An Equation of State for Active Matter

Abstract

We derive a microscopic expression for the mechanical pressure P in a system of spherical active Brownian particles at density ρ. 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) PðρÞ 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. Much recent research addresses the statistical physics of active matter, whose constituent particles show autonomous dissipative motion (typically self-propulsion), sustained by an energy supply. Progress has been made in understanding spontaneous flow [1] and phase equilibria in active matter [2-6], but as yet there is no clear thermodynamic framework for these systems. Even the definition of basic thermodynamic variables such as temperature and pressure is problematic. While "effective temperature" is a widely used concept outside equilibrium [7], the discussion of pressure P in active matter has been neglected until recently [8][9][10][11][12][13][14]. At first sight, because P can be defined mechanically as the force per unit area on a confining wall, its computation as a statistical average looks unproblematic. Remarkably, though, it was recently shown that for active matter the force on a wall can depend on details of the wall-particle interaction so that P is not, in general, a state function [15].Active particles are nonetheless clearly capable of exerting a mechanical pressure P on their containers. (When immersed in a space-filling solvent, this becomes an osmotic pressure [8,10].) Less clear is how to calculate P; several suggestions have been made [9][10][11][12] whose interrelations are, as yet, uncertain. Recall that for systems in thermal equilibrium, the mechanical and thermodynamic definitions of pressure [force per unit area on a confining wall, and −ð∂F =∂VÞ N for N particles in volume V, with F the Helmholtz free energy] necessarily coincide. Accordingly, various formulas for P (involving, e.g., the density distribution near a wall [16], or correlators in the bulk [17,18]) are always equivalent. This ceases to be true, in general, for active particles [11,15].In this Letter we adopt the mechanical definition of P. We first show analytically that P is a state function, independent of the wall-particle interaction, for one important and well-studied class of systems: spherical active Brownian particles (ABPs) with isotropic repulsions. By definition, such ABPs undergo overdamped motion in response to a force that combines an arbitrary pair interaction with an external forcing term of constant magnitude along a...

Local stress and pressure in an inhomogeneous system of spherical active Brownian particles

“…As a step toward such a relation, the existence of an equation of state for systems of ABPs has been addressed 24,28,31 . Various simulation and theoretical studies confirm that indeed such a relation, including the active pressure, exists for spherical ABPs [22][23][24][25][31][32][33] . However, for (anisotropic) active particles which experience a torque, pressure is in general not a state function anymore 28,34 .…”

Virial pressure in systems of spherical active Brownian particles

“…14,[21][22][23][24][25] Pressure in active fluids attracted considerable attention lately, because a mechanical pressure as force per area can be defined far from equilibrium and might be useful in the strive to derive a nonequilibrium equation of state. 13,21,[26][27][28] Interestingly, simulation studies reveal such an equation of state, with a van der Waals-type pressure-volume phase diagram typical for gas-liquid coexistence. 13,21,26,27 In thermal equilibrium, pressure can equivalently be defined in various ways.…”

Dynamics of flexible active Brownian dumbbells in the absence and the presence of shear flow

“…7,15,16,20,49 Moreover, it is a valuable theoretical model to gain insight into the nonequilibrium statistical properties of active Brownian systems. [50][51][52][53][54][55][56][57][58] Excluded-volume interactions are of paramount importance for collective phenomena in motility assays and active gels (cytoskeleton). The high aspect ratio of actin filaments or microtubules leads to the alignment of interacting units.…”