Gaussian memory in kinematic matrix theory for self-propellers

We extend the kinematic matrix (“kinematrix”) formalism [Phys. Rev. E 89, 062304 (2014).], which via simple matrix algebra accesses ensemble properties of self-propellers influenced by uncorrelated noise, to treat Gaussian correlated noises. This extension brings into reach many real-world biological and biomimetic self-propellers for which inertia is significant. Applying the formalism, we analyze in detail ensemble behaviors of a 2D self-propeller with velocity fluctuations and orientation evolution driven by an Ornstein-Uhlenbeck process. On the basis of exact results, a variety of dynamical regimes determined by the inertial, speed-fluctuation, orientational diffusion, and emergent disorientation time scales are delineated and discussed.

Figure 1

The coupling of deterministic and stochastic elements can give rise to many kinds of distinctive motion and spreading patterns, such as a rectilinear swimmer with speed fluctuation [13, 53] and persistent turning [31, 32], a nanorotor with flipping [42], E. coli circle swimming [7, 54] or magnetotactic bacteria with occasional velocity reversal [55, 56, 57, 58].

Figure 2

The disorientation time for a self-propeller to forget its initial orientation depends on the ratio of inertial ${\tau }_{{}_{\xi }}$ and orientational diffusion $D_{\text{o}}^{-1}$ time scales. If the ${\tau }_{{}_{\xi }}\ll D_{\text{o}}$ we are close to the white noise limit and the self-propeller disorients over an orientational diffusion time scale ${\tau }_{\theta }\sim D_{\text{o}}^{-1}$. On the other hand, when the inertial time is much larger than the orientational time scale, the disorientation time is the geometric average of inertial and orientational diffusion time scales, ${\tau }_{\theta }\sim {\left({\tau }_{{}_{\xi }}{D}_{\text{o}}^{-1}\right)}^{1/2}$.

Figure 3

Effective diffusion coefficient $D_{\text{eff}}$ of the linear self-propeller with orientational Gaussian noise characterized by correlation time ${\tau }_{{}_{\xi }}$, speed fluctuations characterized by correlation time ${\tau }_v$ and asymptotic orientational diffusion coefficient $D_{\text{o}}$. (a) The mean-speed and fluctuation speed make contributions to $D_{\text{eff}}$ proportional to $\Phi (D_{\text{o}}{\tau }_{{}_{\xi }},\infty )$ and $\Phi (D_{\text{o}}{\tau }_{{}_{\xi }},D_{\text{o}}{\tau }_v)$, respectively. (b) $\Phi$ shows two major regimes depending upon whether the speed correlation time is smaller or larger than the disorientation time ${\tau }_{\theta }$. In the former case, the diffusion is essentially determined by speed fluctuations and in the latter by orientational wandering. The disorientation time ${\tau }_{\theta }$ is proportional to ${\left(D_{\text{o}}^{-1}{\tau }_{{}_{\xi }}\right)}^{1/2}$ when $D_{\text{o}}\ll {\tau }_{{}_{\xi }}$, and saturates to $\sim 1$ as $D_{\text{o}}{\tau }_{{}_{\xi }}\to 0$. (c) $D_{\text{eff}}$ may contain several crossovers as a result of the relative sizes and individual crossovers of the mean-speed and fluctuation-speed components. Note that for $D_{\text{o}}{\tau }_{{}_{\xi }}\gg [\mathrm{var}\left(v_s\right)/{\overline{v}}_s^2]{\left(D_{\text{o}}{\tau }_v\right)}^2$, the mean-speed component always dominates. (d) Slices through $D_{\text{eff}}$ in the other direction, normalized to the $D_{\text{o}}{\tau }_{{}_{\xi }}=0$ value.

Figure 4

Mean-squared displacement of the linear self-propeller with Gaussian orientational noise of correlation time ${\tau }_{{}_{\xi }}$, signed-speed fluctuations with correlation time ${\tau }_v$, and asymptotic orientational diffusion coefficient $D_{\text{o}}$. If the self-propeller disorients faster than signed-speed changes value, ${\tau }_{\theta }\ll {\tau }_v$, (upper curves) we observe a single crossover from ballistic to diffusive regimes at ${\tau }_{\theta }$; with increase in inertial time ${\tau }_{{}_{\xi }}$ the crossover happens later. In the opposite regime ${\tau }_{\theta }\gg {\tau }_v$ (lower curves) for large signed-speed fluctuations $\mathrm{var}\left(v_s\right)/{\overline{v}}_s^2\gg 1$ the signed-speed fluctuation contribution shows a one-dimensional ballistic to diffusive crossover about ${\tau }_v$. Combined with mean signed-speed part (behaving like the ${\tau }_{\theta }\ll {\tau }_v$ case), this produces three crossovers.