Brownian systems with spatially inhomogeneous activity

We generalize the Green-Kubo approach, previously applied to bulk systems of spherically symmetric active particles [J. Chem. Phys. 145, 161101 (2016)], to include spatially inhomogeneous activity. The method is applied to predict the spatial dependence of the average orientation per particle and the density. The average orientation is given by an integral over the self part of the Van Hove function and a simple Gaussian approximation to this quantity yields an accurate analytical expression. Taking this analytical result as input to a dynamic density functional theory approximates the spatial dependence of the density in good agreement with simulation data. All theoretical predictions are validated using Brownian dynamics simulations.

Figure 1

Average orientation per particle in the ${\widehat{\mathit{e}}}_z$ direction for ${\rho }_b=0.2$ in (a) and (b) and ${\rho }_b=0.6$ in (c) and (d). The activity field is sinusoidal with amplitude $v_a$ and the angular frequency $\omega =n{\omega }_0$, where ${\omega }_0=2\pi /L$ and $n$ is a parameter. In (a) and (c), $\omega =2{\omega }_0$, whereas in (b) and (d), $\omega =3{\omega }_0$. The circles correspond to $v_a=10$ and the squares to $v_a=20$. The thick lines correspond to the theoretical prediction of Eq. (17). The numerically measured $\mathit{p}\left(z\right)$ is in very good agreement with the theoretical prediction for low density. At higher density, the theory overestimates the average orientation. Nonlinear deviations are apparent e.g., in (a) and (b) where higher harmonics contribute to the average orientation per particle.

Figure 2

Relative change in density $(\rho -{\rho }_b)/{\rho }_b$ for ${\rho }_b=0.2$ in (a) and ${\rho }_b=0.6$ in (b). The dotted line shows a sinusoidal activity with $\omega =2{\omega }_0$ with arbitrary amplitude. The circles correspond to $v_a=10$ and the squares to $v_a=20$. Particles accumulate at the nodes of the activity and the change in density is asymmetric. The thick lines correspond to the theoretical prediction of Eq. (15) with Eq. (17) as input.