Two-dimensional active motion

TL;DR

This paper derives exact analytical expressions for the diffusion behavior of two-dimensional active particles with arbitrary motion patterns, revealing how direction distribution influences transport properties and connecting to generalized diffusion equations.

Contribution

It provides a comprehensive analytical framework for understanding two-dimensional active particle diffusion with arbitrary direction distributions, including effects of persistence and circular motion.

Findings

Exact formulas for particle position distribution and mean-square displacement.

Connection established between direction distribution moments and diffusion properties.

Analysis of persistence and circular motion effects on diffusion behavior.

Abstract

The diffusion in two dimensions of non-interacting active particles that follow an arbitrary motility pattern is considered for analysis. Accordingly, the transport equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided; and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. For this, it is shown that only the first trigonometric moments of the distribution of the scattered direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.

Equations

$$\begin{gathered} \frac{\partial }{\partial t}\mathcal{P}(\mathbf{x},\phi ,t)+v_0\hat{\mathbf{v}}\cdot \nabla \mathcal{P}(\mathbf{x},\phi ,t)=D_T{\nabla }^2\mathcal{P}(\mathbf{x},\phi ,t) \\ +{\int }_{-\pi }^{\pi }d{\phi }^{\prime }{K}_A\left(\phi |{\phi }^{\prime }\right)\mathcal{P}(\mathbf{x},{\phi }^{\prime },t), \end{gathered}$$

$$\mathcal{P}(\mathbf{x},\phi ,t)=\int d^2{x}^{\prime }{G}_{D_T}(\mathbf{x}-{\mathbf{x}}^{\prime },t)P({\mathbf{x}}^{\prime },\phi ,t),$$

$$\begin{gathered} \frac{\partial }{\partial t}P(\mathbf{x},\phi ,t)+v_0\hat{\mathbf{v}}\cdot \nabla P(\mathbf{x},\phi ,t)= \\ {\int }_{-\pi }^{\pi }d{\phi }^{\prime }Q\left(\phi ,{\phi }^{\prime }\right)P(\mathbf{x},{\phi }^{\prime },t) \\ -\left[{\int }_{-\pi }^{\pi }d{\phi }^{\prime }Q\left({\phi }^{\prime },\phi \right)\right]P(\mathbf{x},\phi ,t), \end{gathered}$$

$$K_A(\phi |{\phi }^{\prime })=Q(\phi ,{\phi }^{\prime })-\delta (\phi -{\phi }^{\prime }){\int }_{-\pi }^{\pi }d{\phi }^{\prime \prime }Q({\phi }^{\prime \prime },\phi ).$$

$$\begin{gathered} \frac{\partial }{\partial t}P(\mathbf{x},\phi ,t)+v_0\hat{\mathbf{v}}\cdot \nabla P(\mathbf{x},\phi ,t)= \\ \Lambda {\int }_{-\pi }^{\pi }d{\phi }^{\prime }\tilde{Q}\left(\phi -{\phi }^{\prime }\right)P(\mathbf{x},{\phi }^{\prime },t) \\ -\Lambda P(\mathbf{x},\phi ,t), \end{gathered}$$

$$\begin{gathered} \frac{\partial }{\partial t}\tilde{P}(\mathbf{k},\phi ,t)+i{v}_0\hat{\mathbf{v}}\cdot \mathbf{k}\tilde{P}(\mathbf{k},\phi ,t)= \\ \Lambda {\int }_{-\pi }^{\pi }d{\phi }^{\prime }\tilde{Q}\left(\phi -{\phi }^{\prime }\right)\tilde{P}(\mathbf{k},{\phi }^{\prime },t) \\ -\Lambda \tilde{P}(\mathbf{k},\phi ,t), \end{gathered}$$

$$\tilde{P}(\mathbf{k},\phi ,t)=\int \frac{d^{2}x}{2\pi }{e}^{-i\mathbf{k}\cdot \mathbf{x}}P(\mathbf{x},\phi ,t),$$

$$\tilde{P}(\mathbf{k},\phi ,t)=\frac{1}{2\pi }\sum _{n=-\infty }^{\infty }{\tilde{p}}_n(\mathbf{k},t)e^{-{\lambda }_{n}t}{e}^{in\phi },$$

$${\tilde{p}}_n(\mathbf{k},t)$$

$$=e^{{\lambda }_{n}t}{\int }_{-\pi }^{\pi }d\phi \tilde{P}(\mathbf{k},\phi ,t)e^{-in\phi }$$

$$\frac{\partial }{\partial t}f(\phi ,t)=\Lambda {\int }_{-\pi }^{\pi }d{\phi }^{\prime }\tilde{Q}\left(\phi -{\phi }^{\prime }\right)f({\phi }^{\prime },t)-\Lambda f(\phi ,t),$$

$${\lambda }_n=\Lambda \left[1-⟨e^{-in\phi }{⟩}_{\tilde{Q}}\right],$$

$$⟨\Phi (\phi ){⟩}_{\tilde{Q}}={\int }_{-\pi }^{\pi }d\phi \tilde{Q}(\phi )\Phi (\phi )$$

$${\Gamma }_n$$

$${\Omega }_n$$

$$\begin{gathered} \tilde{P}(\mathbf{k},\phi ,t)=\frac{1}{2\pi }{\tilde{p}}_0(\mathbf{k},t)+ \\ \frac{1}{2\pi }\sum _{\begin{array}{c}n=-\infty ,\\ n\ne 0\end{array}}^{\infty }{\tilde{p}}_n(\mathbf{k},t)e^{-{\Gamma }_{n}t}{e}^{-i{\Omega }_{n}t}{e}^{in\phi }, \end{gathered}$$

$$\begin{gathered} \frac{d}{dt}{\tilde{p}}_n(\mathbf{k},t)=-\frac{v_0}{2}ik{e}^{{\lambda }_{n}t}[e^{-i\theta }{e}^{-{\lambda }_{n-1}t}{\tilde{p}}_{n-1}(\mathbf{k},t) \\ +e^{i\theta }{e}^{-{\lambda }_{n+1}t}{\tilde{p}}_{n+1}(\mathbf{k},t)], \end{gathered}$$

$$\varrho (\mathbf{x},t)=\frac{1}{2\pi }{p}_0(\mathbf{x},t)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }d\phi P(\mathbf{x},\phi ,t).$$

$$j_x(\mathbf{x},t)$$

$$=\frac{1}{\pi }{\int }_{-\pi }^{\pi }d\phi \cos \phi P(\mathbf{x},\phi ,t),$$

$$j_y(\mathbf{x},t)$$

$$=\frac{1}{\pi }{\int }_{-\pi }^{\pi }d\phi \sin \phi P(\mathbf{x},\phi ,t).$$

$${\mathbb{W}}_{xx}(\mathbf{x},t)$$

$$=\frac{e^{-{\Gamma }_{2}t}}{\pi }\text{Re}\left[p_2(\mathbf{x},t)e^{-i{\Omega }_{2}t}\right]$$

$$=\frac{1}{\pi }{\int }_{-\pi }^{\pi }d\phi \cos 2\phi P(\mathbf{x},\phi ,t),$$

$${\mathbb{W}}_{xy}(\mathbf{x},t)$$

$$=\frac{e^{-{\Gamma }_{2}t}}{\pi }\text{Im}\left[p_{-2}(\mathbf{x},t)e^{i{\Omega }_{2}t}\right]$$

$$=\frac{1}{\pi }{\int }_{-\pi }^{\pi }d\phi \sin 2\phi P(\mathbf{x},\phi ,t).$$

$$\mathbb{R}(\mathbf{x},t)=$$

$$=$$

$$\tan 2\Theta (\mathbf{x},t)=\frac{{\mathbb{W}}_{xy}(\mathbf{x},t)}{{\mathbb{W}}_{xx}(\mathbf{x},t)}.$$

$$\tilde{P}(\mathbf{k},\hat{\mathbf{v}},t)=\tilde{\varrho }(\mathbf{k},t)+\hat{\mathbf{v}}\cdot \tilde{\mathbf{j}}(\mathbf{k},t)+\hat{\mathbf{v}}\cdot \tilde{\mathbb{W}}(\mathbf{k},t)\cdot \hat{\mathbf{v}}+\dots ,$$

$${\tilde{p}}_0(\mathbf{k},ϵ)=\int \frac{d^{2}x}{2\pi }{e}^{-i\mathbf{k}\cdot \mathbf{x}}{p}_0(\mathbf{k},ϵ)$$

$${\tilde{p}}_0(\mathbf{k},ϵ)={\tilde{p}}_0^{(0)}(\mathbf{k})\frac{1}{ϵ+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_1+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_2+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_3+\ddots }}}+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_1^{\ast }+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_2^{\ast }+\frac{(v_0/2{)}^2{\mathbf{k}}^2}{ϵ+{\lambda }_3^{\ast }+\ddots }}}},$$

$${\tilde{p}}_0(\mathbf{k},ϵ)=\frac{{\tilde{p}}_0^{(0)}(\mathbf{k})}{ϵ+(v_0/2{)}^2{\mathbf{k}}^2\tilde{\mathfrak{D}}(\mathbf{k},ϵ)},$$

$$ϵ{\tilde{p}}_0(\mathbf{k},ϵ)-{\tilde{p}}_0^{(0)}(\mathbf{k})=-{\left(\frac{v_0}{2}\right)}^2{\mathbf{k}}^2\tilde{\mathfrak{D}}(\mathbf{k},ϵ){\tilde{p}}_0(\mathbf{k},ϵ),$$

Full text

DOI: 10.1103/PhysRevE.101.022608

I Introduction

The intensive study of the out-of-equilibrium systems called active matter, has allowed to set up a firm basis for the understanding of a variety of out-of-equilibrium phenomena. Even at the individual level of description, the intrinsic nonequilibrium nature of active motion leads to diverse phenomena not observed in particles that move passively. Furthermore, the great diversity of the patterns of self-propelled motion observed in biological organisms (see the introductory section in Refs. Taktikos et al. (2014); Detcheverry (2015)) or in artificially designed active particles Bechinger et al. (2016), enriches the variety of effects exhibited by these systems .

A salient feature of active motion is that it is persistent, a characteristic that explicitly depends on the specific pattern of motion performed by the particle. The effects of persistence are well known, for instance, when active particles are confined to move under the effects of either an external potential or hard-walls, to lead to stationary distributions that differ from the ones of passive particles. In the case of confined motion by trapping potentials, the effects of persistence lead to distributions that differ from the one given by Boltzmann and Gibbs Pototsky and Stark (2012); Cates and Tailleur (2013); Solon et al. (2015); Stark (2016); Caprini Lorenzo et al. (2019).

On the other hand, among the many models that describe active motion Wu and Libchaber (2000); Szamel (2014); Fodor et al. (2016); Maggi Claudio et al. (2015); Farage et al. (2015); Paoluzzi et al. (2016); Sevilla and Castro-Villarreal (2019), theoretical comparative studies that consider the free diffusion of two of the more studied patterns of active motion –active Brownian motion, where the orientation of motion undergoes rotational diffusion, and run-and-tumble motion, which alternates running events with instantaneous, temporally uncorrelated tumbling events– reveal that, although there are important quantitative differences between them in the intermediate-time regime, they behave similarly in the long-time regime, namely, the exhibit normal diffusion, and have the same behavior in the short-time regime, that is, they move ballistically.

It is precisely in the intermediate-time regime, i.e., for times of the order of the persistence time, when conspicuous differences are revealed between both patterns of motion Kurzthaler et al. (2018). Based on these findings, I present in this paper an analysis of the statistics of active motion of particles that follows an arbitrary, however Markovian and spatially local, orientational dynamics. I focus the analysis to the intermediate-time regime, where the difference among specific patters of motion are conspicuous, as shown in the following sections, naturally, Gaussian normal diffusion is observed in the long-time regime and non-Gaussian ballistic superdiffusion in the short-time one.

Hence, to have at our disposal a theoretical framework that incorporates an arbitrary pattern of motion (circular, run-and-reverse, run-and-flick, etc.) of active swimmers it is highly desirable. In this paper, I present a theoretical framework of two-dimensional motion of active swimmers for a family of patterns of motion characterized by constant speed and an arbitrary probability distribution of the turning angle (scattered angle) of the swimming direction. Another important theoretical framework based on the continuous time random walks of single particles has been known in the literature, however this focuses on a family of patterns of motion characterized by the Poissonian or non-Poissonian statistics between turning events Taktikos et al. (2014); Detcheverry (2015, 2017).

On the basis of the transport equation Duderstadt and Martin (1979); Porra et al. (1997), I introduce in section II such a framework, and present the corresponding Fokker-Planck equation for the probability density that at time $t$, a particle is located at $\bm{x}$ and moving along the direction $\hat{\bm{v}}$. Although this theoretical framework considers the case of a spatially local Markovian dynamics of a single active particle, this is susceptible to be generalized not only to incorporate a more complex dyamics (as suggested by recent experiments on Escherichia Coli in Ref. Figueroa-Morales et al. (2018)), like memory effects in the swimming dynamics Sevilla et al. (2019), but also, to include many-body interactions, which in combination with a specific pattern of active motion, have important consequences in the collective dynamics Zhang et al. (2010); Thutupalli et al. (2015); Ginot F. et al. (2018); Mahault et al. (2018); Jeckel et al. (2019); Liu et al. (2019). The general solution of such a Fokker-Planck equation is presented in Sec. III. The marginal probability distribution of finding a swimmer at $\bm{x}$ at time $t$, independently of the direction of motion is of great interest, and in Sec. III I provide an exact solution, whose physical consequences are analyzed. A connection with a generalized diffusion equation is also unveiled. In Sec IV, generalities, applications and predictions of the framework are presented for some general families of patterns of motion. Finally I give my concluding remarks in Sec. V.

II The two-dimensional active transport equation

The starting point is the two-dimensional equation for the probability density, $\mathcal{P}({\bm{x}},\varphi,t)$, of a single particle being at position $\bm{x}$, moving at constant speed $v_{0}$ along a direction given by the angle $\varphi$ at time $t$, that is,

[TABLE]

where the unit vector $\hat{\bm{v}}$ is defined by $(\cos\varphi,\sin\varphi)$, $\varphi$ being the angle between the particle direction of motion and the horizontal axis of a given Cartesian reference frame. $D_{T}$ is the translational diffusion coefficient that gives account of the thermal fluctuations exerted by the surrounding medium. The transition rate of the direction of motion, $K_{A}(\varphi|\varphi^{\prime})$, gives the probability rate of the transition from the direction of motion $\varphi^{\prime}$ to $\varphi$, and encompasses the detailed information of a specific pattern of active motion considered. In this paper, I focus on the broad case in which $K_{A}(\varphi|\varphi^{\prime})$ is independent of time, of the swimming speed and of the particle position. However, such dependences must be considered in the more general case, as for instance to describe the motion of the bacterium Pseudomonas putida, whose swimming speed depends on the selected direction of motion after a transition Theves et al. (2013), or the motion of E. coli, for which a large variability in its motility behavior has been observed Figueroa-Morales et al. (2018).

I refer to Eq. (1) as the active-transport equation. The fact that passive fluctuations exerted on the particle motion are separated from the active ones allows to write (see the Appendix)

[TABLE]

where $G_{D_{{T}}}(\bm{x},t)$ denotes the two-dimensional Gaussian propagator of the diffusion equation with diffusion coefficient $D_{{T}}$, given explicitly by $\exp\{-\bm{x}^{2}/4D_{{T}}t\}/4\pi D_{{T}}t$. The active part of motion is entailed by the probability density $P(\bm{x},\varphi,t)$, which satisfies the gain-loss equation

[TABLE]

when $K_{A}(\varphi|\varphi^{\prime})$ is written in terms of the distribution of scattering-angle $Q(\varphi,\varphi^{\prime})$ as

[TABLE]

A further simplification can be realized by considering a rotationally invariant transition rate function, i.e., $Q(\varphi,\varphi^{\prime})=Q(\varphi-\varphi^{\prime})$. In such a case we can write Porra et al. (1997)

[TABLE]

where $\Lambda\equiv\int_{-\pi}^{\pi}d\varphi^{\prime}Q(\varphi^{\prime})$ is the inverse of the timescale that measures the average time between transitions, and $\widetilde{Q}(\varphi)=Q(\varphi)/\Lambda$.

III The general solution to the active transport equation

We are interested in the analytical solutions, $P(\bm{x},\varphi,t)$, if any, of Eq. (5), with the initial condition $P({\bm{x}},\varphi,0)=\delta^{(2)}({\bm{x}})/2\pi$, which corresponds to the case of an ensemble of independent active particles that depart from the origin in a Cartesian system of coordinates, and propagates in a random direction of motion drawn from the uniform distribution in $[-\pi,\pi]$, $\delta^{(2)}({\bm{x}})$ being the two dimensional Dirac’s delta function.

Due to the assumed spatial isotropy of the system, I apply the Fourier transform to Eq. (5) and obtain

[TABLE]

where

[TABLE]

denotes the symmetric Fourier transform of $P(\bm{x},\varphi,t)$ and ${\bm{k}}=(k_{x},k_{y})$, denotes the system’s wave-vector. The following Fourier series expansion,

[TABLE]

is suitable since it fulfills the periodicity condition of the probability density, $\widetilde{P}({\bm{k}},\varphi,t)=\widetilde{P}({\bm{k}},\varphi+2\pi,t)$.

The coefficients $\widetilde{p}_{n}(\bm{k},t)$ in the expansion (8) are obtained by the use of the standard orthogonality relation among the Fourier basis functions $\left\{e^{in\varphi}\right\}$, explicitly

[TABLE]

and satisfy the identity $\widetilde{p}_{-n}^{*}(\bm{k},t)=\widetilde{p}_{n}(\bm{k},t)$, since the probability density $P(\bm{x},\varphi,t)$ is a real function. The factors $e^{-\lambda_{n}t}$ in the expansion (8), correspond to the coefficients, $c_{n}(t)$, of the expansion in Fourier series of $f(\varphi,t)$ that solves the equation

[TABLE]

with $\lambda_{n}$ a complex number given by

[TABLE]

where

[TABLE]

denotes the average of the $\varphi$-dependent quantity $\Phi(\varphi)$ computed by the use of the scattering-angle distribution $\widetilde{Q}(\varphi)$.

Accordingly, the main features of a particular pattern of active motion are encoded in the distribution of scattered angles $\widetilde{Q}(\varphi)$, which entails the particular orientation dynamics of the swimming direction. Such features are equivalently inherited in the trigonometric moments:

[TABLE]

which correspond to the real and imaginary part of $\lambda_{n}$, respectively, thus $\lambda_{n}=\Gamma_{n}+i\,\Omega_{n}$. These quantities fully characterize the statistical properties of active motion (see for instance Refs. Viswanathan et al. (2005); Bartumeus et al. (2008) where only $\langle\cos\varphi\rangle_{\widetilde{Q}}$ is considered for their analysis of two-dimensional correlated random walks).

A series of properties for $\Gamma_{n}$ and $\Omega_{n}$ can be deduced in a straightforward way. From the normalization of $\widetilde{Q}$ we have that $\Gamma_{0}=\Omega_{0}=0$, and since $\widetilde{Q}(\varphi)$ is a real-valued function, we have that the complex conjugate of $\lambda_{n}$ is given by $\lambda_{n}^{*}=\lambda_{-n}$, which implies $\Gamma_{n}=\Gamma_{-n}$ and $\Omega_{n}=-\Omega_{-n}$. From this property one can show that the coefficients $\widetilde{p}_{n}(\bm{k},t)$ of the expansion (15) satisfy $\widetilde{p}_{-n}(\bm{-k},t)=\widetilde{p}^{*}_{n}(\bm{k},t)$. Notice further that $0\leq\Gamma_{n}\leq 2\Lambda$ and that $-\Lambda\leq\Omega_{n}\leq\Lambda$. With this observations, the expansion (8) can be explicitly split as

[TABLE]

where $\Gamma_{n}$ expresses the damping rate of the contribution of the $n$-th Fourier mode in the expansion (8). $P(\bm{x},\varphi,t)$ tends asymptotically to $p_{0}(\bm{x},t)/2\pi$ as time goes by, since $\Gamma_{0}=0$ and $\Omega_{0}=0$.

III.1 The coefficients $p_{n}(\bm{x},t)$

After substitution of Eq. (8) into Eq. (6), and use of the orthogonality of the Fourier basis functions, a set of coupled ordinary differential equations for the coefficients $\widetilde{p}_{n}({\bm{k}},t)$ is obtained, namely Sevilla and Nava (2014); Sevilla and Sandoval (2015); Sevilla (2016)

[TABLE]

where $\theta$ and $k$ correspond to the polar coordinates of the two-dimensional Fourier vector $\bm{k}$, i.e., $k_{x}\pm ik_{y}=ke^{\pm i\theta}$. Equations (16) are complemented by the initial conditions $\widetilde{p}_{n}^{(0)}(\bm{k})=(2\pi)^{-1}\delta_{n,0}$, which are obtained straightforwardly from the initial distribution considered, i.e., $P(\bm{x},\varphi,0)=\delta^{(2)}(\bm{x})/2\pi$.

Notice that the first coefficient $p_{0}(\bm{x},t)$ is related to the probability density

[TABLE]

The next coefficients, $p_{\pm 1}(\bm{x},t)$, are related to the first-rank tensor $\bm{j}(\bm{x},t)$ whose components are given by

[TABLE]

These, give the average direction of motion at position $\bm{x}$ at time $t$ and from which the probability density current $\bm{J}(\bm{x},t)=\frac{v_{0}}{2}\bm{j}(\bm{x},t)$ is introduced. Re[z] and $\text{Im}[z]$ denote for the real and imaginary part of $z$ respectively.

The coefficients $p_{\pm 2}(\bm{x},t)$ define the traceless, symmetric, 2$\times$2 second-rank tensor $\mathbb{W}(\bm{x},t)$, whose entries are given by

[TABLE]

The matrix

[TABLE]

corresponds to the two-dimensional matrix representation of the reflection transformation about the direction \hat{\bm{r}}(\bm{x},t)=\bigl{(}\cos\Theta(\bm{x},t),\sin\Theta(\bm{x},t)\bigr{)}, where $\Theta(\bm{x},t)$ is given by

[TABLE]

With $\varrho(\bm{x},t)$, $\bm{j}(\bm{x},t)$, $\mathbb{W}(\bm{x},t)$ and so on, it is customarily to rewrite $\widetilde{P}(\bm{k},\varphi,t)$ in the form

[TABLE]

where the second term in the right-hand side gives the contribution to $P(\bm{x},\hat{\bm{v}},t)$ due to the projection of this average direction of motion along the direction of motion $\hat{\bm{v}}$. This term decays exponentially at the rate $\Gamma_{1}$, whose inverse characterizes the persistence time of active motion. The third term in the right-hand side of (24), gives a contribution to $P(\bm{x},\hat{\bm{v}},t)$ proportional to the projection of the reflected direction of motion $\mathbb{R}(\bm{x},t)\hat{\bm{v}}$ [about the axis $\hat{\bm{r}}(\bm{x},t)$], along $\hat{\bm{v}}$. This term decays exponentially at the rate $\Gamma_{2}$. As is shown afterwards in the following sections, $\Gamma_{1}^{-1}$, $\Gamma_{2}^{-1}$, $\Omega_{1}$ and $\Omega_{2}$, are necessary to give a minimal comprehensive description of the statistical properties of active motion.

III.2 The probability density $p_{0}(\bm{x},t)$

As in previous studies, the probability density of finding a particle at position $\bm{x}$, independently of its direction of motion, $p_{0}(\bm{x},t)$, is of interest. After transforming the time domain to the Laplace domain, an exact solution for

[TABLE]

can be obtained from Eq. (16) in the form of continuous fractions (see the Appendix Appendix), namely

[TABLE]

where the explicit dependence on the variable $\epsilon$ conveys that the Laplace transform [$f(\epsilon)=\int_{0}^{\infty}dt\,e^{-\epsilon t}f(t)$] has been carried out, and $\widetilde{p}_{0}^{(0)}(\bm{k})$ denotes the initial distribution $\widetilde{p}_{0}(\bm{k},t=0)$. The solution (26) is a generalization of the kind of solution obtained in Ref. Mehraeen et al. (2008) for the probability distribution for a semiflexible polymer (modeled as an inextensible thread with a linear-elastic bending energy subjected to thermal fluctuations, known as a wormlike chain), that starts at the origin and ends at $\bm{x}$, independently of its orientation, as has been also pointed out in Ref. Detcheverry (2014) in the context of random walks.

In the present paper, the meaning of the solution (26) can be elucidated after rewritten it as

[TABLE]

or equivalently as

[TABLE]

which can be recognized as the Fourier-Laplace transform of the spatially-non-local generalized diffusion equation,

[TABLE]

introduced in Ref. Kenkre and Sevilla (2007) and obtained in the context of animal motion with internal states in Ref. Giuggioli et al. (2009). The integral over the spatial coordinates is computed over the whole two-dimensional plane. The connecting function $\mathfrak{D}(\bm{x},t)$ is given explicitly in the Fourier-Laplace domain by

[TABLE]

We introduce the recursive relations

[TABLE]

for $n\geq 0$, to write Eq. (30) in a simplified form as

[TABLE]

In the asymptotic limit, i.e., in the long-time regime, $\epsilon\rightarrow 0$, and in the short-wave-vector limit, $k=|\bm{k}|\rightarrow 0$, the connecting function is given by the zeroth order approximant, $\widetilde{\mathfrak{D}}^{(0)}(\epsilon)$, obtained after evaluating $\widetilde{\mathfrak{D}}(\bm{k},\epsilon)$ at $\bm{k}=\bm{0}$, i.e.,

[TABLE]

This implies a spatially-local connecting function, that exhibits oscillations of frequency $\Omega_{1}$, exponentially damped with relaxation time $\Gamma_{1}^{-1}$, namely

[TABLE]

With this approximation of the connecting function, we have that Eq. (29) can be rewritten in the form

[TABLE]

which corresponds to a generalization of the telegrapher equation in that it incorporates the effects of an effective torque that gives rise to circular motion of angular speed $\Omega_{1}$. If $\Omega_{1}=0$, Eq. (35) reduces to the standard telegrapher’s equation Goldstein (1951)

[TABLE]

where the diffusion coefficient due to the persistence of the swimming direction, $D_{\text{pers}}=v_{0}^{2}/2\Gamma_{1}$, is apparent. As before, I identify $\Gamma_{1}^{-1}$ with the persistence time. In the temporal asymptotic limit we have, from (26), that $\widetilde{p}_{0}(\bm{k},\epsilon)\sim\left[\epsilon+(v_{0}/2)^{2}\bm{k}^{2}\left(\lambda_{1}^{-1}+{\lambda_{1}^{*}}^{-1}\right)\right]^{-1}$, which can be inverted straightforwardly to the spatial and temporal variables to give the Gaussian $G_{D_{\text{eff}}}(\bm{x},t)$, i.e.,

[TABLE]

where the effective diffusion coefficient, $D_{\text{eff}}$, due to active motion is defined by $D_{\text{eff}}=D_{\text{pers}}/(1+\Omega_{1}^{2}/\Gamma_{1}^{2})$, which reduces to $D_{\text{pers}}$ when $\Omega_{1}$ vanishes.

On the other hand, in the short time regime ($|\epsilon|\gg|\lambda_{n}|$ for all $n$), we have that $\widetilde{p}_{0}(\bm{k},\epsilon)$ can be approximated by $\widetilde{p}_{0}^{(0)}(\bm{k})\left[1/\epsilon-2(v_{0}/2)^{2}\bm{k}^{2}/\epsilon^{3}+\ldots\right]$. After taking the inverse Laplace transformation we obtain

[TABLE]

wich results from identifying the first two terms of the power series of the zeroth-order Bessel function of the first kind, $J_{0}(x)=1-(x/2)^{2}+\ldots$. For the initial distribution considered, we obtain the radial pulse:

[TABLE]

wich propagates at speed $v_{0}$ free of the wakes exhibited by the solution of the approximated description given by the telegrapher’s equation in the short time regime Sevilla and Nava (2014), where $x=\|\bm{x}\|$.

The next order approximant of $\widetilde{\mathfrak{D}}(\bm{k},\epsilon)$ is of particular interest since it leads to a connecting function coupled in the spatial and temporal variables. Some models of stochastic motion consider memory functions that couple space and time, as is the case for the family of stochastic motion known as Lévy walks –described within the formalism of continuous time random walks– where the transition probability density that connects two distinct points in space at different times is constrained by the condition that the walker moves at constant speed Zaburdaev et al. (2015). In our case the first order approximant, $\widetilde{\mathfrak{D}}^{(1)}(\bm{k},\epsilon)$, is obtained from $\widetilde{\mathfrak{D}}(\bm{k},\epsilon)$ after evaluating $\Delta_{1}(\bm{k},t)$ and $\overline{\Delta}_{1}(\bm{k},\epsilon)$ at $\bm{k}=\bm{0}$, which leads to

[TABLE]

In the time regime for which $|\epsilon|\ll|\lambda_{2}|$, an explicit expression for $\mathfrak{D}^{(1)}(\bm{x},t)$ in spatial and temporal coordinates is obtained, namely

[TABLE]

Due to the explicit appearance of the Gaussian $G_{v_{0}^{2}/4\Gamma_{2}}(\bm{x},t)$, the connecting function (41) gives a major contribution to those spatial positions $\bm{x}$, $\bm{x}^{\prime}$, whose separation is less or of the order of the distance $\sqrt{v_{0}^{2}t/\Gamma_{2}}$, and decays quickly to zero for pairs of points whose distance is larger than this. It is expected that the Gaussian nonlocality of (41), is a consequence of the approximation made, and that a connecting function that vanishes for pair of points whose distance is larger than $v_{0}t$ is more appropriate. Note that (41) reduces to the long-time approximation given by (34), by taking the limit $\Omega_{2},\Gamma_{2}\rightarrow 0$.

III.3 The connecting function $\widetilde{\mathfrak{D}}(\bm{k},\epsilon)$ and the moments of $p_{0}(\bm{x},t)$

For the initial condition considered, we have that the solution given in Eq. (27) is a rotationally symmetric function that depends solely on $\bm{k}^{2}$, and we simply write $\widetilde{p}_{0}(k,\epsilon)$. Likewise, we can write $p_{0}(\bm{x},\epsilon)=(2\pi)^{-1}p_{0}(x,\epsilon)$, where $x$ denotes the magnitude of $\bm{x}$, and the explicit appearance of the Laplace variable $\epsilon$ indicates that the Laplace transform is considered. The mentioned rotational symmetry allows to write (25) as

[TABLE]

i.e., $\widetilde{p}_{0}(k,\epsilon)=\langle J_{0}(kx)\rangle_{\text{rad}}$,

where $\langle z[x(\epsilon)]\rangle_{\text{rad}}$ denotes the average of $z(x)$ over the radial distribution $x\,p_{0}(x,\epsilon)$, thus, after use of the power series representation of the Bessel function J_{0}(x)=\sum_{l=0}^{\infty}\bigl{[}(-1)^{l}/(l!)^{2}\bigr{]}(x/2)^{2l} we have that

[TABLE]

where $\langle x^{2n}(\epsilon)\rangle_{\text{rad}}$ are the rotationally symmetric moments given by

[TABLE]

These can also be obtained directly from $\widetilde{p}_{0}(k,\epsilon)$, if this is known, from the formula

[TABLE]

III.3.1 The mean-square displacement

The mean-square displacement is defined by $\langle\bm{x}^{2}(t)\rangle$, which coincides with $\langle x^{2}(t)\rangle_{\text{rad}}$. It follows straightforwardly from (45) with $n=1$, that the Laplace transform of the mean-square displacement is given by

[TABLE]

By substitution of expression (27) for the probability density independent of the direction of motion, $\widetilde{p}_{0}(k,\epsilon)$, in the last equation, with the initial condition $\widetilde{p}_{0}^{(0)}(k)=1/2\pi$, we have that (see the Appendix .3)

[TABLE]

where $\left.\widetilde{\mathfrak{D}}(\bm{k},\epsilon)\right|_{k=0}$ corresponds to the zeroth-order approximant $\widetilde{\mathfrak{D}}^{(0)}(\epsilon)$ of $\widetilde{\mathfrak{D}}(\bm{k},\epsilon)$, given in (33). After inverting the Laplace transform, the exact time dependence of the mean-square displacement is given by

[TABLE]

which reduces to the well-known expression

[TABLE]

when $\Omega_{1}$ vanishes.

It is noticed, from expression (48), that the regime for which the particle motion is dominantly ballistic, $\langle\bm{x}^{2}(t)\rangle\rightarrow v_{0}^{2}t^{2}$, is obtained in the short-time regime, $\Gamma_{1}t\ll 1$, for arbitrary $\Omega_{1}$ (see Fig. 1). In contrast, in the long-time regime, $\Gamma_{1}t\gg 1$, we get the standard linear dependence in time of the mean-square displacement

[TABLE]

with $D_{\text{eff}}$, given as before, as $D_{\text{pers}}/(1+\Omega_{1}^{2}/\Gamma_{1}^{2})$. As is well-known Larralde (1997); Weber et al. (2011), the effective diffusion coefficient reaches its maximum value

[TABLE]

at the ratio $\Omega_{1}/\Gamma_{1}=1$. It can be clearly noticed from Eq. (48), that the time dependence of the mean-square displacement depends only on the ratio $\Omega_{1}^{2}/\Gamma_{1}^{2}$, whose explicit value depends on the particular transition probability density $\widetilde{Q}(\varphi)$. Thus, the crossover from the ballistic regime to the normal diffusion one is sensitive to the particular details of the pattern of active motion, entailed in $\Omega_{1}/\Gamma_{1}$ through $\widetilde{Q}(\varphi)$ as is shown in Fig. 1. For large values of the ratio $\Omega_{1}/\Gamma_{1}$, the particle get self-trapped in the intermediate-time regime due to the circular motion induced by the particular choice of $\widetilde{Q}(\varphi)$, and revealed by the corresponding oscillations of the mean-square displacement (see the solid-black line in Fig. 1 for $\Omega_{1}/\Gamma_{1}=100$).

III.3.2 The kurtosis

The non-Gaussian feature of the probability density $p_{0}(\bm{x},t)$ can be characterized by its kurtosis $\kappa$, which as a matter of convenience, the definition given by Mardia Mardia (1974) is used, namely

[TABLE]

where $\bm{x}^{\text{T}}$ denotes the transpose of the vector $\bm{x}$ and $\Sigma$ is the $2\times 2$ matrix defined by the average of the dyadic product \bigl{(}\bm{x}(t)-\langle\bm{x}(t)\rangle\bigr{)}^{\text{T}}\cdot\bigl{(}\bm{x}(t)-\langle\bm{x}(t)\rangle\bigr{)}. For the circularly symmetric case, the one considered in this paper, Eq. (52) reduces to

[TABLE]

From Eq. (45) we have that the Laplace transform of the time dependence of the fourth-moment is given by

[TABLE]

Notice that the first term in the last equation depends solely on $\lambda_{1}$, $\lambda_{1}^{*}$ since $\left[\widetilde{\mathfrak{D}}(\bm{k},\epsilon)\right]^{2}_{k=0}=\left[\widetilde{\mathfrak{D}}^{(0)}(\epsilon)\right]^{2}=\left[(\epsilon+\lambda_{1})^{-1}+(\epsilon+\lambda_{1}^{*})^{-1}\right]^{2}$ , while the second term carries information about $\lambda_{2}$, $\lambda_{2}^{*}$ (and therefore about $\Gamma_{2}$ and $\Omega_{2}$) since, as can be shown straightforwardly from (30) and (31) (see the Appendix .3),

[TABLE]

Thus the fourth moment in Laplace domain is explicitly given by

[TABLE]

The general explicit time dependence of the fourth moment is too involved to be discussed at this point. Besides, the values of $\Gamma_{1}$, $\Omega_{1}$, $\Gamma_{2}$ and $\Omega_{2}$ are not independent among them, but they are related through the transition probability density $\widetilde{Q}(\varphi)$. Thus, an analysis of the time dependence of the kurtosis is presented in the next section for particular cases of $\widetilde{Q}(\varphi)$. Notwithstanding this, the short- and long-time regimes can be discussed straightforwardly.

In the long-time regime ($|\epsilon|\ll|\lambda_{1}|,\,|\lambda_{2}|$), the second and third terms in the squared brackets of Eq. (56) can be neglected with respect to the first one, and thus, it is the first term that mainly contributes in the long-time regime. In such regime the fourth moment is independent of $\lambda_{2}$ and $\lambda_{2}^{*}$, and the inversion of the Laplace transform can be done straightforwardly, which gives

[TABLE]

and from this, we deduce that $\kappa\sim 8$, which uniquely characterizes the two-dimensional Gaussian distribution. Unlike this case, in the short-time regime ($|\epsilon|\gg|\lambda_{1}|,\,|\lambda_{2}|$), we have that all the terms in Eq. (56) contribute, and such an expression reduces to $4!v_{0}^{4}/\epsilon^{5}$, which can be inverted to give $v_{0}^{4}t^{4}$ (independent of $\lambda_{1}$, $\lambda_{2}$ and their complex conjugates), and thus $\kappa\simeq 4$ which characterizes the distortionless propagation of the sharp pulse $\delta(x-v_{0}t)/(2\pi x)$ Sevilla and Nava (2014); Sevilla and Sandoval (2015).

The effects of $\lambda_{2}$, $\lambda_{2}^{*}$ can be observed only in the intermediate-time regime, where the particle positions distribution suffers of the important effects of persistence, as has been anticipated in Sect. III.1 as is discussed in the following sections.

IV Persistence time, natural period of rotation and other time-scales

As has been already introduced in Sec. III, the persistence time of the swimming direction, $\Gamma_{1}^{-1}$, and the natural period of the circular motion, $\Omega_{1}^{-1}$, correspond to the relevant time-scales that define the diffusive regime of the active motion [see Eq. (50)]. $\Gamma_{1}^{-1}$ is closely related to the persistence time introduced by Wu et al. in Ref. i Wu et al. (2000), and by Bartumeus et al. in Ref. Bartumeus et al. (2008) in the modeling and analysis of animal motion in two dimensions as correlated random walks. All the other time-scales that appear in the present analysis [see, for instance, the expansion Eq. (15)], namely, $\Gamma_{n}^{-1}$, $\Omega_{n}^{-1}$, with $n>1$, determine precisely the statistical properties of active motion at all time regimes. These depend on the particular choice of the scattering-angle distribution $\widetilde{Q}(\varphi)$.

The simplest scattering-angle distribution may correspond to the case when $\widetilde{Q}(\varphi)$ is uniform in $[-\pi,\pi]$, i.e., $\widetilde{Q}(\varphi)=(2\pi)^{-1}$. This has been used to model the paradigmatic two-dimensional pattern of active motion called run-and-tumble Schnitzer (1993); Porra et al. (1997); Solon et al. (2015); Hancock and Baskaran (2015), for which $\Gamma_{n}=\Lambda$ for all $n$, i.e., $\Lambda{{}^{-1}}$ is the unique time-scale that defines the dynamics of the swimming direction, meaning that all Fourier modes in the series (8) decay at the same pace $\Lambda$.

Moreover, many scattering-angle distributions can be built on by wrapping out a standard single-variate distribution, $\rho(\eta)$, with support on the interval $(-\infty,\infty)$, to the unitary circle, namely

[TABLE]

One important set of scattering-angle distributions obtained in this manner, is got from the well-known Lévy $\alpha$-stable distributions with index $0<\alpha\leq 2$, $\rho_{\alpha;\sigma,\phi,\beta}(\eta)$, whose characteristic function is given by

[TABLE]

being $\sigma>0$ the width, $\phi$ the mode, and $\beta$ the skewness, $\Phi$ equals $\tan(\pi\alpha/2)$ if $\alpha\neq 1$ and $-2\ln|\kappa|/\pi$ if $\alpha=1$. The cases $\alpha=2$, $\alpha=1$, with $\beta=0$; and $\alpha=1/2$, with $\beta=1$, are of interest, since these cases correspond to the wrapped Gaussian and the wrapped Lorentz (Cauchy) distributions in the first cases and to the wrapped Lévy distribution in the second one. For the Lévy $\alpha$-stable distributions (59), it is possible to obtain explicit expressions for $\Gamma_{n}$ and $\Omega_{n}$, we have for $n>1$ that

[TABLE]

Another important family of scattering-angle distributions is the one given by the angle distribution of Jones and Pewsey, $\widetilde{Q}_{\text{JP},\sigma,\phi,\psi}(\varphi)$ Jones and Pewsey (2005), with parameters: $\sigma>0$, $\phi$, and $\psi\in(-\infty,\infty)$, which correspond respectively to the distribution width, the location of the unique mode, and the shape parameter. It has the explicit representation

[TABLE]

where $P_{\gamma}(z)$ is the associated Legendre function of the first kind of degree $\gamma$. The distribution (61) contains as particular cases Jones and Pewsey (2005): the angle distribution of von Misses ($\psi=0$)

[TABLE]

the cardioid distribution ($\psi=-1$)

[TABLE]

and the wrapped Cauchy distribution ($\psi=1$)

[TABLE]

The distribution (61) has also been used in the analysis of two-dimensional correlated random walks Bartumeus et al. (2008).

Although the number of possibilities to make a choice of the turning-angle distribution is vast, we focus our analysis on two wide-enough classes of the scattering functions: a class of unimodal distributions and one of bimodal ones. Subclasses will be defined by features such as the symmetry with respect the turning angle zero, and will endow with specific properties, to the quantities $\Gamma_{n}$ and $\Omega_{n}$.

IV.1 Unimodal angular distributions

Lets first consider the case of unimodal distributions, which splits into two wide categories: the symmetric scattering-angle distributions around the instantaneous swimming direction, i.e., the distributions $\widetilde{Q}(\varphi)$ for whose single one mode is centered about $\varphi=0,$ or $\pm\pi$; and the asymmetric ones, whose mode is located at some value on the interval $[-\pi,\pi]$, except [math] or $\pi$.

IV.1.1 Symmetric scattering-angle distributions

Smooth-enough unimodal distributions, $\widetilde{Q}_{\text{S}}(\varphi)$, that are symmetrically distributed around the mode $\phi=0$ or around the mode $\phi=\pm\pi$, are of great interest since there is a variety of biological organisms, and artificially designed particles too, that follow this pattern (strategy) of motion (see for instance Ref. Bechinger et al. (2016) for a variety of microswimmers—like Janus particles, E. coli, etc—that can be described by a dynamics of the particle reorientation along the forward direction of motion, and Refs. Zhang et al. (2010); Liu et al. (2019) for organisms that exhibits dynamics of the particle reorientation along the backward direction of motion). When the scattered angle is distributed around the forward direction of motion ($\phi=0$), the motion is highly persistent, and it is perhaps the most ubiquitous pattern of active motion observed. This type of dynamics is widely known as rotational diffusion dynamics Bechinger et al. (2016). On the contrary, motion becomes highly antipersistent if the distribution of scattered angles is centered around $\pm\pi$, a pattern of motion known as run-and-reverse, exhibited by the bacteria Myxococcus xanthus Liu et al. (2019) and a variety of other microorganisms [][andreferencestherein]GrossmannNJP2016. In both cases we have that $\Omega_{n}=0$ for all $n$, since $\widetilde{Q}(\varphi)$, being an even function of $\varphi$ or $\varphi\pm\pi$, makes $\langle\sin\varphi\rangle_{\widetilde{Q}}$ to vanish in this case.

When the direction of motion is frequently scattered forwardly, i.e., around the instantaneous direction of motion, the mode of $\widetilde{Q}_{\text{S}}(\varphi)$ is located at $\phi=0$ and it can be shown that $0<\Gamma_{n}\leq\Gamma_{m}$ whenever $n<m$ (see Fig. 2 for some specific symmetric unimodal distributions). Particularly, we have that $\Gamma_{2}/\Gamma_{1}\geq 1$, or $\Gamma_{2}^{-1}\leq\Gamma_{1}^{-1}$, and the effects of this are revealed by the kurtosis during times before the persistence time, $\Gamma_{1}t\lesssim 1$, (see dashed-dotted-blue and dashed-double-dotted-magenta lines in Fig. 3 for $\Gamma_{2}/\Gamma_{1}=10$ and 100, respectively). In such a period of time, the initial sharp pulse diminishes from its characteristic value $\kappa=4$ giving rise to wakes, which are characteristic of wavelike propagation in two dimensions (see Ref. Sevilla and Nava (2014)). This can be appreciated in Fig. 3, where the transit from the initial sharp pulse ($\kappa=4$) to the Gaussian distribution ($\kappa=8$) is not monotonic.

On the contrary, the inequality $\Gamma_{2}/\Gamma_{1}\leq 1$ is satisfied for scattered angles that frequently occur around the contrary direction to the instantaneous direction of motion, i.e., when the mode $\phi$ is located at $\pi$. Thus, these effects are revealed in the kurtosis at times $t$ for which $\Gamma_{1}t\gtrsim 1$ (see dashed-red line in Fig. 3). There are no wakes in the propagation pulse in short-time regime, but now the distribution becomes conspicuously leptokurtic (more acute than Gaussian for which $\kappa>8$) asymptotically tending to the Gaussian.

These inequalities give a clear insight of the role of the properties of the scattering-angles distribution, $\widetilde{Q}(\varphi)$, on the time evolution of the “shape”of $p_{0}(\bm{x},t)$, characterized by its kurtosis $\kappa(t)$.

The case $\Gamma_{2}/\Gamma_{1}=1$ is of some interest and leads to a simple expression for the kurtosis, namely,

[TABLE]

where we have written $\Gamma_{1}=\Gamma_{2}=\Gamma$. This particular case has as an instance, the well-known pattern of active motion called run-and-tumble, for which $\widetilde{Q}_{\text{S}}(\varphi)=(2\pi)^{-1}$ and therefore $\Gamma_{n}=\Lambda$ for all $n$. Notice that the “shape”of $p_{0}(\bm{x},t)$ changes from the initial sharp pulse to the Gaussian distribution in a monotonic way, as can be deduced from the monotonic-nondecreasing time dependence of the kurtosis ($4\leq\kappa(t)\leq 8$ at all instants). This monotonic growth is representative of many patterns of active motion for which the direction of motion is slightly scattered from the instantaneous one. For comparison purposes, I have included in Fig. 3 the time dependence of the kurtosis for active Brownian motion Sevilla and Nava (2014) (solid-golden line), for which the persistence of the direction of motion is lost by rotational diffusion.

IV.1.2 Asymmetric scattering-angle distributions

Unimodal distributions that consider a frequent scattering of the swimming direction towards directions of motion different from the forward one, or the reverse one, i.e., those that have a mode at angles $\phi\neq 0$, $\pi$, lead naturally to circular motion. This is expected even in the case of forward or reverse scattering, however circular motion emerges as consequence of the skewness of the distribution $\widetilde{Q}(\varphi)$, i.e., when angles are more frequently scattered clockwise or anticlockwise. These statistical considerations allow to describe the motion of circular swimmers, which are ubiquitous in nature and have been observed in a variety of biological organisms and of artificially designed swimmers Crenshaw (1993); Lauga et al. (2006); Shenoy et al. (2007); Schmidt et al. (2008); Friedrich and Jülicher (2008); Marine et al. (2013); Kümmel et al. (2013); Takagi et al. (2013); Gomez-Solano et al. (2016); Narinder et al. (2018), these swimmers have been of theoretical interest leading to diverse models that describe their motion van Teeffelen and Löwen (2008); Friedrich and Jülicher (2009); Ohta and Ohkuma (2009); Weber et al. (2011); Wittkowski and Löwen (2012); Ledesma-Aguilar et al. (2012); Löwen (2016); Kurzthaler and Franosch (2017).

The specific physical processes underlying the stationary scattering-angle distribution $\widetilde{Q}(\varphi)$, define the mode $\phi$. We consider the effects of $\phi$ on the values of $\Gamma_{1}$, $\Gamma_{2}$, $\Omega_{1}$ and $\Omega_{2}$, (whose variations are not independent among them). For the particular case of the wrapped Gaussian ($\alpha=2$) with fixed scale parameter $\sigma=1/4$ and 1/10, and zero skewness, the ratios $\Gamma_{2}/\Gamma_{1}$, $\Omega_{1}/\Gamma_{1}$ and $\Omega_{2}/\Gamma_{1}$ are shown in Fig. 4 (dark-thick lines correspond to $\sigma=0.25$, fuzzy-thin lines to $\sigma=0.1$) to be nonmonotonous functions of $\phi$.

The ratio $\Gamma_{2}/\Gamma_{1}$ (thick-solid-black line) is symmetric about $\phi=0$ and reaches its maximum and minimum values [see Eqs. (91) and (92) in the appendix] at $\phi=0$ and $\phi=\pm\pi$ respectively. The ratio $\Omega_{1}/\Gamma_{1}$ (thick-dashed-red line), which is antisymmetric about $\phi=0$ and gives the frequency of circular motion in units of $\Gamma_{1}$ induced by the distribution asymmetry, has a unique maximum value [given by Eq. (93) in the appendix] at the mode $\phi=\arccos e^{-\sigma^{\alpha}}.$ This mode departs rapidly from the origin as $\sigma$ gets larger (for any $0<\alpha\leq 2$), saturating asymptotically at the value $\pi/2$. Thus, the larger frequency of active-circular motion is found at modes for which the scattered-angles is less than $\pi/2$. The ratio $\Omega_{2}/\Gamma_{1}$ (thick-dashed-dotted-blue line), is also antisymmetric about $\phi=0$ and exhibits a maximum and a minimum in the interval $[0,\pi];$ this occurs due to the two branches of the function $\cos 2\phi/\cos^{3}\phi$ in Eq. (90c) (see appendix) that determines the extrema values of the ratio $\Omega_{2}/\Gamma_{1}$.

As has been pointed out in previous sections, the kurtosis of the particle-position distribution carries information of the ratios $\Gamma_{2}/\Gamma_{1}$, $\Omega_{1}/\Gamma_{1}$ and $\Omega_{2}/\Gamma_{1}$, and, likewise, these ratios carry information about the asymmetry of the unimodal scattering-angle distribution (58), induced by the wrapped Lévy $\alpha$-stable distributions (59). In Fig. 5, the time dependence of the kurtosis is shown for the values of the ratios obtained at the mode $\phi$, that makes $\Omega_{2}/\Gamma_{1}$ to have its maximum and minimum value: $\phi_{\text{max}}\approx 0.316$, $\phi_{\text{min}}\approx 0.242$ for $\sigma=0.25$ (thin-red lines); and $\phi_{\text{max}}\approx 0.139$, $\phi_{\text{min}}\approx 2.238$ for $\sigma=0.1$ (thick-blue lines).

IV.2 Bimodal scattering-angle distributions

It has been observed a variety of organisms that exhibit a bimodal distribution of scattering angles in their pattern of motion Duffy and Ford (1997); Theves et al. (2013), and this bimodality has profound consequences on the spatial distributions of the particles. For the sake of clarifying this and in spite of the general analysis that can be carried out from our formalism for arbitrary scattered-angle distribution, we consider the limit case that corresponds to the bimodal distribution of scattered angles, with modes at the angles $\varphi_{1}$, $\varphi_{2}$, and of zero width, i.e.,

[TABLE]

where $0<\nu<1$ gives a weighing factor to each mode of the distribution.

IV.2.1 Symmetrically distributed modes

Consider the bimodal scattering-angle distribution of zero width

[TABLE]

where the modes are located symmetrically with respect to the forward direction at $\pm\varphi_{0}$ with $0<\varphi_{0}<\pi$; and $0<\nu<1$ gives the weight of each mode making the scattering-angle distribution asymmetric if $\nu\neq 1/2$. It can be noticed from Eq. (14a) that $\Gamma_{n}$ is independent of $\nu$ for all $n$, having $\Lambda(1-\cos n\varphi_{0})$ as its value for given $n$ and $\varphi_{0}$. Also notice that the persistence time, $\Gamma^{-1}$, becomes arbitrarily large as $\varphi_{0}$ vanishes. In contrast, $\Omega_{n}$ does explicitly depend on $\nu$ as $\Lambda(2\nu-1)\sin n\varphi_{0}$.

The ratios $\Gamma_{2}/\Gamma_{1}$, $\Omega_{1}/\Gamma_{1}$ and $\Omega_{2}/\Gamma_{1}$, that give the full characterization of the kurtosis of the particle position distribution, can be calculated explicitly giving

[TABLE]

From these expressions, several diffusive properties in terms of the parameters $\varphi_{0}$ and $\nu$ are obtained. First, after setting $\Omega_{1}/\Gamma_{1}=1$ in Eq. (68b), the maximum value of the effective diffusion coefficient [see Eq. (51)] is obtained whenever $\nu=\left[1+\tan(\varphi_{0}/2)\right]/2$, with $0<\varphi_{0}<\pi/2$. The contour lines, $\nu=\nu(\varphi_{0})$, defined by fixing the ratio $\Omega_{1}/\Gamma_{1}$ to a constant $\chi$ are shown in Fig. 6.

In Fig. 7, the time dependence of the kurtosis is shown as function of the dimensionless time $\Gamma_{1}t$, for the values of the ratios $\Gamma_{2}/\Gamma_{1}$ and $\Omega_{2}/\Gamma_{1}$, that correspond to the values of $\varphi_{0}^{*}$ that makes $\nu=1$ for a given ratio of $\chi=\Omega_{1}/\Gamma_{1}$: Oscillations are observed for times smaller or of the order of the persistence time for $\chi=10$, 2 (dashed-dotted lines), for which $\Gamma_{2}/\Gamma_{1}=3.96$, 3.2 and $\Omega_{2}/\Gamma_{1}=19.6$, 2.4 respectively. A smooth transition from a sharp pulse and the Gaussian distribution is observed for $\chi=1$ (maximum effective diffusion coefficient marked by the solid line), for which $\Gamma_{2}/\Gamma_{1}=2$ and $\Omega_{2}/\Gamma_{1}=0$. Such a transition is still smooth for $\chi=0.5$ (thick-dashed line, $\Gamma_{2}/\Gamma_{1}=0.8$, $\Omega_{2}/\Gamma_{1}=-0.6$). The transition between the sharp pulse and the Gaussian distribution becomes nonmonotonic again, but now for times larger than the persistence time, when $\chi=0.1$ (thin-dashed line), for which $\Gamma_{2}/\Gamma_{1}=0.0396$ and $\Omega_{2}/\Gamma_{1}=-0.196$.

IV.2.2 Run-and-reverse

Another instance of a simple bimodal distribution that can be analyzed to some detail, is given by the pattern of motion called run-and-reverse. This pattern considers the scattering of the direction of motion along the forward and backward direction, thus having modes at $\phi=0$ and $\pi$, respectively. In the case of zero width distribution, it can be written as

[TABLE]

Notice that $\Gamma_{n}$ vanishes for even $n$, and gives $2\Lambda(1-\nu)$ for all odd $n$, while $\Omega_{n}$ vanishes for all $n$. With this, the expansion (15) can be written as

[TABLE]

from which a particular dynamics can be noticed, namely, there is a highly directional dependence in the long-time regime, as is evidenced by the fact that the second term in (70) contributes to $\widetilde{P}(\bm{k},\varphi,t)$ in such regime. This clearly contrasts with other patterns of active motion, for which $\widetilde{p}_{0}(\bm{k},t)$ gives the only contribution in the long-time regime as has been discussed in the previous sections. $\Gamma_{1}=2\Lambda(1-\nu),$ denotes the value of $\Gamma_{n}$, $n$ being odd.

The time dependence of the kurtosis can be obtained explicitly in this case,

[TABLE]

and is shown in Fig. 8. In the asymptotic limit, the mean-squared displacement is linear in time with effective diffusion coefficient $v_{0}^{2}/4\Lambda(1-\nu)$, while the kurtosis of the spatial distribution of the active particles goes to the value 12 (see Fig. 8), which differs conspicuously from the value 8 that characterizes the two-dimensional Gaussian distribution. This scenario illustrates another instance of a diffusive process called “anomalous, yet Brownian, diffusion” Wang et al. (2009); Wang Bo et al. (2012); Bhattacharya et al. (2013), which has been addressed theoretically in different one-dimensional models Cressoni et al. (2012); Chubynsky and Slater (2014); Wang et al. (2016) and in a three-dimensional study of chiral active motion Sevilla (2016).

For the scattering-angle distribution (69) we have that the connecting function (30) acquires the simple form

[TABLE]

which manifestly exhibits the singular role of the persistence time as the only timescale present in the dynamics and of the fact that in the long-time regime ($\epsilon\rightarrow 0$) no approximant of the connecting function is possible. If the width at the modes of Eq. (69) is made finite, the Gaussian distribution is recovered in the long-time regime since, the whole hierarchy of the $\Gamma_{n}$ being recovered, the uniform scattering of the direction of motion in such a regime is assured.

V Conclusions

I have presented a theoretical framework for the statistical analysis of the two-dimensional motion of active swimmers. This framework generalizes existent ones in that considers an arbitrary navigating strategy, that also takes into account circular motion, embedded in the arbitrary distribution of scattered angles of the particle’s swimming direction. The framework is susceptible for generalizations, indeed, the transition rate of the direction of motion $K_{A}(\varphi|\varphi^{\prime})$ in Eq. (1) can take into account a spatial and a temporal dependence, and also complements others that focus on the time distribution between fixed turning events. The method of solution presented, allowed for an exact analytical expression for the marginal probability distribution of finding a swimmer at $\bm{x}$ at time $t$, independently of the direction of motion. Such a solution can be cast as the exact solution of the generalized diffusion equation (29), and an explicit expression for the time-space dependent memory function is presented. This result opens the door to consider the generalized diffusion equation (29) as a well-founded framework to analyze the motion of active swimmers.in particular, to consider time-space coupled memory function to describe other variety of patterns of active motion, as the ones described by Lévy walks .

I also presented exact calculations for the time dependence of the mean-square displacement, which depends only on the ratio of the frequency of the circular motion induced by the specific scattering-angle distribution that embeds the pattern of active motion, $\Omega_{1}$, to the persistence time $\Gamma_{1}$. Certainly, there are plenty of patterns of motion that lead to the same ratio $\Omega_{1}/\Gamma_{1}$, and as such, the mean-square displacement is typical of many of them. However, the differences among different patterns of motion are unveiled in the intermediate-time regime if more information of the pattern of motion is considered (as analyzed experimentally and theoretically for active Brownian motion and run-and-tumble particles in Ref. Kurzthaler et al. (2018)), and not only those related to $\langle\cos\varphi\rangle_{\widetilde{Q}}$ and $\langle\sin\varphi\rangle_{\widetilde{Q}}$, as is the case for the mean-square displacement.

It was shown that consideration of $\Gamma_{2}$ and $\Omega_{2}$, besides $\Gamma_{1}$ and $\Omega_{1}$, is enough to distinguish some features among different patterns of motion. Certainly, knowledge of these quantities allows the exact calculation of the time dependence of the kurtosis, which gives information about the “shape” of the particle’s position distribution. Some patterns of motion induce a smooth transition with time, from the initial sharp pulse, to the Gaussian of the long-time regime. Others deviate from this behavior and transit, from the initial sharp pulse to the Gaussian distribution, in a rather complex way characterized by oscillations.

Finally, although exact solutions to the Fokker-Planck equation (5) are known for the particular case of the uniform scattering-angle distribution $\widetilde{Q}(\varphi)=(2\pi)^{-1}$ Martens et al. (2012) (and for the three-dimensional active Brownian motion Kurzthaler Christina et al. (2016)), the analysis presented in this paper provides a broad understanding of the influence of an arbitrary pattern of motion on the statistical properties of the active swimmers, and encourages the development of more general theoretical frameworks of active motion that allow the incorporation of more general conditions.

Acknowledgements.

The author kindly acknowledges Juan Manuel Pérez Peña for his interest during the initial part of the research presented in this paper. This work was supported by UNAM-PAPIIT IN114717.

Appendix

.1 The convolved solution (2)

Since the transport equation (1) describes the diffusion process of an active particle in unconfined space, the Fourier transform can be applied to it, thus

[TABLE]

The solutions of the last equation admits the separation form

[TABLE]

where $\widetilde{P}(\bm{k},\varphi,t)$ satisfies the equation

[TABLE]

and $\widetilde{G}_{D_{T}}(\bm{k},t)=e^{-D_{T}\bm{k}^{2}t}$. Notice that expression (2) corresponds to the inverse Fourier transform of the convolved solution (74), Eq. (75) corresponds to an equivalent form of the Fourier transform of Eq. (3), and the Fourier inverse $\widetilde{G}_{D_{T}}(\bm{k},t)$, $G_{D_{T}}(\bm{x},t)$ satisfies the diffusion equation

[TABLE]

.2 Derivation of Eq. (26)

The explict and exact solution for $\widetilde{p}_{0}(\bm{k},\epsilon)$ given by Eq. (26) is obtained as follows. After taking the Laplace transform of (16) we have that this can be written as

[TABLE]

with $\widetilde{p}_{n}^{(0)}(\bm{k})=(2\pi)^{-1}\delta_{n,0}$ are the initial conditions. Notice that the corresponding Laplace argument of $\widetilde{p}_{n\pm 1}(\bm{k},\epsilon)$ is shifted by $\lambda_{n\pm 1}-\lambda_{n}$ respectively.

For $n=0$ we have that

[TABLE]

where we have used that $\lambda_{0}=0$. By use of the recurrence relation (77), $\widetilde{p}_{\pm 1}(\bm{k},\epsilon+\lambda_{\pm 1})$ can be written in terms of $\widetilde{p}_{0}(\bm{k},\epsilon)$ and $\widetilde{p}_{\pm 2}(\bm{k},\epsilon+\lambda_{\pm 2})$ and the last equation can cast into

[TABLE]

In turn, $\widetilde{p}_{\pm 2}(\bm{k},\epsilon+\lambda_{\pm 2})$ can be written in terms of $\widetilde{p}_{0}(\bm{k},\epsilon)$ and $\widetilde{p}_{\pm 3}(\bm{k},\epsilon+\lambda_{\pm 3})$ by use of the recurrence relation (77), and thus Eq. (79) can turn into

[TABLE]

and in turn, $\widetilde{p}_{\pm 3}(\bm{k},\epsilon+\lambda_{\pm 3})$ can be written in terms of $\widetilde{p}_{0}(\bm{k},\epsilon)$ and $\widetilde{p}_{\pm 4}(\bm{k},\epsilon+\lambda_{\pm 4})$ by use of the recurrence relation (77), and so on. Thus the factor of $\widetilde{p}_{0}(\bm{k},\epsilon)$ corresponds to the denominator of Eq. (26).

.3 The second and fourth moments of $p_{0}(\bm{x},t)$

By use of the Eq. (46), the Laplace transform of the mean-squared displacement is obtained by substitution of the probability density $\widetilde{p}_{0}(\bm{k},\epsilon)$ given by Eq. (27) with the initial condition $\widetilde{p}_{0}^{(0)}(k)=1/2\pi$, we have that

[TABLE]

After evaluation of the second-order derivative and evaluating at $k=0$ all terms proportional to $k$ and $k^{2}$ vanish, thus getting $\left[v_{0}^{2}\mathfrak{D}(,\epsilon)/\epsilon^{2}\right]_{k=0}$ which corresponds to Eq. (47). With Eq. (33) we get explicitly that

[TABLE]

which can be inverted by the use of the convolution theorem of the Laplace transform, to have

[TABLE]

By writing $\lambda_{1}=\Gamma_{1}+i\Omega_{1}$ and after evaluation of the elementary integrals we get Eq. (48).

Analogously, the Laplace transform of the fourth moment is obtained from Eq. (45) with $n=2$, i.e. from

[TABLE]

After evaluating the fourth-order derivative the only terms that do not vanish are given in Eq. (54). Finally, Eq. (55) is obtained as follows: From Eq. (32), we have that

[TABLE]

From the definitions (31), $\Delta_{0}(\bm{k},\epsilon)$ [and analogously $\overline{\Delta}_{0}(\bm{k},\epsilon)$] can be written as

[TABLE]

and thus,

[TABLE]

where $\partial_{k}$ denotes the partial derivative with respect $k$. A similar expression is obtained for $\left(\partial^{2}/\partial k^{2}\right)\overline{\Delta}_{0}(\bm{k},\epsilon)$. Thus, by evaluating these results at $k=0$, the expression

[TABLE]

is obtained. From (31) we have that $\Delta_{1}(\bm{k},\epsilon)|_{k=0}=1/(\epsilon+\lambda_{2})$, and $\overline{\Delta}_{1}(\bm{k},\epsilon)|_{k=0}=1/(\epsilon+\lambda_{2}^{*})$, and the Eq. (55) follows.

.4 Extrema of the ratios $\Gamma_{2}/\Gamma_{1}$, $\Omega_{1}/\Gamma_{1}$ and $\Omega_{2}/\Gamma_{1}$ for the Lévy alpha-stable

distributions

From Eqs. (60) we have the ratios

[TABLE]

in function of the mode $\phi$ for the distribution Lévy alpha-stable distribution (59) with vanishing skewness [$\beta=0$ in Eq. (59)]. $\Gamma^{-1}$ gives account of the persistence time. The ratios (89) are nonmotonous functions of $\phi$ and their corresponding extrema values are obtained from the equations

[TABLE]

respectively.

The maximum value of $\Gamma_{2}/\Gamma_{1}$,

[TABLE]

occurs at $\phi=0$, while the minimum,

[TABLE]

occurs at $\phi=\pm\pi$ as is shown in Fig. 4 for $\alpha=2$ and $\sigma=1/4,$ 1/10.

The ratio $\Omega_{1}/\Gamma_{1}$ has its maximum value

[TABLE]

at the sole extreme $\phi=\arccos e^{-\sigma^{\alpha}}$.