Generalized Ornstein-Uhlenbeck model for active motion

We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E 90, 012111 (2014)]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.

Figure 1

(a) Examples of trajectories $x\left(t\right)$ evolving according to the generalized Ornstein-Uhlenbeck model (1), with memory kernel given by (18), for $\alpha =0.9$ and different values of the memory time $\tau$. From bottom to top: $\tau =0.1,0.2,0.4,0.8,1.6,3.2,6.4,12.8,25.6,400$. Inset: Expanded view the active trajectory with $\tau =12.8$. (b) Corresponding velocity autocorrelation function for different values of $\tau$, same color code as in (a). The values of $\tau$ increase from left to right. Inset: Expanded view for $\tau =400$. (c) Mean-squared displacements of the trajectories shown in (a). The values of $\tau$ increase from bottom to top. Inset: Expanded view of the intermediate regime around ${\tau }_R$. (d) Dependence on the relaxation time $\tau$ of the frequency of the damped oscillations, which emerge only between ${\tau }_{+}$ ($▵$) and ${\tau }_{-}$ ($▿$), for different values of $\alpha$ increasing from inner to outer curves: $\alpha =0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$. (e) Velocity autocorrelation function and (f) mean-squared displacement, obtained from the analytical expression, for the same parameters plotted in (b) and (c), respectively.

Figure 2

(a) Examples of trajectories $x\left(t\right)$ evolving according to the generalized Ornstein-Uhlenbeck model (1) with power-law memory kernel (36) and fractional Brownian noise (37) in the absence of thermal fluctuations for different values of the Hurst parameter increasing from bottom to top: $H=0.525,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95,0.975$. (b) Velocity autocorrelation function and (c) corresponding mean-squared displacement for the different values of the Hurst parameter in (a), same color code. In (b) and (c), the values of $H$ increase from inner to outer curves and from top to bottom, respectively. The insets in (b) and (c) corresponds to the velocity autocorrelation function and the mean-squared displacement for $H=0.975$ and different values of ${\gamma }_0/{\tau }_R$; from left to right and bottom to top, respectively: ${\gamma }_0/{\tau }_R=4,2,1,0.5,0.25$. (d) Exponent $\beta$ of the long-time behavior ($t\gg {\tau }_R$) of the mean-squared displacement as a function of $H$. The symbols ($\square$) mark the values obtained from the numerical solutions, whereas the solid line represents $2-2H$. Inset: Frequency of the velocity oscillations for $H=0.975$ as a function of ${\gamma }_0/{\tau }_R$. The dashed line represents Eq. (42). (e) Velocity autocorrelation function and (f) mean-squared displacement, directly computed from the analytical expressions (40) and (43), respectively, for the same values of $H$ as those shown in (b) and (c).

Figure 3

(a) Velocity autocorrelation function for the active Ornstein-Uhlenbeck model with the power-law memory kernel (36) and fractional Brownian noise at $H=0.975$. (b) Resulting active trajectories for different translational diffusion coefficients from top to bottom: $D_T=0,{10}^{-3},{10}^{-2},{10}^{-1},1,10$. (c) Corresponding mean-squared displacements. Same color code as in Fig. 3. The values of $D_T$ increase from bottom to top. The solid lines represent Eq. (45).