Pressure and flow of exponentially self-correlated active particles

Microscopic swimming particles, which dissipate energy to execute persistent directed motion, are a classic example of a nonequilibrium system. We investigate the noninteracting Ornstein-Uhlenbeck Particle (OUP), which is propelled through a viscous medium by a force which is correlated over a finite time. We obtain an exact expression for the steady-state phase-space density of a single OUP confined by a quadratic potential, and use the result to explore more complex geometries, both through analytical approximations and numerical simulations. In a “Casimir”-style setup involving two narrowly spaced walls, we describe a particle-trapping phenomenon, which leads to a repulsive effective interaction between the walls, while in a two-dimensional annulus geometry, we observe net stresses which resemble the Laplace pressure.

Figure 1

Net current $J$ as a function of the correlation time (both measured in convenient units) for an OUP in a periodic, asymmetrical potential in one dimension. Solid lines with markers show simulation results for several degrees of asymmetry; dashed lines show the approximate prediction described in the text. For these data, the height of the potential $U_0/T=1$, meaning the approximation described in the text is not fully applicable; yet it still captures the general behavior. Inset: Contours of phase-space density. The solid straight lines show $-f\left(x\right)$. Current lines are sketched and adorned with arrows.

Figure 2

Pressure as a function of the dimensionless correlation time $\frac{k\tau }{\zeta }$, for several bulk widths $L$. The prediction for $L=0$ is shown as a dashed line (obscured by data), and the prediction for $L\to \infty$ is a constant. The pressure exerted by an ideal gas of passive particles, $P^{\mathrm{passive}}$ is calculated by substituting the Boltzmann distribution into Eq. (6).

Figure 3

Plots of two sets of data. The pressure on the inner and outer portions of the Casimir potential (circles, solid lines, left ordinate axis) and the total probability of finding the OUP in each region (triangles, dashed lines, right ordinate axis), both as a function of the dimensionless correlation time $\frac{k\tau }{\zeta }$. Here $P_{\text{in}}>P_{\text{out}}$ and $M_{\text{in}}>M_{\text{out}}$; the height of the potentials is $T/2$ and their half-width is $\sqrt{T/k}$. Upper inset: A representative probability distribution of $\eta$ between the walls, which is narrower than the distribution in the large bulk $\left[p_{\text{bulk}}^{\text{large}}\left(\eta \right)\propto exp\left(-\frac{\tau }{2T\zeta }{\eta }^2\right)\right]$. Lower inset: Sketch of the piecewise-quadratic potential, whose the walls are penetrable for OUPs with sufficiently high $\eta$.

Figure 4

The pressure difference $(P_{\text{out}}^{\text{OUP}}-P_{\text{in}}^{\text{OUP}})/(P_{\text{out}}^{\text{flat}}\sqrt{\frac{k\tau }{\zeta }})$ for an annular potential, as a function the wall position $R$ and for several values of the dimensionless correlation time $\frac{k\tau }{\zeta }$. (We divide the pressure difference by the pressure for a flat wall in order to fix normalization as $R$ changes, and we also divide by the free-particle persistence length $\sqrt{\frac{k\tau }{\zeta }}$ for better comparison of curves.) The bulk is of zero width and located at $r=R$; and the line $1/R$ is indicated by dots. Inset: Schematic of the annular potential in 3D, with the foot of the wall indicated.

Figure 5

(a): Density distribution in $(x,\eta )$ phase space. Elliptical level lines illustrate the exact solution (3). (b): Currents in $(x,\eta )$ phase space. Arrows represent velocity, while the contours are magnitude of current.

Figure 6

(a) Schematic of the original Casimir potential from the main text (solid blue line), and the approximation to it considered here (green, dashed). (b) The mass $M$ in the narrow region of the Casimir potential, as a function of the stiffness ratio $k/K$, for several values of the potential height $U_0/T$.

Figure 7

The net repulsive pressure $P$ on the interior walls of the Casimir potential, as a function of the potential-stiffness ratio $k/K$, for several values of the potential height $U_0/T$.

Figure 8

A sample trajectory trace in the annular geometry. Shaded according to time, with later times shaded darker. The solid black circle marks $r=R$.