††thanks<EMAIL_ADDRESS>
# Chiral run-and-tumble walker: transport and optimizing search

Rahul Mallikarjun and Arnab Pal  The Institute of Mathematical Sciences, CIT
Campus, Taramani, Chennai 600113, India & Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We study the statistical properties of a non-Markovian chiral run-and-tumble
particle (CRTP) in two dimensions in continuous space and time. In our model,
the possible orientations of the particle correspond to the four cardinal
directions. The particle can reorient by turning left, right or reversing its
direction of motion at different rates. We show how chirality manifests itself
in the transport properties like the spatial moments of the marginal position
distribution and the first-passage properties of a CRTP. Interestingly, we
find that the chirality leads to enhanced diffusion and a looping tendency in
the trajectory space. Furthermore, our results show that chirality plays a
pivotal role in the improvement of the search strategy – notably, there exists
an optimal bias in tumbling that minimizes the mean search time. This key
observation can play a crucial role in determining how living systems
efficiently search under non-equilibrium conditions.

††preprint: APS/123-QED

##  I Introduction

In recent years, a great deal of research has been, and is still being,
devoted to studying active matter systems [1, 2, 3]. A distinguishing feature
of these non-equilibrium systems is continuous energy consumption from the
environment by individual units forming the system to achieve self-propelled
motion [4]. This active behaviour is exemplified in diverse contexts such as
active gel description of the cytoskeleton [5, 6], living liquid crystals [7]
and synthetic structures mimicking biological microswimmers [8]. Active
systems show a variety of interesting emergent behaviour such as persistent
autonomous motility [9], large scale synchronization [10], pattern formation
and flocking [11, 2, 12, 8]. Run-and-tumble particles (RTPs) serve as
minimalistic models for active systems [13, 14]. The rich behaviour shown by
even an individual RTP has made them one of the most studied models. Exact
results for the transport and first-passage properties of a single one-
dimensional RTP in free space and subjected to a confining potential were
calculated in [15, 16, 17] with some results being extended to higher
dimensions in [18, 19] and with resetting dynamics [20]. Furthermore, it was
shown in [21] that the search time for a single-target search problem in a
bounded domain could be minimized for an RTP searcher, with orientation a
continuous variable in two and three dimensions, by tuning the persistence
time.

In the similar vein, we study the statistical properties of a four-state run-
and-tumble particle where the possible orientations of the particle correspond
to the four cardinal directions and focus on a less explored aspect of the
dynamics, i.e., chirality. This chirality, or broken parity symmetry, is an
inherent feature of the dynamics of the CRTP, and an adjustable parameter in
our model. This modification to the dynamics leads to the statistical
properties of a chiral run-and-tumble particle (CRTP) differing noticeably
from its non-chiral counterpart [18].

Figure 1: Schematic representation of a two-dimensional CRTP (initially moving
rightward) with four possible orientations $\theta\in\\{0,\pi/2,\pi,3\pi/2\\}$
that can reorient by tumbling left, right or reversing its direction with
unequal rates $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ respectively.

We study and distinguish the transport and first passage properties of a CRTP
in two dimensions from that of an RTP. To this end, we arrange the paper in
the following way. In Sec. II we describe the model of the CRTP and introduce
the relevant parameters. In Sec. II.1 we calculate the spatial moments of the
marginal position distribution in free space for isotropic and anisotropic
initial conditions. We show how chirality leads to enhanced diffusion in the
long-time limit, and how anisotropy of initial conditions give rise to
characteristic average trajectories. In Sec. II.2 we calculate the
autocorrelation of effective noise governing the dynamics of the CRTP and show
how the looping tendency of the CRTP plays a key role in altering it. Using
the autocorrelation function we also compute the power spectrum of this
process in Sec. II.3. Next, we turn our attention to discuss the first-passage
properties of the CRTP. In particular, we explore the effects of chirality on
the mean first-passage time to fixed boundaries in two dimensions.
Specifically, we ask the following: Does there exist an optimal value of the
bias in the rates of turning left and right that minimizes the mean first-
passage time to these boundaries? In other words, can chirality induce a
reduction in the search time? In Sec. III, we will address these questions. We
conclude our paper with a summary and a future outlook in Sec. IV.

##  II CRTP: Model and spatial properties

We consider a four-state random walker that moves in continuous two-
dimensional space $(x,y)$, and there are four possible orientations
$\theta\in\\{0,\pi/2,\pi,3\pi/2\\}$ associated with directions
$\\{\rightarrow,\uparrow,\leftarrow,\downarrow\\}$ that the particle can take.
The particle moves in a direction with a constant speed $v_{0}$ before
randomly changing its direction of motion and can reorient by turning left,
right or reversing (flipping) its direction of motion. The time $\tau$ for
subsequent tumble to be left, right or flip is exponentially distributed as
$\Gamma_{1}e^{-\Gamma_{1}\tau}$, $\Gamma_{3}e^{-\Gamma_{3}\tau}$ and
$\Gamma_{2}e^{-\Gamma_{2}\tau}$ (see Fig. 1). Since the rates of tumbling left
and right are unequal, we define this asymmetry as chirality. Moreover, the
movement of the particle is restricted to only four direction which is a
direct generalization of an one-dimensional run-and-tumble particle to two
dimensions, and hence, the name chiral run-and-tumble particle.

###  II.1 Spatial Moments of Position

Since the particle persists moving in a direction before randomly reorienting,
the model described above is non-Markovian. Nevertheless, it can be reduced to
an effective Markovian model by decomposing the motion into each possible
direction as we will see in Eq. (1). Let $P_{i}(x,y,t)$ be the probability
distribution of the CRTP being at $(x,y)$ and in state
$i\in\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$ at time $t$. Then the
full probability density is $P(x,y,t)=\sum_{i}P_{i}(x,y,t)$ and the $P_{i}$-s
satisfy the following coupled partial differential equations (see Appendix A
for derivation)

$\displaystyle\partial_{t}P_{\rightarrow}$
$\displaystyle=\Gamma_{1}P_{\downarrow}\
+\Gamma_{2}P_{\leftarrow}+\Gamma_{3}P_{\uparrow}\ -\gamma
P_{\rightarrow}-v_{0}\partial_{x}P_{\rightarrow},$ (1)
$\displaystyle\partial_{t}P_{\uparrow}$
$\displaystyle=\Gamma_{1}P_{\rightarrow}+\Gamma_{2}P_{\downarrow}\
+\Gamma_{3}P_{\leftarrow}-\gamma P_{\uparrow}\
-v_{0}\partial_{y}P_{\uparrow},$ $\displaystyle\partial_{t}P_{\leftarrow}$
$\displaystyle=\Gamma_{1}P_{\uparrow}\
+\Gamma_{2}P_{\rightarrow}+\Gamma_{3}P_{\downarrow}\ -\gamma
P_{\leftarrow}+v_{0}\partial_{x}P_{\leftarrow},$
$\displaystyle\partial_{t}P_{\downarrow}$
$\displaystyle=\Gamma_{1}P_{\leftarrow}+\Gamma_{2}P_{\uparrow}\
+\Gamma_{3}P_{\rightarrow}-\gamma P_{\downarrow}\
+v_{0}\partial_{y}P_{\downarrow},$

where $\gamma=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}$.

It will be useful to define a chirality parameter $\epsilon$ as the measure of
bias in rates of turning left and right. Since this bias is reflected only in
the absolute difference of $\Gamma_{1}$ and $\Gamma_{3}$, we can simply take
$\Gamma_{1}=\epsilon$ or $\Gamma_{3}=\epsilon$ and set the other of the two to
be equal to $1$ by choosing the appropriate scale.

To compute the spatial moments, we first apply the Fourier transform for the
spatial variables and then apply the Laplace transform for the time variable
in Eq. (1) where the transforms are defined as

$\displaystyle\bar{P}(\bm{k},t)$
$\displaystyle=\int_{-\infty}^{\infty}P(\bm{x},t)e^{-i\bm{k}.\bm{x}}d\bm{x},$
(2) $\displaystyle\text{and\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }\widetilde{P}(\bm{k},s)$
$\displaystyle=\int_{0}^{\infty}\bar{P}(\bm{k},t)e^{-st}dt.$ (3)

We can solve the resulting set of equations algebraically to find
$\widetilde{P}(k_{x},k_{y},s)$ (see Appendix A). Using
$\widetilde{P}(k_{x},k_{y},s)$, we next find the spatial moments of $P(x,y,t)$
since moments quantify the key parameters such as the location and shape of a
distribution. For present purposes, we solve for a particle that moves freely
in space and use natural boundary conditions, i.e., $P\to 0$ as
$x,y\to\pm\infty$.

First, we use the isotropic initial condition
$P_{\rightarrow}(\bm{k},0)=P_{\leftarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)=1/4$.
For $\langle x^{n}(t)\rangle$ we set $\bm{k}=k_{x}$. Due to the symmetry of
the initial conditions, the $\langle y^{n}(t)\rangle$ would be the same as
$\langle x^{n}(t)\rangle$. The moments in the Laplace space are simply given
by

$\langle\widetilde{x}^{n}(s)\rangle=i^{n}\bigg{[}\partial^{n}\widetilde{P}(k_{x},s)/\partial{k_{x}^{n}}\bigg{]}_{k_{x}=0}.$
(4)

Figure 2: Plot of MSD $\langle x^{2}(t)\rangle$ for a 2D CRTP for different
values of $\Gamma_{3}$ for isotropic initial conditions. The values of other
parameters $v_{0}$, $\Gamma_{1}$ and $\Gamma_{2}$ are kept constant. We have
also plotted $\langle x^{2}(t)\rangle$ for a one dimensional RTP for
comparison.

In particular, the mean squared deviation reads

$\langle
x^{2}(t)\rangle=\frac{v_{0}^{2}[-2abe^{-at}sin(bt)+(a^{2}-b^{2})e^{-at}cos(bt)]}{(a^{2}+b^{2})^{2}}\\\
+\frac{v_{0}^{2}[a(a^{2}+b^{2})t+(b^{2}-a^{2})]}{(a^{2}+b^{2})^{2}},$ (5)

where $a=\gamma+\Gamma_{2}=1+\epsilon+2\Gamma_{2}$ and
$b=\Gamma_{1}-\Gamma_{3}=1-\epsilon$. Here we’ve chosen $\Gamma_{1}=1.0$ and
$\Gamma_{3}=\epsilon$. Switching the values of $\Gamma_{1}$ and $\Gamma_{3}$
does not affect the result. At very short times, the probability of the CRTP
reorienting to a direction other than the initial one is low. This negligible
probability of tumbling in turn implies that the position of the particle is
known with almost certainty, and this certainty in position is reflected in
the position distribution of the CRTP. For the purpose of illustration, let us
focus on the marginal position distribution $P(x,t)$ when the probability of
starting out in the four possible directions is the same. For some very short
time $t$, $P(x,t)$ will be dominated by three $\delta$ functions; two of them
at $x=\pm v_{0}t$ and a third one at $x=0$. The former two, also referred to
as ballistic fronts, correspond to the position at time $t$ after no tumbling,
and the maximum distance the CRTP can move in $t$ . The third corresponds to
movement in the perpendicular $y$ direction while $x$ remains zero. These
three $\delta$ functions are exponentially decaying as the certainty in the
position is drowned in the randomness of the motion exponentially fast. The
existence of the delta functions in the position distribution is a general
feature and does not arise specifically out of chirality. The marginal
position distribution of a four state RTP was calculated in [18] with the
three $\delta$ functions shown explicitly in the expression. In Fig. 3 we see
the numerically evaluated marginal position distribution $P(x,t)$. The
decaying $\delta$ functions are clearly visible at $t=5.0$. As $t$ increases
we see that the spikes have vanished.

Figure 3: Numerical plot of marginal distribution $P(x,t)$. The value of the
parameters chosen for the simulation are: $v_{0}=1.0$,
$\Gamma_{1}=\epsilon=0.4$, $\Gamma_{2}=0.0,\Gamma_{3}=1.0$. The three spikes
in the distribution at $t=5.0$ represent the decaying $\delta$ functions that
mark the fact that the position of the particle is known with greater
certainty at short times. This certainty is drowned out in the randomness of
the motion as time progresses. This is shown in the distributions at $t=10.0$
and $t=15.0$ which do not have any pronounced spikes.

Now, at very short times when the position distribution is dominated by the
three $\delta$ functions, the majority of contribution to $\langle
x^{2}(t)\rangle$ comes from the $\delta$ functions at $x=\pm v_{0}t$. The
$\delta$ function at $x=0$ can not contribute to $\langle x^{2}(t)\rangle$. If
we look at the expression for $\langle x^{2}(t)\rangle$ for the CRTP in Eq.
(5), one thing that is immediately noticeable is the oscillatory behavior at
short times $t<<a$. It stands to reason that this oscillation is seen in the
heights of the ballistic fronts of the position distribution $P(x,y,t)$. The
reason for this oscillation can be attributed to the looping tendency of the
CRTP. If there is a bias in the rates of tumbling left and right, the CRTP
tumbles in one direction more often than the other and that leads to a cycling
trajectory of the particle. The Taylor expansion of $\langle x^{2}(t)\rangle$
in this limit gives

$\langle x^{2}(t)\rangle\approx v_{0}^{2}t^{2}/2+\mathcal{O}(t^{3}).$ (6)

Thus, at short times the CRTP shows ballistic behaviour. For $t>>a$, we find
that $\langle x^{2}(t)\rangle$ evolves as

$\langle x^{2}(t)\rangle=\frac{v_{0}^{2}a}{a^{2}+b^{2}}t,$ (7)

which essentially tells that the CRTP behaves like a diffusing particle at
large time (similar to 1D RTP [16, 18]) with an effective diffusion constant
$\frac{av_{0}^{2}}{2(a^{2}+b^{2})}$. It is unfeasible to write the full
expression for $\langle x^{4}(t)\rangle$, but it should be noted that apart
from the oscillatory behaviour of the ballistic fronts for $t<<a$ as in the
case of the $\langle x^{2}(t)\rangle$, for $t>>a$ one finds

$\langle x^{4}(t)\rangle=\frac{3v_{0}^{4}a^{2}t^{2}}{(a^{2}+b^{2})^{2}}.$ (8)

Evidently, for large $t$ we have

$\frac{\langle x^{4}(t)\rangle}{(3\langle x^{2}(t)\rangle)^{2}}\to 1.$ (9)

Thus, $P(x,y,t)$ tends to a bivariate Gaussian distribution in the large time
limit and the marginal distribution $P(x,t)$ (or $P(y,t)$) tends to a Gaussian
distribution (see Fig. 3).

It maybe pointed out that since the tumbling rates are unequal, the growth of
$\langle x^{2}(t)\rangle$ is now a function of the parameters
$\\{\Gamma_{2},\epsilon\\}$ (assuming fixed $v_{0}$) through $a$ and $b$ [see
Eq.(5)]. As Fig. 2 shows, $\langle x^{2}(t)\rangle$ evidently grows noticeably
faster for certain values of $\epsilon$ for a fixed $\Gamma_{2}$. Indeed, [22]
also reported enhanced diffusion for a two-dimensional chiral random walker
where the orientation of the particle $\theta$ is a continuous variable and
the step lengths are all equal. A larger $\langle x^{2}\rangle$ indicates a
larger area explored, and as we will show in Sec. III.2, this has interesting
implications in the first passage properties of the CRTP. Finally, we note
that all the odd raw moments are zero as expected because of the symmetry.

One specific behaviour that a CRTP is characterized by is its spiral mean
trajectory when there is an asymmetry in the initial conditions. Let
$\langle.\rangle_{i}$ indicate the average of an observable for a particle
initially moving from the origin in the direction indicated by the subscript.
To illustrate we choose $i$ to be $\rightarrow$. As before, $\langle
x(t)\rangle_{\rightarrow}$ and $\langle y(t)\rangle_{\rightarrow}$ can be
found using the non-isotropic initial condition $P_{\rightarrow}(\bm{k},0)=1$
and
$P_{\leftarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)=0$
which read

$\displaystyle\langle x(t)\rangle_{\rightarrow}$
$\displaystyle=v_{0}\frac{a-e^{-at}[acos(bt)-bsin(bt)]}{a^{2}+b^{2}},$ (10)
$\displaystyle\langle y(t)\rangle_{\rightarrow}$
$\displaystyle=v_{0}\frac{b-e^{-at}[asin(bt)+bcos(bt)]}{a^{2}+b^{2}}.$

Figure 4: Plot showing spiral average trajectories characteristic of a CRTP
for anisotropic intial conditions. Each spiral trajectory corresponds to a
CRTP starting at the origin at $t=0.0$ in the direction indicated by the
associated arrow. The trajectories are plotted for a time long enough for each
spiral to converge to a single point. In this case, the trajectories are
plotted till $t=10.0$.

One can similarly calculate the average trajectories of the CRTP for different
anisotropic initial conditions (see Appendix A). The spiral nature of the
trajectories is shown in Fig. 4. From Eq. (10), for $t>>a$, the trajectory
converges to a single point. This is also reflected in Fig. 4. Such spiral
trajectories have been experimentally observed, for example, in chiral active
colloids [23, 24].

###  II.2 Correlations

Similar to the master equations for the state probabilities in Eq. (1), one
can also write stochastic equations of motion for the CRTP. The latter prove
to be useful in computing the correlations. These Langevin equations read

$\displaystyle\dot{x}(t)=v_{0}cos(\theta(t))=v_{0}\sigma_{x}(t),$ (11)
$\displaystyle\dot{y}(t)=v_{0}sin(\theta(t))=v_{0}\sigma_{y}(t),$

where $\sigma_{x}$ and $\sigma_{y}$ play the role of effective noise. In what
follows, we will calculate the two-time autocorrelation functions
$\langle\sigma_{x}(t^{\prime})\sigma_{x}(t)\rangle$ &
$\langle\sigma_{y}(t^{\prime})\sigma_{y}(t)\rangle$, and further show how
chirality plays a role in their characteristics.

Integrating $P_{i}(x,y,t)$ over the spatial variables for the entire two-
dimensional space gives $p_{i}(t)$. Thus, $p_{i}(t)$ denotes the probability
that the particle is in the state $i$ at time $t$. We require $p_{i}(t)$ to
calculate the correlation time for the effective noise and to understand in
what aspect the autocorrelation of the effective noise for a CRTP differs from
that of an RTP. Using the same line of reasoning used to arrive at the
equation for the evolution of $P_{i}(x,y,t)$, we can write the equation
governing the evolution of $p_{i}(t)$ as

$\frac{d}{dt}\begin{bmatrix}p_{\rightarrow}\\\ p_{\leftarrow}\\\
p_{\uparrow}\\\
p_{\downarrow}\end{bmatrix}=\begin{bmatrix}-\gamma&\Gamma_{2}&\Gamma_{3}&\Gamma_{1}\\\
\Gamma_{2}&-\gamma&\Gamma_{1}&\Gamma_{3}\\\
\Gamma_{1}&\Gamma_{3}&-\gamma&\Gamma_{2}\\\
\Gamma_{3}&\Gamma_{1}&\Gamma_{2}&-\gamma\end{bmatrix}\begin{bmatrix}p_{\rightarrow}\\\
p_{\leftarrow}\\\ p_{\uparrow}\\\ p_{\downarrow}\end{bmatrix}.$ (12)

Without any loss of generality, one can alter the notation to write $p_{i}(t)$
as $p(\theta_{i},t|\theta_{p},t^{\prime})$ where $\theta_{i}$ explicitly marks
the orientation at time $t$ given that the orientation of the particle at at
time $t^{\prime}$ was $\theta_{p}$. Following Appendix B, one then finds the
correlation to be

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{n}\sum_{\theta_{i}}\sum_{\theta_{p}}[cos(\theta_{i})cos(\theta_{p})p(\theta_{i},t|\theta_{p},t^{\prime})],$
(13)

which yields [see Appendix B]

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{2}e^{-a|t^{\prime}-t|}cos[b(t^{\prime}-t)],$
(14)

where $a$ is the same as in Eq. (5) but now we’ve taken $\Gamma_{1}=\epsilon$
and $\Gamma_{3}=1.0$ again emphasizing that it is the absolute difference of
$\Gamma_{1}$ and $\Gamma_{3}$ that matters here. We can carry out similar
steps to find
$\langle\sigma_{y}(t)\sigma_{y}(t^{\prime})\rangle=\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle$.
This is because in Eq. (13) we summed over $\theta_{p}$, and therefore,
assumed isotropic initial conditions. Eq. (14) differs notably from the result
for $\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle$ for non-chiral four-
state RTP [18] by multiplication with the factor
$cos[(\Gamma_{1}-\Gamma_{3})(t^{\prime}-t)]$ (see Fig. 5). This additional
factor can be explained as follows: If $\Gamma_{1}=\Gamma_{3}$ then, on an
average, the particle tumbles left and right equally from its current state.
But if $\Gamma_{3}>\Gamma_{1}$, then there is bias in turning left and right
leading to a looping tendency. The CRTP turns right from its current state
more frequently than left and thus cycles around a given
$\theta\in\\{0,\pi/2,\pi,3\pi/2\\}$. Since
$\theta\in\\{0,\pi,\pi/2,3\pi/2\\}$, there is no value of $\theta$ in this
model when $\cos\theta$ and $\sin\theta$ are simultaneously non-zero. This
means that the cross correlation
$\langle\sigma_{x}(t)\sigma_{y}(t^{\prime})\rangle$ is zero. The position
autocorrelation functions $\langle x(t)x(t^{\prime})\rangle$, $\langle
y(t)y(t^{\prime})\rangle$, etc., can also be computed directly using Eqs. (11)
and (14) as we have shown in the Appendix B. These correlation functions
become instrumental in the study of the power spectrum properties that follow
next.

Figure 5: Plot comparing the autocorrelation of the effective noise
$\langle\sigma_{x}(t)\sigma_{x}(0)\rangle$ for a 2D RTP
($\Gamma_{1}=\Gamma_{3}=1.0\ \&\ \Gamma_{2}=0.0$) and CRTP. The exponentially
decaying oscillations of $\langle\sigma_{x}(t)\sigma_{x}(0)\rangle$ is not
exhibited by a non-chiral RTP.

###  II.3 Power spectrum

The power spectrum of an ensemble of long trajectories contains more detailed
information on the particle dynamics than the simple average or the mean
squared displacement [25, 26]. Moreover, the power spectrum often attains a
simple form, where individual terms characterize various underlying mechanisms
involved. By definition the marginal power spectrum $S_{xx}(\omega)$ of the
CRTP is given as

$S_{xx}(\omega)=\lim_{\tau\to\infty}\frac{1}{\tau}\Biggl{\langle}\Biggl{|}\int_{0}^{\tau}x(t)e^{i\omega
t}dt\Biggr{|}^{2}\Biggr{\rangle}=\\\
\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}\int_{0}^{\tau}\langle
x(t_{1})x(t_{2})\rangle e^{i\omega(t_{1}-t_{2})}dt_{1}dt_{2}.$ (15)

The autocorrelation function $\langle x(t_{1})x(t_{2})\rangle$ is calculated
in Appendix B for isotropic initial conditions. Substituting the expression
for $\langle x(t_{1})x(t_{2})\rangle$ in Eq. (12) and carrying out the
integral we find

$S_{xx}(\omega)=\frac{av_{0}^{2}}{a^{2}+b^{2}}\bigg{[}\frac{4}{\omega^{2}}-\frac{(a^{2}-3b^{2}+\omega^{2})}{[(a^{2}+b^{2})^{2}+2(a^{2}-b^{2})\omega^{2}+\omega^{4}]}\bigg{]}.$
(16)

In the low frequency limit $\omega\to 0$ (which is $t\to\infty$ limit), we
have

$S_{xx}(\omega)\approx\frac{4av_{0}^{2}}{(a^{2}+b^{2})\omega^{2}},$ (17)

which has the form of the power spectrum of one-dimensional Brownian motion as
expected with diffusion coefficient (see Eq. (7))

$D=\frac{av_{0}^{2}}{2(a^{2}+b^{2})}.$ (18)

In the intermediate range of values of $\omega$, a difference between the
power spectrum of an RTP and a CRTP can be observed due to the finite bias in
tumbling rates (see Fig. (6). In the opposite limit of large $\omega$ the
second term in Eq. (16), which differentiates the power spectrum of an RTP
from that of a Brownian particle, also contributes to $S_{xx}(\omega)$. Since
the autocorrelation function $\langle x(t_{1})x(t_{2})\rangle$ is calculated
for isotropic initial conditions, by symmetry we have
$S_{xx}(\omega)=S_{yy}(\omega)$.

We make some closing remarks concerning the spatial moments of CRTP and how
they are connected to the odd diffusivity named in the same spirit as odd
viscosity [27]. Chiral active fluids exhibit broken time-reversal and parity
symmetries giving rise to odd (or Hall) viscosity. Like how odd viscosity
gives rise to a flow perpendicular to applied pressure, odd diffusivity
generates fluxes perpendicular to concentration gradients. By deriving the
Green-Kubo relations for odd diffusivity, and using the expressions for
$\langle x\rangle_{\rightarrow}$ and $\langle y\rangle_{\rightarrow}$ (as in
Eq. (10)), it was shown in [28] how odd diffusivity exists as a consequence of
broken time-reversal and parity symmetries at a microscopic level.

Figure 6: Comparison of the power spectrum between a simple 1D Brownian
particle, a CRTP and an RTP. For the RTP: $\Gamma_{1}=\Gamma_{3}=1.0$ and
$\Gamma_{2}=0.0$. For the CRTP: $\Gamma_{1}=1.0$, $\Gamma_{2}=0.0$ and
$\Gamma_{3}=0.6$. For the RTP and the CRTP, we choose $v_{0}=1.0$. The
diffusion constant for the Brownian particle used to calculate the power
spectrum is equal to that of the CRTP in the diffusive limit as given in Eq.
(17).

##  III First-Passage Properties

We now turn our attention to the first passage properties of a CRTP. Broadly,
first passage properties are of importance since they characterize the
performance of search processes in various contexts such as foraging [29, 30],
chemical reactions [31], protein-DNA interaction [32], computer algorithm
optimization and complexity [33] and many others [32, 34, 35, 36, 37, 38, 39].
The most intuitive and important observable is the first-passage time (FPT)
for the particle to reach, say, the origin. The FPT is itself a stochastic
quantity and the probability distribution function of the FPT has its own rich
characteristics. The optimization of the search efficiency usually amounts to
minimizing the search time to the targets. A precise measure of this is the
mean first-passage time (MFPT). Thus, the MFPT is the average time it takes
for a searcher to reach a specified site for the first time. One of the main
goals here is to optimize the MFPT with respect to the parameters of the
model. In doing so, we can shed light on the way living or mechanical systems
can optimize their search.

It is useful to first describe the geometry on which the CRTP conducts its
search. We consider a topography where the CRTP is free to move along the
$y$-direction, however its motion is bounded by two absorbing boundaries which
are placed at $x=\pm L$. The boundaries act as targets and we will look into
the time statistics when the CRTP finds one of these targets. Similar set-up
has also been used in experiments with Brownian particles where the absorbing
targets are virtually placed on the 2D glass plate [40]. In what follows, we
first construct the backward Fokker-Planck formalism for the survival
probability and then compute them for the chiral random walker by using the
appropriate boundary conditions. Next, we compute the mean first passage time
which is directly related to the survival probability and finally, we discuss
the optimization.

Figure 7: Schematic of the boundary conditions satisfied by the CRTP with
orientations $\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$. The CRTP can
only be absorbed in specific orientations (black arrows). The boundary
conditions for other orientations marked with red crosses remain unspecified.

###  III.1 Survival Probability

Let $S_{i}(x,y,t)$ be the probability distribution for a particle initially
starting at $(x,y)$ with a speed $v_{0}$ in the direction
$i\in\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$, indicated by the
subscript, that survives being absorbed at the specified boundaries up to time
$t$. In a manner similar to how the evolution equations for $P_{i}$ were
derived, we can arrive at the set of evolution equations that the survival
probabilities satisfy, but in contrast to $P_{i}$, $S_{i}$ is a function of
the initial position of the CRTP rather than the final position. One can then
write the following backward equations for the survival probability

$\displaystyle\partial_{t}S_{\rightarrow}$
$\displaystyle=[\Gamma_{1}S_{\downarrow}\
+\Gamma_{2}S_{\leftarrow}+\Gamma_{3}S_{\uparrow}]\
+[v_{0}\partial_{x}-\gamma]S_{\rightarrow},$ (19)
$\displaystyle\partial_{t}S_{\leftarrow}$
$\displaystyle=[\Gamma_{1}S_{\uparrow}\
+\Gamma_{2}S_{\rightarrow}+\Gamma_{3}S_{\downarrow}]\
-[v_{0}\partial_{x}+\gamma]S_{\rightarrow},$
$\displaystyle\partial_{t}S_{\uparrow}$
$\displaystyle=[\Gamma_{1}S_{\rightarrow}+\Gamma_{2}S_{\downarrow}\
+\Gamma_{3}S_{\leftarrow}]+[v_{0}\partial_{y}-\gamma]S_{\uparrow},$
$\displaystyle\partial_{t}S_{\downarrow}$
$\displaystyle=[\Gamma_{1}S_{\leftarrow}+\Gamma_{2}S_{\uparrow}\
+\Gamma_{3}S_{\rightarrow}]-[v_{0}\partial_{y}+\gamma]S_{\downarrow},$

with the initial conditions
$\\{S_{\rightarrow}(x,y,0)=S_{\leftarrow}(x,y,0)=S_{\uparrow}(x,y,0)=S_{\downarrow}(x,y,0)$
$=1\\}$. Taking the Laplace transform of the set of Eqs. (19) for the time
variable, and introducing the shift

$\widetilde{S}_{i}(x,y,s)=\frac{1}{s}+U_{i}(x,y,s),$ (20)

followed by some simple algebraic manipulation of the resulting set of
equations, one arrives at

$\displaystyle\partial_{x}Q_{-}$ $\displaystyle=\alpha P_{+}+\beta Q_{+},$
(21) $\displaystyle\partial_{x}Q_{+}$ $\displaystyle=\chi P_{-}-\delta Q_{-},$
$\displaystyle\partial_{y}P_{-}$ $\displaystyle=\alpha Q_{+}+\beta P_{+},$
$\displaystyle-\partial_{y}P_{+}$ $\displaystyle=\chi Q_{-}+\delta P_{-}.$

where the new variables have been defined in the following way:
$Q_{+}=U_{\rightarrow}+U_{\leftarrow}$,
$Q_{-}=U_{\rightarrow}-U_{\leftarrow}$, $P_{+}=U_{\uparrow}+U_{\downarrow}$
and $P_{-}=U_{\uparrow}-U_{\downarrow}$. Furthermore, for convenience, we have
introduced a new set of parameters

$\displaystyle\alpha$ $\displaystyle\equiv(\Gamma_{1}+\Gamma_{3})/(-v_{0}),$
$\displaystyle\beta$ $\displaystyle\equiv(\Gamma_{2}-s-\gamma)/(-v_{0}),$
$\displaystyle\chi$ $\displaystyle\equiv(\Gamma_{3}-\Gamma_{1})/(-v_{0}),$
$\displaystyle\delta$ $\displaystyle\equiv(\Gamma_{2}+s+\gamma)/(-v_{0})$ (22)

This set of coupled linear first order partial differential equations for the
four functions $Q_{\pm}$ and $P_{\pm}$ can be reduced (see Appendix C) to the
higher order partial differential equation

$[\partial_{yy}+\beta\delta][\partial_{xx}+\beta\delta]f=[\alpha^{2}\chi^{2}-\beta^{2}\chi^{2}+\alpha^{2}\delta^{2}]f.$
(23)

where $f$ can be $Q_{\pm}$ or $P_{\pm}$. We can take a trial solution of the
form $e^{\lambda(x+y)}$ to solve the above equation.

However, before proceeding with the calculations, it is important to discuss
the boundary conditions. Recall that we have introduced two virtual infinite
boundaries at $x=-L$ and $x=L$ representing a CRTP confined to an effective
channel. As opposed to an ordinary particle, the run-and-tumble particle with
the states $\\{\rightarrow,\uparrow,\leftarrow,\downarrow,\\}$ that we are
studying can only be absorbed in specific orientations (see Fig. 7). In the
particular case of the two infinite vertical boundaries, the CRTP can be
absorbed only when in state $\leftarrow$ when at $x=-L$ and in state
$\rightarrow$ when at $x=L$. Considering these possibilities, the boundary
conditions read

$\displaystyle\widetilde{S}_{\leftarrow}(x=-L,y,s)=0\ \forall\ y,$ (24)
$\displaystyle\widetilde{S}_{\rightarrow}(x=L,y,s)=0\ \forall\ y,$

which are interpreted from their real time counterpart.

Figure 8: (a) The mean first-passage time (MFPT) for a CRTP starting at
$-10.0\leq x\leq 10.0$ in the direction $\leftarrow$ at $t=0$ to the absorbing
boundaries placed at $x=-10$ and $x=10$. The numerical markers (red dots) are
compared to the analytic prediction (dashed black line). (b) The MFPT for a
CRTP for the same initial conditions as (a) but different values of the
parameter $\Gamma_{3}=\epsilon$. (c) The maximum value that $\langle
T\rangle_{\leftarrow}$ attains for $-10.0\leq x\leq 10.0$ for a specific
$\epsilon$ is plotted against different values of $\epsilon$. Additional
values of $\epsilon$ are taken compared to (b). The plot shows that this
$max(\langle T\rangle_{\leftarrow})$ is minimized for
$\epsilon=\epsilon^{*}\approx 0.42$. This essentially indicates that there
exists an optimal value of the chirality parameter that minimizes the MFPT
(see main text for further details).

In particular, if we stick with the boundary conditions introduced in Eq.
(24), then $S_{i}$-s no longer depend on the $y$ coordinate of the starting
position of the CRTP as for a fixed $x$ all $y$ coordinates are equivalent.
This implies that $U_{i}$ (Eq. (20)) or their linear combinations as in
$Q_{\pm}$ and $P_{\pm}$ do not depend on the $y$ coordinate either for the
boundaries specified in Eq. (24). Thus, in Eq. (21) the partial derivatives
with respect to $y$ are zero and we get

$\displaystyle P_{+}$ $\displaystyle=\frac{-\alpha}{\beta}Q_{+},$ (25)
$\displaystyle P_{-}$ $\displaystyle=\frac{-\chi}{\delta}Q_{-}.$

and subsequently Eq. (23) becomes

$\beta\delta[\partial_{xx}+\beta\delta]g=[\alpha^{2}\chi^{2}-\beta^{2}\chi^{2}+\alpha^{2}\delta^{2}]g,$
(26)

where $g$ denotes $Q_{\pm}$ since $\partial_{yy}g$ vanishes. As the $y$
dependence is now removed, the trial solution for Eq. (26) becomes $e^{\lambda
x}$ and substituting the same into Eq. (26) gives

$\lambda=\pm\sqrt{\frac{(\alpha^{2}-\beta^{2})(\chi^{2}+\delta^{2})}{\beta\delta}}.$
(27)

We then have the formal solutions for $Q_{\pm}$

$\displaystyle Q_{-}$ $\displaystyle=A_{1}e^{-\lambda x}+A_{2}e^{\lambda x},$
(28) $\displaystyle Q_{+}$ $\displaystyle=B_{1}e^{-\lambda x}+B_{2}e^{\lambda
x},$

along with Eq. (25). The next task is to evaluate the four constants in Eq.
(28). To do so, we return to the original set of coupled first-order linear
partial differential Eqs. (21) that the solutions Eq. (25) and Eq. (28) must
also satisfy. This puts constraints on the constants $B_{1}$ and $B_{2}$,
i.e.,

$\displaystyle B_{1}$
$\displaystyle=\frac{-\beta\lambda}{\beta^{2}-\alpha^{2}}A_{1},$ (29)
$\displaystyle B_{2}$
$\displaystyle=\frac{\beta\lambda}{\beta^{2}-\alpha^{2}}A_{2}.$

Finally, the constants $A_{1}$ and $A_{2}$ are fixed using the boundary
conditions from Eq. (24). Recall that

$\displaystyle\widetilde{S}_{\leftarrow}$
$\displaystyle=\frac{1}{s}+U_{\leftarrow}=\frac{1}{s}+\frac{Q_{+}-Q_{-}}{2},$
(30) $\displaystyle\widetilde{S}_{\rightarrow}$
$\displaystyle=\frac{1}{s}+U_{\rightarrow}=\frac{1}{s}+\frac{Q_{+}+Q_{-}}{2},$

and now using Eqs. (28) & (29) we can write $\widetilde{S}_{\leftarrow}(x,s)$
and $\widetilde{S}_{\rightarrow}(x,s)$ in terms of the remaining two constants
$A_{i}$-s. We then apply the boundary conditions from Eq. (24) to get

$\begin{bmatrix}\upsilon&\omega\\\
-\omega&-\upsilon\end{bmatrix}\begin{bmatrix}A_{1}\\\
A_{2}\end{bmatrix}=\begin{bmatrix}-2/s\\\ -2/s\end{bmatrix},$ (31)

where $\upsilon=(-\beta\lambda+\beta^{2}-\alpha^{2})e^{-\lambda
L}/(\beta^{2}-\alpha^{2})$ and
$\omega=(\beta\lambda+\beta^{2}-\alpha^{2})e^{\lambda
L}/(\beta^{2}-\alpha^{2})$. Thus, one can find $A_{1}$ and $A_{2}$ by solving
the above equation which in turn, provides us exact expression for the
survival probabilities $\widetilde{S}_{i}(x,s)$ in Laplace space – see [41]
for details.

###  III.2 Mean first-passage time and optimization

The mean first-passage time estimates the time it takes for the CRTP to hit
any of the boundaries. Formally, this can be written as

$\langle T_{i}(x)\rangle=\int_{0}^{\infty}tF_{i}(x,t)dt,$ (32)

where $x$ is the initial coordinate and
$i\in\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$ indicates the initial
direction. Here, $F_{i}(x,t)$ is the first-passage time distribution which is
related to the survival probability in the following way
$F_{i}(x,t)=-\frac{\partial S_{i}(x,t)}{\partial t}$ [35]. Using this
relation, one can represent the mean first passage time in terms of the
survival function directly in the Laplace space i.e.,

$\langle T_{i}(x)\rangle=\widetilde{S}_{i}(x,s)|_{s\to 0},$ (33)

where $\widetilde{S}_{i}(x,s)$-s have been computed in the previous
subsection.

While it is not feasible to explicitly present $\langle T_{i}(x)\rangle$ as a
function of the parameters $\\{v_{0},\epsilon,\Gamma_{2}\\}$ here, one has, in
principle, the exact solution (see [41]). Using that, one can still study the
behaviour by plugging in the values of the parameters and using the plots for
the MFPT. Fig. 8 (a) shows a comparison of theory and numerical simulation for
the MFPT for a CRTP starting at $x$, where $x\in(-10,10)$, in the direction
$\leftarrow$ at $t=0$. The absorbing boundaries are located at $x=-10$ and
$x=10$. Recall that the first passage quantities no longer depend on the $y$
coordinate of the starting position as all the $y$ coordinates are equivalent
for a fixed $x$. This is solely due to the choice of the topographical
construction.

As was shown in Sec. II.1, $\langle x^{2}(t)\rangle$ grows noticeably faster
for certain values of the parameter $\epsilon$ while keeping $\Gamma_{2}$
fixed. Fig. 8 (b) shows that there indeed exists as certain value of
$\epsilon$ that minimizes the MFPT of the CRTP for all starting positions
between the boundaries. And to aid visualization, in Fig. 8 (c) the maximum
value that $\langle T\rangle_{\leftarrow}$ attains for $-10.0\leq x\leq 10.0$
for a specific $\epsilon$ is plotted against different values of $\epsilon$.
While the value of $x$ for which $\langle T\rangle_{\leftarrow}$ achieves its
maximum is not precisely the same for every $\epsilon$, the values of $x$ are
close enough for the general comparison. It can be clearly seen that there
exist an optimal value $\epsilon^{*}$ that minimizes the MFPT for all starting
positions between the boundaries.

On a technical note this behaviour can be explained in a straightforward
manner: for $t>>a$, $\langle x^{2}(t)\rangle=[v_{0}^{2}Dt]$ with
$D=a/(a^{2}+b^{2})$. $D$ is a function of $\epsilon$ through $a$ and $b$ and
can be maximized with respect to $\epsilon$. A larger $D$ implies more area
explored in a fixed amount of time and leads to a smaller MFPT. On a more
intuitive note this behaviour can be explained in the following manner: The
trajectory of the CRTP is interspersed with segments between tumbling that are
much longer than the average length of segments between tumbles. These
unusually long segments take the particle quite far from its starting
position. Consider one extreme case when there is an absence of a bias in the
rates of turning left and right ($\Gamma_{1}=\Gamma_{3}=\epsilon=1.0$). On an
average, these long segments occur with equal frequency in opposing
directions. If the particle has made significant advancement towards one
boundary after one long segment, the next long segment is more likely to take
it away from the boundary than closer, and essentially nullifies the
advancement made previously.

Consider the other extreme case when $\Gamma_{1}>\Gamma_{3}=\epsilon=0.0$, so
that the CRTP never tumbles right. After the two subsequent left tumbles the
particle starts moving in the opposite direction. Again, the long segments are
more likely to be in the opposite direction and nullify each other’s effect.

But there exists an optimal value of $\epsilon$ when the particle turns left
more frequently than right, and on an average, the unusually long segments are
more likely to be in the same direction than the opposite. The advancement
made towards a boundary by one long segment is reinforced by another. Thus,
the time to the boundaries is minimized.

We conclude with a few final remarks: (1) The minimum $\langle T\rangle$ can
be computed when upper bounds of $\\{\Gamma_{i}\\}$ are known. We have chosen
all $\Gamma_{j}\leq 1$. (2) Readjustment of $\epsilon$ and $\Gamma_{2}$ based
on environmental cues can lead to an optimal search strategy. A non-zero
$\Gamma_{2}$ significantly increases the MFPT since by reversing its direction
of motion the CRTP confines its exploration. $\Gamma_{2}\neq 0$ is thus
beneficial when the resources are abundant in the vicinity of the CRTP.

##  IV Concluding remarks

In this paper, we have studied motion of a chiral run-and-tumble particle in
two dimensions. The particle can move in four directions with different
tumbling rates. We have shown how chirality manifests itself in the transport
and first-passage properties of a CRTP. Signatures of chirality were clearly
observed in the moments of the position distribution and the correlation
functions. We have shown how tuning the bias in the tumbling rates, or the
chirality, leads to enhanced diffusion. We then looked into the confined
motion of the particle in the presence of absorbing boundaries and
investigated the first-passage properties by computing the mean first-passage
time (MFPT). Interestingly, we observed that the minimization of MFPT occurs
if one suitably adjusts the bias in the rates of tumbling left and right. This
key observation turns out to be in stark contrast to that of the simple non-
chiral particle.

Active particles carry signatures of living systems. These microswimmers
propel themselves with directed motion through constant consumption and
dissipation of energy resulting in non-equilibrium activity. Several living
entities such as the E. coli bacteria are known to perform such active motion.
In a similar spirit, we believe that the one-dimensional projection of this
two-dimensional motion of the chiral run-and-tumble particle can serve as a
baseline model for organisms such as the Daphnia [42], motion of which often
consists of Lévy type walks with rests and tumbles. As shown in Fig. 2, the
spatial extent of exploration for a CRTP at a given time is significantly
reduced compared to a one-dimensional RTP with pauses between ballistic hops.
Thus, the CRTP motion can serve as a better model for the aforementioned
living organisms. Delving deeper into this direction remains a potential
research avenue.

##  V Acknowledgements

RM gratefully acknowledges IMSc for the technical support and research
facilities through the Visiting Student Programme.

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##  Appendix A Moments of marginal position distribution

In this appendix section, we derive Eq. (1) of the main text and calculate the
first four moments of the marginal position distribution $P(x,t)$. We
reiterate that $P_{i}(x,y,t)$ denotes the probability distribution of being in
state $i\in\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$ at $(x,y)$ at
$t$, and the full probability distribution is $P(x,y,t)=\sum_{i}P_{i}(x,y,t)$.
The evolution equations for $P_{i}$ can be found by writing the balance
equation for the evolution over the two intervals $[0,\Delta t]$ and $[\Delta
t,t+\Delta t]$. Consider $P_{\rightarrow}(x,y,t)$: in the interval $\Delta t$
the CRTP in state $\rightarrow$ can continue in the same direction from
$x-\Delta x$ to $x$, tumble to a new state
$i\in\\{\leftarrow,\uparrow,\downarrow\\}$, or the CRTP initially in a
different state can tumble to state $\rightarrow$ (see Fig. 9). Thus,
considering all the possibilities, we have

$P_{\rightarrow}(x,y,t+\Delta t)=[1-\gamma\Delta t]P_{\rightarrow}(x-\Delta
x,y,t)+\Delta
t[\Gamma_{1}P_{\downarrow}(x,y,t)+\Gamma_{2}P_{\leftarrow}(x,y,t)+\Gamma_{3}P_{\uparrow}(x,y,t)],$

where $\Delta x=v_{0}\Delta t$ and $\gamma=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}$.
After expanding in Taylor series for small $\Delta t$, taking the limit
$\Delta t\to 0$, retaining only first order derivatives, and repeating the
exercise for the other directions, we arrive at Eq. (1) of the main text.

Figure 9: Schematic representation of the transitions that can take place in a
small time interval $\Delta t$. To aid clarity the contribution from the
reversal of direction has been suppressed.

Similarly, for all the probabilities originating from different directions,
one can write

$\displaystyle\partial_{t}P_{\rightarrow}(x,y,t)$
$\displaystyle=\Gamma_{1}P_{\downarrow}(x,y,t)\
+\Gamma_{2}P_{\leftarrow}(x,y,t)+\Gamma_{3}P_{\uparrow}(x,y,t)\ -\gamma
P_{\rightarrow}(x,y,t)-v_{0}\partial_{x}P_{\rightarrow}(x,y,t),$ (34)
$\displaystyle\partial_{t}P_{\uparrow}(x,y,t)$
$\displaystyle=\Gamma_{1}P_{\rightarrow}(x,y,t)+\Gamma_{2}P_{\downarrow}(x,y,t)\
+\Gamma_{3}P_{\leftarrow}(x,y,t)-\gamma P_{\uparrow}(x,y,t)\
-v_{0}\partial_{y}P_{\uparrow}(x,y,t),$
$\displaystyle\partial_{t}P_{\leftarrow}(x,y,t)$
$\displaystyle=\Gamma_{1}P_{\uparrow}(x,y,t)\
+\Gamma_{2}P_{\rightarrow}(x,y,t)+\Gamma_{3}P_{\downarrow}(x,y,t)\ -\gamma
P_{\leftarrow}(x,y,t)+v_{0}\partial_{x}P_{\leftarrow}(x,y,t),$
$\displaystyle\partial_{t}P_{\downarrow}(x,y,t)$
$\displaystyle=\Gamma_{1}P_{\leftarrow}(x,y,t)+\Gamma_{2}P_{\uparrow}(x,y,t)\
+\Gamma_{3}P_{\rightarrow}(x,y,t)-\gamma P_{\downarrow}(x,y,t)\
+v_{0}\partial_{y}P_{\downarrow}(x,y,t).$

Applying the Fourier and Laplace transforms to the spatial and time variables
respectively, one has

$\displaystyle(s+\gamma)\widetilde{P}_{\rightarrow}(\bm{k},s)-\Gamma_{1}\widetilde{P}_{\downarrow}(\bm{k},s)-\Gamma_{2}\widetilde{P}_{\leftarrow}(\bm{k},s)-\Gamma_{3}\widetilde{P}_{\uparrow}(\bm{k},s)+ik_{x}v_{0}\widetilde{P}_{\rightarrow}(\bm{k},s)$
$\displaystyle=P_{\rightarrow}(\bm{k},0),$ (35)
$\displaystyle(s+\gamma)\widetilde{P}_{\uparrow}(\bm{k},s)-\Gamma_{1}\widetilde{P}_{\rightarrow}(\bm{k},s)-\Gamma_{2}\widetilde{P}_{\downarrow}(\bm{k},s)-\Gamma_{3}\widetilde{P}_{\leftarrow}(\bm{k},s)+ik_{y}v_{0}\widetilde{P}_{\uparrow}(\bm{k},s)$
$\displaystyle=P_{\uparrow}(\bm{k},0),$
$\displaystyle(s+\gamma)\widetilde{P}_{\leftarrow}(\bm{k},s)-\Gamma_{1}\widetilde{P}_{\uparrow}(\bm{k},s)-\Gamma_{2}\widetilde{P}_{\rightarrow}(\bm{k},s)-\Gamma_{3}\widetilde{P}_{\downarrow}(\bm{k},s)-ik_{x}v_{0}\widetilde{P}_{\leftarrow}(\bm{k},s)$
$\displaystyle=P_{\leftarrow}(\bm{k},0),$
$\displaystyle(s+\gamma)\widetilde{P}_{\downarrow}(\bm{k},s)-\Gamma_{1}\widetilde{P}_{\leftarrow}(\bm{k},s)-\Gamma_{2}\widetilde{P}_{\uparrow}(\bm{k},s)-\Gamma_{3}\widetilde{P}_{\rightarrow}(\bm{k},s)-ik_{y}v_{0}\widetilde{P}_{\downarrow}(\bm{k},s)$
$\displaystyle=P_{\downarrow}(\bm{k},0).$

Adding the set of Eqs. (35) gives

$s\widetilde{P}(k_{x},k_{y},s)=1-iv_{0}[k_{x}(\widetilde{P}_{\rightarrow}(k_{x},k_{y},s)-\widetilde{P}_{\leftarrow}(k_{x},k_{y},s))+k_{y}(\widetilde{P}_{\uparrow}(k_{x},k_{y},s)-\widetilde{P}_{\downarrow}(k_{x},k_{y},s))].$
(36)

The set of Eqs. (35) can be cast into a matrix form

$\begin{bmatrix}s+\gamma+\Gamma_{2}&\Gamma_{1}-\Gamma_{3}&ik_{x}v_{0}&0\\\
\Gamma_{3}-\Gamma_{1}&s+\gamma+\Gamma_{2}&0&ik_{y}v_{0}\\\
ik_{x}v_{0}&0&s+\gamma-\Gamma_{2}&-\Gamma_{1}-\Gamma_{3}\\\
0&ik_{y}v_{0}&-\Gamma_{1}-\Gamma_{3}&s+\gamma-\Gamma_{2}\end{bmatrix}\begin{bmatrix}\widetilde{P}_{\rightarrow}(\bm{k},s)-\widetilde{P}_{\leftarrow}(\bm{k},s)\\\
\widetilde{P}_{\uparrow}(\bm{k},s)-\widetilde{P}_{\downarrow}(\bm{k},s)\\\
\widetilde{P}_{\rightarrow}(\bm{k},s)+\widetilde{P}_{\leftarrow}(\bm{k},s)\\\
\widetilde{P}_{\uparrow}(\bm{k},s)+\widetilde{P}_{\downarrow}(\bm{k},s)\end{bmatrix}=\begin{bmatrix}P_{\rightarrow}(\bm{k},0)-P_{\leftarrow}(\bm{k},0)\\\
P_{\uparrow}(\bm{k},0)-P_{\downarrow}(\bm{k},0)\\\
P_{\rightarrow}(\bm{k},0)+P_{\leftarrow}(\bm{k},0)\\\
P_{\uparrow}(\bm{k},0)+P_{\downarrow}(\bm{k},0)\end{bmatrix},$ (37)

from which we find the spatial moments of $P(x,y,t)$ in the following.

###  A.1 For isotropic initial conditions

We first use the isotropic initial condition
$P_{\rightarrow}(\bm{k},0)=P_{\leftarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)=\frac{1}{4}$
in Eq. (37). For the generic moment $\langle x^{n}\rangle$ we set
$\bm{k}=k_{x}$ and get

$\widetilde{P}(k_{x},s)=\frac{2[s+2(\gamma-\Gamma_{2})][(s+\gamma+\Gamma_{2})^{2}+(\Gamma_{1}-\Gamma_{3})^{2}]+k_{x}^{2}v_{0}^{2}(s+\gamma+\Gamma_{2})}{2s[s+2(\gamma-\Gamma_{2})][(s+\gamma+\Gamma_{2})^{2}+(\Gamma_{1}-\Gamma_{3})^{2}]+2k_{x}^{2}v_{0}^{2}[(s+\gamma)^{2}-\Gamma_{2}^{2}]}.$
(38)

The moments in the Laplace space are then given by

$\langle\widetilde{x}(s)\rangle=i\frac{\partial\widetilde{P}(k_{x},s)}{\partial{k_{x}}}\bigg{|}_{k_{x}=0}=0,$

$\langle\widetilde{x}^{2}(s)\rangle=-\frac{\partial^{2}\widetilde{P}(k_{x},s)}{\partial{k_{x}}^{2}}\bigg{|}_{k_{x}=0}=\frac{v_{0}^{2}(\gamma+\Gamma_{2}+s)}{s^{2}[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]},$

$\langle\widetilde{x}^{3}(s)\rangle=-i\frac{\partial^{3}\widetilde{P}(k_{x},s)}{\partial{k_{x}}^{3}}\bigg{|}_{k_{x}=0}=0,$

$\langle\widetilde{x}^{4}(s)\rangle=\frac{\partial^{4}\widetilde{P}(k_{x},s)}{\partial{k_{x}}^{4}}\bigg{|}_{k_{x}=0}=\frac{12v_{0}^{4}(s+\gamma+\Gamma_{2})^{2}(s+\gamma-\Gamma_{2})}{s^{3}[(\Gamma_{1}-\Gamma_{3})^{2}+(s+\gamma+\Gamma_{2})^{2}]^{2}(s+2\gamma-2\Gamma_{2})}.$

After Laplace inversion we can get the exact expression for the spatial
moments. Some of these have been mentioned in the main text. We do not
explicitly write $\langle y^{n}\rangle$ since the symmetry of the initial
conditions will lead to exactly the same results as that for $\langle
x^{n}\rangle$.

###  A.2 For anisotropic initial conditions

In this subsection, we will compute the spatial moments for the anisotropic
initial conditions. Let $\langle.\rangle_{\rightarrow}$ indicate average for
on the particle initially moving from the origin in the direction indicated by
the subscript.

For $\langle x\rangle_{\rightarrow}$ and $\langle y\rangle_{\rightarrow}$: We
use the non-isotropic initial condition $P_{\rightarrow}(\bm{k},0)=1$ and
$P_{\leftarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)=0$
in Eq. (37).

$\widetilde{P}(k_{x},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}-ik_{x}v_{0}(\gamma+\Gamma_{2}+s)]}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{x}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{x}(s)\rangle_{\rightarrow}=i\frac{\partial\widetilde{P}(k_{x},s)}{\partial{k_{x}}}\bigg{|}_{k_{x}=0}=\frac{v_{0}\left(\gamma+\Gamma_{2}+s\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
x(t)\rangle_{\rightarrow}=v_{0}\frac{a-e^{-at}[acos(bt)-bsin(bt)]}{a^{2}+b^{2}}.$

$\widetilde{P}(k_{y},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}-ik_{y}v_{0}(\Gamma_{1}-\Gamma_{3})]-k_{y}^{2}v_{0}^{2}(s+\gamma+\Gamma_{2})}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{y}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{y}(s)\rangle_{\rightarrow}=i\frac{\partial\widetilde{P}(k_{y},s)}{\partial{k_{y}}}\bigg{|}_{k_{y}=0}=\frac{v_{0}\left(\Gamma_{1}-\Gamma_{3}\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
y(t)\rangle_{\rightarrow}=v_{0}\frac{b-e^{-at}[asin(bt)+bcos(bt)]}{a^{2}+b^{2}}.$

For $\langle x\rangle_{\leftarrow}$ and $\langle y\rangle_{\leftarrow}$: We
use the non-isotropic initial condition
$P_{\rightarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)=0$
and $P_{\leftarrow}(\bm{k},0)=1$ in Eq. (37).

$\widetilde{P}(k_{x},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}+ik_{x}v_{0}(\gamma+\Gamma_{2}+s)]}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{x}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\displaystyle\langle\widetilde{x}(s)\rangle_{\leftarrow}=i\frac{\partial\widetilde{P}(k_{x},s)}{\partial{k_{x}}}\bigg{|}_{k_{x}=0}=\frac{-v_{0}\left(\gamma+\Gamma_{2}+s\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
x(t)\rangle_{\leftarrow}=-v_{0}\frac{a-e^{-at}[acos(bt)-bsin(bt)]}{a^{2}+b^{2}}.$

$\widetilde{P}(k_{y},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}+ik_{y}v_{0}(\Gamma_{1}-\Gamma_{3})]-k_{y}^{2}v_{0}^{2}(s+\gamma+\Gamma_{2})}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{y}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{y}(s)\rangle_{\leftarrow}=i\frac{\partial\widetilde{P}(k_{y},s)}{\partial{k_{y}}}\bigg{|}_{k_{y}=0}=\frac{-v_{0}\left(\Gamma_{1}-\Gamma_{3}\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
y(t)\rangle_{\leftarrow}=-v_{0}\frac{b-e^{-at}[asin(bt)+bcos(bt)]}{a^{2}+b^{2}}.$

For $\langle x\rangle_{\uparrow}$ and $\langle y\rangle_{\uparrow}$: We use
the non-isotropic initial condition
$P_{\rightarrow}(\bm{k},0)=P_{\leftarrow}(\bm{k},0)=P_{\downarrow}(\bm{k},0)$
and $P_{\uparrow}(\bm{k},0)=1$ in Eq. (37).

$\widetilde{P}(k_{x},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}+ik_{x}v_{0}(\Gamma_{1}-\Gamma_{3})]-k_{x}^{2}v_{0}^{2}(s+\gamma+\Gamma_{2})}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{x}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{x}(s)\rangle_{\uparrow}=i\frac{\partial\widetilde{P}(k_{x},s)}{\partial{k_{x}}}\bigg{|}_{k_{x}=0}=\frac{-v_{0}\left(\Gamma_{1}-\Gamma_{3}\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
x(t)\rangle_{\uparrow}=v_{0}\frac{-b+e^{-at}[asin(bt)+bcos(bt)]}{a^{2}+b^{2}}.$

$\widetilde{P}(k_{y},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}-ik_{y}v_{0}(\gamma+\Gamma_{2}+s)]}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{y}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{y}(s)\rangle_{\uparrow}=i\frac{\partial\widetilde{P}(k_{y},s)}{\partial{k_{y}}}\bigg{|}_{k_{y}=0}=\frac{v_{0}\left(\gamma+\Gamma_{2}+s\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
y(t)\rangle_{\uparrow}=v_{0}\frac{a-e^{-at}[acos(bt)-bsin(bt)]}{a^{2}+b^{2}}.$

For $\langle x\rangle_{\downarrow}$ and $\langle y\rangle_{\downarrow}$: We
use the non-isotropic initial condition
$P_{\rightarrow}(\bm{k},0)=P_{\leftarrow}(\bm{k},0)=P_{\uparrow}(\bm{k},0)=0$
and $P_{\downarrow}(\bm{k},0)=1$ in Eq. (37).

$\widetilde{P}(k_{x},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}-ik_{x}v_{0}(\Gamma_{1}-\Gamma_{3})]-k_{x}^{2}v_{0}^{2}(s+\gamma+\Gamma_{2})}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{x}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{x}(s)\rangle_{\downarrow}=i\frac{\partial\widetilde{P}(k_{x},s)}{\partial{k_{x}}}\bigg{|}_{k_{x}=0}=\frac{v_{0}\left(\Gamma_{1}-\Gamma_{3}\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
x(t)\rangle_{\downarrow}=v_{0}\frac{b-e^{-at}[asin(bt)+bcos(bt)]}{a^{2}+b^{2}}.$

$\widetilde{P}(k_{y},s)=\frac{(2\gamma-2\Gamma_{2}+s)[(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}+ik_{y}v_{0}(\gamma+\Gamma_{2}+s)]}{s[2\gamma-2\Gamma_{2}+s][(\Gamma_{1}-\Gamma_{3})^{2}+(\gamma+\Gamma_{2}+s)^{2}]+k_{y}^{2}v_{0}^{2}[(\gamma+s)^{2}-\Gamma_{2}^{2})]},$

$\langle\widetilde{y}(s)\rangle_{\downarrow}=i\frac{\partial\widetilde{P}(k_{y},s)}{\partial{k_{y}}}\bigg{|}_{k_{y}=0}=\frac{-v_{0}\left(\gamma+\Gamma_{2}+s\right)}{s\left[\left(\Gamma_{1}-\Gamma_{3}\right)^{2}+\left(\gamma+\Gamma_{2}+s\right)^{2}\right]},$

$\langle
y(t)\rangle_{\downarrow}=-v_{0}\frac{a-e^{-at}[acos(bt)-bsin(bt)]}{a^{2}+b^{2}}.$

##  Appendix B Correlation functions

In this section we show the in detail the steps to arrive at the correlation
function for the effective noise as given in Eq. (14) of the main text. As in
Appendix A, the evolution equations for the probability $p_{i}(t)$ (and not
the entire distribution) can be found by writing the balance equation for the
evolution over the two intervals $[0,\Delta t]$ and $[\Delta t,t+\Delta t]$.
We can write the equation governing the evolution of $p_{\rightarrow}(t)$ as

$p_{\rightarrow}(t+\Delta t)=p_{\rightarrow}(t)+\Delta
t[\Gamma_{3}p_{\uparrow}(t)+\Gamma_{1}p_{\downarrow}(t)+\Gamma_{2}p_{\leftarrow}(t)-\gamma
p_{\rightarrow}(t)].$

Similar evolution equations can be written for the other directions as well.
Taking the limit $\Delta t\to 0$ in above, we recover Eq. (12) of the main
text. The same can be written as a matrix equation as follows

$\dot{\bm{p}}=\bm{V}\bm{p}.$ (39)

The solution to Eq. (39) is $p_{i}(t)=\sum_{k}\alpha_{k}v_{ik}e^{\psi_{k}t}$
where again $i\in\\{\rightarrow,\leftarrow,\uparrow,\downarrow\\}$, $v_{ik}$
are entries of the matrix of eigenvectors of $\bm{V}$, $\psi_{k}$ are the
associated eigenvalues, and $\alpha_{k}$ are constants determined using the
knowledge of $p_{i}$ at some other time $t^{\prime}$.

Of course, Eq. (39) tells us that
$\alpha_{k}=e^{-\psi_{k}t^{\prime}}\sum_{m}v^{-1}_{km}p_{m}(t^{\prime})$ for
some $t^{\prime}<t$. Substituting $\alpha_{k}$ back in solution $p_{i}(t)$ we
get

$p_{i}(t)=\sum_{k}e^{\psi_{k}(t-t^{\prime})}v_{ik}\sum_{m}v^{-1}_{km}p_{m}(t^{\prime}).$

As in the main text, we change the notation to write $p_{i}(t)$ as
$p(\theta_{i},t)$. If the orientation of the particle at $t^{\prime}$ is
$\theta=\theta_{p}$, i.e, we know its orientation with certainty at
$t^{\prime}$, then the probability of the CRTP to be in a state $m\neq p$ at
$t^{\prime}$ is then zero. Therefore, we get

$p(\theta_{i},t|\theta_{p},t^{\prime})=\sum_{k}e^{\psi_{k}(t-t^{\prime})}v_{ik}v^{-1}_{kp}.$
(40)

The eigenvalues of $\bm{V}$ are

$\displaystyle\psi_{1}$
$\displaystyle=-\gamma-\Gamma_{2}-i(\Gamma_{1}-\Gamma_{3}),$
$\displaystyle\psi_{2}$
$\displaystyle=-\gamma-\Gamma_{2}+i(\Gamma_{1}-\Gamma_{3}),$
$\displaystyle\psi_{3}$ $\displaystyle=-\gamma+\Gamma_{2}-\
(\Gamma_{1}+\Gamma_{3}),$ $\displaystyle\psi_{4}$
$\displaystyle=-\gamma+\Gamma_{2}+\ (\Gamma_{1}+\Gamma_{3}).$

The eigenvector matrix $\bm{v}$ is

$\bm{v}=\begin{bmatrix}i&-i&-1&1\\\ -i&i&-1&1\\\ -1&-1&1&1\\\
1&1&1&1\end{bmatrix},$

where the columns are eigenvectors and column $j$ corresponds to eigenvalue
$\psi_{j}$. The the inverse matrix is

$\bm{v}^{-1}=\frac{1}{4}\begin{bmatrix}-i&i&-1&1\\\ i&-i&-1&1\\\ -1&-1&1&1\\\
1&1&1&1\end{bmatrix}.$

It is important to note that once the matrices $\bm{v}$ and $\bm{v}^{-1}$ are
explicitly written, each row of $\bm{v}$ and each column of $\bm{v}^{-1}$ are
associated to a specific orientation as decided by the Eq. (12) of the main
text. We have

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{4}\sum_{i}\sum_{p}[cos(\theta_{i})cos(\theta_{p})p(\theta_{i},t|\theta_{p},t^{\prime})].$

Substituting $p(\theta_{i},t|\theta_{p},t^{\prime})$ from Eq. (40) gives

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{4}\Big{[}\sum_{k}e^{\psi_{k}(t-t^{\prime})}\sum_{i}\sum_{p}[v_{ik}v^{-1}_{kp}cos(\theta_{i})cos(\theta_{p})]\Big{]}.$

Using the eigenvalues and eigenvectors of $\bm{V}$ the expression reduces to

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{2}e^{-(\gamma+\Gamma_{2})(t-t^{\prime})}cos[(\Gamma_{1}-\Gamma_{3})(t-t^{\prime})].$
(41)

Note that $t^{\prime}<t$ need not be true. Thus, we can write more generally

$\langle\sigma_{x}(t)\sigma_{x}(t^{\prime})\rangle=\frac{1}{2}e^{-(\gamma+\Gamma_{2})|t^{\prime}-t|}cos[(\Gamma_{1}-\Gamma_{3})(t^{\prime}-t)].$
(42)

No modulus for $cos$ was added since it $cos(-x)=cos(x)$. Now that Eq. (42)
has been derived, one obvious way to test its validity is to compute $\langle
x^{2}(t)\rangle$ and see if it matches with what was calculated previously.
From Eq. (11) of the main text we have

$x(t)=v_{0}\int_{0}^{t}\sigma_{x}(s)ds.$

This means that

$\langle
x^{2}(t)\rangle=v_{0}^{2}\int_{0}^{t}ds^{\prime}\int_{0}^{t}\langle\sigma_{x}(s)\sigma_{x}(s^{\prime})\rangle
ds.$ (43)

Using Eq. (42) in Eq. (43) one arrives at the correct expression for $\langle
x^{2}(t)\rangle$. Similarly, we can find the autocorrelation function to be
used in calculating the power spectrum of the stochastic process

$\langle
x(t_{1})x(t_{2})\rangle=v_{0}^{2}\int_{0}^{t_{1}}ds^{\prime}\int_{0}^{t_{2}}\langle\sigma_{x}(s)\sigma_{x}(s^{\prime})\rangle
ds.$ (44)

Using the two-time noise correlation function, we arrive at the following
expression for the position correlation function

$\begin{split}\langle
x(t_{1})x(t_{2})\rangle=\frac{v_{0}^{2}}{2(a^{2}+b^{2})^{2}}\Big{[}2abe^{-a|t_{2}-t_{1}|}sin[b|t_{2}-t_{1}|]+(b^{2}-a^{2})e^{-a|t_{2}-t_{1}|}cos[b(t_{2}-t_{1})]-2abe^{-at_{2}}sin(bt_{2})+\\\
(a^{2}-b^{2})e^{-at_{2}}cos(bt_{2})-2abe^{-at_{1}}sin(bt_{1})+(a^{2}-b^{2})e^{-at_{1}}cos(bt_{1})+2a(a^{2}+b^{2})min(t_{1},t_{2})+(b^{2}-a^{2})\Big{]},\end{split}$
(45)

which was used to compute the power spectral function.

##  Appendix C Survival probability

In this section we show how to arrive at Eq. (23) of the main text and thus,
find find the mean first-passage time to specified boundaries for the RTP. For
the sake of completeness and to aid the understanding of the long derivation,
we start with Eq. (19) of the main text. We repeat that $S_{i}(x,y,t)$ is the
probability distribution for a particle that initially starting at $(x,y)$
with a speed $v_{0}$ in the direction
$i\in\\{\leftarrow,\rightarrow,\downarrow,\uparrow\\}$ indicated by the
subscript survives being absorbed at the specified boundaries up to time $t$.

$\displaystyle\partial_{t}S_{\rightarrow}$
$\displaystyle=[\Gamma_{1}S_{\downarrow}\
+\Gamma_{2}S_{\leftarrow}+\Gamma_{3}S_{\uparrow}]\
+[v_{0}\partial_{x}-\gamma]S_{\rightarrow},$
$\displaystyle\partial_{t}S_{\leftarrow}$
$\displaystyle=[\Gamma_{1}S_{\uparrow}\
+\Gamma_{2}S_{\rightarrow}+\Gamma_{3}S_{\downarrow}]\
-[v_{0}\partial_{x}+\gamma]S_{\rightarrow},$
$\displaystyle\partial_{t}S_{\uparrow}$
$\displaystyle=[\Gamma_{1}S_{\rightarrow}+\Gamma_{2}S_{\downarrow}\
+\Gamma_{3}S_{\leftarrow}]+[v_{0}\partial_{y}-\gamma]S_{\uparrow},$
$\displaystyle\partial_{t}S_{\downarrow}$
$\displaystyle=[\Gamma_{1}S_{\leftarrow}+\Gamma_{2}S_{\uparrow}\
+\Gamma_{3}S_{\rightarrow}]-[v_{0}\partial_{y}+\gamma]S_{\downarrow}.$

Taking the Laplace transform of this set of equations for the time variable

$\widetilde{S}_{i}(\bm{x},s)=\int_{0}^{\infty}S_{i}(\bm{x},t)e^{-st}dt,$

using the initial conditions
$\\{S_{\rightarrow}(x,y,0)=S_{\leftarrow}(x,y,0)=S_{\uparrow}(x,y,0)=S_{\downarrow}(x,y,0)=1\\}$
and introducing the shift

$\widetilde{S}_{i}(x,y,s)=\frac{1}{s}+U_{i}(x,y,s),$

we have

$\displaystyle-v_{0}\partial_{x}U_{\rightarrow}$
$\displaystyle=\Gamma_{1}U_{\downarrow}\
+\Gamma_{2}U_{\leftarrow}+\Gamma_{3}U_{\uparrow}\ -(s+\gamma)U_{\rightarrow},$
$\displaystyle v_{0}\partial_{x}U_{\leftarrow}$
$\displaystyle=\Gamma_{1}U_{\uparrow}\
+\Gamma_{2}U_{\rightarrow}+\Gamma_{3}U_{\downarrow}\
-(s+\gamma)U_{\leftarrow},$ $\displaystyle-v_{0}\partial_{y}U_{\uparrow}$
$\displaystyle=\Gamma_{1}U_{\rightarrow}+\Gamma_{2}U_{\downarrow}\
+\Gamma_{3}U_{\leftarrow}-(s+\gamma)U_{\uparrow},$ $\displaystyle
v_{0}\partial_{y}U_{\downarrow}$
$\displaystyle=\Gamma_{1}U_{\leftarrow}+\Gamma_{2}U_{\uparrow}\
+\Gamma_{3}U_{\rightarrow}-(s+\gamma)U_{\downarrow}.$

After some straightforward algebraic manipulation we arrive at Eq. (21) of the
main text

$\ \ \partial_{x}Q_{-}=\alpha P_{+}+\beta Q_{+},$ (46a) $\ \
\partial_{x}Q_{+}=\chi P_{-}-\delta Q_{-},$ (46b) $\ \
\partial_{y}P_{-}=\alpha Q_{+}+\beta P_{+},$ (46c) $-\partial_{y}P_{+}=\chi
Q_{-}+\delta P_{-},$ (46d)

where the new variables are defined as follows

$\displaystyle Q_{+}$ $\displaystyle\equiv U_{\rightarrow}+U_{\leftarrow},$
$\displaystyle Q_{-}$ $\displaystyle\equiv U_{\rightarrow}-U_{\leftarrow},$
$\displaystyle P_{+}$ $\displaystyle\equiv U_{\uparrow}\ +U_{\downarrow},$
$\displaystyle P_{-}$ $\displaystyle\equiv U_{\uparrow}\ -U_{\downarrow}.$

And for convenience we have used

$\displaystyle\alpha$
$\displaystyle\equiv\frac{\Gamma_{1}+\Gamma_{3}}{-v_{0}},$
$\displaystyle\beta$ $\displaystyle\equiv\frac{\Gamma_{2}-s-\gamma}{-v_{0}},$
$\displaystyle\chi$ $\displaystyle\equiv\frac{\Gamma_{3}-\Gamma_{1}}{-v_{0}},$
$\displaystyle\delta$ $\displaystyle\equiv\frac{\Gamma_{2}+s+\gamma}{-v_{0}}.$

Using algebraic manipulation, the set of linear coupled partial differential
Eqs. (46a)-(46d) can be reduced to a single higher order partial differential
equation for one of the functions. One particular way is shown here.

Using Eq. (46a) and Eq. (46c) and eliminating one function we get

$Q_{+}=\frac{\beta\partial_{x}Q_{-}-\alpha\partial_{y}P_{-}}{\beta^{2}-\alpha^{2}},$
(47a)
$P_{+}=\frac{-\alpha\partial_{x}Q_{-}+\beta\partial_{y}P_{-}}{\beta^{2}-\alpha^{2}}.$
(47b)

Taking partial derivative of Eq. (47b) with respect to $y$ and using Eq. (46d)
one simply gets

$[\beta\partial_{yy}+\delta(\beta^{2}-\alpha^{2})]P_{-}=[\partial_{xy}-\chi(\beta^{2}-\alpha^{2})]Q_{-}.$
(48)

Similarly, using Eq. (46b) and Eq. (46d) and eliminating one function we get

$Q_{-}=\frac{\delta\partial_{x}Q_{+}+\chi\partial_{y}P_{+}}{-(\chi^{2}+\delta^{2})},$
(49a)

$P_{-}=\frac{\chi\partial_{x}Q_{+}-\delta\partial_{y}P_{+}}{\chi^{2}+\delta^{2}}.$
(49b)

Taking partial derivative of Eq. (49b) with respect to $y$ and using Eq. (46c)
one gets

$[\chi\partial_{xy}-\alpha(\chi^{2}+\delta^{2})]Q_{+}=[\delta\partial_{yy}+\beta(\chi^{2}+\delta^{2})]P_{+}.$
(50)

We rewrite Eqs. (48) and (50) as

$D_{1}P_{-}=D_{2}Q_{-},$ (51a) $D_{3}Q_{+}=D_{4}P_{+}.$ (51b)

with

$\displaystyle D1$
$\displaystyle\equiv[\beta\partial_{yy}+\delta(\beta^{2}-\alpha^{2})],$
$\displaystyle D2$
$\displaystyle\equiv[\alpha\partial_{xy}-\chi(\beta^{2}-\alpha^{2})],$
$\displaystyle D3$
$\displaystyle\equiv[\chi\partial_{xy}-\alpha(\chi^{2}+\delta^{2})],$
$\displaystyle D4$
$\displaystyle\equiv[\delta\partial_{yy}+\beta(\chi^{2}+\delta^{2})].$

Evidently, the operators $D_{1}$, $D_{2}$, $D_{3}$ and $D_{4}$ commute since
$\alpha$, $\beta$, $\chi$ and $\delta$ are constant. Substituting expressions
for $Q_{-}$ and $P_{-}$ from Eq. (49a) and (49b) in Eq. (51a) and using Eq.
(51b) gives

$[\partial_{y}(dD_{1}-cD_{2})D_{3}-\partial_{x}(cD_{1}+dD_{2})D_{4}]Q_{+}=0.$
(52)

And substituting expressions for $Q_{+}$ and $P_{+}$ from Eq. (47a) and (47b)
in Eq. (51b) and using Eq. (51a) gives

$[\partial_{y}(aD_{3}+bD_{4})D_{2}-\partial_{x}(aD_{4}+bD_{3})D_{1}]Q_{-}=0.$
(53)

It can easily be proved that $Q_{+}$ and $P_{+}$ satisfy the same differential
equation and it can be shown that $Q_{-}$ and $P_{-}$ satisfy the same
differential equation. Substituting the operators $D_{1}$, $D_{2}$, $D_{3}$
and $D_{4}$ defined in Eqs. (51a) and (51b) we recover the Eq. (23) of the
main text.