Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies

To generate long-time correlated trajectories we consider two statistically independent fBm processes, $B_{1}^{H}(t)$ and $B_{2}^{H^{\prime}}(t)$, characterized by Hurst exponents $0<H,H^{\prime}<1$, respectively. These are Gaussian stochastic processes with $\langle B^{H}(t)\rangle=0$ and autocorrelation function

where $D_{H}$ (with units of [length]${}^{2}\times$[time]^-2H) measures its amplitude. Eq. (3) reduces to the auto-correlation of the Wiener process $2D_{1/2}\textrm{min}(t,s)$ when $H=\frac{1}{2}$ and gives $\langle[B^{H}(t)]^{2}\rangle=D_{H}\,t^{2H}$ at equal times for all $H$ in $(0,1)$. The fBm process is also defined as the integral of fractional Gaussian noise $\xi^{H}(t)$, as $B^{H}(t)=\int_{0}^{t}ds\,\xi^{H}(s)$, $\xi^{H}(t)$ is a stationary stochastic process with autocorrelation function $\langle\xi(t)\xi(0)\rangle=2HD_{H}t^{2H-1}[(2H-1)t^{-1}+\delta(t)]$ Qian (2003), from which is recognized a time-dependent diffusion coefficient $D(t)=2HD_{H}t^{2H-1}$.

We are interested in the case for which $H=H^{\prime}$, however an extension to the anisotropic case is straightforward. The increments of the fBms during the time interval $\Delta t=t-t^{\prime}$, $t>t^{\prime}$, $\delta B_{1}^{H}(\Delta t)\equiv B_{1}^{H}(t)-B_{1}^{H}(t^{\prime})$ and $\delta B_{2}^{H}(\Delta t)\equiv B_{2}^{H}(t)-B_{2}^{H}(t^{\prime})$ (corresponding to the increments $\delta U(\Delta t)$ and $\delta V(\Delta t)$ respectively), are stationary, long-time correlated, and not independent if $H\neq 0.5$. In the case $0<H<\frac{1}{2}$ they are negatively correlated, while they are positively correlated if $\frac{1}{2}<H<1$ Biagini et al. (2008). In this case the increment $\delta\boldsymbol{r}_{n}$ is given by $\delta B_{1}^{H}(\Delta t)\,\hat{\boldsymbol{u}}_{n}+\delta B_{2}^{H}(\Delta t)\,\hat{\boldsymbol{v}}_{n}$, with $\{\hat{\boldsymbol{u}}_{n},\hat{\boldsymbol{v}}_{n}\}$ an orthonormal vector basis for $\mathbb{T}_{\boldsymbol{r}_{n}}\mathbb{S}^{2}$.

We found that the long-time correlations of fBm ($H\neq\frac{1}{2}$) collude with the specific implementation of the updating rule (1) through the choice of the orthonormal basis $\{\hat{\boldsymbol{u}}_{n},\hat{\boldsymbol{v}}_{n}\}$, which defines a kind of navigation strategy that drives the particle motion to give rise to a rich variety of statistical patterns of motion on the sphere. On the contrary, uncorrelated motion (as is the standard Brownian motion, $H=1/2$) is insensitive to the choice of such strategies [see (a.2) and (b.2) for a qualitative comparison in the right panel in Fig. 1].

We consider for our analysis two contrasting physically-motivated navigation strategies, one is observed in many biological organisms which may have a distinct body axis (head-tail axis) defining a preferred direction of motion sometimes referred as heading. In such a case the particle heading $\hat{\boldsymbol{t}}_{n}$, together with the normal vector $\hat{\boldsymbol{n}}_{n}$, form the orthonormal basis for the updating rule (1) that carry the long-term correlated motion. Notice that with $\hat{\boldsymbol{r}}_{n}$ (the binormal vector), these three vectors form the well-known Frenet-Serret system. Three sample trajectories are shown: (b.1), (b.2) and (b.3), in the second column of the right panel of Fig. 1, for highly anticorrelated motion, $H=0.1$ (b.1), uncorrelated motion, $H=0.5$ (b.2), and highly correlated motion, $H=0.9$ (b.3).

The other navigation strategy corresponds to the case for which the orthonormal basis for the updating rule, $\{\hat{\boldsymbol{\theta}}_{n},\hat{\boldsymbol{\phi}}_{n}\}$, forms a kind of “laboratory reference frame”, which for our analysis is chosen to be the one induced by the spherical coordinates of $\mathbb{S}^{2}$, and consequently attached to each point of the sphere surface, namely, $\hat{\boldsymbol{\theta}}_{n}=(\cos\theta_{n}\cos\phi_{n},\cos\theta_{n}\sin\phi_{n},-\sin\theta_{n})$, oriented along the sphere meridians, and $\hat{\boldsymbol{\phi}}_{n}=$ $(-\sin\theta_{n}$ $\sin\phi_{n},$ $\sin\theta_{n}\cos\phi_{n},0)$ orthogonally oriented along the parallels (see left panel in Fig. 1). As a consequence, the long-time correlated trajectories describe a distinct pattern as is shown by the trajectories shown in the right panel of Fig. 1 for $H=0.1$ (a.1) and $H=0.9$ (a.3). Self-confinement is observed for $H\neq 0.5$, each particle excursion is limited to a sector of the sphere around well-defined time-average values $\overline{\theta}$, $\overline{\phi}$ that depend on $H$. For $0<H<0.5$, particles concentrate forming an island around $\overline{\theta}=\pi/2$ (equator) with $\overline{\phi}$ randomly chosen; while for $0.5<H<1$ distinctively, the particles concentrate their excursions around the poles, i.e. $\overline{\theta}=0,\pi$, wandering between them as is shown in Fig. 1(a.3). For this navigation strategy, ensemble averages strikingly differs from time averages giving rise to weak ergodicity breaking Thiel and Sokolov (2014), especially for $0<H<0.5$, for which single trajectories get trapped diffusing around a randomly chosen sector of the sphere, thus, the time average using a single long trajectory will differ from the ensemble average which samples uniformly those sectors; the equivalence between ensemble averages and time averages is recovered for $H=1/2$ [see a trajectory generated in Fig. 1(a.2)].

The other pattern of stochastic motion we address in this paper refers to sBm. The updating rule (1) in this case considers two independent Gaussian processes $G_{1}(t)$ and $G_{2}(t)$, with statistically independent increments $\delta G_{1}(\Delta t_{n})=\sqrt{2{D_{H}\bigl{[}{t_{n}}^{2H}-{t_{n-1}}^{2H}\bigr{]}}}\,\Delta W_{1}$, and $\delta G_{2}(\Delta t_{n})=\sqrt{2{D_{H}\bigl{[}{t_{n}}^{2H}-{t_{n-1}}^{2H}\bigr{]}}}\,\Delta W_{2}$, for the time increment $\Delta t_{n}=t_{n}-t_{n-1}$, $n=1,\,2\,\ldots$; $\Delta W_{1}$, $\Delta W_{2}$ are two independent Wiener processes of vanishing mean and unitary variance; and $t_{n}=\sum_{k=1}^{n}\Delta t_{k}$ being the total time elapsed up to step $n$. Scaled Brownian motion is highly non-stationary due to the time-dependence of the diffusion coefficient, a feature that is implemented by considering $\delta\boldsymbol{r}_{n}=\sqrt{2D_{H}{\Delta t}^{2H}\bigl{[}n^{2H}-(n-1)^{2H}\bigr{]}}\Bigl{[}\Delta W_{1}\,\hat{\boldsymbol{u}}_{n}+\Delta W_{2}\,\hat{\boldsymbol{v}}_{n}\Bigr{]}$, where we have assumed homogeneous time increments $\Delta t$, and the pair of orthonormal vectors $\{\hat{\boldsymbol{u}}_{n},\hat{\boldsymbol{v}}_{n}\}$ is an arbitrary vector basis for $\mathbb{T}_{\boldsymbol{r}_{n}}\mathbb{S}^{2}$. It is the statistical independence among the increments what guaranties that the updating rule is independent of the basis choice, as occurs for uncorrelated standard Brownian motion (or fBm with $H=0.5$). Three sBm trajectories are shown in the third column of the right panel of Fig. 1 for $H=0.1$ (c.1), $H=0.5$ (c.2) and $H=0.9$ (c.3), which are contrasted with those for the two navigation strategies considered for the fractional Brownian motion case (trajectories in the first and second columns in the same figure).

The corresponding modification of the Euclidean Fokker-Planck equation that considers that sBm occurs on $\mathbb{S}^{2}$ is

where $\boldsymbol{r}=(r\sin{\theta}\cos{\phi},r\sin{\theta}\sin{\phi},r\cos{\theta})$ denotes the particle’s position in spherical coordinates, $r$ being the radius of $\mathbb{S}^{2}$. The solution of (4) is given by

the coefficients $p_{l}^{m}$ are determined from the initial distribution, and $Y_{l}^{m}(\theta,\phi)$ are the standard spherical harmonics. If the initial condition corresponds to a localized distribution at the north pole we get

with $\textrm{P}_{n}(\cos\theta)$ the Legendre polynomials. The absence of $\phi$ in (6) implies the azimuthal symmetry of the process, which leads to the moments of the polar angle $\theta$, $\langle\theta^{n}\rangle_{\textrm{sBm}}=\int d\Omega\,\theta^{n}P_{\textrm{sBm}}(\theta,\phi,t)$ given by

where $g_{\theta^{n}}(l)=\int_{0}^{\pi}d\theta\,\theta^{n}\,(\sin\theta/2)\,\textrm{P}_{l}(\cos\theta)$. It is clear from (6) that for each $H$, the marginal stationary distribution of the polar angle, $P_{eq}(\theta)=\sin\theta/2$, is attained in the long-time regime, indicating the uniform distribution of the particle positions on $\mathbb{S}^{2}$.

In addition the auto-correlation function of the particle positions can be computed straightforwardly and is given by

Notice that the time dependence of the quantities computed from Eq. (4), namely $\langle\theta^{n}\rangle_{\textrm{sBm}}$ and $\langle\hat{\boldsymbol{r}}(t)\cdot\hat{\boldsymbol{r}}(0)\rangle_{\textrm{sBm}}$, become $H$-invariant under the time-scaling transformation $d\tau_{H}=2HD_{H}\,t^{2H-1}\,dt$. Such transformation makes Eq. (4) $H$-invariant leading to dynamical scaling as can be checked straightforwardly.

We present an statistical analysis of a set of ensembles of trajectories generated with the method described in the previous paragraphs: Two subsets corresponding to fBm, one for each of the navigation strategies considered; and one subset corresponding to sBm. In the case of fBm each ensemble consisted of 5$\times 10^{3}$ trajectories for each of the nine values of the $H$ considered, namely $H=0.1,0.2,\ldots,0.9$ for which we have set $D_{H}=0.1$ in arbitrary units. For sBm the absence of long-time correlations allow to consider ensembles of 1.2$\times 10^{5}$ trajectories.

Each ensemble of trajectories was generated with fixed initial position at the sphere’s north pole $\boldsymbol{r}_{0}=(0,0,1)$, this choice allows us to assume an azimuthal symmetry in the ensemble average and therefore to focus on the frequency distribution of the polar angle $\theta$ only. From each ensemble, the frequency histograms of the polar angle $\overline{P}(\theta,t|\theta_{0},t_{0})$, with $\theta_{0}=0$ at $t_{0}=0$, is computed at each time step giving rise to the time evolution shown in Figs. 2 and 3; in Fig. 2 we show the case for fractional Brownian motion, for which long-time correlations make manifest the role of the navigation strategy through the basis system chosen at each $\mathbb{T}_{\boldsymbol{r}}\mathbb{S}^{2}$. In Fig. 3 we show the corresponding histograms for the case of sBm.

For the Frenet-Serret navigation strategy $\{\hat{\boldsymbol{t}},\hat{\boldsymbol{n}}\}$ (second column of Fig. 2), the system attains the stationary distribution $P_{eq}(\theta)$ for each $H$, this is shown in the evolution of the frequency histograms in the plots of the second column of Fig. 2 for $H=0.1$ (b.1), 0.5 (b.2) and 0.9 (b.3) respectively ($P_{eq}$ is depicted by the dashed-red line). Notice that the relaxation times towards $P_{eq}(\theta)$ depend on $H$, being shorter the smaller $H$ is [compare the time scales considered from (b.1) to (b.3) histograms in the second column of Fig. 2]. In contrast, for the navigation strategy $\{\hat{\boldsymbol{\theta}},\hat{\boldsymbol{\phi}}\}$, the system attains a non-equilibrium stationary state $P_{\textrm{ness}}(\theta)$ (except for the uncorrelated case $H=0.5$, for which the attained stationary state is independent of the navigation strategy and coincides with $P_{eq}(\theta)$ shown in the histograms of (a.2) of Fig. 2). For anti-correlated motion ($0<H<0.5$) the frequency histograms settle into a unimodal distribution, highly peaked around the equator line ($\theta=\pi/2$) the smaller the value of $H$ [see histograms in (a.1) of Fig. 2 for $H=0.1$], the distribution transforms continuously into $P_{eq}(\theta)$ as $H\rightarrow 0.5$ (a.2). For correlated motion ($0.5<H<1$) the stationary distribution settles into a bimodal distribution with modes at the sphere poles [see histograms in (a.3) of Fig. 2 for $H=0.9$], such stationary distribution is determined by the specific basis chosen only, and does not depend on the initial distribution. Thus, the poles can be interpreted as being repulsive if $0<H<0.5$, and being attractive if $0.5<H<1$. These non-equilibrium stationary distributions, when $H\neq 0.5$, are attained independently of the initial distribution.

Analogous effects to the ones observed with the basis induced by the spherical coordinates have been observed in confined fBm, specifically as result of the interplay of long-time correlations and hard-walls boundaries Wada and Vojta (2018); Guggenberger et al. (2019); Vojta et al. (2020), where depleted regions for the probability distribution of the particle positions are observed near the boundaries for anticorrelated motion ($0<H<0.5$), while accreted regions near the boundaries are observed for $0.5<H<1$.

In Fig. 3 shows $\overline{P}(\theta,t|\theta_{0},t_{0})$ for sBm. $P_{eq}$ is attained for all $H$ and the cases $H=0.1$ (c.1), $0.5$ (c.2) and $0.9$ (c.3) are shown. Excellent agreement is observed except for the short-time regime of $H=0.1$ and 0.2, where there is some discrepancy due to numerical stability.