Pressure and Phase Equilibria in Interacting Active Brownian Spheres

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...

Destabilization of a flow focused suspension of magnetotactic bacteria

“…As for the idea of generalizing the surface tension in active systems, it has only emerged very recently [31]. Indeed, the mere notion of active pressure was recently introduced [32][33][34] and here, with field oriented swimmers, active pressure anisotropy and active surface tensions might be expected. However, the practical meaning and usefulness of these different variables are still a matter of extensive theoretical research [35] that precludes its current use for drawing generalization from passive systems instabilities.…”

An Equation of State for Active Matter

“…The equivalence of (12), (14), and (16) is proven analytically in the Supplemental Material [39] and confirmed numerically in Fig. 1 for ABP simulations performed as in Refs.…”

“…ρ 0 is a nearclose-packed density at which vðρÞ vanishes andρ is the threshold density above which P D > P S . See the Supplemental Material [39] for details. PRL 114, 198301 (2015) P…”

“…All four functions are plotted in the Supplemental Material [39]; the two distinct routes indeed predict different binodals at high Pe (see Fig. 2) [43].…”

Sufficient conditions for the additivity of stall forces generated by multiple filaments or motors

“…Thus, as far as the translational movement of the motors against an applied force is concerned, we can conceptually think of the system to be in thermodynamic equilibrium at stall with respect to the translational degree of freedom -the chemical coordinate is simply orthogonal to the translation coordinate. This seems quite analogous to the case of non-interacting active Brownian particles, which exert pressure on the confining walls similar to an ideal gas in equilibrium, but with a renormalised temperature due to the free energy consuming activity [83]. Nevertheless, these arguments cannot be claimed to be true for every model for motors or filaments.…”