Pressure is the mechanical force per unit area that a confined system exerts on its container. In thermal equilibrium, it depends only on bulk properties-such as density and temperature-through an equation of state. Here we show that in a wide class of active systems the pressure depends on the precise interactions between the active particles and the confining walls. In general, therefore, active fluids have no equation of state. Their mechanical pressure exhibits anomalous properties that defy the familiar thermodynamic reasoning that holds in equilibrium. The pressure remains a function of state, however, in some specific and well-studied active models that tacitly restrict the character of the particle-wall and/or particle-particle interactions.

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...

Abstract. Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both self-propel at fixed speed v along a body-axis u that reorients either through slow angular diffusion (ABPs) or sudden complete randomisation (RTPs). We compare the physics of these two model systems both at microscopic and macroscopic scales. Using exact results for their steady-state distribution in the presence of external potentials, we show that they both admit the same effective equilibrium regime perturbatively that breaks down for stronger external potentials, in a model-dependent way. In the presence of collisional repulsions such particles slow down at high density: their propulsive effort is unchanged, but their average speed along u becomes v(ρ) < v. A fruitful avenue is then to construct a mean-field description in which particles are ghost-like and have no collisions, but swim at a variable speed v that is an explicit function or functional of the density ρ. We give numerical evidence that the recently shown equivalence of the fluctuating hydrodynamics of ABPs and RTPs in this case, which we detail here, extends to microscopic models of ABPs and RTPs interacting with repulsive forces.

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of "active matter" in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano-and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics.

We show that the flocking transition in the Vicsek model is best understood as a liquid-gas transition, rather than an order-disorder one. The full phase-separation observed in flocking models with Z2 rotational symmetry is however replaced by a micro-phase separation leading to a smectic arrangement of traveling ordered bands. Remarkably, continuous deterministic descriptions do not account for this difference, which is only recovered at the fluctuating hydrodynamics level. Scalar and vectorial order parameter indeed produce different types of number fluctuations, which we show to be essential in selecting the inhomogeneous patterns. This highlights an unexpected role of fluctuations in the selection of flock shapes.Many of the phenomena heretofore only invoked to illustrate the many facets of active matter are now being investigated in careful experiments, and more and more sophisticated models are built to account for them. For flocking alone, by which we designate the collective motion of active agents, spectacular results have been obtained on both biological systems [1-9] and man-made self-propelled particles [10][11][12]. Nevertheless, it is fair to say that the current excitation about flocking takes place while our understanding of the simplest situations remains unsatisfactory. This is true even for idealized selfpropelled particles interacting only via local alignment rules, as epitomized by the Vicsek model (VM) [2] which stands out for its minimality and popularity. Twenty years after the introduction of this seminal model for the flocking transition, and despite the subsequent extensive literature [14], we still lack a global understanding of the transition to collective motion.It took a decade to show that the transition to collective motion in the VM, initially thought to be critical [2], was discontinuous [3]: upon increasing the density or reducing the noise strength, high-density bands of spontaneously aligned particles form suddenly [3] (Fig 1). The homogeneous ordered "Toner-Tu" phase [16] is only observed after a second transition at significantly lower noise/higher density [3]. Since then, hydrodynamiclevel deterministic descriptions have been established and shown to support band-like solutions [17][18][19], but it was recently proved [20] that many such different solutions generically coexist. In fact, the connection of these results to microscopic models remains elusive. More generally, we lack a unifying framework encompassing the two transitions (disorder-bands, bands-Toner-Tu phase). bias along one arbitrarily fixed direction ±u x , the sign being given by the local magnetization (see [31] for a detailed definition). The emergence of flocking in this model is akin to a liquid-gas transition between an ordered liquid and a disordered gas. Unlike the traveling bands of the VM, inhomogeneous profiles in the AIM are fully phase-separated, with a single macroscopic liquid domain traveling in the gas (Fig 1). More generally, the symmetry difference between the two models questions the rele...

Motility-induced phase separation (MIPS) leads to cohesive active matter in the absence of cohesive forces. We present, extend and illustrate a recent generalized thermodynamic formalism which accounts for its binodal curve. Using this formalism, we identify both a generalized surface tension, that controls finite-size corrections to coexisting densities, and generalized forces, that can be used to construct new thermodynamic ensembles. Our framework is based on a nonequilibrium generalization of the Cahn-Hilliard equation and we discuss its application to active particles interacting either via quorum-sensing interactions or directly through pairwise forces.

We study, from first principles, the pressure exerted by an active fluid of spherical particles on general boundaries in two dimensions. We show that, despite the non-uniform pressure along curved walls, an equation of state is recovered upon a proper spatial averaging. This holds even in the presence of pairwise interactions between particles or when asymmetric walls induce ratchet currents, which are accompanied by spontaneous shear stresses on the walls. For flexible obstacles, the pressure inhomogeneities lead to a modulational instability as well as to the spontaneous motion of short semi-flexible filaments. Finally, we relate the force exerted on objects immersed in active baths to the particle flux they generate around them.Active forces have recently attracted much interest in many different contexts [1]. In biology, they play crucial roles on scales ranging from the microscopic, where they control cell shape and motion [2] to the macroscopic, where they play a dominant role in tissue dynamics [3,4]. More generally, active systems offer novel engineering perspectives, beyond those of equilibrium systems. In particular, boundaries have been shown to be efficient tools for manipulating active particles. Examples range from rectification of bacterial densities [5,6] and optimal delivery of passive cargoes [7] to powering of microscopic gears [8,9]. Further progress, however, requires a predictive theoretical framework which is currently lacking for active systems. To this end, simple settings have been at the core of recent active matter research.Understanding the effect of boundaries on active matter starts with the mechanical pressure exerted by an active fluid on its containing vessel. This question has been recently studied extensively for dry systems [10][11][12][13][14][15][16][17], revealing a surprisingly complex physics. For generic active fluids, the mechanical pressure is not a state variable [13]. The lack of equation of state, through a dependence on the wall details, questions the role of the mechanical pressure in any possible thermodynamic description of active systems [18,19]; it also leads to a richer phenomenology than in passive systems by allowing more general mechanical interplays between fluids and their containers.Interestingly, for the canonical model of self-propelled spheres of constant propelling forces, on which neither walls nor other particles exert torques, the pressure acting on a solid flat wall has been shown to admit an equation of state [11,12,19]. While the physics of this model does not clearly differ from other active systems, showing for instance wall accumulation [20] and motility-induced phase separation [21][22][23], the mechanical pressure exerted on a flat wall satisfies an equation of state, even in the presence of pairwise interactions [11,12,19]. One might thus hope that the intuition built on the rheology of equilibrium fluids extends to this case. Derived in a particular setting, the robustness of this equation of state however remains an open question. For...

Motility-induced phase separation (MIPS) arises generically in fluids of self-propelled particles when interactions lead to a kinetic slowdown at high densities. Starting from a continuum description of scalar active matter akin to a generalized Cahn-Hilliard equation, we give a general prescription for the mean densities of coexisting phases in flux-free steady states that amounts, at a hydrodynamics scale, to extremizing an effective free energy. We illustrate our approach on two well-known models: self-propelled particles interacting either through a density-dependent propulsion speed or via direct pairwise forces. Our theory accounts quantitatively for their phase diagrams, providing a unified description of MIPS.