Memory effects in active particles with exponentially correlated propulsion

The Ornstein-Uhlenbeck particle (OUP) model imagines a microscopic swimmer propelled by an active force which is correlated with itself on a finite time scale. Here we investigate the influence of external potentials on an ideal suspension of OUPs, in both one and two spatial dimensions, with particular attention paid to the pressure exerted on “confining walls.” We employ a mathematical connection between the local density of OUPs and the statistics of their propulsion force to demonstrate the existence of an equation of state in one dimension. In higher dimensions we show that active particles generate a nonconservative force field in the surrounding medium. A simplified far-from-equilibrium model is proposed to account for OUP behavior in the vicinity of potentials. Building on this, we interpret simulations of OUPs in more complicated situations involving asymmetrical and spatially curved potentials, and characterize the resulting inhomogeneous stresses in terms of competing active length scales.

Figure 1

Trajectories in $(x,\eta )$ for the diffusion-free approximation in the wall region. Each line corresponds to a different ${\eta }_0$. Insets show the resulting phase space density (darker shading means higher density). (a) Linear potential $U=x$. (b) Quadratic potential $U=\frac{1}{2}{x}^2$.

Figure 2

Contours: Density $\rho (x,\eta )$ for an OUP confined by quadratic walls with a finite bulk of zero force in between (this region is demarcated by vertical lines). Vectors: Velocity field. Curves: Projected spatial density $n\left(x\right)$, which varies in the bulk, and the quantity $〈{\eta }^2〉\left(x\right)n\left(x\right)$ of Eq. (20), which is indeed constant in the bulk.

Figure 3

Main figure: Pressure (circles, solid) and total probability (triangles, dashed) on either side of the penetrable inner wall, plotted against $\alpha \equiv k\tau /\zeta$. The small bulk experiences larger pressure and accumulates more particles. Upper inset: $\eta$ distribution in the bulk of the two regions. The small bulk has a narrower distribution. Lower inset: A sketch of the potential.

Figure 4

Main figure: Projected density $n(x,y)$, with the bulk region on the left and darker color corresponding to higher density. Inset: Potential (periodic along $y$, with period $\lambda$), where darker color corresponds to higher potential.

Figure 5

Slices of density at fixed coordinate $y$, from the convex apex ($y=0.5\lambda$) to the concave apex ($y=0.75\lambda$), where $\lambda$ is the period of the wall. Peaks move progressively to the right with increasing $y$, as one would expect. The $x$ position of the wall is indicated by a dashed line of matching color.

Figure 6

Stress in the $x$ direction as a function of $y$ [defined in Eq. (28)]. The $\alpha =0$ pressure is constant, and lines progress monotonically with increasing $\alpha$. Note logarithmic ordinate.

Figure 7

Spatial density $n\left(r\right)$ for annular geometry, with $U\left(r\right)=\frac{1}{2}{(r-5)}^2$ and $\alpha =5$. The OUP density is more sharply peaked than the passive particle density, and the average OUP position is offset from the bulk in the $+r$ direction, signaling a tendency for OUPs to collect in geometrically concave wall regions.

Figure 8

Pressure difference $(P_{\mathrm{out}}-P_{\mathrm{in}})/P_{\mathrm{out}}^{\mathrm{passive}}$ plotted against $\alpha$ for several bulk radii $R$. The potential is the radially symmetrical $U\left(r\right)=\frac{1}{2}{(r-R)}^2$, where the bulk has zero width. For a given $\alpha$, the pressure difference diminishes with $R$. Note that $\mathrm{\Delta }P(\alpha =0)\ne 0$ when $R$ is small enough that the inner wall is substantially penetrable. Note also that $\mathrm{\Delta }P$ continues to increase slowly with $\alpha$ for $R\ge 10$.

Figure 9

Analogous to Fig. 8, but where the bulk has finite width.

Figure 10

Distribution of $\eta$ (normalized) in a large bulk and a small bulk. The potential at the cusp of the interior wall is 0.125 (in units of $T$) and the force magnitude is 0.5 (in units of $\sqrt{Tk}$).

Figure 11

Pressure on either side of a penetrable inner wall (as in Fig. 3) along with the pressure on the confining walls (dashed triangles). The net pressure in the $-x$ direction is plotted in black squares connected by dots. This is the same data as for Fig. 3 in the main text.

Figure 12

Coordinate system for the radially symmetric geometry.

Figure 13

Pressure on the outer (solid lines) and inner (dashed) walls as a function of $R$ for several values of $\alpha$, in the annular geometry with zero bulk ($R=S$). The pressure is not normalized.

Figure 14

Pressure as a function of $R$ for several values of $\alpha$, in a disk geometry where there is no inner wall. The pressure is normalized by the passive value, and is higher for higher $\alpha \equiv k\tau /\zeta$ (for a fixed $R$).

Figure 15

(a) Spatial density for an OUP confined in a piecewise linear potential (sketched as dashed black lines). (b) Pressure as a function of $\alpha$ for the linear potential pictured in the inset. Note that the $L\to \infty$ line coincides with the $L=0$ line.