (24)
involve the superposition of both non-equilibrium and thermal contributions,
resulting in the expression (31). This implies that, for a high Péclet number,
there is a crossover between the active subdiffusive motion and the long-time
thermal subdiffusive dynamics. The time scale on which this crossover occurs,
denoted as $\tau_{sub}$, is not easily determined and crucially depends on the
microscopic parameters of the model, as illustrated in Fig.1. Situations may
arise in which $\tau_{sub}\simeq\tau_{A}$, causing the contribution in (54) to
become less apparent. For the sake of clarity in our analysis, we focus on
cases where $\tau_{sub}\gg\tau_{A}$.

The transition to thermal motion was also observed in the analysis presented
in Ref.s [108, 109]. This crossover time was defined as $\tau_{R}$, signifying
Rouse’s time in [108] and the relaxation time in the case of semiflexible
chains [109]. Both interpretations could be seen as thresholds marking the
transition to the Browinan linear regime. As we are dealing with infinite
systems, $\tau_{R}\to\infty$ in our context. Consequently, as the influence of
the non-equilibrium OU active drive diminishes ($t\gg\tau_{A}$), the thermal
dynamics predominates ($t\gg\tau_{sub}$).

The analysis of the MSD of probes other than AOUP is complex and will be
addressed elsewhere. However, even without a detailed analytical derivation,
some important conclusions can be drawn. As discussed in Sec.4.2, the dynamics
of regions in the elastic system close to the AOUP are primarily influenced by
the active non-equilibrium force up to $\tau_{sub}$, with an initial thermal
subdiffusive motion. For $t\gg\tau_{sub}$, the MSD of the probes is still
governed by fractional Gaussian noise (fGn).

The initial thermal regime becomes more pronounced as one considers regions
progressively farther from the AOUP. Consequently, the non-equilibrium
component of the MSD, denoted as
$\langle\delta^{2}h\left(\vec{x}^{,}t\right)\rangle_{OUN}$, diminishes. In the
limit of very distant regions satisfying
$\tau(|\vec{x}-\vec{x}^{|})\gg\tau_{sub}$, the diffusion of the probes is
entirely dictated by thermal fluctuations, rendering the impact of any non-
equilibrium driving force negligible. It’s worth highlighting that this
diffusional scenario precisely aligns with the findings from the numerical
simulations in Ref. [108].

The numerical implementation of GEM with an active particle can be pursued by
the numerical integration of the entire system governing Eq.(4).
Alternatively, one could just simulate the AOUP FLE (24) or the distant probe
FLE (17). As stated in [110] both the description offer the same level of
complexity, although the FLE framework provides a simpler and faster way to
delve into the dynamical regimes of a single particle.Furthermore, the
universal nature of the FLE equation allows for the definition of universality
classes, categorized by the value of $\beta$. The most effective method to
simulate the FLE is to utilize the $(k,\omega)$ solution as given in Equation
(11) and then invert it.

Overall, these results may help to elucidate in vivo dynamics observed in
experiments [90, 65].

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