Pressure and Phase Equilibria in Interacting Active Brownian Spheres

We derive a microscopic expression for the mechanical pressure $P$ in a system of spherical active Brownian particles at density $\rho$. Our exact result relates $P$, defined as the force per unit area on a bounding wall, to bulk correlation functions evaluated far away from the wall. It shows that (i) $P(\rho )$ is a state function, independent of the particle-wall interaction; (ii) interactions contribute two terms to $P$, one encoding the slow-down that drives motility-induced phase separation, and the other a direct contribution well known for passive systems; and (iii) $P$ is equal in coexisting phases. We discuss the consequences of these results for the motility-induced phase separation of active Brownian particles and show that the densities at coexistence do not satisfy a Maxwell construction on $P$.

Figure 1

Numerical measurements of $P_0+P_I$, $P_S$, and $P_D$ in single-phase ABP simulations at Péclet number $\mathrm{Pe}\equiv 3v_0/(D_r\sigma )=40$, where $\sigma$ is the particle diameter. Expressions (12), (14), and (16) for $P_0+P_I$ and $P_S$ show perfect agreement. Also shown are data for $\mathrm{Pe}=20$, unscaled and rescaled by factor 2. This confirms that $P_S=P_0+P_I$ is almost linear in Pe; small deviations arise from the Pe dependence of the correlators. In red is $P_D$ for $\mathrm{Pe}=20,40$, with no rescaling. Pe was varied using $D_r$, at fixed $v_0$ and with $D_t=D_r{\sigma }^2/3$. Solid lines are fits to piecewise parabolic ($P_S$) and exponential ($P_D$) functions used in the semiempirical equation of state. ${\rho }_0$ is a near-close-packed density at which $v(\rho )$ vanishes and $\tilde{\rho }$ is the threshold density above which $P_D>P_S$. See the Supplemental Material [39] for details.

Figure 2

Simulated coexistence curves (binodals) for ABPs (red) and those calculated via the Maxwell construction (black) on the mechanical pressure $P$ using the semiempirical equation of state for $P_S$ and $P_D$ fitted from Fig. 1. Dashed lines: predicted high Pe asymptotes for the binodals calculated via $f$ or $P_f$ (lower line) and calculated via $P$ or $f_P$ (upper line). Inset: measured binodal pressures and densities (diamonds) fall on the equation-of-state curves but do not match the MC values (horizontal dashed lines). Stars show the $P(\rho )$ relation across the full density range from simulations at $\mathrm{Pe}=40$ and $\mathrm{Pe}=100$. The latter includes two metastable states at low density (high ${\rho }_0/\rho$) that are yet to phase separate.