\section{Introduction}
\label{Introduction}
Self-propelled particles undergo active Brownian motion by
consuming energy, obtained either from internal or external
sources. Examples range from the transport of motor proteins
on cytoskeletal filaments \cite{Ross08} which is a biologically
relevant system, to the motion of self-motile colloidal
particles \cite{Howse07} as a nonliving realization. The
particles are powered by the hydrolysis of ATP in the former
case, whereas they use a chemical reaction catalyzed on their
surface to swim in the latter example. The directed propulsion
subject to fluctuations has been described by persistent
random walks \cite{Howse07,Peruani07,Shaebani14}, where a
tendency to move along the previous direction is taken into
account. A strong self-propulsion overcomes the stochastic
fluctuations and directs the motion, which renders it ballistic
for short time scales \cite{Howse07,Shaebani14}. Nevertheless,
the interplay between self-propulsion and random motion in
general may lead to various scenarios of anomalous diffusive
dynamics on varying time scales \cite{Shaebani14}. The influence
of self-propulsion diminishes over time and eventually a
crossover to an asymptotic diffusive regime occurs.

Even in the absence of self-propulsion, the stochastic
motion of particle may remain complicated because in
general a random walker can perform steps with arbitrary
turning angles and variable step lengths. Moreover, there
can be a relation between the step size and the turning angle
of each step. Generalized random walks had been studied
e.g.\ in the context of animal and cell movements as a
Markovian process \cite{OldRefs,Codling08}, i.e.\
by considering the motion as a series of independent draws
from the step-length and turning-angle distributions for
each step. While the focus of prior studies has been more
on the asymptotic diffusion coefficient of such random walks,
the short-time behavior is neither thoroughly investigated
nor completely understood.

In the persistent random walk model that we study here a particle
moves straight in continuous space in a randomly chosen direction
over a randomly chosen distance and then changes direction by a
randomly chosen turning angle. The typical trajectories of the
random walker depend strongly on the chosen turning angle and
distance distributions. For small values of the angular
change the new direction will be strongly correlated with the old
direction, introducing a directional memory into the model without
changing the Markov property of the process. The emerging
intermittent directional bias is controlled by the characteristics
(mean, width, asymmetry, etc.) of the turning angle distribution
and the probability with which the direction is unchanged, i.e.\
the processivity. The bias decays with time after a few turns
and the directions of the particle motion become asymptotically
randomized.

Memory effects have also been included in other random walk
models. In fractional Brownian motion \cite{FBM}, sub or
superdiffusive motion is observed asymptotically. In continuous
time random walk (CTRW) models \cite{CTRW}, for example, true
non-Markovian effects can be implemented via broad waiting time
distributions which lead to a subdiffusive dynamics of the
walker. Although these approaches are conceptually very exciting,
a direct comparison to experimental results is sometimes
difficult. In case of intracellular transport, for example,
spatial confinement as well as a finite observation time imply
that the asymptotic behavior of the walker is not accessible.

Here, we develop a general analytical framework to study
persistent random walks over the whole range of time scales.
The goal is to clarify and disentangle the combined effects
of self-propulsion $p$ and the stepping strategy of the
walker, consisting of its step-length $\mathcal{F}(\ell)$ and
turning-angle $R(\phi)$ distributions. The method enables
us to analytically determine the time evolution of arbitrary
moments of displacement. Using this approach, the second
moment, i.e.\ the mean square displacement, has been
recently analyzed \cite{Shaebani14} revealing a variety
of signatures of anomalous diffusion on short time scales
even in the absence of viscoelasticity, traps, or overcrowding.
These elements were frequently identified in the nature
of the biological environments and received considerable
attention as the possible sources of subdiffusion \cite{Reviews}.

An alternative way to interpret the model and its outcome
is to consider the motion on complex networks, such as motor
proteins on the cytoskeleton. These motors have an effective
processivity $p$ (i.e.\ a tendency to move along the same
filament \cite{ProcesivityRefs1,Vershinin07,ProcesivityRefs2})
and may switch to a new filament at the intersections of
the network. From a coarse-grained perspective, one can
describe such motion by a persistent random walk on the nodes
of the network, thus, $\mathcal{F}(\ell)$ and $R(\phi)$
represent the distributions of the segment-length $\ell$
between neighboring intersections and the angle $\phi$
between intersecting filaments, respectively. Note that
the cytoskeleton is a dynamic network due to the underlying
growth and shrinkage of filaments, thus, the structure
on which the transport takes place often changes. This
justifies the relevance of the proposed stochastic approach,
where the network structure is always implicitly given.

Our analytical approach provides a recipe to obtain any
arbitrary moment of displacement, yet we extend the
calculations to the third and fourth moments, which
are of particular interest, e.g.\ in the evaluation
of the skewness and kurtosis, or to obtain the variance
$\sigma_{r^2}{=}\langle r^4 \rangle {-} \langle r^2
\rangle^2$ (and thus the standard error) of the mean
square displacement (MSD) of a persistent random walker.
$\sigma_{r^2}$ is also a useful quantity for estimating
the first moment $\langle r \rangle$ of the net
displacement: In the absence of an exact analytical
solution for $\langle r \rangle$, an approximate
expression $\langle r \rangle \approx \sqrt{\langle
r^2 \rangle} (1{-}\frac18 \frac{\sigma_{r^2}}{\langle
r^2 \rangle^2})$ was proposed \cite{McCulloch89} by means
of a Taylor expansion of the square root function. The
asymptotic behavior is however shown to follow $\langle
r \rangle = \frac12 \sqrt{\pi \langle r^2 \rangle}$
\cite{Bovet88}.

We first introduce the general approach to obtain arbitrary
moments of displacement. Next, an analytical expression is
derived for the special case of the mean square displacement
in two dimensions. This part is the full exposition and
expansion of the results presented in \cite{Shaebani14}.
Then we clarify the similarities and differences between
the results in two and three dimensions for persistent
walks which are symmetric around the previous direction
of motion, and briefly discuss the case of asymmetric
turning-angle distributions in two dimensions. Finally,
the calculations are extended to higher moments and
cumulants of displacement, and the probability density
of the position of the random walker and its time evolution
are investigated. The paper is organized in the following
manner: Section~\ref{Model-2D} contains the description
of the master equation formalism for the persistent
motion of random walkers in two dimensions. In
Sec.~\ref{Model-3D}, we discuss symmetric persistent
motions in three dimensions. The analytical predictions
for MSD are compared with simulation results in
Sec.~\ref{Results}. The calculations are extended to
higher moments and cumulants of displacement in
Sec.~\ref{HigherMoments}, and the results are compared
with simulations. We investigate the parameter dependence
and time evolution of the probability distribution of
the net distance from the origin in Sec.~\ref{Probabilities}.
Moreover, the coupling between longitudinal and
perpendicular transport is briefly discussed, and
an analytical expression is obtained for the
probability that the direction of motion after
$n$ steps makes an angle $\alpha$ with the current
direction of motion. Section \ref{Summary} concludes
the paper.

\section{Model}
\label{Model-2D}
We consider the motion of a self-propelled particle as a
persistent random walk consisting of steps of different
lengths and orientations. Here, a two-dimensional motion
is introduced and the extension to three dimensions is
discussed in the next section. The stochastic motion is
described in continuous space and discrete time as follows:
At each time step, the particle takes a step of length $l$
along either its previous direction with probability $p$
or a newly chosen direction with probability $s{=}1{-}p$,
as shown in Fig.~\ref{Fig1}. Thus, $p$ represents the
self-propulsion of the particle. We assume probability
distributions $R(\phi)$ and $\mathcal{F}(\ell)$ for
the rotation angle $\phi{=}\theta{-}\gamma$ and the
step length $l$ of the walker, respectively. The following
master equation expresses the evolution of the probability
density $P_{n}(x,y|\theta)$ for the particle to arrive
at position $(x,y)$ along the direction $\theta$ at
time step $n$:
\begin{equation}
\begin{aligned}
&P_{n+1}(x,y|\theta) = p \! \int \!\! d\ell \, \mathcal{F}
(\ell) \, P_{n}\big(x{-}\ell \text{cos}(\theta),y{-}\ell
\text{sin}(\theta) \big|\theta\big)\\
&+ s \!\! \int \!\! d\ell \, \mathcal{F}(\ell) \!
\int_{-\pi}^{\pi} \!\!\!\! d\gamma \,R(\theta{-}\gamma)
\, P_{n}\big(x{-}\ell \text{cos}(\theta),y{-}\ell
\text{sin}(\theta)\big|\gamma\big).
\end{aligned}
\label{Eq:MasterEquation}
\end{equation}
The terms on the right hand side of the above equation
correspond to persistent motion with probability $p$
and turning with probability $s$. One can obtain
arbitrary moments of displacement since they are
accessible by the derivatives of the Fourier transform
of $P_{n}(x,y|\theta)$, which is defined as
\begin{eqnarray}
P_{n}(\bm\omega|m) \equiv \! \int_{-\pi}^{\pi} \!\!\!\!\!
d\theta \, e^{im\theta} \! \int \!\! dy \int \!\! dx \,\,
e^{i\bm\omega\cdot\bm r} P_{n}(x,y|\theta).
\label{Eq:FourierDefinition}
\end{eqnarray}
The arbitrary moment $\langle x^{k_{1}} y^{k_{2}} \rangle$
is given by
\begin{eqnarray}
\begin{aligned}
\langle &x^{k_{1}} y^{k_{2}} \rangle_n \equiv \int \!\! d\theta \int
\!\! dy \int \!\! dx \,\, x^{k_{1}} y^{k_{2}}  P_{n}(x,y|\theta) \\
&= \left. (-i)^{k_{1}+k_{2}} \frac{\partial^{k_{1}+k_{2}}
P_{n}(\omega_{x},\omega_{y} |m {=} 0)}{\partial \omega_{x}^{k_{1}}
\partial \omega_{y}^{k_{2}}} \right|_{(\omega_{x},\omega_{y})=(0,0)}.
\end{aligned}
\label{Eq:Moments}
\end{eqnarray}
To study the diffusive behavior of particles one deals
with the first and second moments of $P_{n}(x,y|\theta)$,
namely $\langle x\rangle$, $\langle y\rangle$, $\langle
x^2\rangle$ and $\langle y^2\rangle$. Thus, we first
focus on the derivation of these quantities in the
following. The same procedure is followed in
Sec.~\ref{HigherMoments} to obtain higher moments of
displacement. A similar Fourier-Z-transform technique
was applied to study diffusive transport of light in
foams \cite{FoamRefs}. The first two moments along the
$x$ direction are given by
\begin{eqnarray}
\begin{aligned}
\langle x \rangle\!_{_n} &\equiv \left. -i \frac{\partial
P_{n}(\omega_{x},\omega_{y} |m {=} 0)}{\partial \omega_{x}
} \right|_{(\omega_{x},\omega_{y})=(0,0)}, \\
\langle x^2 \rangle\!_{_n} &\equiv \left. (-i)^2 \frac{\partial^2
P_{n}(\omega_{x},\omega_{y} |m {=} 0)}{\partial \omega_{x}^2
} \right|_{(\omega_{x},\omega_{y})=(0,0)}.
\end{aligned}
\label{Eq:Moments12}
\end{eqnarray}
Similar expressions can be written for $\langle y\rangle$
and $\langle y^2\rangle$. Fourier transforming
Eq.~(\ref{Eq:MasterEquation}), we find
\begin{eqnarray}
\begin{aligned}
&P_{n+1}(\omega, \alpha |m) = \sum_{k{=}-\infty}^{\infty}
\Bigg[ i^k e^{-ik\alpha} \times \\
&P_{n}(\omega, \alpha |m{+}k) \big(p+s \,\mathcal{R}(m{+}k)
\big) \! \int \!\! d\ell \, \mathcal{F}(\ell) J_k(\omega\ell)
\Bigg],
\end{aligned}
\label{Eq:FourierMasterEq11}
\end{eqnarray}
where $J_k(z) {=} \frac{1}{2\pi i^{k}} \int_{-\pi}^{\pi}
\! d\theta \,\, e^{iz\cos\theta} e^{-ik\theta}$ is the $k$th
order Bessel's function and $\mathcal{R}(m) {=} \int_{-{\pi}}^{{\pi}}
\! d\phi \,\, e^{i m \phi} R(\phi)$ is the
Fourier transform of the rotation angle distribution
$R(\phi)$. One can expand $P_{n}(\omega, \alpha |m)$
as a Taylor series
\begin{eqnarray}
\begin{aligned}
P_{n}(\omega, \alpha |m)
&= Q_{0,n}(\alpha|m) + i \omega \, \langle \ell \rangle \,
Q_{1,n}(\alpha|m) \\
& - \frac{1}{2} \omega^2 \, \langle \ell^2 \rangle \,
Q_{2,n}(\alpha|m)+ \cdot \cdot \cdot,
\end{aligned}
\label{Eq:TaylorExpPw}
\end{eqnarray}
with $\langle \ell \rangle$ and $ \langle \ell^2 \rangle$
being the first and second moments of the step-length
distribution $\mathcal{F}(\ell)$. From Eqs.~(\ref{Eq:Moments12})
and (\ref{Eq:TaylorExpPw}) we obtain the moments in terms
of the Taylor expansion coefficients $Q_{i,n}(\alpha|m)$:
\begin{figure}[t]
\centering
\includegraphics[scale=1.05,angle=0]{Figure1.eps}
\caption{(color online). Trajectory of the walker during
two successive steps.}
\label{Fig1}
\end{figure}
\begin{eqnarray}
\begin{aligned}
\langle x \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \, \ell
\,\, Q_{1,n}(0|0) \,\,\,\,\, = \langle \ell \rangle \, Q_{1,n}(0|0),\\
\langle y \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \, \ell
\,\, Q_{1,n}\big(\frac{\pi}{2}\big|0\big) \,\, = \langle \ell
\rangle \, Q_{1,n}\big(\frac{\pi}{2}\big|0\big),\\
\langle x^2 \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \,
\ell^2 \,\, Q_{2,n}(0|0) \,\,\, = \langle \ell^2 \rangle \, Q_{2,n}(0|0),\\
\langle y^2 \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \,
\ell^2 \,\, Q_{2,n}\big(\frac{\pi}{2}\big|0\big) = \langle
\ell^2 \rangle \, Q_{2,n}\big(\frac{\pi}{2}\big|0\big).
\end{aligned}
\label{Eq:MeanMoments}
\end{eqnarray}
By Taylor expansion of both sides of Eq.(\ref{Eq:FourierMasterEq11}),
one can collect all terms with the same power in $\omega$, leading
to the following recursion relations for the Taylor coefficients
$Q_{0,n}(\alpha|m)$, $Q_{1,n}(\alpha|m)$, and $Q_{2,n}(\alpha|m)$:
\begin{eqnarray}
\hspace{-25mm} Q_{0,n+1}(\alpha|m) = Q_{0,n}(\alpha|m) \big(p+s
\, \mathcal{R}(m)\big),
\label{Eq:omega0terms3}
\end{eqnarray}

\begin{eqnarray}
\begin{aligned}
\hspace{-8mm} Q_{1,n+1}(\alpha|m) &= Q_{1,n}(\alpha|m) \big(p+s
\, \mathcal{R}(m)\big)\\
&+ \frac{1}{2} \bigg[ e^{i\alpha} Q_{0,n}(\alpha|m{-}1) \big(p+s
\, \mathcal{R}(m{-}1)\big)\\
&+ e^{-i\alpha} Q_{0,n}(\alpha|m{+}1)
\big(p+s \, \mathcal{R}(m{+}1)\big) \bigg],
\end{aligned}
\label{Eq:omega1terms4}
\end{eqnarray}
\begin{eqnarray}
\begin{aligned}
Q_{2,n+1}&(\alpha|m) {=}
\bigg[ \frac{1}{2} Q_{0,n}(\alpha|m) + Q_{2,n}(\alpha|m) \bigg]
\big(p+s \, \mathcal{R}(m)\big) \\
&+ \frac{\langle \ell \rangle^2}{\langle \ell^2 \rangle} \bigg[
e^{i\alpha} Q_{1,n}(\alpha|m{-}1) \big(p+s \,
\mathcal{R}(m{-}1)\big)\\
&+ e^{-i\alpha} Q_{1,n}(\alpha|m{+}1)
\big(p+s \, \mathcal{R}(m{+}1)\big) \bigg]\\
&+ \frac{1}{4} e^{2i\alpha} Q_{0,n}(\alpha|m{-}2) \big(p+s \,
\mathcal{R}(m{-}2)\big)\\
&+ \frac{1}{4} e^{-2i\alpha} Q_{0,n}(\alpha|m{+}2) \big(p+s \,
\mathcal{R}(m{+}2)\big).
\end{aligned}
\label{Eq:omega2terms4}
\end{eqnarray}

The coupled linear equations (\ref{Eq:omega0terms3}),
(\ref{Eq:omega1terms4}) and (\ref{Eq:omega2terms4})
can be solved by means of the $z$-transform technique.
The $z$-transform ${G}(z)$ of a function $G_n$ of a
discrete variable $n=0, 1, 2, \cdot\cdot\cdot$ is
defined as
\begin{equation}
{G}(z)=\sum_{n=0}^{\infty} G_n z^{-n}.
\label{Eq:zTransformDef1}
\end{equation}
By applying the $z$-transform to
Eqs.(\ref{Eq:omega0terms3})-(\ref{Eq:omega2terms4}),
one obtains a set of algebraic equations for
$Q_{0}(z,\alpha|m)$, $Q_{1}(z,\alpha|m)$ and
$Q_{2}(z,\alpha|m)$ quantities (see Appendix
\ref{Appendix1}).

\subsection{The quantities of interest}
\label{QuantitiesOfInterest}
The main goal is to evaluate $\langle x \rangle\!_{_n}$,
$\langle y \rangle\!_{_n}$, $\langle x^2 \rangle\!_{_n}$,
and $\langle y^2 \rangle\!_{_n}$, which are given in
terms of $Q_{i,n}(\alpha|m)$ quantities in
Eq.~(\ref{Eq:MeanMoments}). Here, we derive $\langle
x \rangle\!_{_n}$ and $\langle x^2 \rangle\!_{_n}$,
and a similar approach can be followed to obtain $\langle
y \rangle\!_{_n}$ and $\langle y^2 \rangle\!_{_n}$.
The $z$-transform of Eq.~(\ref{Eq:MeanMoments})
leads to the following expressions in $z$-space
\begin{eqnarray}
\begin{aligned}
\langle x \rangle (z) &{=} \sum_{n=0}^{\infty} z^{-n} \langle
\ell \rangle \, Q_{1,n}(0|0) {=} \langle \ell \rangle \, Q_{1}(z,0|0),\\
\langle x^2 \rangle (z)&{=} \sum_{n=0}^{\infty} z^{-n} \langle
\ell^2 \rangle \, Q_{2,n}(0|0) {=} \langle \ell^2 \rangle \, Q_{2}(z,0|0).
\end{aligned}
\label{Eq:zTransformMeanMoments}
\end{eqnarray}
From Eqs.(\ref{Eq:zTransformMeanMoments}),
(\ref{Eq:Q1zTransform5}), and (\ref{Eq:Q2zTransform4}),
one obtains the first and second moments of $x$ in the
$z$-space (See Eqs.(\ref{Eq:zTransformMeanX-2}) and
(\ref{Eq:zTransformMeanX2-2}) in Appendix \ref{Appendix2}).
The last step to get the moments $\langle x \rangle$
and $\langle x^2 \rangle$ in real time is the inverse
$z$-transforming of the $z$-space moments [i.e.\
Eqs.(\ref{Eq:zTransformMeanX-2}) and (\ref{Eq:zTransformMeanX2-2})].
By introducing $A_i{=}p{+}s\,\mathcal{R}(i)$,
the resulting moments are:

\begin{eqnarray}
&\hspace{3mm}\langle x \rangle\!_{_n} = \,\,\langle \ell \rangle \, Q_{1,n{=}0}(0|0)
\hspace{35mm} \\ \nonumber
&{+} \frac{\langle \ell \rangle}{2} Q_{0,n{=}0}(0|-1) A_{_-1}
\frac{1-A_{_-1}^n}{1-A_{_-1}} \hspace{1mm} \\ \nonumber
&{+} \frac{\langle \ell \rangle}{2} Q_{0,n{=}0}(0|1) A_{_1}
\frac{1-A_{_1}^n}{1-A_{_1}}, \hspace{7.5mm}
\label{Eq:InverseZTransform-X-2}
\end{eqnarray}

\begin{widetext}
\begin{eqnarray}
\begin{aligned}
\hspace{-15mm}\langle x^2 \rangle\!_{_n} &= \,\,\,\langle \ell^2 \rangle Q_{2,n{=}0}(0|0)
+ n \frac{\langle \ell^2 \rangle}{2} Q_{0,n{=}0}(0|0)\\
&\,\,\,\,\,+ \!\langle \ell \rangle^2 Q_{1,n{=}0}(0|-1) A_{_-1}
\frac{1-A_{_-1}^n}{1-A_{_-1}} +\langle \ell \rangle^2 Q_{1,n{=}0}(0|1) A_{_1}
\frac{1-A_{_1}^n}{1-A_{_1}}\\
&\,\,\,\,\,+ \!\frac{\langle \ell \rangle^2}{2} Q_{0,n{=}0}(0|-2) \,A_{_-1} \, A_{_-2} \times
\frac{A_{_-1}^{n}A_{_-2} +A_{_-2}^{n}\big(A_{_-1}{-}1\big) + A_{_-2} -
A_{_-1}}{\big(A_{_-1}{-}A_{_-2}\big)
\, \big(A_{_-2}{-}1\big) \, \big(1{-}A_{_-1}\big)}\\
&\,\,\,\,\,+ \!\frac{\langle \ell \rangle^2}{2} Q_{0,n{=}0}(0|2)\,A_{_1} \, A_{_2} \times
\frac{A_{_1}^{n}A_{_2} +A_{_2}^{n}\big(A_{_1}{-}1\big) + A_{_2} -
A_{_1}}{\big(A_{_1}{-}A_{_2}\big)
\, \big(A_{_2}{-}1\big) \, \big(1{-}A_{_1}\big)} \\
&\,\,\,\,\,+ \!\frac{\langle \ell \rangle^2}{2} Q_{0,n{=}0}(0|0) \bigg[\, A_{_-1} \,
\frac{A_{_-1}^{n} - A_{_-1}n + n-1}{(1{-}A_{_-1})^2}
+ \, A_{_1} \,
\frac{A_{_1}^{n} - A_{_1}n + n-1}{(1{-}A_{_1})^2}\bigg]\\
&\,\,\,\,\,+ \!\frac{\langle \ell^2 \rangle}{4} Q_{0,n{=}0}(0|-2)  A_{_-2}
\frac{1{-} A_{_-2}^n}{1{-} A_{_-2}}
+ \frac{\langle \ell^2 \rangle}{4} Q_{0,n{=}0}(0|2)  A_{_2}
\frac{1{-} A_{_2}^n}{1{-} A_{_2}}.
\end{aligned}
\label{Eq:InverseZTransform-X2-2}
\end{eqnarray}
\end{widetext}

\subsection{Isotropic initial condition}
\label{IsotropicInitialCondition}
For the isotropic initial condition $P_{0}(x,y|\theta) {=}
\frac{1}{2\pi} \delta(x) \delta(y)$, one finds
from Eq.~(\ref{Eq:FourierDefinition}) that $P_{0}(\bm\omega|m)
{=} \frac{1}{2\pi} \int_{-\pi}^{\pi} d\theta \,
e^{im\theta} {=} \frac{\text{sin}(m\pi)}{m\pi}$. Then, using
the expansion equation (\ref{Eq:TaylorExpPw}) for $\omega{=}0$,
it can be seen that the only nonzero $Q$ quantity is
$Q_{0,n{=}0}(\alpha|0){=}1$ (for $m {=} 0$). Therefore,
Eq.~(\ref{Eq:InverseZTransform-X-2}) leads to
\begin{equation}
\langle x \rangle\!_{_n} = 0,
\label{Eq:InverseZTransform-X-IsotropicInit}
\end{equation}
and Eq.~(\ref{Eq:InverseZTransform-X2-2}), after replacing $s$
with $1{-}p$, reads
\begin{eqnarray}
\begin{aligned}
&\langle x^2 \rangle\!_{_n} = \\
&n \frac{\langle \ell \rangle^2}{2}
\bigg[\lambda + \frac{\big(p{+}\mathcal{R}(1)
{-}p\mathcal{R}(1)\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{R}(1)\big)} + \frac{\big(p{+}
\mathcal{R}(-1){-}p\mathcal{R}(-1) \big)}{
\big(1{-}p\big)\big(1{-}\mathcal{R}(-1)\big)}\bigg] \\
&\!\!+ \!\frac{\langle \ell \rangle^2}{2} \frac{\big(p {+}
\mathcal{R}(1){-} p\mathcal{R}(1)
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{R}(1)
\big)^2}\bigg[\big(p{+}\mathcal{R}(1) {-}
p\mathcal{R}(1)\big)^n{-}1\bigg]\\
&\!\!+ \!\frac{\langle \ell \rangle^2}{2} \frac{\big(p {+}
\mathcal{R}(-1) {-} p\mathcal{R}(-1)
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{R}(-1)
\big)^2} \bigg[\big(p{+}\mathcal{R}(-1) {-}
p\mathcal{R}(-1)\big)^n{-}1\bigg],
\end{aligned}
\label{Eq:InverseZTransform-r2-IsotropicInit}
\end{eqnarray}
with $\lambda{=}\langle \ell^2 \rangle / \langle
\ell \rangle^2$ being the relative variance of the
step-length distribution. The $y$-component of the
mean square displacement $\langle y^2 \rangle_n$
has the same form as shown in
Eq.~(\ref{Eq:InverseZTransform-r2-IsotropicInit})
due to symmetry. Therefore, one obtains
$\langle r^2 \rangle_n {=} \langle x^2 \rangle_n
{+} \langle y^2 \rangle_n {=} 2 \langle x^2 \rangle_n$.

\subsection{Long-time behavior}
\label{LongTimeBehavior}
From Eq.~(\ref{Eq:InverseZTransform-r2-IsotropicInit}) in
the limit of long time (i.e.\ $n {\rightarrow} \infty$)
one obtains the asymptotic mean square displacement as
\begin{equation}
\begin{split}
&\langle r^2 \rangle_n / \langle \ell \rangle^2 \simeq \\
&n \bigg[\lambda + \frac{\big(p{+}\mathcal{R}(1)
{-}p\mathcal{R}(1)\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{R}(1)\big)} + \frac{\big(p{+}
\mathcal{R}(-1){-}p\mathcal{R}(-1) \big)}{
\big(1{-}p\big)\big(1{-}\mathcal{R}(-1)\big)}\bigg] \\
&- \frac{\big(p {+}
\mathcal{R}(1){-} p\mathcal{R}(1)
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{R}(1)
\big)^2} - \frac{\big(p {+}
\mathcal{R}(-1) {-} p\mathcal{R}(-1)
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{R}(-1)
\big)^2}.
\end{split}
\label{Eq:r2-longtime}
\end{equation}
Assuming that the particle moves with a constant
speed $v$ during the ballistic parts of motion, the
elapsed time after $n$ steps is $\tau = n \langle
\ell \rangle /v$. The diffusion constant $D$ is
related to $\langle r^2 \rangle_n$ as
\begin{eqnarray}
\langle r^2 \rangle_n &=& 4 D \tau,
\label{Eq:DiffusionConstantDef}
\end{eqnarray}
thus, we find
\begin{equation}
D {=} \frac{v\langle \ell \rangle}{4} \bigg[\lambda
+ \frac{\big(p{+}\mathcal{R}(1)
{-}p\mathcal{R}(1)\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{R}(1)\big)} + \frac{\big(p{+}
\mathcal{R}(-1){-}p\mathcal{R}(-1) \big)}{
\big(1{-}p\big)\big(1{-}\mathcal{R}(-1)\big)}\bigg].
\label{Eq:D}
\end{equation}

\subsection{Turning with left-right symmetry}
\label{LRsymmetry}
The distribution $R(\phi)$ reflects to what extent the
directions of the successive steps are correlated. The
analytical method presented in this section allows us to
handle an arbitrary function $R(\phi)$, however, we are
particularly interested in the distributions with equal
probabilities to turn clockwise or anticlockwise. This
implies that $\mathcal{R}(1) {=} \mathcal{R}(-1) ({\equiv}
\mathcal{R})$. The asymmetry of the turning angle with
respect to the arrival direction is quantitatively
reflected in the value of $\mathcal{R}$ which ranges
between $-1$ and $1$, with zero denoting a uniform
case and negative (positive) values corresponding
to a higher chance of motion to the near backward
(forward) directions in the next step. When left-right
symmetry holds the imaginary part of $\mathcal{R}(m)$
vanishes, thus, $\mathcal{R}$ becomes
\begin{equation}
\mathcal{R} = \int_{-{\pi}}^{{\pi}} \!\!\! d\phi \,\,
\cos(\phi) \,\, R(\phi),
\label{Eq:R}
\end{equation}
and Eq.~(\ref{Eq:InverseZTransform-r2-IsotropicInit})
reduces to
\begin{eqnarray}
\begin{aligned}
\langle x^2 \rangle_n = &\frac12 n \langle \ell \rangle^2
\bigg[\lambda + \frac{2\big(p{+}\mathcal{R}
{-} p\mathcal{R}\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{R}\big)} \bigg] \\
&\!\!+ \!\langle \ell \rangle^2 \frac{\big(p {+}
\mathcal{R}{-} p\mathcal{R}
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{R}
\big)^2}\bigg[\big(p{+}\mathcal{R} {-}
p\mathcal{R}\big)^n{-}1\bigg].
\end{aligned}
\label{Eq:r2-LRsymmetry}
\end{eqnarray}

\section{Extension to three dimensions}
\label{Model-3D}
\begin{figure}[b]
\centering
\includegraphics[scale=0.65,angle=0]{Figure2.eps}
\caption{(color online). Illustration of symmetric
rotations with respect to the incoming direction,
in two (left) and three (right) dimensions.}
\label{Fig2}
\end{figure}
The analytical approach of Sec.~\ref{Model-2D} can
be straightforwardly generalized to the persistent
motion in three dimensions by introducing the
probability density $P_{n}(x,y,z|\phi,\varphi)$
for the particle to arrive at position $(x,y,z)$
at time step $n$ along the direction characterized
by the azimuthal and polar angles $\phi$ and
$\varphi$, even though the calculations for the
general motion in three dimensions are quite
lengthy. However, the processes with symmetric
turning-angle distribution are of particular
importance since usually the rotational symmetry
holds in biological applications. Thus, we
restrict ourselves in this section to the
turning-angle distributions with cylindrical
symmetry with respect to the incoming direction.
This is the three dimensional analogue to those
processes in two dimensions which obey left-right
symmetry.

\begin{figure}[t]
\centering
\includegraphics[scale=0.37,angle=0]{Figure3.eps}
\caption{(color online). Time evolution of the mean square
displacement $\langle r^2 \rangle$ scaled by $n$, for
different values of $\lambda$, $p$, and $\mathcal{R}$. The
solid lines correspond to analytical predictions via
Eq.~(\ref{Eq:r2-LRsymmetry}), and the symbols denote
simulation results.}
\label{Fig3}
\end{figure}

Similar to the 2D case, we introduce $R(\phi)$ as the
probability of turning with angle $\phi$ with respect
to the incoming direction (see Fig.~\ref{Fig2}). The
polar angle $\varphi$ is supposed to be uniformly
distributed over the range $[0,2\pi]$. Note that, in
contrast to the 2D case, the normalization condition
of $R(\phi)$ in 3D requires an integration over all
possibilities of $\varphi$ i.e.\ $\int_{0}^{{\pi}} \! R(\phi) \,
\sin(\phi) \, d\phi {=} 1$. Here again the corresponding
Fourier transform of $R(\phi)$ is real. Since every
two successive steps lie in a plane, as shown in
Fig.~\ref{Fig2}, one can intuitively write the
same two-dimensional master equation
[Eq.~(\ref{Eq:MasterEquation})] to describe the
motion in three dimensions. By solving this master
equation, one gets a similar expression for $\langle
x^2 \rangle$ as the 2D solution presented in
Eq.~(\ref{Eq:InverseZTransform-r2-IsotropicInit})
in the case of $\mathcal{R}(1){=}\mathcal{R}(-1)$,
even though with different prefactors:
\begin{eqnarray}
\begin{aligned}
\langle x^2 \rangle_n = \, &n \frac{\langle \ell
\rangle^2}{3} \bigg[\lambda + \frac{2\big(p{+}\mathcal{E}
{-}p\,\mathcal{E}\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{E}\big)} \bigg] \\
&\!\!+ \!\frac{2\langle \ell \rangle^2}{3} \frac{\big(p {+}
\mathcal{E}{-} p\,\mathcal{E}
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{E}
\big)^2}\bigg[\big(p{+}\mathcal{E} {-}
p\,\mathcal{E}\big)^n{-}1\bigg].
\end{aligned}
\label{Eq:x2-3D}
\end{eqnarray}
Here, $\mathcal{E}$ is the real part of the Fourier
transform of the rotation-angle distribution
\begin{eqnarray}
\mathcal{E}= \int_{0}^{{\pi}} \!\!\! d\phi \,\,
\cos(\phi) \,\, R(\phi) \, \sin(\phi).
\label{Eq:E}
\end{eqnarray}
Finally, one can obtain the total mean square displacement
$\langle r^2 \rangle_n = \langle x^2 \rangle_n
{+} \langle y^2 \rangle_n {+} \langle z^2
\rangle_n$, which has the same form as in 2D, only
$\mathcal{R}$ is replaced with $\mathcal{E}$:
\begin{eqnarray}
\begin{aligned}
\langle r^2 \rangle_n = &n \langle \ell \rangle^2
\bigg[\lambda + \frac{2\big(p{+}\mathcal{E}
{-} p\mathcal{E}\big)}{\big(1{-} p\big)\big(1
{-}\mathcal{E}\big)} \bigg] \\
&\!\!+ \!\langle \ell \rangle^2 \frac{2\big(p {+}
\mathcal{E}{-} p\mathcal{E}
\big)}{\big(1{-}p\big)^2\big(1{-}\mathcal{E}
\big)^2}\bigg[\big(p{+}\mathcal{E} {-}
p\mathcal{E}\big)^n{-}1\bigg].
\end{aligned}
\label{Eq:r2-LRsymmetry-3D}
\end{eqnarray}

\section{Simulation Results for MSD}
\label{Results}
In this section we compare the analytical predictions
with the results of extensive Monte Carlo simulations
obtained from the same step-length $\mathcal{F}(\ell)$
and turning-angle $R(\phi)$ distributions, and
self-propulsion $p$. The formalism introduced in
Secs.~\ref{Model-2D} and \ref{Model-3D} enables us to
handle any arbitrary function for $R(\phi)$ and
$\mathcal{F}(\ell)$, nevertheless, we restrict $R(\phi)$
in this section to symmetric distributions along the
incoming direction for simplicity and because of its
practical applications in biological systems.

\begin{figure}[t]
\centering
\includegraphics[scale=0.4,angle=0]{Figure4.eps}
\caption{(color online). (a) The crossover time $n\!_{_c}$
to the asymptotic diffusive regime versus self-propulsion
$p$. (b) The asymptotic diffusion coefficient $D$ in terms
of self-propulsion $p$ (light gray curves) or anisotropy $\mathcal{R}$
(dark gray curves). The solid, dashed, and dotted lines correspond
to $\lambda{=}1,2,5$, respectively.}
\label{Fig4}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[scale=0.42,angle=0]{Figure5.eps}
\caption{(color online). Time evolution of $\langle r^2
\rangle$ for two different distributions $\mathcal{F}(\ell)$
with the same moments $\langle \ell \rangle{=}2$ and
$\langle \ell^2 \rangle{=}5$ ($\lambda{=}1.25$). Inset:
Comparison between the exponential distribution $\mathcal{F}(\ell){=}
e^{1{-}\ell}$ ($\ell\in[1,\infty]$) and the uniform
distribution $\mathcal{F}(\ell){=}\frac{H(\ell{-}
\ell_\text{min}){+}H(\ell_\text{max}{-}\ell){-}1}
{\ell_\text{max}{-}\ell_\text{min}}$ ($\ell_\text{min}{=}0.268$,
$\ell_\text{max}{=}3.732$).}
\label{Fig5}
\end{figure}

We first investigate the overall behavior of the mean square
displacement for different values of $\lambda$, $p$, and
$\mathcal{R}$. $\lambda$ is a measure of the heterogeneity
of the network structure or the diversity of the step sizes,
and $\mathcal{R}$ quantifies the anisotropy of the structure
or the asymmetry of the turning angles of the walker. The
characteristics of $\mathcal{F}(\ell)$ and $R(\phi)$ distributions
can be considered as the stepping strategy of the random
walker which may be tunable externally (e.g.\ by controlling
the external agitation imposed on a driven granular system
\cite{GranularRefs}) or internally (by controlling the strength,
density, and spatial arrangement of obstacles in the system,
or by adjusting the underlying structure of the environment
such as a porous medium \cite{deAnna13}). $p$ is the self-propulsion
of the particle (equivalently, the processivity or persistency
of the walker). The case $p{=}\mathcal{R}{=}0$ and $\lambda{=}1$
corresponds to a simple diffusion (see Fig.~\ref{Fig3}). When
$p$ and $\mathcal{R}$ are both positive, they cooperate to
send the walker to the near forward directions more frequently,
resulting in superdiffusion at short time scales. If
$\mathcal{R}$ is negative, it competes against $p$ which
may lead to sub, normal, or superdiffusion. At the extreme
negative value of $\mathcal{R}$ (i.e.\ $\mathcal{R} {\rightarrow} -1$),
an oscillatory phase can be observed where the particle
experiences a nearly back and forth motion \cite{Shaebani14}.

\begin{figure}[t]
\centering
\includegraphics[scale=0.38,angle=0]{Figure6.eps}
\caption{(color online). (a) Mean square displacement
$\langle r^2 \rangle$ vs $n$ for three different
distributions $R(\phi)$ with $\mathcal{R}{=}0$ given
in the text. The step-length distribution is chosen
to be either $\mathcal{F}(\ell){=}\delta(\ell{-}1)$
(with $p{=}0$) or $\mathcal{F}(\ell){=}e^{1{-}\ell}$
(with $p{=}0.9$). Insets: The possible directions of
motion at the next step for each $R(\phi)$. (b) A
few sample distributions $R(\phi)$ with $\mathcal{R}
{\simeq}0.4$ (top) and $\mathcal{R}{\simeq}{-}0.4$
(bottom). The dotted lines show the arrival direction
and the arrows indicate the possible directions in
the next step, with length being proportional to the
probability.}
\label{Fig6}
\end{figure}

It can be seen from Fig.~\ref{Fig3} that the asymptotic
behavior of all curves is diffusive. This is due to the
fact that there is no preferred direction in the system
and the effective correlations which exist between
successive step angles are short-range. The crossover
time $n_c$ to asymptotic diffusion can be estimated by
balancing the linear and exponential terms in
Eq.(\ref{Eq:r2-LRsymmetry}). In Fig.~\ref{Fig4}(a),
$n_c$ is shown as a function of self-propulsion for
several values of $\mathcal{R}$ and $\lambda$.
Increasing $p$ and/or $\mathcal{R}$ delays the
crossover, while the walker gets randomized more
quickly for strong heterogeneities. The asymptotic
diffusion coefficient varies by several orders of
magnitude with control parameters $p$, $\mathcal{R}$,
and $\lambda$ [see Fig.~\ref{Fig4}(b)].

\begin{figure}[t]
\centering
\includegraphics[scale=0.9,angle=0]{Figure7.eps}
\caption{(color online). Typical trajectories at $p{=}0$ and
$\lambda{=}1$ for isotropic (left), forward symmetric (middle),
and forward asymmetric (right) turning-angle distributions,
after the same number of steps. The arrows show possible
directions of motion in the next step.}
\label{Fig7}
\end{figure}

A remarkable outcome of the analytical formalism is that the
anomalous diffusive motion of the particle is fully described
by the self-propulsion, and the characteristics of the step-length
and turning-angle distributions, namely the first two moments
$\langle \ell \rangle$ and $\langle \ell^2 \rangle$ of
$\mathcal{F}(\ell)$ and the Fourier transform of $R(\phi)$.
Therefore, one expects that stepping with different distributions
but with the same key characteristics mentioned above should lead
to the same results, independent of the functional form of the
distributions. To verify this finding by simulations, we first
choose an isotropic distribution $R(\phi){=}\frac{1}{2\pi}$ and
compare two different step-length distributions with the same
$\langle \ell \rangle$ and $\langle \ell^2 \rangle$ moments.
As shown in Fig.~\ref{Fig5}, the simulation results match
remarkably for an exponential function $\mathcal{F}(\ell){=}
e^{1{-}\ell}$ and a uniform distribution $\mathcal{F}(\ell)
{=}\frac{H(\ell{-}\ell_\text{min}){+}H(\ell_\text{max}{-}\ell){-}1}
{\ell_\text{max}{-}\ell_\text{min}}$ ($H(x)$ is the Heaviside
step function), both with $\langle \ell \rangle{=}2$ and $\langle
\ell^2 \rangle{=}5$.

Next, we choose a given step-length distribution (either
$\mathcal{F}(\ell){=}\delta(\ell{-}1)$ or $\mathcal{F}
(\ell){=}e^{1{-}\ell}$) and compare three different
turning-angle distributions: a uniform function $R_1(\phi)
{=}\frac{1}{2\pi}$, a motion restricted to left or right
directions $R_2(\phi){=}\frac12 (\delta(\phi{-}\pi{/}2){+}
\delta(\phi{-}\pi{/}2))$, and a motion restricted to
forward or backward directions $R_3(\phi){=}\frac12
(\delta(\phi){+}\delta(\phi{-}\pi))$. All these
examples correspond to $\mathcal{R}{=}0$ i.e., on
average, they have no preference for forward or backward
motion. Figure~\ref{Fig6}(a) reveals that there is a
perfect agreement between the simulation results
obtained for these different turning-angle distributions.
One can also generate positive or negative values of
$\mathcal{R}$ from different $R(\phi)$ distributions.
Several examples are shown in Fig.~\ref{Fig6}(b) for
$\mathcal{R}{\simeq}0.4$ (or $\mathcal{R}{\simeq}{-}0.4$),
which all lead to the same diffusive motion.

So far, only symmetric distributions are studied. Now, we
briefly investigate turning-angle distributions which are
asymmetric with respect to the incoming direction. Let us
consider two-dimensional walks for simplicity. An asymmetric
$R(\phi)$ in this case means that the left-right symmetry
of turning is broken, leading to (anti-) clockwise spiral
trajectories. A comparison is made in Fig.~\ref{Fig7}
between the trajectories obtained from three different
uniform distributions over the range $(\phi_\text{min},
\phi_\text{max})$: $R_1(\phi)$ is an isotropic function
corresponding to a normal diffusion ($\mathcal{R}{\equiv}
\mathcal{R}({+}1){=}\mathcal{R}({-}1){=}0$), $R_2(\phi)$
is a symmetric function ($\phi_\text{min}{=}{-}\pi{/}6$,
$\phi_\text{max}{=}\pi{/}6$) which results in $\mathcal{R}
{\equiv}\mathcal{R}({+}1){=}\mathcal{R}({-}1){\simeq}0.95$,
and $R_3(\phi)$ is an asymmetric distribution over the
range $[{-}\pi{/}6,\pi{/}3]$ which creates clockwise spirals.
Here $\mathcal{R}({+}1){\simeq}0.87{+}\text{i}\, 0.23$
and $\mathcal{R}({-}1){\simeq}0.87{-}\text{i}\, 0.23$,
thus, $\mathcal{R}({+}1){\neq}\mathcal{R}({-}1)$. The
asymptotic diffusion coefficient [Eq.~(\ref{Eq:D})] is
however a real number $D/v\langle \ell \rangle{=}\frac14
\big[\lambda{+} 2 \frac{A{-}A^2 {-}B^2}{(1{-}A)^2{+}B^2}\big]$
in the absence of self-propulsion, with $A$ and $B$ being
the real and imaginary parts of $\mathcal{R}({\pm}1)$.
We obtain $D/ (v\langle \ell \rangle/4) \simeq 1.0$,
$43.4$, and $2.7$ for $R_1(\phi)$, $R_2(\phi)$, and
$R_3(\phi)$, respectively, in agreement with the
simulation results.

\section{Higher moments and cumulants}
\label{HigherMoments}
The procedure described in Sec.~\ref{Model-2D} enables one
to obtain any arbitrary moment of displacement. To better
clarify the proposed recipe, we extend the calculations
to the third and fourth moments in this section, which are
sufficient to derive up to the fourth cumulants of displacement
and obtain the skewness and kurtosis of a persistent
random walk which are measures for the asymmetry and
peakedness of the probability distribution, respectively.
We also compare the analytical predictions with Monte Carlo
simulation results.

From Eq.~(\ref{Eq:Moments}), the third and fourth moments
of the displacement are given by
\begin{eqnarray}
\begin{aligned}
\langle x^3 \rangle\!_{_n} &\equiv \left. (-i)^3 \frac{\partial^3
P_{n}(\omega_{x},\omega_{y} |m {=} 0)}{\partial \omega_{x}^3
} \right|_{(\omega_{x},\omega_{y})=(0,0)},
\end{aligned}
\label{Eq:Moments3}
\end{eqnarray}
and
\begin{eqnarray}
\begin{aligned}
\langle x^4 \rangle\!_{_n} &\equiv \left. (-i)^4 \frac{\partial^4
P_{n}(\omega_{x},\omega_{y} |m {=} 0)}{\partial \omega_{x}^4
} \right|_{(\omega_{x},\omega_{y})=(0,0)}.
\end{aligned}
\label{Eq:Moments4}
\end{eqnarray}
Moreover, by expanding $P_{n}(\omega, \alpha |m)$ up to the forth order
terms in $\omega$ one finds
\begin{eqnarray}
\begin{aligned}
P_{n}(\omega, \alpha |m)
&= Q_{0,n}(\alpha|m) + i \omega \, \langle \ell \rangle \,
Q_{1,n}(\alpha|m) \\
& - \frac{1}{2} \omega^2 \, \langle \ell^2 \rangle \,
Q_{2,n}(\alpha|m) \\
& - \frac{i}{6} \omega^3 \, \langle \ell^3 \rangle \,
Q_{3,n}(\alpha|m) \\
& + \frac{1}{24} \omega^4 \, \langle \ell^4 \rangle \,
Q_{4,n}(\alpha|m)+ \cdot \cdot \cdot,
\end{aligned}
\label{Eq:TaylorExpPw4thOrder}
\end{eqnarray}
which results in the following relations between $\langle x^3 \rangle$
or $\langle x^4 \rangle$ and the Taylor expansion coefficients
\begin{eqnarray}
\begin{aligned}
\langle x^3 \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \,
\ell^3 \,\, Q_{3,n}(0|0) \,\,\, = \langle \ell^3 \rangle \, Q_{3,n}(0|0), \\
\langle x^4 \rangle\!_{_n} &= \! \int \!\! d\ell \, \mathcal{F}(\ell) \,
\ell^4 \,\, Q_{4,n}(0|0) \,\,\, = \langle \ell^4 \rangle \, Q_{4,n}(0|0).
\end{aligned}
\label{Eq:MeanMomentsX3X4}
\end{eqnarray}
Then, by following the procedure introduced in Sec.~\ref{Model-2D}
one obtains recursion relations for $Q_{3,n}(\alpha|m)$ and
$Q_{4,n}(\alpha|m)$ as
\begin{widetext}
\begin{eqnarray}
\nonumber
Q_{3,n+1}(\alpha|m) &{=}& \\ \nonumber
&\,\,& \bigg[ \frac32 \frac{\langle \ell \rangle \langle \ell^2
\rangle}{\langle \ell^3 \rangle} Q_{1,n}(\alpha|m) {+}
Q_{3,n}(\alpha|m) \bigg] \big(p{+}s \, \mathcal{R}(m)\big) \\ \nonumber
&+&\bigg[ \frac{\langle \ell \rangle \langle \ell^2 \rangle}
{\langle \ell^3 \rangle} Q_{2,n}(\alpha|m{+}1) + \frac14
Q_{0,n}(\alpha|m{+}1) \bigg] \, \times \frac32 e^{-i\alpha} \big(p{+}s \, \mathcal{R}(m{+}1)\big) \hspace{3.2cm} \\ \nonumber
&+&\bigg[ \frac{\langle \ell \rangle \langle \ell^2 \rangle}
{\langle \ell^3 \rangle} Q_{2,n}(\alpha|m{-}1) + \frac14
Q_{0,n}(\alpha|m{-}1) \bigg] \, \times \frac32 e^{i\alpha} \big(p{+}s \, \mathcal{R}(m{-}1)\big) \\ \nonumber
&+&\bigg[ \frac{\langle \ell \rangle \langle \ell^2 \rangle}
{\langle \ell^3 \rangle} Q_{1,n}(\alpha|m{+}2) \bigg] \frac34 e^{-2i\alpha}
\big(p{+}s \, \mathcal{R}(m{+}2)\big) \\ \nonumber
&+&\bigg[ \frac{\langle \ell \rangle \langle \ell^2 \rangle}
{\langle \ell^3 \rangle} Q_{1,n}(\alpha|m{-}2) \bigg] \frac34 e^{2i\alpha}
\big(p{+}s \, \mathcal{R}(m{-}2)\big) \\ \nonumber
&+& \frac18 e^{-3i\alpha} Q_{0,n}(\alpha|m{+}3) \big(p{+}s \,
\mathcal{R}(m{+}3)\big) \\
&+& \frac18 e^{3i\alpha} Q_{0,n}(\alpha|m{-}3) \big(p{+}s \,
\mathcal{R}(m{-}3)\big),
\label{Eq:omega3terms}
\end{eqnarray}
and
\begin{eqnarray}
\nonumber
Q_{4,n+1}(\alpha|m) &{=}& \\ \nonumber
&\,\,& \bigg[ \frac38 Q_{0,n}(\alpha|m) {+} 3 \frac{\langle
\ell^2 \rangle^2}{\langle \ell^4 \rangle} Q_{2,n}(\alpha|m) {+}
Q_{4,n}(\alpha|m) \bigg] \big(p{+}s \, \mathcal{R}(m)\big) \\ \nonumber
&+& e^{-i\alpha} \bigg[ 2 \frac{\langle \ell \rangle \langle \ell^3 \rangle}
{\langle \ell^4 \rangle} Q_{3,n}(\alpha|m{+}1) {+} \frac32
\frac{\langle \ell \rangle \langle \ell^3 \rangle}{\langle \ell^4 \rangle}
Q_{1,n}(\alpha|m{+}1)\bigg] \big(p{+}s \, \mathcal{R}(m{+}1)\big) \\ \nonumber
&+& e^{i\alpha} \bigg[ 2 \frac{\langle \ell \rangle \langle \ell^3 \rangle}
{\langle \ell^4 \rangle} Q_{3,n}(\alpha|m{-}1) {+} \frac32
\frac{\langle \ell \rangle \langle \ell^3 \rangle}{\langle \ell^4 \rangle}
Q_{1,n}(\alpha|m{-}1)\bigg] \big(p{+}s \, \mathcal{R}(m{-}1)\big) \\ \nonumber
&+& e^{-2i\alpha} \bigg[ \frac32 \frac{\langle \ell^2 \rangle^2}{\langle
\ell^4 \rangle} Q_{2,n}(\alpha|m{+}2) + \frac14 Q_{0,n}(\alpha|m{+}2)
\bigg] \big(p{+}s \, \mathcal{R}(m{+}2)\big) \\ \nonumber
&+& e^{2i\alpha} \bigg[ \frac32 \frac{\langle \ell^2 \rangle^2}{\langle
\ell^4 \rangle} Q_{2,n}(\alpha|m{-}2) + \frac14 Q_{0,n}(\alpha|m{-}2)
\bigg] \big(p{+}s \, \mathcal{R}(m{-}2)\big) \\ \nonumber
&+& \frac12 e^{-3i\alpha} \frac{\langle \ell \rangle \langle \ell^3 \rangle}
{\langle \ell^4 \rangle} Q_{1,n}(\alpha|m{+}3) \big(p{+}s \,
\mathcal{R}(m{+}3)\big)
+ \frac12 e^{3i\alpha} \frac{\langle \ell \rangle \langle \ell^3 \rangle}
{\langle \ell^4 \rangle} Q_{1,n}(\alpha|m{-}3) \big(p{+}s \,
\mathcal{R}(m{-}3)\big) \\
&+& \frac{1}{16} e^{-4i\alpha} Q_{0,n}(\alpha|m{+}4) \big(p{+}s \,
\mathcal{R}(m{+}4)\big)
+ \frac{1}{16} e^{4i\alpha} Q_{0,n}(\alpha|m{-}4) \big(p{+}s \,
\mathcal{R}(m{-}4)\big).
\label{Eq:omega4terms}
\end{eqnarray}
\end{widetext}
The corresponding algebraic equations for $Q_{3}(z,\alpha|m)$ and
$Q_{4}(z,\alpha|m)$ after the $z$-transform are given in Appendix
\ref{Appendix1}. From Eq.~(\ref{Eq:MeanMomentsX3X4}), the third
and fourth moments of $x$ in the $z$-space can be obtained as
\begin{eqnarray}
\begin{aligned}
\langle x^3 \rangle (z) &{=} \sum_{n=0}^{\infty} z^{-n} \langle
\ell^3 \rangle \, Q_{3,n}(0|0) {=} \langle \ell^3 \rangle \, Q_{3}(z,0|0),\\
\langle x^4 \rangle (z)&{=} \sum_{n=0}^{\infty} z^{-n} \langle
\ell^4 \rangle \, Q_{4,n}(0|0) {=} \langle \ell^4 \rangle \, Q_{4}(z,0|0).
\end{aligned}
\label{Eq:zTransform-Moments3,4}
\end{eqnarray}
By inserting $Q_{3}(z,0|0)$ and $Q_{4}(z,0|0)$ from Eqs.~(\ref{Eq:Q3zTransform4})
and (\ref{Eq:Q4zTransform4}), and inverse $z$-transforming
of the $z$-space moments, we finally obtain $\langle x^3
\rangle\!_{_n}$ and $\langle x^4 \rangle\!_{_n}$, which are
very lengthy equations. However, they fortunately reduce to
simpler forms when we consider the most interesting cases.
For example, the isotropic initial condition leads to
\begin{equation}
\langle x^3 \rangle\!_{_n} = 0,
\label{Eq:InverseZTransform-X3-IsotropicInit}
\end{equation}
and if we further limit the motion to a constant
step size, $\mathcal{F}(\ell){=}\delta(\ell{-}L)$,
and turning with left-right symmetry, the resulting
$\langle x^4 \rangle\!_{_n}$ reads
\begin{widetext}
\begin{eqnarray}
\nonumber
\displaystyle\langle x^4 \rangle\!_{_n} = &\,\,\,\displaystyle\frac{-3}{8} L^4 \Bigg[
-\displaystyle\frac{2 (A_{_1}{+}A_{_2})^2 A_{_2}^{n{+}1}}{(1{-}A_{_2})^2 (A_{_1}{-}A_{_2})^2} +
\displaystyle\frac{4 A_{_1}^{n+1} (n{+}1) \Big(A_{_1} (3 A_{_1}{+}1)-A_{_2}(A_{_1}{+}3)
\Big)}{(1{-}A_{_1})^3 (A_{_1}{-}A_{_2})} \hspace{50mm} \\
\nonumber
&+\displaystyle\frac{4 A_{_1}^{n+1} \Big(-8 (2{+}A_{_1}) A_{_1}^2 A_{_2}{+}(A_{_1}
(7 A_{_1}{+}4){+}1) A_{_1}^2{+}\big(A_{_1} (3 A_{_1}{+}8){+}1\big) A_{_2}^2
\Big)}{(1{-}A_{_1})^4 (A_{_1}{-}A_{_2})^2} \hspace{58mm} \\
\nonumber
&+\displaystyle\frac{(n{+}1)(1{+}A_{_1}) \Big(-3 A_{_1}^2+(A_{_1}{-}3)
(5 A_{_1}{+}3) A_{_2}{+}12 A_{_1}{+}7\Big)}{(1{-}A_{_1})^3 (1{-}A_{_2})}
-\displaystyle\frac{2 (n+1) (n+2) (1{+}A_{_1})^2}{(1{-}A_{_1})^2} \hspace{36.5mm} \\
&+ \displaystyle\frac{2 A_{_1} A_{_2} \left(A_{_1}^3{+}18 A_{_1}{+}24\right)
{-}A_{_2}^2 (A_{_1}{+}1) \Big(A_{_1} \big((A_{_1}{-}5) A_{_1}{+}23\big){+}5\Big)
{+}A_{_1} (A_{_1}{+}2) \big((A_{_1}{-}6) A_{_1}{-}10\big){+}10 A_{_2}{-}3}{(1{-}A_{_1})^4 (1{-}A_{_2})^2}
\Bigg], \hspace{8mm}
\label{Eq:4thMoment-general}
\end{eqnarray}
\end{widetext}
with
\begin{eqnarray}
\nonumber
A_{_1}&{=}p{+}s\,\mathcal{R}(1){=}p{+}s\displaystyle\!\!
\int_{-{\pi}}^{{\pi}} \!\!\!\!\! d\phi \,\cos(\phi) \, R(\phi), \hspace{30mm} \\
\nonumber
A_{_2}&{=}p{+}s\,\mathcal{R}(2){=}p{+}s\displaystyle\!\!
\int_{-{\pi}}^{{\pi}} \!\!\!\!\! d\phi \,\cos(2\phi) \,
R(\phi). \hspace{30mm}
\end{eqnarray}
The analytical prediction for $\langle x^4 \rangle\!_{_n}$ via
Eq.~(\ref{Eq:4thMoment-general}) is in agreement with the
simulation results as shown in Fig.~\ref{Fig8}. It can be seen
that $\langle x^4 \rangle\!_{_n}$ contains similar information
as the MSD concerning the anomalous diffusive motion of the
persistent walker.

\begin{figure}[b]
\centering
\includegraphics[scale=0.45,angle=0]{Figure8.eps}
\caption{(color online). Time evolution of the fourth moment
of displacement for different values of $\lambda$, $p$, and
$\mathcal{R}$. The solid lines correspond to analytical
predictions and the symbols denote simulation results.}
\label{Fig8}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[scale=0.7,angle=0]{Figure9.eps}
\caption{The 4th cumulant of $x$ (in units of
$\langle \ell^4 \rangle$) in terms of $n$ for a random walk
with $p{=}A_{_1}{=}A_{_2}{=}0$. Inset: The corresponding 4th
cumulant of the net displacement $r$ vs.\ $n$.}
\label{Fig9}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.99,angle=0]{Figure10.eps}
\caption{The kurtosis measure
$\beta_2$ in terms of the step number $n$ for different
turning-angle distributions. The lines correspond to
analytical predictions via Eq.~(\ref{Eq:SkewnessKurtosis})
and the symbols denote simulation results.}
\label{Fig10}
\end{figure}

The cumulants are often used in the statistical analysis
as an alternative to the moments of the distribution. In
general, the following relations hold between the $n$-th
cumulant $\kappa_n$ and the moments (shown up to the 4th
cumulant):
\begin{equation}
\begin{aligned}
\kappa_1 &= \langle x \rangle, \\
\kappa_2 &= \langle x^2 \rangle {-} \langle x \rangle^2, \\
\kappa_3 &= \langle x^3 \rangle {-} 3 \langle x^2 \rangle
\langle x \rangle {+} 2 \langle x \rangle^3, \\
\kappa_4 &= \langle x^4 \rangle {-} 4 \langle x^3 \rangle
\langle x \rangle {-} 3 \langle x^2 \rangle^2 {+} 12 \langle
x^2 \rangle \langle x \rangle^2 {-} 6 \langle x \rangle^4.
\end{aligned}
\label{Eq:CumulantsMoments1}
\end{equation}
If the walker starts from the origin with the isotropic initial
condition, the odd moments equal zero and the cumulant-moment
relations reduce to
\begin{equation}
\begin{aligned}
\kappa_1 &= 0, \\
\kappa_2 &= \langle x^2 \rangle, \\
\kappa_3 &= 0, \\
\kappa_4 &= \langle x^4 \rangle {-} 3 \langle x^2 \rangle^2.
\end{aligned}
\label{Eq:CumulantsMoments2}
\end{equation}
In the case of an ordinary random walk with $\mathcal{F}(\ell)
{=}\delta(\ell{-}L)$, we have $p{=}A_{_1}{=}A_{_2}{=}0$,
$\langle x^2 \rangle / L^2{=}\frac12 n$, and $\langle x^4
\rangle / L^4 {=}\frac34 n^2 {-} \frac38 n$, which lead to $\kappa_4 / L^4 {=}-\frac38 n$
(see Fig.~\ref{Fig9} for comparison with simulation).
Thus, from Eqs.~(\ref{Eq:r2-LRsymmetry}), (\ref{Eq:4thMoment-general}),
and (\ref{Eq:CumulantsMoments2}) one can calculate
the cumulants, from which other useful quantities
such as the skewness $\beta_1$ and kurtosis $\beta_2$
measures can be obtained as
\begin{equation}
\begin{aligned}
\beta_1 &= \frac{\kappa_3}{\kappa_2^{3{/}2}}=0, \\
\beta_2 &= \frac{\kappa_4}{\kappa_2^2}=\frac{\langle x^4 \rangle}{\langle
x^2 \rangle^2} - 3.
\end{aligned}
\label{Eq:SkewnessKurtosis}
\end{equation}
In Fig.~\ref{Fig10}, the time evolution of kurtosis is shown
for different turning-angle distributions. For a simple random
walk, $\beta_2$ decreases as $-3/2n$. Moreover, in anomalous
diffusive cases, $\beta_2$ asymptotically converges to zero
since the long-term behavior is diffusion.

It is notable that the higher moments are influenced
by the details of the shape of the turning-angle
distribution $R(\phi)$. While the MSD depends only on
$\mathcal{R}{\equiv}\mathcal{R}(1)$ (i.e.\ $\langle \cos
\phi \rangle$), $\langle x^4 \rangle$ is a function of
both $\mathcal{R}(1)$ and $\mathcal{R}(2)$ (i.e.\ $\langle
\cos \phi \rangle$ and $\langle \cos 2\phi \rangle$).
Thus, looking at the behavior of the higher moments would
reveal the underlying differences between turning-angle
distributions, which are not visible from the MSD results.
For example, $\mathcal{R}$ equals to zero for the three
different distributions $R(\phi)$ introduced in Fig.~\ref{Fig6}(a),
thus, their MSD is the same. However, their $\mathcal{R}(2)$
is $0$, $-1$, and $1$ for $R_1(\phi)$, $R_2(\phi)$, and
$R_3(\phi)$, respectively. Therefore, one obtains different
analytical expressions for their higher moments such as
$\langle x^4 \rangle$.

Besides the components of the displacement, the net
distance $r$ of the walker from the origin is also
a quantity of interest. For a persistent walk with an
arbitrary turning-angle distribution, so far, there has
been no exact closed-form expression for $\langle r \rangle$.
For approximate expressions $\langle r \rangle {\approx}
\sqrt{\langle r^2 \rangle} (1{-}\frac18 \frac{\sigma_{r^2}}
{\langle r^2 \rangle^2})$ (short-time \cite{McCulloch89})
and $\langle r \rangle {\approx} \frac12 \sqrt{\pi \langle
r^2 \rangle}$ (asymptotic \cite{Bovet88}), one deals with
the calculation of $\langle r^2 \rangle$ and $\langle
r^4 \rangle$. The second moment $\langle r^2 \rangle$ can
be obtained from Eq.~(\ref{Eq:r2-LRsymmetry}) since $\langle
r^2 \rangle{=}\langle x^2 {+} y^2 \rangle{=} \langle x^2
\rangle {+} \langle y^2 \rangle$. The fourth moment reads
$\langle r^4 \rangle{=} \langle (x^2 {+} y^2)^2 \rangle
{=} \langle x^4 \rangle{+}2\langle x^2y^2 \rangle {+}
\langle y^4 \rangle$. In general, $\langle x^2 y^2
\rangle {\neq} \langle x^2 \rangle \langle y^2 \rangle$ [see
Fig.~\ref{Fig11}(a)]. In order to calculate $\langle
x^2 y^2 \rangle$, one can start from Eq.~(\ref{Eq:Moments})
and follow the analytical procedure as explained for
arbitrary moments $\langle x^i \rangle$. For example,
a simple random walk with $p{=}A_{_1}{=}A_{_2}{=}0$
and $\lambda{=}1$ leads to $\langle x^2 y^2 \rangle
{=} \frac14 n^2 {-} \frac18 n$ vs.\ $\langle x^2
\rangle \langle y^2 \rangle {=} \frac14 n^2$. Finally,
the analytical form of $\langle r^4 \rangle$ for a walker
with constant step size $L$ and isotropic initial
conditions is obtained as
\begin{widetext}
\begin{eqnarray}
\nonumber
\displaystyle\langle r^4 \rangle\!_{_n} = \frac14 L^4 \bigg[ \!\! &\!& \frac{(3
A_{_1}{+}1) n^2{+}(A_{_1}{+}1) (n{-}1) n}{1{-}A_{_1}}
- 4\frac{A_{_1} A_{_2} \Big(A_{_2} (A_{_1}{+}A_{_2})
\left(A_{_1}{-}A_{_1}^{n{-}1}\right){+}(A_{_2}{-}A_{_1})
\left(A_{_2}{-}A_{_2}^n\right)\Big)}{(1{-}A_{_1})
(1{-}A_{_2}) (A_{_2}{-}A_{_1})^2} \\
\nonumber
&{-}& 4\frac{A_{_1} \left(1{-}A_{_1}^n\right) n {+}A_{_1}^2
\left(3 n{-}n^2\right){-}2 A_{_1}^2 \left(A_{_1}^n{+}1\right)
{+}A_{_1} (n{-}1)+0.5 A_{_1} (A_{_1}{+}1) (n^2{-}n
{-}2)}{(1{-}A_{_1})^2} \\
\nonumber
&{+}& 4\frac{{-}2 A_{_1}^{n{+}2} n {+}
A_{_1}^n{+}2 A_{_1}^4{-}A_{_1}^3 \left(n^2{+}
3 n{-}9\right){-}A_{_1}^2 (n{-}2){+}A_{_1} (n{-}2)
n}{(1{-}A_{_1})^3} + \frac{2 (A_{_1}{+}1) A_{_2}
(n{-}1)}{(1{-}A_{_1}) (1{-}A_{_2})}\\
\nonumber
&{+}& 4 \frac{A_{_1}^2 \Big(A_{_1} \big(2 A_{_1}
(A_{_1}{+}2){+}2 n{-}1\big){-}2 n{+}5\Big){-}
\Big(A_{_1} \left(2 A_{_1}^3{+}4 A_{_1}{+}3\right)
{+}1\Big) A_{_1}^n}{(1{-}A_{_1})^4} - \frac{2
(A_{_1}{+}1) \left(A_{_2}^2{-}A_{_2}^{n{+}
1}\right)}{(1{-}A_{_1}) (1{-}A_{_2})^2}\\
\nonumber
&{+}& 8 \frac{\Big(2 A_{_2}{-}A_{_1} (A_{_2}{+}1)\Big)
\left(A_{_1}^3{-}A_{_1}^{n{+}1}\right) n{-}
A_{_1}^3 (n{-}1) \Big(2 A_{_2}{-}A_{_1}
(A_{_2}{+}1)\Big)}{(1{-}A_{_1})^3 (A_{_2}{-}A_{_1})} \\
\nonumber
&{-}&\frac{4 A_{_1} A_{_2} (A_{_1}{+}1)
\Big((1{-}A_{_1}) (n{-}2){-}\left(A_{_1}{-}
A_{_1}^{n{-}1}\right)\Big)}{(1{-}A_{_1})^3 (A_{_2}{-}1)}
+ 4 \frac{2 \Big(2 A_{_2}{-}A_{_1} (A_{_2}{+}1)\Big)
A_{_1}^{n{+}1}{+}A_{_1} A_{_2} \left(A_{_1}
{-}A_{_1}^n\right)}{(1{-}A_{_1})^2 (A_{_2}{-}A_{_1})} \\
&{+}&\frac{8 A_{_1} \Big(2 A_{_2}{-}A_{_1} (A_{_2}{+}1)\Big)
\left(A_{_1}^3{-}A_{_1}^{n{+}2}\right)}{(1{-}A_{_1})^4
(A_{_2}{-}A_{_1})} + \frac{4 A_{_1} A_{_2}^2 (A_{_1}{+}A_{_2})
\left(A_{_2}{-}A_{_2}^{n-1}\right)}{(1{-}A_{_2})^2 (A_{_2}{-}A_{_1})^2} \bigg],
\label{Eq:4thMoment-r}
\end{eqnarray}
\end{widetext}
\begin{figure*}[t]
\centering
\includegraphics[scale=1.6,angle=0]{Figure11.eps}
\caption{(color online). (a) Comparison between
$\langle x^2 y^2 \rangle$ (solid lines) and $\langle
x^2 \rangle \langle y^2 \rangle$ (dashed lines) (both
presented in units of $\langle\ell^4 \rangle$), obtained
from the simulations where the short-time motion is sub
($\mathcal{R}{=}-0.9$), normal ($\mathcal{R}{=}0$), or
superdiffusion ($\mathcal{R}{=}0.9$). (b) $\langle
r^4 \rangle$ vs.\ $n$ from simulations (symbols) or
via Eq.~(\ref{Eq:4thMoment-r}) (solid lines).}
\label{Fig11}
\end{figure*}
which is confirmed by the simulation data, as shown
in Fig.~\ref{Fig11}(b). When $p$ is set to zero,
Eq.~(\ref{Eq:4thMoment-r}) reduces to the expression
recently proposed in \cite{Cheung10}, even though our
formalism allows for obtaining $\langle r^4 \rangle$
in the more general case of $p{\neq}0$ and even $\lambda
{\neq}1$ (i.e.\ variable step lengths) and anisotropic
initial conditions. One can similarly calculate the
cumulants and relative cumulants such as the kurtosis
for the net displacement $r$. For example, one finds
that $-\kappa_4$ grows as $n^2{+}n$ in the simple case
of $p{=}A_{_1}{=}A_{_2}{=}0$, as shown in Fig.~\ref{Fig9}
(inset).

\section{Probability distributions of the net distance
and turning angle after $\bm n$ steps}
\label{Probabilities}
In this section, we show how the probability densities of the
position of the random walker and its orientation evolve with
time. First, we study the probability distribution $P(r)$ of
the distance $r$ of the persistent random walker from the
origin in simulations. For a simple random walk in 2D, the
shape of the distribution at step $n$ approaches
\begin{eqnarray}
\displaystyle
P(r) \simeq \frac{2\,r}{\alpha D n} e^{-\frac{r^2}{\alpha D n}}
\label{Eq:Pr}
\end{eqnarray}
in the large
$n$ limit ($D$ is the diffusion constant and $\alpha{=}4\langle
\ell \rangle{/}v$). However, the anomalous motion of the persistent
walker at short times alters the shape of $P(r)$ as well
as its propagation speed. In figure \ref{Fig12}(a), $P(r)$
is plotted at different values of $n$ for $p{=}0$ and three
turning-angle distributions with $\mathcal{R}=-0.9, 0,$ and
$0.9$. From Eq.~(\ref{Eq:Pr}) one expects that all normal-diffusion
data collapse onto a universal curve when $P(r){\cdot}\sqrt{n}$
is plotted versus $r{/}\sqrt{n}$ (see the inset). However, the
distributions of sub and superdiffusion do not follow such
a master curve at short times. Indeed, $P(r)$ is narrower
and the peak shifts to the left (right) for subdiffusion
(superdiffusion) [see Fig.~\ref{Fig12}(b) (left)]. In the extreme
limit of localization or ballistic motion, $P(r)$ will be a delta
function at $r{=}0$ or $r{=}n \langle \ell \rangle$, respectively.
A similar comparison at long times reveals that $P(r)$ broadens
slower (faster) than a simple random walk in the case of subdiffusion
(superdiffusion). The shapes are, however, expected to
follow Eq.~(\ref{Eq:Pr}), as the asymptotic motion is
diffusive with different diffusion coefficients obtained
from Eq.~(\ref{Eq:D}). When scaled by $\sqrt{Dn}$, one finds
\begin{eqnarray}
\displaystyle
P(r){\cdot}\sqrt{Dn} \simeq \frac{2\,r}{\alpha\sqrt{Dn}}
e^{-\frac{r^2}{\alpha D n}},
\label{Eq:Pr2}
\end{eqnarray}
thus, we achieve a data collapse for $P(r){\cdot}\sqrt{Dn}$
vs.\ $r{/}\sqrt{Dn}$ in the asymptotic regime of the persistent
walks when the motility is purely diffusive, as shown in
Fig.~\ref{Fig12}(b) (right). For random walks in 3D,
Eq.~(\ref{Eq:Pr}) should be replaced with $P(r) {\simeq}
\frac{4}{\sqrt{\pi}} \frac{r^2}{(\alpha D n{/}2)^{3{/}2}}
e^{-\frac{r^2}{\alpha D n}}$.

\begin{figure*}
\centering
\includegraphics[scale=1.2,angle=0]{Figure12.eps}
\caption{(color online). (a) Probability distribution
of the distance $r$ (in units of $\langle \ell \rangle$)
from the origin at $n{=}60, 190, 1400,$ and $4400$
(solid, dashed, dash-dotted, and dotted lines, respectively), separately
shown for persistent walks with short-time sub (left), normal
(middle), and superdiffusive motion (right). Insets: Collapse
of $P(r)\cdot\sqrt{n}$ vs.\ $r{/}\sqrt{n}$. (b) Comparison
between persistent walks with short-time sub (dashed lines),
normal (dash-dotted lines), and superdiffusive motion (dotted lines)
at the early stages of the walk (left) and after a long time (right).
Insets: $P(r)\cdot\sqrt{D \, n}$ in terms of $r{/}\sqrt{D \, n}$,
where $D$ is the asymptotic diffusion coefficient. The solid
lines are obtained from Eq.~(\ref{Eq:Pr2}).}
\label{Fig12}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[scale=1.8,angle=0]{Figure13.eps}
\caption{(a) Evolution of the angular distribution
$f\!_{_n}\!(\alpha)$ of the direction of motion $\alpha$ in the
lab frame. The walker initially arrives along the $+x$ direction.
A comparison is made between motions with short-time sub
($\mathcal{R}{=}{-}0.9$), normal ($\mathcal{R}{=}0$), and
superdiffusion ($\mathcal{R}{=}0.9$). The lines are
obtained from Eq.~(\ref{Eq:f_alpha2}) and symbols denote
simulation results. The gray (dotted) lines are guides to eye.
(b) Probability $f\!_{_n}\!(\pm \frac{\pi}{2})$ of turning
to a perpendicular direction after $n$ steps, from simulations
(symbols) or via Eq.~(\ref{Eq:f_alpha2}) (dashed lines).}
\label{Fig13}
\end{figure*}

Finally, we investigate how the introduced angular correlations
between successive steps weaken over time. This is reflected
e.g.\ in the evolution of the shape of the probability
$f\!_{_n}\!(\alpha)$ that the direction of motion after
$n$ steps makes an angle $\alpha$ with the current
direction of motion. An ordinary random walk is memoryless
meaning that the walker gets randomized immediately, thus,
$f\!_{_n}\!(\alpha)$ remains isotropic over the whole range
of time. However, $f\!_{_n}\!(\alpha)$ is expected to exhibit
anisotropic shapes at short times for persistent walks, with
a gradual transition towards isotropic distributions in the
limit of large $n$. Let us consider the case $p{=}0$ for
simplicity, i.e.\ the walker turns to a new direction at
each step. The probability $f\!_{_n}\!(\alpha)$ reads
\begin{equation}
\displaystyle
f\!_{_n}\!(\alpha) {=} \!\! \int \!\! {\cdot\!\cdot\!\cdot} \!\!
\int \!\! d\phi_1 {\cdot\!\cdot\!\cdot} d\phi_n \,
R(\phi_1) {\cdot\!\cdot\!\cdot} R(\phi_n) \delta(\phi_1{+}
{\cdot\!\cdot\!\cdot}{+}\phi_n{-}\alpha),
\label{Eq:f_alpha1}
\end{equation}
where $\phi_i$ denotes the turning angle at step $i$. Using
the discrete Fourier transform of the delta function
$\delta(x{-}\alpha){=}\frac{1}{2\pi}\sum_{k{=}-\infty}^{\infty}
e^{ik(x{-}\alpha)}$ one obtains the following expression
for the walks which are symmetric with respect to the
arrival direction
\begin{equation}
\displaystyle
f\!_{_n}\!(\alpha) {=} \frac{1}{2\pi} \Big[ 1{+}2 \!
\sum_{k{=}1}^{\infty} \mathcal{R}^n(k) \, \cos(k\alpha) \Big].
\label{Eq:f_alpha2}
\end{equation}
In the case of $\alpha{=}{\pm}\pi{/}2$, $f\!_{_n}\!({\pm}
\pi{/}2)$ reflects the chance of turning to a perpendicular
direction after $n$ steps, which can be considered as a
measure of the coupling between longitudinal and perpendicular
transport. Figure \ref{Fig13} shows that Eq.~(\ref{Eq:f_alpha2})
is in agreement with simulation results.

\section{Summary and outlook}
\label{Summary}
A persistent random walk model was introduced to study
the stochastic motion of self-propelled particles. By
developing a general master equation formalism and a
Fourier-Z-transform technique it was shown that analytical
exact expressions can be obtained for the time evolution
of arbitrary moments of displacement. The combination of
self-propulsion and characteristics of step-size and
turning-angle distributions lead to a rich transport
phase diagram at short times. The long-time behavior
is, however, diffusive since the successive step angles
in the proposed master equation are only indirectly
correlated over a few number of steps. This defines
a time-scale between two arbitrarily chosen steps
beyond which the steps are practically independent
of each other. It will be interesting to enhance the
correlation range e.g.\ by introducing (anti-)cross
correlations between processivity, step sizes, and
turning angles. For particular functional forms of
(anti-)cross correlations, one could even obtain a
stationary increment for the mean square displacement
(either sub or superdiffusion) over finite time scales
as observed for the motion in viscoelastic environments.

\begin{acknowledgments}
This work was funded by the Deutsche Forschungsgemeinschaft
(DFG) through Collaborative Research Centers SFB 1027 (Projects
A7 and A3).
\end{acknowledgments}