Effects of cross-correlated noises on the transport of active Brownian particles

The transport properties of active Brownian particles driven by cross-correlated noises are investigated. Using the Langevin and Fokker-Planck approaches, the theoretical analysis of the model is presented. It is found that correlated noises can produce a net velocity, which stems from the symmetric breaking of the system induced by the correlation between noises. The mean velocity is negative for positive correlation but positive for negative correlation. The mean velocity increases while the effective diffusion decreases as the absolute value of the correlation between the noises increases. Both the mean velocity and the effective diffusion show a nonmonotonic dependence on the multiplicative noise, but a monotonically decreasing dependence on the additive noise.

Figure 1

(a) and (b) The numerical simulation of the first moment $\langle x\left(t\right)\rangle$ and variance of the displacement $\langle \Delta x^2\left(t\right)\rangle$ of ABP as a function of time $t$ for different correlation strength $\lambda$ between noises. (c) The mean velocity $\langle v\rangle$ vs. $\lambda$. (d) The effective diffusion $D_{\mathrm{eff}}$ as a function of $\lambda$. Symbols correspond to numerical simulations from a direct integration of Eq. (3) and lines are the theoretical results of Eqs. (10) and (11). The parameters are $D_1=0.2,D_2=0.2$.

Figure 2

(a) The probability distribution $P_{\text{st}}\left(v\right)$ for different $\lambda$. (b) The contour plot of $P_{\text{st}}\left(v\right)$ as a function of $v$ and $\lambda$. The black-to-white grayscale represents the lowest value 0 to the highest value $1.6$. The parameters are $D_1=0.5,D_2=0.2$.

Figure 3

(a) Dependence of the mean velocity $|\langle v\rangle |$ on the multiplicative noise strength $D_1$ for different $\lambda$ with $D_2=0.2$. (b) Dependence of $|\langle v\rangle |$ on the additive noise strength $D_2$ for different $\lambda$ with $D_1=0.2$. From bottom to the top: $\lambda =0,\pm 0.25,\pm 0.5,\pm 0.75,\pm 0.9$. Symbols correspond to numerical simulations from a direct integration of Eq. (3) and lines are the theoretical results of Eq. (10).

Figure 4

(a) Dependence of the effective diffusion $D_{\mathrm{eff}}$ on the multiplicative noise strength $D_1$ for different $\lambda$ with $D_2=0.2$. (b) Dependence of $D_{\mathrm{eff}}$ on the additive noise strength $D_2$ for different $\lambda$ with $D_1=0.2$. From top to the bottom: $\lambda =\pm 0.25,\pm 0.5,\pm 0.75,\pm 0.9$. Inset: The theoretical results of the contour plot of the effective diffusion $D_{\mathrm{eff}}$ as a function of $D_1$ and $D_2$ for $\lambda =0.75$ and each square marks the optimal $D_1$ for a given $D_2$. The black-to-white grayscale represents the lowest value 1 to the highest value $3.5$. Symbols correspond to numerical simulations from a direct integration of Eq. (3) and lines are the exact result of Eq. (11).

Figure 5

The rectified velocity potential $\Phi \left(v\right)$ of Eq. (8) as a function of $v$. (a) $\Phi \left(v\right)$ vs. $v$ for different noise correlation strength $\lambda$ with $D_1=0.5,D_2=0.2$. (b) $\Phi \left(v\right)$ vs. $v$ for different multiplicative noise strength $D_1$ with $D_2=0.2,\lambda =0.9$. (c) $\Phi \left(v\right)$ vs. $v$ for different additive noise strength $D_2$ with $D_1=0.2,\lambda =0.9$.

Figure 6

(a) The potential gap $\Delta \Phi$ as a function of $D_1$ for different $\lambda$ with $D_2=0.2$. (b) The potential gap $\Delta \Phi$ as a function of $D_2$ for different $\lambda$ with $D_1=0.2$. From bottom to the top: $\lambda =0.25(-0.25),0.5(-0.5),0.75(-0.75),0.9(-0.9)$. Inset: The potential gap $\Delta \Phi$ as a function of $\lambda$.