# Run-and-tumble particle with saturating rates

Kavita Jain(1,*) and Sakuntala Chatterjee(2,*)  (1) Theoretical Sciences Unit,
Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064,
India
(2) Department of Physics of Complex Systems, S.N. Bose National Center for
Basic Research, Salt Lake Sector 3, Kolkata 700106, India
* equal contribution

###### Abstract

We consider a run-and-tumble particle whose speed and tumbling rate are space-
dependent on an infinite line. Unlike most of the previous work on such
models, here we make the physical assumption that at large distances, these
rates saturate to a constant. For our choice of rate functions, we show that a
stationary state exists, and the exact steady state distribution decays
exponentially or faster and can be unimodal or bimodal. The effect of
boundedness of rates is seen in the mean-squared displacement of the particle
that displays qualitative features different from those observed in the
previous studies where it approaches the stationary state value monotonically
in time; in contrast, here we find that if the initial position of the
particle is sufficiently far from the origin, the variance in its position
either varies nonmonotonically or plateaus before reaching the stationary
state. These results are captured quantitatively by the exact solution of the
Green’s function when the particle has uniform speed but the tumbling rates
change as a step-function in space; the insights provided by this limiting
case are found to be consistent with the numerical results for the general
model.

##  I Introduction

Active matter comprises a wide class of natural systems ranging from flock of
birds to bacterial colony as well as artificially prepared systems like
engineered E.coli bacteria [1, 2, 3] or artificial Janus colloids [4, 5, 6,
7], and have received a lot of attention lately [8, 9, 10, 11]. In such
systems, each particle consumes energy from its environment and performs an
activity or active motion which is not symmetric under time-reversal [12, 13,
14] and hence active systems are intrinsically out-of-equilibrium even at the
level of a single particle.

One of the most well-studied active motion is run-and-tumble motion [15, 16,
17, 18] which is seen among various different kinds of microorganisms
including prokaryotic cells like E. coli, S. typhimurium, B. subtilis, R.
sphaeroides and S. marcescens [19, 20, 21, 22, 23] or eukaryotic cells such as
C. rheinhartii or T. foetus [24, 25]. During a run, a directed motion takes
place while during a tumble, the direction of motion is randomized, and by
repeatedly switching between run mode and tumble mode, an organism can
navigate in its environment. Experiments have shown that the run durations are
exponentially distributed [26] which means that the noise present in run-and-
tumble motion due to abrupt changes in the run direction is not the Gaussian
white noise, but is exponentially correlated in time.

A run-and-tumble motion is defined by specifying the velocity or run speed and
the tumbling rate. These rates can be intrinsic to the active particle or they
can be influenced by the environment; for example, during bacterial
chemotaxis, E.coli bacteria uses run-and-tumble motility to migrate towards
regions with higher nutrient concentration [19]. Experiments suggest that
while the run speed of the cell generally remains uniform [27, 28], the
tumbling rate depends on the local concentration gradient of the nutrient in
such a way that uphill runs are extended and downhill runs are shortened [29,
30, 31, 32], and therefore, by changing the nutrient environment, tumbling
rates can be modified. Recently, it was found that even the run speed can be
changed; some examples include an engineered strain of E.coli bacteria which
responds to the intensity of light by modulating its run speed [1, 2, 3, 33,
34, 35, 36] and wild-type bacteria that changes its speed on sensing the
ligand [37].

The above mentioned experiments suggest that, in general, the active particle
responds to the changes in its environment by modifying both the run speed and
tumbling rate. Furthermore, it is expected that a strong signal from the
environment will elicit a strong response from the particle; for example, a
steep chemical gradient will bring about a large change in the tumbling rate,
or by shining a high intensity light, the particle velocity will also show a
larger increase. However, neither the tumbling rate nor the velocity can
increase in an unbounded manner with the strength of the input signal and
beyond a limit, the response of any physical system is expected to saturate.
Yet most models assume that either the speed or tumbling rate (and not both)
are inhomogeneous and, importantly, these rates increase without bound in
space [18, 38, 39, 40, 41].

Motivated by the above physical considerations, in this work, we consider a
simple model of a run-and-tumble particle in one dimension with position-
dependent velocity and tumbling rate which saturate at large distance from the
origin. To the best of our knowledge, this is the first ever study of run-and-
tumble motion with spatially varying but bounded rates. As described in Sec.
II, the particle moves with rightward velocity $v_{+}(x)$ and leftward
velocity $v_{-}(x)$, while the corresponding tumbling rates are
$\gamma_{+}(x)$ and $\gamma_{-}(x)$, respectively. The rates $v_{\pm}(x)$ and
$\gamma_{\pm}(x)$ given by (3) and (4), respectively, saturate with $x$ for
large $|x|$ to zero or a nonzero constant. In Sec. III, we first identify the
parameter region where the stationary state exists, and find the exact
position distribution which decays exponentially or faster, and it may be
unimodal centered around the origin, or bimodal with two peaks symmetrically
placed on both sides of the origin.

In Sec. IV, to understand the dynamics of the active particle far from its
steady state, we focus on the mean-squared displacement (MSD) of the particle
as a function of time, starting from a given initial position. Our Monte Carlo
simulations show that if the particle starts close to the origin, the MSD
increases monotonically towards the stationary state value. But on starting
sufficiently far from the origin, the dynamics of the MSD fall in three
distinct dynamical regimes: at short times, it grows with time as a power law
with an exponent $2$ or $3$, depending on the initial orientation of the
particle; at intermediate times, it reaches a plateau or a peak, depending on
whether or not the tumbling rates vanish at large distances; and at large
times, the MSD relaxes exponentially fast to its steady state value. To the
best of our knowledge, a plateau or peak in the MSD at intermediate times has
not been reported earlier for a run-and-tumble motion, and is one of the most
striking effects of saturating rates. We obtain an analytical understanding of
these results in a limiting case where the speeds are homogeneous in space and
the tumbling rates vary as a step-function at the origin by calculating an
exact expression of the Green’s function.

##  II Description of the model

We consider a run-and-tumble particle (RTP) in one dimension which is
described by its position $-\infty<x<\infty$ and internal states, viz., $+$
(right-mover) and $-$ (left-mover) in continuous time. In time $dt\to 0$, the
right-mover at position $x$ at time $t$ either moves rightward at rate
$v_{+}(x)$ or changes its internal state to $-$ at rate $\gamma_{+}(x)$;
similarly, the left-mover either moves leftward or reverses its direction at
rate $v_{-}(x)$ and $\gamma_{-}(x)$, respectively. Then the probability
distribution $P_{+}(x,t)$ and $P_{-}(x,t)$, respectively, for the right- and
left-mover evolve according to [15]

$\displaystyle\frac{\partial P_{+}(x,t)}{\partial t}$ $\displaystyle=$
$\displaystyle-[v_{+}(x)P_{+}(x,t)]^{\prime}+\gamma_{-}(x)P_{-}(x,t)-\gamma_{+}(x)P_{+}(x,t)$
(1) $\displaystyle\frac{\partial P_{-}(x,t)}{\partial t}$ $\displaystyle=$
$\displaystyle-[-v_{-}(x)P_{-}(x,t)]^{\prime}+\gamma_{+}(x)P_{+}(x,t)-\gamma_{-}(x)P_{-}(x,t)$
(2)

where prime denotes the derivative with respect to $x$. The probability
distribution of the particle’s position is then given by
$P(x,t)=P_{+}(x,t)+P_{-}(x,t)$ with $\int_{-\infty}^{\infty}dxP(x,t)=1$ at all
times.

We now assume the following speed and tumble rates that are space-dependent,
bounded and asymmetric in the two directions,

$\displaystyle v_{\pm}(x)$ $\displaystyle=$ $\displaystyle v_{0}\mp
v_{1}\tanh(ux)$ (3) $\displaystyle\gamma_{\pm}(x)$ $\displaystyle=$
$\displaystyle\gamma_{0}\pm\gamma_{1}\tanh(gx)$ (4)

where $v_{0}\geq v_{1}>0,\gamma_{0}\geq\gamma_{1}>0$ as the rates must be non-
negative. In the above rates, there are six free parameters, and although we
can reduce them to four by scaling time and space, we do not do so as it is
helpful to see their explicit dependence. In general, the parameters $u$ and
$g$ can take either sign, but here we assume that $u\geq 0,g\geq 0$. For
$u=g=0$, we obtain the RTP model with homogeneous rates which has been
extensively studied [15]. For the parameters $u\neq 0,g=0$, we obtain a RTP
with uniform speed and tumbling rate subject to an external force
$f(x)=v_{1}\tanh(ux)$, or external potential

$\displaystyle V(x)$
$\displaystyle=-\int_{0}^{x}dx^{\prime}f(x^{\prime})=\frac{v_{1}}{u}\ln(\cosh(ux))$
$\displaystyle\sim\begin{cases}\frac{v_{1}u}{2}x^{2}~{}&,~{}u|x|\ll 1\\\
v_{1}|x|~{}&,~{}u|x|\gg 1\end{cases}$ (5)

For $u\to\infty$ where the external potential changes discontinuously at the
origin, the stationary state has been investigated [42, 43]. For $u=0,g\neq
0$, the stationary state has been obtained [44, 45], and some aspects of the
dynamics for $g\to\infty$ have been studied [40].

Furthermore, unlike in most previous work where the RTP is initially at the
origin and the initial density of the left- and right-mover is symmetric, here
we assume that

$P_{\pm}(x,0)=a_{\pm}\delta(x-x_{0})$ (6)

where $a_{+}+a_{-}=1,0\leq a_{\pm}\leq 1$. Without loss of generality, below
we also assume that the initial location $x_{0}\geq 0$.

We studied the model described above analytically using (1) and (2), and also
conducted extensive Monte Carlo simulations in which during a time interval
$dt$, depending on its internal state, the particle at position $x$ either
switches its direction with probability $\gamma_{\pm}(x)dt$ or persists in the
same direction with velocity $v_{\pm}(x)$ and moves to the new position $x\pm
v_{\pm}(x)dt$. For the initial condition (6), multiple stochastic trajectories
$\\{x(t)\\}$ for the particle’s position were generated, and its mean
$\overline{x}(t)$ and MSD,
$\sigma^{2}(x_{0},t)={(\overline{x(t)-x_{0})^{2}}}-(\overline{x}(t)-x_{0})^{2}$
(where the bar denotes an average over independent particle trajectories) were
computed. The stationary distribution [see (7) below] was also verified for
representative set of parameters in simulations (data not shown). In the
following sections, we first describe the stationary state and then turn to
the dynamics.

##  III Stationary state of the model

###  III.1 Stationary state distribution

For the rates in (3) and (4), we first find the regions in the $u-g$ plane
where the stationary state is expected to exist. When $u=g=0$, all the rates
are constant in space and we have the telegrapher’s equation for which there
is no stationary state [15]. But when $u>0,g=0$, the stationary state may
exist as the speed of the right (left)-mover decreases with increasing
(decreasing) $x$ which prevents the particle to drift away to
$x\to\infty(-\infty)$. If now the tumbling rate is also nonuniform and $g>0$,
the right (left)-mover reverses its direction at a higher rate with increasing
(decreasing) $x$ which further supports the confinement of the particle. On
the other hand, for $u<0,g<0$, as the right (left)-mover travels faster and
reverses its direction less often with increasing (decreasing) $x$, the
particle can escape and the stationary state does not exist. For $u>0,g<0$,
while the speed of the right (left)-mover decreases with increasing $x$, it
also tumbles less often, and the steady state is expected if the particle’s
speed slows down fast enough to counter the persistent motion; a similar
argument applies to $u<0,g>0$ regime.

In this work, we consider non-negative $u$ and $g$ only, and show below that a
stationary state exists for all $u,g\geq 0$ except when $u=g=0$. As described
in Appendix A, the stationary state distribution of the particle’s position
for the model described in Sec. II is given by

$P(x)=\frac{1}{{\cal
N}}\left[\frac{1}{v_{+}(x)}+\frac{1}{v_{-}(x)}\right]e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(7)

provided the speeds $v_{\pm}(x)$ are nonzero everywhere and the normalization
constant

${\cal
N}=\int_{-\infty}^{\infty}dx\left[\frac{1}{v_{+}(x)}+\frac{1}{v_{-}(x)}\right]e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(8)

is finite [32, 44]. Therefore, except for $u\to\infty,v_{0}=v_{1}$ for which
the speeds can vanish, we first write ${\cal
N}=\int_{-\infty}^{\infty}dxH(x)e^{I(x)}$ where

$\displaystyle H(x)$ $\displaystyle=$
$\displaystyle\frac{1}{v_{+}(x)}+\frac{1}{v_{-}(x)}=\frac{2v_{0}}{v_{0}^{2}-v_{1}^{2}\tanh^{2}(ux)}$
(9) $\displaystyle I(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy=-2\int_{0}^{x}dy\frac{\gamma_{0}v_{1}\tanh(uy)+\gamma_{1}v_{0}\tanh(gy)}{v_{0}^{2}-v_{1}^{2}\tanh^{2}(uy)}$
(10)

As $H$ and $I$ are symmetric about the origin, it is sufficient to determine
if $\int_{0}^{\infty}dxH(x)e^{I(x)}$ is finite. For $u>0$ and large, positive
$x$, we can write

$\displaystyle H(x)\stackrel{{\scriptstyle x\gg 1}}{{\sim}}$
$\displaystyle\frac{2v_{0}}{v_{0}^{2}-v_{1}^{2}}~{}$ , $v_{0}\neq v_{1}$ (11a)
$\displaystyle H(x)\stackrel{{\scriptstyle x\gg 1}}{{\sim}}$
$\displaystyle\frac{e^{2ux}}{2}~{}$ , $v_{0}=v_{1}$ (11b)

and

$\displaystyle I(x)\stackrel{{\scriptstyle x\gg 1}}{{\sim}}$
$\displaystyle-\frac{2(\gamma_{0}v_{1}+\gamma_{1}v_{0}(1-\delta_{g,0}))}{v_{0}^{2}-v_{1}^{2}}x~{}$
, $v_{0}\neq v_{1}$ (12a) $\displaystyle I(x)\stackrel{{\scriptstyle x\gg
1}}{{\sim}}$
$\displaystyle-\frac{\gamma_{0}+\gamma_{1}(1-\delta_{g,0})}{4uv_{0}}e^{2ux}~{}$
, $v_{0}=v_{1}$ (12b)

which show that $e^{I(x)}$ decays much faster than $H(x)$ grows at large $x$
and therefore, the normalization constant (8) is finite. For $u=0$,
$H=\frac{2}{v_{0}},I=-\frac{2\gamma_{1}}{v_{0}}\int_{0}^{x}dy\tanh(gy)\stackrel{{\scriptstyle
x\gg 1}}{{\to}}-\frac{2\gamma_{1}}{v_{0}}x(1-\delta_{g,0})$ and therefore, the
stationary state exists if $g>0$. The above discussion also shows that the
steady distribution is always unbounded and has exponential or double-
exponential tail.

We now consider the small-$x$ behavior of $P(x)$; on expanding the function
$H(x)$ in (9) to quadratic order and the integral $I(x)$ in (10) to linear
order in $x$ about the origin, we find that

$\frac{P(x)}{P(0)}\stackrel{{\scriptstyle x\to
0}}{{\approx}}1-\frac{g\gamma_{1}v_{0}+uv_{1}(\gamma_{0}-uv_{1})}{v_{0}^{2}}x^{2}$
(13)

which means that the distribution has a minimum at the origin if

$v_{1}^{2}u^{2}-\gamma_{0}v_{1}u-g\gamma_{1}v_{0}>0$ (14)

and suggests that the distribution can be multimodal. On taking the derivative
of (7) with respect to $x$ and setting it equal to zero, we find that the
stationary state distribution has turning point(s) at $x^{*}$ that obey the
following exact condition:

$\displaystyle\cosh^{2}(ux^{*})[\gamma_{1}v_{0}\tanh(gx^{*})+\gamma_{0}v_{1}\tanh(ux^{*})]$
$\displaystyle=$ $\displaystyle uv_{1}^{2}\tanh(ux^{*})$ (15)

The above equation shows that $x^{*}=0$ is always a mode of the stationary
state distribution and for $u=0$, the stationary distribution has a single
mode; unfortunately, it does not seem possible to find a closed form
expression for $x^{*}$ for arbitrary $u,g$ except for some special parameters
in the $u-g$ plane. These special cases (discussed in Sec. III.2) suggest that
(15) has at most three solutions and the parameters where the transition
between the unimodal and bimodal distribution occurs is consistent with the
criterion (14). Therefore, if we now assume that there are at most two maxima,
from (14), we find that the distribution is bimodal if

$(u-u^{*}_{+})(u-u^{*}_{-})>0~{},~{}u_{\pm}^{*}=\frac{\gamma_{0}\pm\sqrt{\gamma_{0}^{2}+4g\gamma_{1}v_{0}}}{2v_{1}}$
(16)

and unimodal otherwise. For $u,g>0$, as $u_{-}^{*}<0$, the distribution has
two maxima if

$u>u_{+}^{*}=\frac{\gamma_{0}+\sqrt{\gamma_{0}^{2}+4g\gamma_{1}v_{0}}}{2v_{1}},v_{1}\neq
0$ (17)

as illustrated in Fig. 1.

Figure 1: Steady state distribution for
$\gamma_{0}=v_{0}=1,\gamma_{1}=v_{1}=1/2,g=1/2$, and $u=1$ (dashed) and $u=5$
(solid) where the (blue) curve in $u-g$ plane shows (17). The $\times$ mark in
red at the origin signifies that the stationary distribution does not exist
for $u=g=0$, and the $\times$ mark in blue stands for $g=0,u\to\infty$ to show
that the distribution is unimodal in this limit (see Sec. III.2.2 for
details).

###  III.2 Special cases

As described in Appendix A, except for $u\to\infty,v_{0}=v_{1}$, the steady
state distribution $P_{\pm}(x)$ of the right- and left-movers is given by

$\displaystyle P_{+}(x)$ $\displaystyle=$ $\displaystyle
P_{+}(0)e^{\int_{0}^{x}\left(\frac{\gamma_{-}(x^{\prime})}{v_{-}(x^{\prime})}-\frac{\gamma_{+}(x^{\prime})}{v_{+}(x^{\prime})}-\frac{v^{\prime}_{+}(x^{\prime})}{v_{+}(x^{\prime})}\right)dx^{\prime}}$
(18) $\displaystyle P_{-}(x)$ $\displaystyle=$ $\displaystyle
P_{-}(0)e^{\int_{0}^{x}\left(\frac{\gamma_{-}(x^{\prime})}{v_{-}(x^{\prime})}-\frac{\gamma_{+}(x^{\prime})}{v_{+}(x^{\prime})}-\frac{v^{\prime}_{-}(x^{\prime})}{v_{-}(x^{\prime})}\right)dx^{\prime}}$
(19)

where $P_{\pm}(0)$ are obtained using
$\int_{-\infty}^{\infty}dx^{\prime}P_{\pm}(x^{\prime})=1/2$. Using (3) and (4)
in the above equations and in (7) for the total distribution,
$P(x)=P_{+}(x)+P_{-}(x)$, a general expression for the stationary state can be
obtained but it involves Gauss hypergeometric function and is not particularly
illuminating. However, simple expressions for the stationary distribution can
be obtained for following special points in the parameter space: $g>0,u=0$
[Sec. III.2.1], $g=0,u>0$ [Sec. III.2.2] and $u=g>0$ [Sec. III.2.3].

####  III.2.1 Uniform speed, variable tumbling rates

For constant speed but inhomogeneous tumbling rates,
$v_{\pm}=v_{0},\gamma_{\pm}(x)=\gamma_{0}\pm\gamma_{1}\tanh(gx)$ with $g>0$,
using (7), (18), (19), we find the following simple expressions,

$\displaystyle P_{+}(x)$ $\displaystyle=$
$\displaystyle\frac{g\Gamma\left(\frac{1}{2}+\frac{\gamma_{1}}{gv_{0}}\right)}{2\sqrt{\pi}\Gamma\left(\frac{\gamma_{1}}{gv_{0}}\right)}\left[\cosh(gx)\right]^{-\frac{2\gamma_{1}}{gv_{0}}}$
(20) $\displaystyle P_{-}(x)$ $\displaystyle=$ $\displaystyle P_{+}(x)$ (21)
$\displaystyle P(x)$ $\displaystyle=$ $\displaystyle 2P_{+}(x)$ (22)

which are independent of $\gamma_{0}$. We verify that the above distributions
are unimodal in agreement with (14) for $u=0$ and the tail of the distribution
is exponential, $P(x)\stackrel{{\scriptstyle g|x|\gg
1}}{{\sim}}\frac{\gamma_{1}}{v_{0}}e^{-\frac{2\gamma_{1}|x|}{v_{0}}}$ (see
also (24) below).

In later discussion in Sec. IV, we will be interested in the limit
$g\to\infty$ where

$\gamma_{\pm}(x)=\gamma_{0}\pm\gamma_{1}\textrm{sgn}(x)$ (23)

so that the tumbling rate is discontinuous at $x=0$. In this case, (22)
reduces to

$P(x)=\frac{\gamma_{1}}{v_{0}}e^{-\frac{2\gamma_{1}|x|}{v_{0}}}~{}~{}(u=0,g\to\infty)$
(24)

which is continuous at the origin but its first derivative is discontinuous at
$x=0$, and gives the variance in the particle’s position to be

$\sigma^{2}=\int_{-\infty}^{\infty}dxx^{2}P(x)=\frac{v_{0}^{2}}{2\gamma_{1}^{2}}~{}~{}(u=0,g\to\infty)$
(25)

We also find that the variance for finite $g$ decreases with increasing $g$,
and approaches (25) as $g^{-1},0<g\ll 1$ and $g^{-2},g\gg 1$. In fact, we will
see in Sec. IV that the full dynamics of the variance for finite $g$ are well
approximated by those for $g\to\infty$ for $g>1$.

####  III.2.2 Variable speed, uniform tumbling rates

Figure 2: Variable speed, uniform tumbling rates: Stationary state
distribution for (a) $v_{0}=3/2,v_{1}=1,\gamma_{0}=1,g=0$ and $u=1/2$ (red),
$1$ (black), $2$ (blue), and $10$ (magenta) and (b)
$v_{0}=v_{1}=1,\gamma_{0}=1,g=0$ and $u=1/2$ (red), $1$ (black), $2$ (blue),
and $4$ (magenta). Due to (30b), the transition between unimodal and bimodal
distribution occurs at $u=1$ for both set of parameters.

For constant tumbling rates but inhomogeneous speed, $v_{\pm}=v_{0}\mp
v_{1}\tanh(ux),\gamma_{\pm}(x)=\gamma_{0}$, we obtain

$\displaystyle P_{\pm}(x)$ $\displaystyle\propto$
$\displaystyle\frac{1}{v_{0}\mp
v_{1}\tanh(ux)}\exp\left[-\frac{\gamma_{0}v_{1}}{u\left(v_{0}^{2}-v_{1}^{2}\right)}\ln\left(\cosh^{2}(ux)-\frac{v_{1}^{2}\sinh^{2}(ux)}{v_{0}^{2}}\right)\right]$
(26) $\displaystyle P(x)$ $\displaystyle\propto$
$\displaystyle\frac{1}{v_{0}^{2}-v_{1}^{2}\tanh^{2}(ux)}\exp\left[-\frac{\gamma_{0}v_{1}}{u\left(v_{0}^{2}-v_{1}^{2}\right)}\ln\left(\cosh^{2}(ux)-\frac{v_{1}^{2}\sinh^{2}(ux)}{v_{0}^{2}}\right)\right]$
(27)

when $v_{0}\neq v_{1}$. At large distances from the origin, these
distributions decay exponentially fast and the decay rate is independent of
$u$, $P(x)\stackrel{{\scriptstyle u|x|\gg
1}}{{\sim}}e^{-\frac{2\gamma_{0}v_{1}|x|}{v_{0}^{2}-v_{1}^{2}}}$. For
$v_{0}=v_{1}$, we have

$\displaystyle P_{\pm}(x)$ $\displaystyle\propto$
$\displaystyle\frac{1}{1\mp\tanh(ux)}e^{-\frac{\gamma_{0}\sinh^{2}(ux)}{uv_{0}}}$
(28) $\displaystyle P(x)$ $\displaystyle\propto$
$\displaystyle\cosh^{2}(ux)e^{-\frac{\gamma_{0}\sinh^{2}(ux)}{uv_{0}}}$ (29)

which show that the distributions decay super-exponentially,
$P(x)\stackrel{{\scriptstyle u|x|\gg
1}}{{\sim}}e^{-\frac{\gamma_{0}e^{2u|x|}}{4u}}$. This rapid decay is a
consequence of the fact that the right-mover’s (left-mover’s) speed approaches
zero as $x\to\infty$ ($x\to-\infty$) and therefore, the particle can not be
found at large distances.

For $g=0$, the condition (15) for multimodality simplifies, and the location
of the modes is determined by $\tanh(ux^{*})=0$ or
$\cosh^{2}(ux^{*})=\frac{uv_{1}}{\gamma_{0}}$ which immediately gives

$\displaystyle x^{*}=$ $\displaystyle 0~{}$ , $0<u<\frac{\gamma_{0}}{v_{1}}$
(30a) $\displaystyle x^{*}=$ $\displaystyle
0,u^{-1}\cosh^{-1}\left(\pm\sqrt{\frac{uv_{1}}{\gamma_{0}}}\right)~{}$ ,
$u>\frac{\gamma_{0}}{v_{1}}$ (30b)

so that the distribution $P(x)$ is unimodal for $0<u<\frac{\gamma_{0}}{v_{1}}$
and is bimodal otherwise with the minimum at the origin. The transition curve
$u=\frac{\gamma_{0}}{v_{1}}$ between unimodal and bimodal distribution is
independent of $v_{0}$ and is also obtained from (17). Note that for
$u>\frac{\gamma_{0}}{v_{1}}$, the location $|x^{*}|$ of the maxima is a non-
monotonic function of $u$ which initially increases as
$\sqrt{u-\frac{\gamma_{0}}{v_{1}}}$ and then decays to zero,
$|x^{*}|\stackrel{{\scriptstyle u\gg 1}}{{\sim}}\frac{\ln u}{u}$ [also see
(31b) below]. The distribution $P(x)$ of the particle is shown in Fig. 2 for
various values of $u$, and describes the features discussed above.

Figure 3: Variable speed, uniform tumbling: Stationary state distribution
$P_{+}(x)$ (dashed) and $P_{-}(x)$ (solid) for
$v_{0}=3/2,v_{1}=1,\gamma_{0}=1,g=0$ when (a) $u=1/2$ (red) and $2$ (blue),
and (b) $u=10$ (magenta) and $u\to\infty$ (gray).

Figure 1 shows that for a given $g\geq 0$, as one increases $u$, the steady
state position distribution changes its shape from unimodal to a bimodal one.
To physically understand why bimodality appears for large $u$, we recall that
for $g=0$, the active particle experiences a confining potential $V(x)$ given
by (5) which has a minimum at the origin. Both left-movers and right-movers
cross the origin with speed $v_{0}$ which allows the particle to traverse some
distance from the origin until the potential cost makes it difficult for the
particle to go further. Indeed, as Fig. 3 shows, the distributions $P_{+}(x)$
and $P_{-}(x)$ display a peak at positive and negative $x$, respectively. If
now $u$ increases, the potential becomes steeper which makes it difficult for
the particle to traverse larger distances after crossing the origin; as a
result, the distributions $P_{\pm}(x)$ become narrower and their peak occurs
closer to the origin - see Fig. 3. Thus $P(x)=P_{+}(x)+P_{-}(x)$ shows a dip
at the origin because both $P_{+}(x)$ and $P_{-}(x)$ fall sharply from their
peak value near the origin when $u$ is large. Interestingly, the decrease in
width of $P_{\pm}(x)$ and the shifting of peak position towards the origin
with increasing $u$ also suggests that for $u\to\infty$, the distributions
$P_{\pm}(x)$ reach their maximum value at the origin and also develop a finite
discontinuity at the origin. Below we discuss this case more explicitly.

For $u\to\infty$, the speed $v_{\pm}(x)=v_{0}\mp v_{1}\text{sgn}(x)$ changes
discontinuously at the origin, and the stationary state distribution is given
by

$\displaystyle P(x)=$
$\displaystyle\frac{\gamma_{0}v_{1}}{v_{0}^{2}-v_{1}^{2}}e^{-\frac{2\gamma_{0}v_{1}|x|}{v_{0}^{2}-v_{1}^{2}}}~{}$
, $v_{0}\neq v_{1}$ (31a) $\displaystyle P(x)=$ $\displaystyle\delta(x)~{}$ ,
$v_{0}=v_{1}$ (31b)

(see Appendix A for (31b)). Note that the large-$x$ behavior of $P(x)$ for
finite $u$ is identical to that in $u\to\infty$ limit; this is simply because
for $|x|\gg u^{-1}$, the inhomogeneous speed becomes constant and behaves the
same way as for $u\to\infty$. For $v_{0}\neq v_{1}$, the distribution $P(x)$
is continuous but not differentiable at $x=0$, that is, there is a finite
discontinuity in the first derivative at the origin. But the distribution of
right- and left-movers is given by

$\displaystyle P_{\pm}(x)$ $\displaystyle=$
$\displaystyle\frac{\gamma_{0}v_{1}}{v_{0}}\frac{e^{-\frac{2v_{1}\gamma_{0}|x|}{v_{0}^{2}-v_{1}^{2}}}}{v_{0}\mp
v_{1}\textrm{sgn}(x)}$ (32)

which shows that these distributions have a finite discontinuity at the
origin, $|P_{\pm}(x\to 0+)-P_{\pm}(x\to
0-)|=\frac{2\gamma_{0}v_{1}^{2}}{v_{0}(v_{0}^{2}-v_{1}^{2})}$ (see Fig. 3b
also). This behavior is unlike that for $u=0,g\to\infty$ limit (see Sec.
III.2.1) where the distributions $P_{\pm}(x)$ are seen to be continuous at the
origin.

Finally, we remark that as mentioned in Sec. II, for $g=0$, the RTP is an
active particle in an external potential $V(x)$ given by (5). But for
$v_{0},\gamma_{0}\to\infty$ with the diffusion constant,
$2D=v_{0}^{2}/\gamma_{0}$ finite, one expects the above distributions reduce
to their equilibrium or passive counterpart [42, 43]. On taking these limits
in (27), we verify that indeed

$P(x)\propto e^{-\frac{V(x)}{D}}=[\cosh(ux)]^{-\frac{1}{uD}}$ (33)

####  III.2.3 Inhomogeneous speed and tumbling rates

For $g=u>0$ in (3) and (4), the stationary state distribution (7) is given by

$\displaystyle P(x)$ $\displaystyle\propto$
$\displaystyle\frac{1}{v_{0}^{2}-v_{1}^{2}\tanh^{2}(ux)}\exp\left[-\frac{\gamma_{0}v_{1}+\gamma_{1}v_{0}}{u\left(v_{0}^{2}-v_{1}^{2}\right)}\ln\left(\cosh^{2}(ux)-\frac{v_{1}^{2}\sinh^{2}(ux)}{v_{0}^{2}}\right)\right]$
(34)

which reduces to (27) for $g=0$. The above distribution also has a transition
between unimodal and bimodal shapes which is found using either (15) or (17)
at $u=\frac{\gamma_{0}v_{1}+\gamma_{1}v_{0}}{v_{1}^{2}}$. As in Sec. III.2.2,
the distribution decays super-exponentially for $v_{0}=v_{1}$ and
exponentially fast otherwise, and the location of the maxima is a non-
monotonic function of $u$. For $u=g\to\infty$, the speeds and the tumbling
rates are discontinuous at the origin, and we obtain the stationary
distribution in this case on replacing $\gamma_{0}v_{1}$ in (31b) by
$\gamma_{0}v_{1}+\gamma_{1}v_{0}$ .

##  IV Dynamics far from steady state

In this section, we focus on the dynamics of the system when the particle is
initially at position $x_{0}$ (see (6)) and, without loss of generality,
assume that $x_{0}\geq 0$. We are interested in the behavior of the mean-
squared displacement (MSD) denoted by $\sigma^{2}(x_{0},t)$ as a function of
time. While in the limit $t\to\infty$, the MSD reaches the stationary state
result, $\sigma^{2}=\int_{-\infty}^{\infty}dxx^{2}P(x)$ where $P(x)$ is given
by (7) and is independent of $x_{0}$, our Monte Carlo simulations show that at
short and intermediate times, $\sigma^{2}(x_{0},t)$ displays interesting
features depending on the initial position of the particle. We find at short
times, the MSD grows with time as a power law for any initial position but at
intermediate times, if $x_{0}$ is sufficiently large, it shows a plateau or a
peak before finally relaxing exponentially fast to the stationary state.

To get an analytical insight into these behavior, we note that as long as the
particle remains sufficiently far on the right side of the origin, the
dynamics are governed by the homogeneous limit in which the rates (3) and (4)
saturate to a constant:

$\displaystyle v_{\pm}(x\to\infty)$ $\displaystyle=$ $\displaystyle v_{0}\mp
v_{1}$ (35) $\displaystyle\gamma_{\pm}(x\to\infty)$ $\displaystyle=$
$\displaystyle\gamma_{0}\pm\gamma_{1}.$ (36)

In Appendix B, we obtain exact expressions for the mean position and MSD for
the above homogeneous rates (note that for these homogeneous rates, the case
of uniform speed or tumbling is obtained by setting $v_{1}=0$ or
$\gamma_{1}=0$, respectively). In the following, we explain in detail how
these results describe the dynamics at short and intermediate times for the
general model in Sec. IV.1 and Sec. IV.2, respectively. However, once the
particle is sufficiently close to the origin or if its initial position is
small enough, it becomes sensitive to the inhomogeneity in rates and the above
homogenous rates approximation is no longer valid, and we describe the long-
time dynamics in Sec. IV.3 using simulations. Although we have not been able
to capture the full dynamics analytically for general parameters, we obtain
some exact results when $u=0,g\to\infty$ which are described in Appendices C-E
and are also discussed below.

###  IV.1 Super-diffusive growth at short times

As shown in the inset of Fig. 4, at short enough times, the MSD grows as a
power law; this is a generic result and holds for any run-tumble motion,
including the case when velocities and tumbling rates are uniform, and can be
seen as follows. For the homogeneous rates (35) and (36), from (B.9), we find
the mean displacement to be

${\bar{x}}_{homo}(t)\approx x_{0}+(\Delta
v_{0}-v_{1})t,\;\;\;\;\;\;\mbox{$t\ll(2\gamma_{0})^{-1}$}$ (37)

where $\Delta=a_{+}-a_{-}$. Also, from (B.10), we find that the MSD is given
by

$\sigma_{homo}^{2}(t)\approx(1-\Delta^{2})v_{0}^{2}t^{2}+\frac{2}{3}[(3\Delta^{2}-1)\gamma_{0}+2\Delta\gamma_{1}]v_{0}^{2}t^{3},\;\;\;\;\;\;\mbox{$t\ll(2\gamma_{0})^{-1}$}$
(38)

which reduces to that obtained in [46] for symmetric, homogeneous rates and
fully asymmetric initial condition on setting $v_{1}=\gamma_{1}=0,\Delta=1$.
The above expression (38) shows that the MSD grows with time as a power law
with an exponent $2$ for $0<a_{+}<1$, while for $a_{+}=1$ and $a_{+}=0$, the
growth law is $t^{3}$. Our simulation data presented in the inset of Fig. 4 is
seen to be in excellent agreement with (38) for times much smaller than the
typical time scale for tumbling. On setting $v_{1}=0$ in (37) and (38), we
find that the resulting expressions also agree with the exact results obtained
in Appendix D for $u=0,g\to\infty,\gamma_{0}=\gamma_{1}$ when $2\gamma_{0}t\ll
1$.

Figure 4: Uniform speeds, variable tumbling rates: The main figure shows the
deviation of the mean-squared displacement (MSD) $\sigma^{2}(x_{0},t)$ from
the stationary state result $\sigma^{2}$ for $\gamma_{0}\neq\gamma_{1}$. The
simulation data (points) and the analytical prediction (44a) shown by solid
line are obtained for $u=0$, $x_{0}=8$, $\Delta=0$, $v_{0}=1$, $\gamma_{0}=1$,
$\gamma_{1}=0.4$ and various different $g$ values shown in figure legends.
Equation (43) gives the peak time $t^{\ast}=16.25$ which is close to the
observed peak time. The inset shows the superdiffusive growth of MSD at short
times obtained using simulations (points) and (38) (solid line) for $u=0$,
$g=1$, $x_{0}=1$, $v_{0}=1$ and $a_{+}=1,0.5$. Figure 5: Uniform speeds,
variable tumbling rates: The main figure shows the mean-squared displacement
(MSD) $\sigma^{2}(x_{0},t)$ for $\gamma_{0}=\gamma_{1}$. The data shown are
for the same parameters as in Fig. 4, except $\gamma_{0}=\gamma_{1}=1$ here.
The solid line shows the analytical prediction (41) for the plateau value, and
the theoretical value of $t^{\ast}=x_{0}/v_{0}=8$ matches with large $g$ data.
The inset shows the distribution $P(x,t|x_{0},0)$ for $g=1$ at different time
slices as mentioned in the legend.

###  IV.2 Plateau or peak at intermediate times

As shown in the inset of Fig. 4, for small $x_{0}$, the MSD increases
monotonically towards the stationary state - at early times, it increases
algebraically, as already explained in Sec. IV.1 and then it approaches the
steady state value exponentially fast, as described later in Sec. IV.3.
However, for large $x_{0}$, we observe an intermediate time regime, where
after the initial super-diffusive growth and before the late time relaxation,
the MSD displays a plateau for $\gamma_{0}=\gamma_{1}$ or a peak for
$\gamma_{0}>\gamma_{1}$.

Figure 6: Variable speeds, variable tumbling rates: The main figure shows the
mean-squared displacement (MSD) $\sigma^{2}(x_{0},t)$ for different $u$ values
as shown in the legend; the other parameters are $v_{0}=1$, $v_{1}=1$,
$\gamma_{0}=2$, $\gamma_{1}=1$, $g=4$, $x_{0}=8$ and $\Delta=0$. The inset
shows the plateau in MSD for $\gamma_{0}=\gamma_{1}=1$, while all the other
parameters remain the same as in the main plot. The solid line in the inset
corresponds to the predicted level of plateau from (40) which shows reasonable
agreement for large $u$.

For $\gamma_{0}=\gamma_{1}$, as shown in Fig. 5 for $u=0$ and various values
of $g$ and the inset of Fig. 6 for nonzero $g$ and different values of $u$,
the MSD reaches a plateau over an intermediate time range. The tumbling rate
$\gamma_{-}(x)$ of left-mover vanishes at large distances from the origin
which means that far from the origin, once the particle turns leftward, it
does not tumble anymore and only moves ballistically with a finite speed
towards left. For such deterministic motion, the MSD which is a measure of
fluctuations, does not increase with time in the intermediate time regime. To
see this effect more explicitly, in the inset of Fig. 5, we show the
conditional probability distribution $P(x,t|x_{0},0)$ obtained from
simulations as a function of $x$ for some time slices. We find that
$P(x,t|x_{0},0)$ varies exponentially with $x$ and shows two delta-function
peaks at the two edges, which correspond to the persistent trajectories where
the particle did not tumble even once. At short times, the peak at the left
edge of the distribution does not diminish with time, which is consistent with
zero tumbling rate. But at large times, the peak height starts decreasing
again when the tumbling rate becomes nonzero.

The behavior described above can be quantitatively captured by the homogeneous
rates (35) and (36); from (B.9) and (B.10), it follows that for
$t\gg(2\gamma_{0})^{-1}$, the mean position and MSD of a particle moving with
these rates are given by

${\bar{x}}_{homo}(t)\approx
x_{0}+\frac{v_{0}(\Delta\gamma_{0}+\gamma_{1})}{2\gamma_{0}^{2}}-\frac{t(\gamma_{0}v_{1}+\gamma_{1}v_{0})}{\gamma_{0}},\;\;\;\;\;\;\mbox{$t\gg(2\gamma_{0})^{-1}$}$
(39)

and

$\sigma_{homo}^{2}(t)\approx\frac{(\gamma_{0}^{2}-\gamma_{1}^{2})v_{0}^{2}t}{\gamma_{0}^{3}}+\frac{(5\gamma_{1}^{2}-2\gamma_{0}^{2}-\Delta^{2}\gamma_{0}^{2}+2\Delta\gamma_{0}\gamma_{1})v_{0}^{2}}{4\gamma_{0}^{4}},\;\;\;\;\;\;\mbox{$t\gg(2\gamma_{0})^{-1}$}.$
(40)

Then it follows from (40) that for $\gamma_{0}=\gamma_{1}$, the MSD indeed
becomes time-independent and stays constant at a value

$\sigma_{homo}^{2}(t)\stackrel{{\scriptstyle\gamma_{0}=\gamma_{1}}}{{=}}\dfrac{(3-\Delta^{2}+2\Delta)v_{0}^{2}}{4\gamma_{0}^{2}}$
(41)

Note that although the homogeneous rates do not allow the system to reach a
steady state, the variance in the particle’s position becomes time-independent
for the reason described above. We compare (41) with our simulations in the
main plot of Fig. 4 and the inset of 6, and find a good agreement.

As a stationary state exists for the rates (3) and (4), we expect that there
is a time scale $t^{*}$ that marks the end of the plateau in the MSD and the
beginning of the relaxation towards the steady state value. To obtain an
insight into this time scale, we consider the limit of $u=0,g\to\infty$ which
means that the particle moves with a uniform velocity and the tumbling rates
vary in a step-function manner. For this case, using the exact Green’s
function given by (C.23) in Appendix C, we calculate the mean and variance in
the position of the particle as a function of time in Appendix D. We find that
the relaxation regime sets in over a time scale
$t^{*}=\max\left(\dfrac{x_{0}}{v_{0}},\dfrac{1}{2\gamma_{0}}\right)$ which may
be understood as follows: we first note that for $2\gamma_{0}t\gg 1$, the mean
position (D.7b) of the particle coincides with (39) when $v_{1}=0$ and
$\gamma_{0}=\gamma_{1}$ which means that the dynamics can be described by the
homogeneous rates model during the intermediate time regime. But the latter
model is not valid when the mean position ${\bar{x}}_{homo}(t)$ vanishes as
the inhomogeneous tumbling rates (23) change their behavior at the origin.
This time is given by $\dfrac{x_{0}}{v_{0}}+\dfrac{(1+\Delta)}{2\gamma_{0}}$
and is consistent with the above expression for $t^{*}$.

To reiterate, for $u=0,g\to\infty$, the time scale $t^{\ast}$ denotes the time
at which the mean position of the particle moving with homogeneous rates
becomes zero. For $t\ll t^{\ast}$, the dynamics do not distinguish between the
step-function tumbling rates and the homogeneous ones while for $t\gg
t^{\ast}$, the inhomogeneity in the tumbling rates is felt and the MSD leaves
the plateau to relax towards the steady state. Generalizing this argument to
arbitrary $u,g$, we expect that $t^{*}$ is determined by

${\bar{x}}_{homo}(t^{*})\approx\min(u^{-1},g^{-1})$ (42)

Our simulation data in Fig. 5 for $u=0$ and finite $g$ show that for $g\ll 1$,
in accordance with (42), the time scale $t^{*}$ is smaller compared to when
$g\to\infty$ but it has a rather weak $g$-dependence for $g\gtrsim 1$. For
inhomogeneous speed and tumbling rates, as shown in Fig. 6, we see a similar
pattern in that the time scale $t^{*}$ decreases with decreasing $u$, as
suggested by (42).

For $\gamma_{0}>\gamma_{1}$ where the tumbling rate never vanishes, instead of
plateauing, the MSD varies in a nonmonotonic fashion as shown in Fig. 4 for
$u=0$ and various different $g$ values, and in Fig. 6 for different $u$ values
and fixed $g$. As in the above discussion for $\gamma_{0}=\gamma_{1}$, the
homogeneous limit considered in Appendix B applies in this case also until a
time $t^{*}$, and (40) shows that the variance increases linearly with time.
To find how long this diffusive growth continues, we again consider the
$u=0,g\to\infty$ case for which our analytical calculations in Appendix E
indicate that the time $t^{\ast}$ where the MSD peaks is consistent with that
obtained from ${\bar{x}}_{homo}(t^{\ast})=0$. More explicitly, for $v_{1}=0$
in (39), we obtain

$t^{\ast}=\frac{x_{0}\gamma_{0}}{\gamma_{1}v_{0}}+\frac{(\Delta\gamma_{0}+\gamma_{1})}{2\gamma_{0}\gamma_{1}}.$
(43)

which is $\approx\frac{x_{0}\gamma_{0}}{\gamma_{1}v_{0}}$ for large $x_{0}$,
as shown in Appendix E. For general parameters, for the same reasons as for
$\gamma_{0}=\gamma_{1}$, (42) is expected to hold, and the simulation data in
Fig. 4 and Fig. 6 are roughly consistent with it.

The above discussion also gives an insight into how large the initial position
be to observe a plateau or peak in the MSD. The intermediate regime, if it
exists, starts when $t>(2\gamma_{0})^{-1}$ and lasts until time $t^{*}$ and
therefore, if $(2\gamma_{0})^{-1}>t^{*}$, the initial position is sufficiently
small and the MSD will increase monotonically towards the stationary state
otherwise a plateau or peak in the MSD dynamics will be observed. For example,
for $\gamma_{0}=\gamma_{1},u=0,g\to\infty$, when
$x_{0}<\dfrac{v_{0}}{2\gamma_{0}}$, the plateau in the MSD is absent while for
$x_{0}\gg\dfrac{v_{0}}{2\gamma_{0}}$, the MSD remains stationary during
$\dfrac{1}{2\gamma_{0}}\ll t\ll\dfrac{x_{0}}{v_{0}}$ and then relaxes towards
the true stationary state.

###  IV.3 Exponential relaxation at large times

For $t>t^{\ast}$, as shown in Fig. 4 for uniform velocity and Fig. 6 for
inhomogeneous velocity, the MSD relaxes towards its steady state value
$\sigma^{2}$ exponentially fast. For $u=0$, $g\to\infty$, in Appendices D and
E, we calculate the relaxation time scale for $\gamma_{0}=\gamma_{1}$ and
$\gamma_{0}\neq\gamma_{1}$, respectively, and find that

$\displaystyle|\sigma^{2}(x_{0},t)-\sigma^{2}|\propto$ $\displaystyle
t^{-1/2}e^{-(\gamma_{0}-\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}})t}~{}$ ,
$\gamma_{0}\neq\gamma_{1}$ (44a)
$\displaystyle|\sigma^{2}(x_{0},t)-\sigma^{2}|\propto$ $\displaystyle
e^{-2\gamma_{0}t}~{}$ , $\gamma_{0}=\gamma_{1}$. (44b)

In Fig. 4 that shows the MSD for $\gamma_{0}\neq\gamma_{1},u=0$ and various
$g$, we verify (44a) for the relaxation time scale when
$\gamma_{0}\neq\gamma_{1}$ and $g\to\infty$; interestingly, we find that at
long times, the data for finite $g>1$ is almost indistinguishable from the
$g\to\infty$ limit. For $g=1/2$, although the time scale over which the system
relaxes appears to be the same as in (44a), we expect the results to differ
significantly as $g\to 0$ since no stationary state exists for $u=g=0$.

For $u\neq 0$, we have not been able to calculate $\sigma^{2}(x_{0},t)$
analytically. But our simulation results shown in Fig. 6 indicate that the MSD
relaxes exponentially fast, and a quantitative comparison between $u\to\infty$
and $u\gtrsim 1$ reveals that the relaxation time scale does not depend
strongly on $u$.

##  V Summary and open questions

In this work, we have considered a run-and-tumble particle in one dimension
with spatially varying velocities and tumbling rates. Close to the origin, the
spatial variation of velocities and tumbling rates is parametrized by $u$ and
$g$, respectively, while at large distances from the origin, the spatial
variation can be neglected and the rates saturate. For non-negative $u$ and
$g$, we show that as long as at least one of these parameters is positive, the
system reaches a stationary state. Even when one of these parameters is
negative, a steady state can be reached, provided the other one has a large
enough positive value; a detailed discussion of the phase diagram of the
general model is beyond the scope of this article and will be discussed
elsewhere. Here, we mainly focused on non-negative $u$ and $g$ and calculated
the exact position distribution in the steady state which is found to be
unimodal or bimodal depending on the choice of the parameters.

The consequence of saturation in rates is seen most clearly in the dynamics of
the particle away from the steady state. Our numerical simulations in this
case show that in some parameter regimes, the MSD can pass through a plateau
or be a nonmonotonic function of time before it finally relaxes to the
stationary state; such unusual behavior is not known from earlier studies for
any run-tumble motion. It would be very interesting to look for these features
in other active systems and in experiments.

An insight into our numerical results mentioned above is obtained by our
analytical calculations for the case of homogeneous speed ($u=0$) and step-
function tumbling rates ($g\to\infty$). This limit has been previously studied
in [40] only for initial position at the origin with an equal density of
right- and left-movers. But here we have obtained an exact expression for the
Green’s function for an arbitrary initial position with arbitrary fraction of
right- and left-movers present there, and calculated some properties of the
MSD analytically. Although our calculations are restricted to the $g\to\infty$
case, we find that they quantitatively describe the simulation data even for
finite $g\gtrsim 1$. In the same vein, it would be useful to obtain analytical
results for the case of uniform tumbling rate ($g=0$) and step-function
velocities ($u\to\infty$), which also maps onto the problem of run-and-tumble
particle with homogeneous speed and tumbling rate in an external potential
$V(x)=|x|$ as our simulations suggest a weak dependence on $u$.

In this study as well as in much of the literature on active systems, the
relaxation dynamics have been investigated, but the stationary state dynamics
in general models, and especially those that exhibit a phase transition in the
steady state, remain to be explored. Finally, here we have considered the
dynamics of a single RTP but clearly, it remains an interesting problem to
investigate the system with many interacting particles [46, 47, 48, 49].

##  VI Acknowledgements

KJ thanks Manasvi Gautam and Lakshita Jindal for many useful discussions
during the early stages of this work.

##  Appendix A Stationary state distribution

For completeness, here we briefly review the derivation of the stationary
state distribution [32, 44]. In the stationary state, the left-hand side (LHS)
of (1) and (2) obeyed by the distributions $P_{\pm}(x,t)$ vanishes,

$\displaystyle-(v_{+}(x)P_{+}(x))^{\prime}+\gamma_{-}(x)P_{-}(x)-\gamma_{+}(x)P_{+}(x)$
$\displaystyle=$ $\displaystyle 0$ (A.1)
$\displaystyle(v_{-}(x)P_{-}(x))^{\prime}+\gamma_{+}(x)P_{+}(x)-\gamma_{-}(x)P_{-}(x)$
$\displaystyle=$ $\displaystyle 0$ (A.2)

As the total current $J(x)=J_{+}(x)+J_{-}(x)$ with $J_{\pm}(x)=\pm
v_{\pm}(x)P_{\pm}(x)$ is uniform in the stationary state and must vanish at
$x\to\pm\infty$, we also have

$J_{+}(x)+J_{-}(x)=v_{+}(x)P_{+}(x)-v_{-}(x)P_{-}(x)=0~{}~{}\forall~{}~{}x$
(A.3)

Using (A.3) in (A.1) and (A.2), we immediately obtain

$\displaystyle P_{+}(x)$ $\displaystyle=$
$\displaystyle\frac{v_{+}(0)P_{+}(0)}{v_{+}(x)}e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(A.4) $\displaystyle P_{-}(x)$ $\displaystyle=$
$\displaystyle\frac{v_{-}(0)P_{-}(0)}{v_{-}(x)}e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(A.5)

provided $v_{\pm}(x)\neq 0$ for $-\infty<x<\infty$ (for a discussion of
stationary state when the speeds vanish, see [43]). Except for
$u\to\infty,v_{0}=v_{1}$ where $v_{\pm}(x)=v_{0}[1\mp\textrm{sgn}(x)]$, as (3)
satisfies this condition, the total stationary state probability distribution,
$P(x)=P_{+}(x)+P_{-}(x)$ can be written as

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle\left[\frac{J_{+}(0)}{v_{+}(x)}-\frac{J_{-}(0)}{v_{-}(x)}\right]e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(A.6) $\displaystyle=$ $\displaystyle
J_{+}(0)\left[\frac{1}{v_{+}(x)}+\frac{1}{v_{-}(x)}\right]e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(A.7)

Due to particle conservation, $P(x)$ must be normalizable and therefore, the
stationary state exists if

$\displaystyle{\cal
N}=J_{+}(0)\int_{-\infty}^{\infty}dx\left[\frac{1}{v_{+}(x)}+\frac{1}{v_{-}(x)}\right]e^{\int_{0}^{x}\left(\frac{\gamma_{-}(y)}{v_{-}(y)}-\frac{\gamma_{+}(y)}{v_{+}(y)}\right)dy}$
(A.8)

is finite. We also note that for the rates given by (3) and (4),
$v_{\pm}(-x)=v_{\mp}(x),\gamma_{\pm}(-x)=\gamma_{\mp}(x)$ using which in (A.4)
and (A.5), it immediately follows that $P_{+}(-x)=P_{-}(x)$ and therefore,
$\int_{-\infty}^{\infty}dx^{\prime}P_{\pm}(x^{\prime})=1/2$.

For the special case where $u\to\infty,v_{0}=v_{1},g\geq 0$, as
$v_{+}(x)=0,x>0$, it follows from (A.3) and (A.1) that $P_{\mp}(x)=0,x>0$;
similarly, as $v_{-}(x)=0,x<0$, we obtain $P_{\pm}(x)=0,x<0$. Taken together,
these imply that $P(x)=\delta(x)$ in this case.

##  Appendix B Uniform asymmetric rates with asymmetric initial condition

To understand the dynamics of the mean-squared displacement, we first note
that for arbitrary rates, on taking the Laplace transform
${\tilde{P}}_{\pm}(x,s)=\int_{0}^{\infty}P_{\pm}(x,t)e^{-st}dt$ of (1) and (2)
on both sides, we have

$\displaystyle s{\tilde{P}}_{+}(x,s)-P_{+}(x,0)$ $\displaystyle=$
$\displaystyle-[v_{+}(x){\tilde{P}}_{+}(x,s)]^{\prime}+\gamma_{-}(x){\tilde{P}}_{-}(x,s)-\gamma_{+}(x){\tilde{P}}_{+}(x,s)$
(B.1) $\displaystyle s{\tilde{P}}_{-}(x,s)-P_{-}(x,0)$ $\displaystyle=$
$\displaystyle[v_{-}(x){\tilde{P}}_{-}(x,s)]^{\prime}+\gamma_{+}(x){\tilde{P}}_{+}(x,s)-\gamma_{-}(x){\tilde{P}}_{-}(x,s)$
(B.2)

where, $P_{\pm}(x,0)$ are the initial distributions given by (6).

As explained in Sec. IV, the short time dynamics can be captured if one
assumes that the rates are given by (35) and (36), $v_{\pm}(x)=v_{0}\mp
v_{1},\gamma_{\pm}(x)=\gamma_{0}\pm\gamma_{1}$ for all $x$. For general
homogeneous rates (that is,
$\gamma_{\pm}(x)=\gamma_{\pm},v_{\pm}(x)=v_{\pm}$), on eliminating
${\tilde{P}}_{-}(x,s)$ from (B.1) and substituting it in (B.2), we obtain a
second order, inhomogeneous differential equation for ${\tilde{P}}_{+}(x,s)$;
in a similar manner, one can find the equation obeyed by
${\tilde{P}}_{-}(x,s)$. Together these lead to an equation for
${\tilde{P}}_{homo}(x,s)={\tilde{P}}_{+}(x,s)+{\tilde{P}}_{-}(x,s)$ which is
given by

$\displaystyle{\tilde{P}}_{homo}^{\prime\prime}(x,s)+\frac{v_{-}(\gamma_{+}+s)-v_{+}(\gamma_{-}+s)}{v_{-}v_{+}}{\tilde{P}}^{\prime}_{homo}(x,s)-\frac{s(\gamma_{-}+\gamma_{+}+s)}{v_{-}v_{+}}{\tilde{P}}_{homo}(x,s)={\tilde{H}}_{homo}(x,s)$
(B.3)

where the inhomogeneous term,

${\tilde{H}}_{homo}(x,s;x_{0})=\frac{[v_{-}P^{\prime}_{+}(x,0)-v_{+}P^{\prime}_{-}(x,0)]-[s+\gamma_{-}+\gamma_{+}]P(x,0)}{v_{-}v_{+}}$
(B.4)

and we have added the subscript in the above expressions to emphasize that
these hold for the homogeneous rates (35) and (36).

To solve the inhomogeneous equation (B.3), we first find the Green’s function
for which $L_{x,s}{\tilde{G}}_{homo}(x,s;a)=\delta(x-a)$ where $L_{x,s}$ is
the differential operator on the LHS of (B.3). If the homogeneous equation
$L_{x,s}{\tilde{G}}_{homo}=0$ has two linearly independent solutions denoted
by $y_{1}(x,s)$ and $y_{2}(x,s)$ where $y_{1}(x,s)\to 0$ as $x\to-\infty$ and
$y_{2}(x,s)\to 0$ as $x\to+\infty$, we obtain

$\displaystyle{\tilde{G}}_{homo}(x,s;a)=$ $\displaystyle
c_{1}(a,s)y_{1}(x,s)~{}$ , $x<a$ (B.5a)
$\displaystyle{\tilde{G}}_{homo}(x,s;a)=$ $\displaystyle
c_{2}(a,s)y_{2}(x,s)~{}$ , $x>a$ (B.5b)

where the constants $c_{1},c_{2}$ are obtained by demanding the continuity of
solutions at $x=a$ and a finite discontinuity in the first derivative of the
Green’s function at $x=a$. We thus obtain
$c_{1}(a,s)=\frac{y_{2}(a,s)}{W(a,s)},c_{2}(a,s)=\frac{y_{1}(a,s)}{W(a,s)}$
where $y_{1}=e^{\alpha_{+}x},y_{2}=e^{\alpha_{-}x}$ and the Wronskian,
$W(a,s)=(\alpha_{-}-\alpha_{+})e^{(\alpha_{-}+\alpha_{+})x_{0}}$, and
$\alpha_{+}>0$ and $\alpha_{-}<0$ are the solutions of the following quadratic
equation,

$\alpha^{2}+\frac{v_{-}(\gamma_{+}+s)-v_{+}(\gamma_{-}+s)}{v_{-}v_{+}}\alpha-\frac{s(\gamma_{-}+\gamma_{+}+s)}{v_{-}v_{+}}=0$
(B.6)

For later reference, we note that for
$v_{\pm}=v_{0},\gamma_{\pm}=\gamma_{0}\pm\gamma_{1}$, the inverse Laplace
transform of ${\tilde{G}}_{homo}(x,s;a)$ is given by

$G_{homo}(x,t;a)=-\frac{v_{0}e^{\frac{-\gamma_{1}(x-a)}{v_{0}}}}{2}e^{-\gamma_{0}t}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{x-a}{v_{0}}\right)^{2}}\right]\Theta(v_{0}t-|x-a|)$
(B.7)

where $I_{n}(z)$ is the modified Bessel function of the first kind.

For general homogeneous rates, using (6) in (B.4), and (B.5b), we find that

$\displaystyle{\tilde{P}}_{homo}(x,s)=\int_{-\infty}^{\infty}da{\tilde{G}}_{homo}(x,s;a){\tilde{H}}_{homo}(x,s;a)=$
$\displaystyle\frac{s+(\gamma_{-}+\gamma_{+})+(v_{+}a_{-}-v_{-}a_{+})\alpha_{+}}{v_{-}v_{+}(\alpha_{+}-\alpha_{-})}\frac{y_{1}(x,s)}{y_{1}(x_{0},s)},$
$x<x_{0}$ (B.8a)
$\displaystyle{\tilde{P}}_{homo}(x,s)=\int_{-\infty}^{\infty}da{\tilde{G}}_{homo}(x,s;a){\tilde{H}}_{homo}(x,s;a)=$
$\displaystyle\frac{s+(\gamma_{-}+\gamma_{+})+(v_{+}a_{-}-v_{-}a_{+})\alpha_{-}}{v_{-}v_{+}(\alpha_{+}-\alpha_{-})}\frac{y_{2}(x,s)}{y_{2}(x_{0},s)},$
$x>x_{0}$ (B.8b)

For the rates (35) and (36), after inverting the Laplace transform, we obtain
the following exact expressions for the mean and variance in the position of
the particle:

$\displaystyle{\bar{x}}_{homo}(t)$ $\displaystyle=$ $\displaystyle
x_{0}-\frac{t(\gamma_{0}v_{1}+\gamma_{1}v_{0})}{\gamma_{0}}+\frac{v_{0}(\Delta\gamma_{0}+\gamma_{1})}{2\gamma_{0}^{2}}-\frac{v_{0}e^{-2\gamma_{0}t}(\Delta\gamma_{0}+\gamma_{1})}{2\gamma_{0}^{2}}$
(B.9) $\displaystyle\frac{\sigma_{homo}^{2}(t)}{v_{0}^{2}}$ $\displaystyle=$
$\displaystyle\frac{(\gamma_{0}^{2}-\gamma_{1}^{2})t}{\gamma_{0}^{3}}+\frac{5\gamma_{1}^{2}-2\gamma_{0}^{2}-\Delta^{2}\gamma_{0}^{2}+2\Delta\gamma_{0}\gamma_{1}}{4\gamma_{0}^{4}}$
(B.10) $\displaystyle+$ $\displaystyle
e^{-2\gamma_{0}t}\left[\frac{\gamma_{0}\left(\left(\Delta^{2}+1\right)\gamma_{0}-4\gamma_{1}t(\Delta\gamma_{0}+\gamma_{1})\right)-2\gamma_{1}^{2}}{2\gamma_{0}^{4}}\right]$
$\displaystyle-$ $\displaystyle
e^{-4\gamma_{0}t}\left[\frac{(\Delta\gamma_{0}+\gamma_{1})^{2}}{4\gamma_{0}^{4}}\right]$

where $\Delta=a_{+}-a_{-}$. Note that the above expression for variance is
independent of $v_{1}$.

##  Appendix C Uniform speed, step-function tumbling rates: exact Green’s
function

Here we calculate the exact Green’s function when $u=0,g\to\infty$ in (3) and
(4) so that
$v_{\pm}(x)=v_{0},\gamma_{\pm}(x)=\gamma_{0}+\gamma_{1}\textrm{sgn}(x)=\gamma_{0}+\gamma_{1}[2\Theta(x)-1]$.
The Laplace transform of $P_{\pm}(x,t)$ obey (B.1) and (B.2), and for uniform
speed but arbitrary, inhomogeneous tumbling rates, it is convenient to work
with ${\tilde{Q}}(x,s)={\tilde{P}}_{+}(x,s)-{\tilde{P}}_{-}(x,s)$ and
${\tilde{P}}(x,s)={\tilde{P}}_{+}(x,s)+{\tilde{P}}_{-}(x,s)$. From (B.1) and
(B.2), we then obtain

$\displaystyle s{\tilde{P}}(x,s)-P(x,0)$ $\displaystyle=$ $\displaystyle-
v_{0}{\tilde{Q}}^{\prime}(x,s)$ (C.1) $\displaystyle s{\tilde{Q}}(x,s)-Q(x,0)$
$\displaystyle=$ $\displaystyle-
v_{0}{\tilde{P}}^{\prime}(x,s)-2\gamma_{0}{\tilde{Q}}(x,s)-2\gamma_{1}(2\Theta(x)-1){\tilde{P}}(x,s)$
(C.2)

Eliminating ${\tilde{Q}}(x,s)$ from (C.2) and plugging it in (C.1), we find
that ${\tilde{P}}(x,s)$ obeys a second order ordinary differential equation,

$\displaystyle{\tilde{P}}^{{}^{\prime\prime}}(x,s)+\frac{2}{v_{0}}[2\gamma_{1}\delta(x){\tilde{P}}(x,s)+\gamma_{1}(2\Theta(x)-1){\tilde{P}}^{{}^{\prime}}(x,s)]-\frac{s(2\gamma_{0}+s)}{v_{0}^{2}}{\tilde{P}}(x,s)$
$\displaystyle=$ $\displaystyle{\tilde{H}}(x,s;x_{0})$ (C.3)

where,

${\tilde{H}}(x,s;x_{0})=\frac{v_{0}Q^{\prime}(x,0)-(2\gamma_{0}+s)P(x,0)}{v_{0}^{2}}=\frac{\Delta}{v_{0}}\delta^{\prime}(x-x_{0})-\frac{2\gamma_{0}+s}{v_{0}^{2}}\delta(x-x_{0})$
(C.4)

on using (6). Throughout the article, we assume that $x_{0}\geq 0$ but from
(C.3) and (C.4), we also note that

${\tilde{P}}(x,s;x_{0},\Delta)={\tilde{P}}(-x,s;-x_{0},-\Delta)$ (C.5)

Note that the coefficients of (C.3) are not smooth functions of $x$. Using the
Green’s function ${\tilde{G}}(x,s;a)$ that obeys (C.8) below, we then have

$\displaystyle\tilde{P}(x,s)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{\tilde{G}}(x,s;a){\tilde{H}}(x,s;a)da$
(C.6) $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\left[\frac{\Delta}{v_{0}}\frac{d{\tilde{G}}(x,s;a)}{da}+\frac{2\gamma_{0}+s}{v_{0}^{2}}{\tilde{G}}(x,s;a)\right]\delta(a-x_{0})da$
(C.7)

where we have carried out an integration by parts to arrive at the last
expression.

For the inhomogeneous equation (C.3), the Green’s function obeys

$\displaystyle{\tilde{G}}^{{}^{\prime\prime}}(x,s)+\frac{2}{v_{0}}[2\gamma_{1}\delta(x){\tilde{G}}(x,s)+\gamma_{1}(2\Theta(x)-1){\tilde{G}}^{{}^{\prime}}(x,s)]-\frac{s(2\gamma_{0}+s)}{v_{0}^{2}}{\tilde{G}}(x,s)$
$\displaystyle=$ $\displaystyle\delta(x-a)$ (C.8)

where we have suppressed the $a$-dependence in ${\tilde{G}}(x,s;a)$ for
brevity. Assuming continuity of the homogeneous solutions of (C.8) at $x=a$
and on integrating (C.8) in the neighborhood of $a$, we get the usual
condition for the discontinuity in its first derivative,

$\text{Lim}_{\epsilon\to
0}{\tilde{G}}^{{}^{\prime}}(a+\epsilon,s)-{\tilde{G}}^{{}^{\prime}}(a-\epsilon,s)=1,a\neq
0$ (C.9)

provided $a$ is nonzero. To deal with the $\delta$-function on the LHS of
(C.8), we integrate it over $x$ on both sides in the neighborhood of origin
and obtain

$\displaystyle\text{Lim}_{\epsilon\to
0}{\tilde{G}}^{{}^{\prime}}(\epsilon,s)-{\tilde{G}}^{{}^{\prime}}(-\epsilon,s)+\frac{4\gamma_{1}}{v_{0}}{\tilde{G}}(0,s)=\delta_{a,0}$
(C.10)

where we have assumed that the Green’s function is continuous but not
differentiable at the origin. For $x\neq 0$, the solution of the homogeneous
equation (C.8) is given by

$\displaystyle{\tilde{G}}(x,s)=$ $\displaystyle
a_{1}e^{\alpha_{+}x}+a_{2}e^{\alpha_{-}x}~{}$ , $x>0$ (C.11a)
$\displaystyle{\tilde{G}}(x,s)=$ $\displaystyle
b_{1}e^{-\alpha_{-}x}+b_{2}e^{-\alpha_{+}x}~{}$ , $x<0$ (C.11b)

where $\alpha_{+}>0,\alpha_{-}<0$ are solutions of
$\alpha^{2}+\frac{2\gamma_{1}}{v_{0}}\alpha-\frac{s(2\gamma_{0}+s)}{v_{0}^{2}}=0$.
For $x_{0}>0$ (as assumed in this article), as (C.7) is nonzero for $a>0$, we
assume this condition for the discussion below. For $a>0$, on using that the
solution ${\tilde{P}}(x,s)$ and therefore, ${\tilde{G}}(x,s)$ must vanish at
$x\to\pm\infty$, we get

$\displaystyle{\tilde{G}}(x,s)=$ $\displaystyle a_{2}e^{\alpha_{-}x}~{}$ ,
$x>a$ (C.12a) $\displaystyle{\tilde{G}}(x,s)=$ $\displaystyle
c_{1}e^{\alpha_{+}x}+c_{2}e^{\alpha_{-}x}~{}$ , $0<x<a$ (C.12b)
$\displaystyle{\tilde{G}}(x,s)=$ $\displaystyle b_{1}e^{-\alpha_{-}x}~{}$ ,
$x<0,x<a$ (C.12c)

Using continuity of ${\tilde{G}}(x,s)$ at $x=0$ and $x=a$, and (C.9) and
(C.10), all four unknowns in the above equations can be determined, and are
given by

$\displaystyle\alpha_{\pm}$ $\displaystyle=$
$\displaystyle\frac{-\gamma_{1}\pm\sqrt{(\gamma_{0}^{2}-\gamma_{1}^{2})(w+2)w}}{v_{0}}$
(C.13) $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{e^{-a\alpha_{-}}}{\alpha_{-}-\alpha_{+}}+\frac{\gamma_{1}e^{-a\alpha_{+}}}{v_{0}(\alpha_{-}-\alpha_{+})\alpha_{+}}$
(C.14) $\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{e^{-a\alpha_{+}}}{\alpha_{-}-\alpha_{+}}$ (C.15)
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{1}e^{-a\alpha_{+}}}{v_{0}(\alpha_{-}-\alpha_{+})\alpha_{+}}$
(C.16) $\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-\frac{e^{-a\alpha_{+}}}{2\alpha_{+}}$ (C.17)

where
$w=\frac{s+\gamma_{0}-\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}}{\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}}$
as in [40].

To find the Green’s function in real space, we need to invert the Laplace
transforms. Using (29.2.14) and (29.3.91) of [50], we immediately obtain

$\displaystyle
L^{-1}[c_{1}e^{\alpha_{+}x}]{=}-\frac{v_{0}}{2}e^{\frac{\gamma_{1}}{v_{0}}(a-x)}e^{-\gamma_{0}t}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a-x}{v_{0}}\right)^{2}}\right]\Theta\left[t-\frac{a-x}{v_{0}}\right]$
(C.18)

where $L^{-1}f(s)$ denotes the inverse Laplace transform of $f(s)$ and
$I_{n}(z)$ is the modified Bessel function of first kind. Using the derivative
of (29.3.91) of [50] with respect to $k$, we first note that

$\displaystyle
L^{-1}[\frac{d}{da}(b_{1}e^{-\alpha_{-}x})]{=}\frac{e^{\frac{\gamma_{1}(x+a)}{v_{0}}}}{2}e^{-\gamma_{0}t}\delta\left(t-\frac{a-x}{v_{0}}\right)+\frac{\frac{a-x}{v_{0}}\Theta(t-\frac{a-x}{v_{0}})I_{1}\left(\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-(\frac{a-x}{v_{0}})^{2}}\right)}{\sqrt{t^{2}-(\frac{a-x}{v_{0}})^{2}}}$
(C.19)

We fix the constant of integration arising due to the integral over $a$ by
demanding that $b_{1}e^{-\alpha_{-}x}$ and its inverse Laplace transform
vanish for $a\to\infty$, and obtain

$\displaystyle L^{-1}[b_{1}e^{-\alpha_{-}x}]$ $\displaystyle{=}$
$\displaystyle-\Theta(x+v_{0}t-a)\frac{v_{0}e^{-\gamma_{0}t}}{2}e^{\frac{\gamma_{1}(x+a)}{v_{0}}}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a-x}{v_{0}}\right)^{2}}\right]$
(C.20) $\displaystyle-$
$\displaystyle\Theta(x+v_{0}t-a)\frac{v_{0}e^{-\gamma_{0}t}}{2}\int_{a}^{x+v_{0}t}da^{\prime}\frac{\gamma_{1}}{v_{0}}e^{\frac{\gamma_{1}(x+a^{\prime})}{v_{0}}}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a^{\prime}-x}{v_{0}}\right)^{2}}\right]$

after an integration by parts. Proceeding in a fashion similar to above, we
also obtain

$\displaystyle L^{-1}[a_{2}e^{\alpha_{-}x}]$ $\displaystyle=$
$\displaystyle-\frac{v_{0}}{2}e^{\frac{\gamma_{1}}{v_{0}}(a-x)}e^{-\gamma_{0}t}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a-x}{v_{0}}\right)^{2}}\right]\Theta(v_{0}t-x+a)$
(C.21) $\displaystyle-$
$\displaystyle\Theta(v_{0}t-x-a)\int_{a}^{v_{0}t-x}da^{\prime}\frac{\gamma_{1}e^{-\gamma_{0}t}e^{\frac{\gamma_{1}(a^{\prime}-x)}{v_{0}}}}{2}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a^{\prime}+x}{v_{0}}\right)^{2}}\right]$
$\displaystyle L^{-1}[c_{2}e^{\alpha_{-}x}]$ $\displaystyle=$
$\displaystyle-\Theta(v_{0}t-x-a)\int_{a}^{v_{0}t-x}da^{\prime}\frac{\gamma_{1}e^{-\gamma_{0}t}e^{\frac{\gamma_{1}(a^{\prime}-x)}{v_{0}}}}{2}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a^{\prime}+x}{v_{0}}\right)^{2}}\right]$
(C.22)

Using (C.18)-(C.22) in (C.12c), we finally arrive at the following exact
expression for the Green’s function for $a>0$ and arbitrary $x$:

$\displaystyle
G(x,t;a){=}-\frac{v_{0}}{2}e^{\frac{\gamma_{1}}{v_{0}}(a-|x|)}e^{-\gamma_{0}t}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a-x}{v_{0}}\right)^{2}}\right]\Theta(v_{0}t-|x-a|)$
(C.23) $\displaystyle-$
$\displaystyle\Theta(v_{0}t-|x|-a)\int_{a}^{v_{0}t-|x|}da^{\prime}\frac{\gamma_{1}e^{-\gamma_{0}t}e^{\frac{\gamma_{1}(a^{\prime}-|x|)}{v_{0}}}}{2}I_{0}\left[\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{t^{2}-\left(\frac{a^{\prime}+|x|}{v_{0}}\right)^{2}}\right]$

Note that in the above expression, the first term on the RHS clicks at all
times but the second term is nonzero only when $t>\frac{x_{0}}{v_{0}}$. As we
will see in Appendix D and Appendix E, the first term in the above Green’s
function captures the dynamics when the particle experiences homogeneous rates
and its mean position is $>0$ (compare the first term above for positive $x$
with (B.7)), while both terms control the long-time relaxation dynamics where
the first term vanishes and second term approaches the stationary state as
$t\to\infty$. Taking the inverse Laplace transform of (C.7) on both sides, we
find that the exact probability distribution $P(x,t;x_{0})$ is given by

$\displaystyle P(x,t;x_{0})=-\frac{\Delta}{v_{0}}\frac{\partial
G(x,t;x_{0})}{\partial
x_{0}}-\frac{2\gamma_{0}}{v_{0}^{2}}G(x,t;x_{0})-\frac{1}{v_{0}^{2}}\frac{\partial
G(x,t;x_{0})}{\partial t}$ (C.24)

where we have used (29.2.4) of [50] and that $G(x,t;x_{0})=0$ due to (C.23).
In the stationary state where the probability distribution is independent of
$t$ and $x_{0}$, on using (24) in (C.24), we obtain the stationary state
Green’s function to be

$G(x)=-\frac{v_{0}e^{-\frac{2\gamma_{1}|x|}{v_{0}}}}{2}$ (C.25)

In Appendix D and Appendix E, it is verified that (C.23) indeed reduces to the
above equation in the steady state. We have also checked that (C.23) and
(C.24) reproduce the result for $x_{0}=0$ obtained in [40].

##  Appendix D Uniform speed, step-function tumbling rates:
$\gamma_{0}=\gamma_{1}$

For $\gamma_{0}=\gamma_{1}$, the Green’s function (C.23) simplifies to

$\displaystyle G(x,t;a)$ $\displaystyle=$
$\displaystyle-\frac{v_{0}}{2}e^{\frac{\gamma_{0}}{v_{0}}(a-|x|)}e^{-\gamma_{0}t}[\Theta(v_{0}t-|x-a|)-\Theta(v_{0}t-|x|-a)]$
(D.1) $\displaystyle+$ $\displaystyle\Theta(v_{0}t-|x|-a)G(x)$

which approaches the stationary state Green’s function $G(x)$ given by (C.25)
when $t\to\infty$.

Using the above expression in (C.24), the exact position probability
distribution and its cumulants can be obtained; in particular, the mean and
variance of the position of the particle are given by

$\displaystyle{\bar{x}}(t)=$ $\displaystyle
x_{0}-v_{0}t+\frac{(1+\Delta)v_{0}(1-e^{-2\gamma_{0}t})}{2\gamma_{0}}~{}$ ,
$t<\frac{x_{0}}{v_{0}}$ (D.2a) $\displaystyle{\bar{x}}(t)=$ $\displaystyle
r_{0}(t)e^{-2\gamma_{0}t}~{}$ , $t>\frac{x_{0}}{v_{0}}$ (D.2b)

and

$\displaystyle\frac{\sigma^{2}(x_{0},t)}{v_{0}^{2}}=$
$\displaystyle\frac{(3-\Delta)(1+\Delta)}{4\gamma_{0}^{2}}+\frac{(\Delta+1)(\Delta-1-4\gamma_{0}t)e^{-2\gamma_{0}t}}{2\gamma_{0}^{2}}-\frac{(\Delta+1)^{2}e^{-4\gamma_{0}t}}{4\gamma_{0}^{2}}~{}$
, $t<\frac{x_{0}}{v_{0}}$ (D.3a)
$\displaystyle\frac{\sigma^{2}(x_{0},t)}{v_{0}^{2}}=$
$\displaystyle\frac{1}{2\gamma_{0}^{2}}+r_{1}(t)e^{-2\gamma_{0}t}-r_{2}(t)e^{-4\gamma_{0}t}~{}$
, $t>\frac{x_{0}}{v_{0}}$ (D.3b)

where,

$\displaystyle r_{0}(t)$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{2\gamma_{0}x_{0}}{v_{0}}}(v_{0}(2\Delta\gamma_{0}t+\Delta+1)-2\Delta\gamma_{0}x_{0})-(\Delta+1)v_{0}}{2\gamma_{0}}$
(D.4) $\displaystyle r_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{e^{\frac{2\gamma_{0}x_{0}}{v_{0}}}(2\Delta
v_{0}(\gamma_{0}t+1)-4\Delta\gamma_{0}x_{0}+v_{0})-2(\Delta+1)(\gamma_{0}tv_{0}+\gamma_{0}x_{0}+v_{0})}{2\gamma_{0}^{2}v_{0}}$
(D.5) $\displaystyle r_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{\left(e^{\frac{2\gamma_{0}x_{0}}{v_{0}}}(2\Delta\gamma_{0}x_{0}-v_{0}(2\Delta\gamma_{0}t+\Delta+1))+(\Delta+1)v_{0}\right)^{2}}{4\gamma_{0}^{2}v_{0}^{2}}$
(D.6)

On setting $v_{1}=0,\gamma_{0}=\gamma_{1}$ in (B.9) and (B.10), we find that
they match (D.2b) and (D.3b), respectively, when $t<\frac{x_{0}}{v_{0}}$ which
shows that the short time dynamics are exactly described by those of the RTP
with homogeneous rates.

The above expressions show that the dynamics depend on whether time is above
or below $\frac{x_{0}}{v_{0}}$ and involve terms that decay exponentially fast
with the slowest time scale being $(2\gamma_{0})^{-1}$. Below we give
approximate results for the mean position and mean-squared displacement
depending on these time scales. For $x_{0}>\frac{v_{0}}{2\gamma_{0}}$, we find
that

$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle x_{0}+\Delta v_{0}t~{}$ ,
$t\ll\frac{1}{2\gamma_{0}}<\frac{x_{0}}{v_{0}}$ (D.7a)
$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle
x_{0}+\frac{(1+\Delta)v_{0}}{2\gamma_{0}}-tv_{0}~{}$ ,
$\frac{1}{2\gamma_{0}}\ll t<\frac{x_{0}}{v_{0}}$ (D.7b)
$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle
r_{0}(t)e^{-2\gamma_{0}t}~{}$ , $t>\frac{x_{0}}{v_{0}}$ (D.7c)

and

$\displaystyle\sigma^{2}(x_{0},t)\approx$
$\displaystyle(1-\Delta^{2})v_{0}^{2}t^{2}~{}$ ,
$t\ll\frac{1}{2\gamma_{0}}<\frac{x_{0}}{v_{0}}$ (D.8a)
$\displaystyle\sigma^{2}(x_{0},t)\approx$
$\displaystyle\frac{(3-\Delta)(\Delta+1)v_{0}^{2}}{4\gamma_{0}^{2}}~{}$ ,
$\frac{1}{2\gamma_{0}}\ll t<\frac{x_{0}}{v_{0}}$ (D.8b)
$\displaystyle\sigma^{2}(x_{0},t)\approx$
$\displaystyle\sigma^{2}+r_{1}(t)e^{-2\gamma_{0}t}~{}$ ,
$t>\frac{x_{0}}{v_{0}}$ (D.8c)

where $\sigma^{2}=\frac{v_{0}^{2}}{2\gamma_{0}^{2}}$ is the stationary state
variance as shown in Sec. III.2.1. The above expressions thus show that the
variance initially increases superdiffusively and becomes a constant at
intermediate times, but for time $t>t^{*}=\frac{x_{0}}{v_{0}}$, these dynamics
are followed by an exponential relaxation at rate $2\gamma_{0}$ towards the
stationary state. The mean displacement, however, decreases monotonically
towards zero - linearly at short times and exponentially fast at longer times.
In contrast, for $x_{0}<\frac{v_{0}}{2\gamma_{0}}$, both mean and variance
approach the steady state in a monotonic fashion,

$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle x_{0}+\Delta v_{0}t~{}$ ,
$t\ll\frac{1}{2\gamma_{0}}<\frac{x_{0}}{v_{0}}$ (D.9a)
$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle v_{0}~{}$ ,
$\frac{1}{2\gamma_{0}}\ll t<\frac{x_{0}}{v_{0}}$ (D.9b)
$\displaystyle{\bar{x}}(t)\approx$ $\displaystyle
r_{0}(t)e^{-2\gamma_{0}t}~{}$ , $t>\frac{x_{0}}{v_{0}}$ (D.9c)

and

$\displaystyle\sigma^{2}(x_{0},t)\approx$
$\displaystyle(1-\Delta^{2})v_{0}^{2}t^{2}~{}$ ,
$t\ll\frac{x_{0}}{v_{0}}<\frac{1}{2\gamma_{0}}$ (D.10a)
$\displaystyle\sigma^{2}(x_{0},t)\approx$
$\displaystyle\sigma^{2}+r_{1}(t)e^{-2\gamma_{0}t}~{}$ ,
$t>\frac{1}{2\gamma_{0}}$ (D.10b)

Thus, in either case, the stationary state is approached over a time scale
$(2\gamma_{0})^{-1}$ but the relaxation regime starts at
$\max(\frac{x_{0}}{v_{0}},\frac{1}{2\gamma_{0}})$. To understand the meaning
of $t^{*}=\frac{x_{0}}{v_{0}}$, from (D.7b), we note that for
$x_{0}\gg\frac{v_{0}}{2\gamma_{0}}$, the mean position ${\bar{x}}(t)\approx
x_{0}-tv_{0}$ is at the origin at time $t^{*}$ and thus the particle is
sensitive to the inhomogeneous tumbling rate beyond this time scale. This
discussion also shows that if this time is much smaller than the typical time
for tumbling to occur, that is, $t^{*}\ll(2\gamma_{0})^{-1}$, the variance
will show monotonic dynamics.

We also mention that in the special case when initially there is only the
left-mover ($\Delta=-1$), (D.8c) and (D.10b) show that the variance remains
zero (as expected) until $\max(\frac{x_{0}}{v_{0}},\frac{1}{2\gamma_{0}})$ and
then increases monotonically towards $\sigma^{2}$.

##  Appendix E Uniform speed, step-function tumbling rates:
$\gamma_{0}\neq\gamma_{1}$

For $\gamma_{0}\neq\gamma_{1}$, although using the exact Green’s function
(C.23) in (C.24), the exact probability distribution can be obtained, it does
not seem possible to obtain the cumulants of mean displacement exactly, and
therefore, below we develop some approximations for them.

At large times, using the asymptotic expansion of
$I_{0}(z)\stackrel{{\scriptstyle z\to\infty}}{{\sim}}\frac{e^{z}}{\sqrt{2\pi
z}}$ in (C.23), we obtain

$\displaystyle G(x,t;a)$ $\displaystyle\approx$
$\displaystyle-\frac{v_{0}}{2}e^{\frac{\gamma_{1}}{v_{0}}(a-|x|)}e^{-\gamma_{0}t}\frac{e^{t\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{1-(\frac{a-x}{v_{0}t})^{2}}}}{\sqrt{2\pi
t\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{1-(\frac{a-x}{v_{0}t})^{2}}}}\Theta(v_{0}t-|x-a|)$
(E.1) $\displaystyle-$
$\displaystyle\Theta(v_{0}t-|x|-a)v_{0}t\int_{\frac{a+|x|}{v_{0}t}}^{1}dw\frac{\gamma_{1}e^{-\gamma_{0}t}e^{\gamma_{1}tw}e^{\frac{-2\gamma_{1}|x|}{v_{0}}}}{2}\frac{e^{t\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{1-w^{2}}}}{\sqrt{2\pi
t\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}\sqrt{1-w^{2}}}}$

In the above equation, as the second term on the RHS begins to contribute when
$t>\frac{x_{0}}{v_{0}}$, we focus on the first term at short times. We first
note that the first term is a nonmonotonic function of $x$ with a maximum at
$x_{*}(a)=a-\frac{\gamma_{1}v_{0}t}{\gamma_{0}}$. Since for sufficiently large
and positive $x_{0}$, the particle is likely to be at $x>0$ at short times, we
approximate the Green’s function $G(x,t;x_{0})$ for
$t<\frac{x_{0}\gamma_{0}}{v_{0}\gamma_{1}}$ by a Gaussian centred about
$x_{*}(x_{0})$, and find the mean and variance in the particle’s position
using (C.24). We obtain

$\displaystyle\bar{x}(t)$ $\displaystyle\approx$ $\displaystyle
x_{0}+\frac{v_{0}(\Delta\gamma_{0}-\gamma_{1})}{2\gamma_{0}^{2}}-\frac{t\gamma_{1}v_{0}}{\gamma_{0}}$
(E.2) $\displaystyle\sigma^{2}(x_{0},t)$ $\displaystyle\approx$
$\displaystyle\frac{(\gamma_{0}^{2}-\gamma_{1}^{2})v_{0}^{2}t}{\gamma_{0}^{3}}$
(E.3)

For the mean displacement, we observe that the sign of third term in (E.2) and
that in (39) for homogeneous rates and $v_{1}=0$ do not match, but this
appears to be a result of the Gaussian approximation as (B.7) also yields
(E.2) under this approximation; however, for the variance, (E.3) above and
(40) for $v_{1}=0$ match each other. These results are thus consistent with
our expectation that the particle with constant speed and step-function
tumbling rates behaves like the RTP with homogeneous rates until the mean
position of the particle becomes zero.

We now turn to a discussion of the late-time behavior to verify that the
stationary state is obtained and to find the relaxation time scale. For this
purpose, we need to consider the second term on the RHS of (E.1); however, the
integral in (E.1) does not seem to be exactly doable. To derive an
approximation, we write
$w=\sin\theta,\gamma_{1}=\gamma_{0}\sin\phi,\gamma=\gamma_{0}\cos\phi$ with
$0<\phi\leq\pi/2$. Then, for $\gamma_{0}t\gg 1$, the integrand can be
approximated by a Gaussian:

$\displaystyle\int_{\frac{a+|x|}{v_{0}t}}^{1}dwe^{-\gamma_{0}t}e^{\gamma_{1}tw}\frac{e^{t\sqrt{(\gamma_{0}^{2}-\gamma_{1}^{2})(1-w^{2})}}}{(1-w^{2})^{1/4}}$
$\displaystyle\approx$
$\displaystyle\int_{-\Phi}^{\infty}d\theta\sqrt{\cos(\theta+\phi)}e^{-\frac{\gamma_{0}t\theta^{2}}{2}}$
(E.4) $\displaystyle\approx$
$\displaystyle\sqrt{\frac{2\pi\cos\phi}{\gamma_{0}t}}-\frac{\sqrt{\cos(\phi-\Phi)}}{\gamma_{0}t\Phi}e^{-\frac{\gamma_{0}t\Phi^{2}}{2}}$
(E.5)

where, $\Phi\approx\phi-\frac{a+|x|}{v_{0}t}$. At large times where $v_{0}t\gg
a+|x|$, using the above approximation in (E.1), we arrive at an expression
similar to (D.1) for $\gamma_{0}=\gamma_{1}$,

$\displaystyle G(x,t;a)$ $\displaystyle\approx$
$\displaystyle-\frac{v_{0}}{2}e^{\frac{\gamma_{1}}{v_{0}}(a-|x|)}\frac{e^{-(\gamma_{0}-\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}})t}}{\sqrt{2\pi
t\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}}}}[\Theta(v_{0}t-|x-a|)-\Theta(v_{0}t-|x|-a)]$
(E.6) $\displaystyle+$ $\displaystyle G(x)\Theta(v_{0}t-|x|-a)$

where, $G(x)$ is the stationary state Green’s function (C.25). At larger
times, the above expression suggests that the stationary state is approached
over a time scale, $(\gamma_{0}-\sqrt{\gamma_{0}^{2}-\gamma_{1}^{2}})^{-1}$
(with a $\ln t$ correction due to $t^{-1/2}$ factor). Although the amplitude
of the mean and variance calculated from the above expression does not match
the simulations, the relaxation time scale is consistent with the numerical
results in Fig. 4 and that obtained in [40].

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