Ideal bulk pressure of active Brownian particles

The extent to which active matter might be described by effective equilibrium concepts like temperature and pressure is currently being discussed intensely. Here, we study the simplest model, an ideal gas of noninteracting active Brownian particles. While the mechanical pressure exerted onto confining walls has been linked to correlations between particles' positions and their orientations, we show that these correlations are entirely controlled by boundary effects. We also consider a definition of local pressure, which describes interparticle forces in terms of momentum exchange between different regions of the system. We present three pieces of analytical evidence which indicate that such a local pressure exists, and we show that its bulk value differs from the mechanical pressure exerted on the walls of the system. We attribute this difference to the fact that the local pressure in the bulk does not depend on boundary effects, contrary to the mechanical pressure. We carefully examine these boundary effects using a channel geometry, and we show a virial formula for the pressure correctly predicts the mechanical pressure even in finite channels. However, this result no longer holds in more complex geometries, as exemplified for a channel that includes circular obstacles.

Figure 1

Channel of width $L$ with hard walls in $x$ direction and periodic boundaries in $y$ direction. (a) Sketch of channel and snapshot of particles close to one wall for reduced inverse channel width $\nu =v_0{\tau }_{\text{r}}/L=0.5$, where arrows indicate the orientation. Note the adsorbed layer at the wall. (b) Density profile and spatially resolved average orientations for $\nu =0.2$. Shown is the number of particles $n\left(x\right)\propto \rho \left(x\right)$ in uniformly spaced bins. The dashed line indicates the average density. The orientation ($p_x$ and $p_y$) is zero everywhere except exactly at the walls (arrows). (c) Conditional second moments ${\langle e_x^2\rangle }_x (\square )$ and ${\langle e_y^2\rangle }_x (\diamond )$ for $\nu =0.5$. The inset shows that $q\left(x\right)=q$ is constant away from the walls.

Figure 2

The order of limits matters. Starting from dynamics with finite damping and finite speed $v_0>0$, the overdamped limit $m\to 0$ (with mass $m$) and the limit ${\ell }_{\text{p}}/L\to 0$ do not commute. Taking the overdamped limit first (left) leads to ABPs in a large system, for which there is an adsorbed layer (dark stripes) and thus local and wall pressure are different. Taking the overdamped limit last (right), one obtains a passive system with uniform density, in which local and wall pressure agree (with effective temperature $T_{\text{eff}}$). Note that the wall pressures for both the active and passive overdamped system coincide.

Figure 3

(a) Fraction $n_{\text{b}}$ of particles adsorbed at each wall (i.e., their orientations point into the wall) as a function of reduced inverse channel width. (b) Reduced pressure. Shown are the numerical results (symbols) for the mechanical pressure [Eq. (22)] and the virial pressure [Eq. (26)] together with the analytical prediction [Eq. (23), line], which show excellent agreement. In contrast, evaluating the virial pressure from Eq. (4) employing absolute positions overestimates the mechanical pressure (see Sec. 4b). In all cases, the ratio $p/p_0$ approaches unity for infinite systems $\nu \to 0$.

Figure 4

Channel bounded by two hard walls. We consider a box with dimensions $L\times L_y$ bounded by two hard walls and periodic boundary conditions in $y$ direction. The two pressures are indicated, the mechanical pressure $p_{\text{wall}}$ exerted on the walls and the local pressure $p_{\text{loc}}$ defined for a small subsystem away from the walls. Sketched is the trajectory of a single particle crossing the boundary of the box. Its position can thus be described by either $\mathbf{r}$ within the periodic box or the absolute position $\mathbf{R}=n_{\text{w}}{L}_y{\mathbf{e}}_y+\mathbf{r}$ with winding number $n_{\text{w}}$.

Figure 5

Channel with obstacles. (a) Snapshot for $\nu =0.5$ and obstacle radius $R/L=0.2$. (b) Reduced pressure for two values of $\nu$ as a function of obstacle radius $R/L$. (c) Conditional second moment for a plane $x=X$ close to the walls. (d) Reduced density $\rho \left(X\right)/\overline{\rho }$.