Self-propelled particles include both self-phoretic synthetic colloids and various micro-organisms. By continually consuming energy, they bypass the laws of equilibrium thermodynamics. These laws enforce the Boltzmann distribution in thermal equilibrium: the steady state is then independent of kinetic parameters. In contrast, self-propelled particles tend to accumulate where they move more slowly. They may also slow down at high density, for either biochemical or steric reasons. This creates positive feedback which can lead to motility-induced phase separation (MIPS) between dense and dilute fluid phases. At leading order in gradients, a mapping relates variable-speed, self-propelled particles to passive particles with attractions. This deep link to equilibrium phase separation is confirmed by simulations, but generally breaks down at higher order in gradients: new effects, with no equilibrium counterpart, then emerge. We give a selective overview of the fast-developing field of MIPS, focusing on theory and simulation but including a brief speculative survey of its experimental implications.

We consider self-propelled particles undergoing run-and-tumble dynamics (as exhibited by E. coli) in one dimension. Building on previous analyses at drift-diffusion level for the one-particle density, we add both interactions and noise, enabling discussion of domain formation by 'self-trapping', and other collective phenomena. Mapping onto detailed-balance systems is possible in certain cases.PACS numbers: 87.10.Mn, 87.17.Jj Several species of bacteria, including Escherichia coli, perform self-propulsion by a sequence of 'runs' -periods of almost straight-line motion at near-constant speed (v) -punctuated by sudden and rapid randomizations in direction, or 'tumbles', occurring stochastically with rate α. It is no surprise that the resulting class of random walk gives a diffusive relaxation of the number density at large scales [1]. The resulting diffusion constant ∼ v 2 /α, is vastly larger than that of non-swimming particles undergoing pure thermal motion at room temperature. Therefore, apart from, e.g., the upper limit it imposes on the duration of a straight run (set by rotational diffusion), true Brownian motion can usually be ignored.Because bacterial diffusion is not thermal, the steadystate probability density cannot be written as, with H a Hamiltonian, even for a single diffuser. The physicist's intuition can easily be led astray: for instance, Refs.[2, 3, 4] address models (of chemotaxis) comprising noninteracting particles in 1D, with no external forces, but v(x), α(x) functions of position x. Instead of a uniform density, as would arise with any force-free detailed-balance dynamics, one findsHere we extend previous analyses of run-and-tumble motion to the many-particle level, addressing the roles of noise and interactions. These determine, for instance, the dynamic correlator of run-and-tumble bacteria, which is measurable by light scattering at low density [5] and at higher density, in principle, by particle-tracking microscopy [6]. Additionally, particles for which v, α depend on the local density (either via thermodynamic interactions such as depletion [7], or kinetic effects such as collision-induced tumbles) could show collective phenomena such as domain-formation or flocking. Such effects have previously been addressed within models where a self-propelled particle responds vectorially to the velocity of its neighbors, by direct sensing or passive hydrodynamics [8,9,10]. Below, we shall find, for run-and-tumble dynamics, similar effects in even simpler cases when only the speed of a particle is density-dependent.In making the transition from a single particle to many, most bacterial modelling approximates the number density by a simple replacement ρ = N p [3,4]. But even for noninteracting particles, ρ (unlike p) is a fluctuating quantity, and a full statistical mechanics must compute noise terms for ρ. As seen below, these are not ad-hoc, but follow from the run-and-tumble dynamics directly.To allow relatively rigorous progress we work in 1-D throughout. For d > 1, although good descriptions...

Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relation between injection and dissipation of energy at the microscopic scale. We consider such a system of interacting particles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics still satisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that, for short persistent times, the entropy production rate vanishes. This endows such systems with an effective fluctuation-dissipation theorem akin to that of thermal equilibrium systems. Last, we show how interacting particle systems with viscous drags and correlated noises can be seen as in equilibrium with a viscoelastic bath but driven out of equilibrium by nonconservative forces, hence providing energetic insight into the departure of active systems from equilibrium.

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.

PACS 05.40.-a -Fluctuation phenomena: statistical physics PACS 87.10.Mn -Stochastic models in biological physics PACS 64.75.Jk -phase separation and segregation in Nanoscale systems Abstract -Active Brownian particles (ABPs, such as self-phoretic colloids) swim at fixed speed v along a body-axis u that rotates by slow angular diffusion. Run-and-tumble particles (RTPs, such as motile bacteria) swim with constant u until a random tumble event suddenly decorrelates the orientation. We show that when the motility parameters depend on density ρ but not on u, the coarse-grained fluctuating hydrodynamics of interacting ABPs and RTPs can be mapped onto each other and are thus strictly equivalent. In both cases, a steeply enough decreasing v(ρ) causes phase separation in dimensions d = 2, 3, even when no attractive forces act between the particles. This points to a generic role for motility-induced phase separation in active matter. However, we show that the ABP/RTP equivalence does not automatically extend to the more general case of u-dependent motilities.

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...

The run-and-tumble dynamics of bacteria, as exhibited by E. coli, offers a simple experimental realization of non-Brownian, yet diffusive, particles. Here we present some analytic and numerical results for models of the ideal (low-density) limit in which the particles have no hydrodynamic or other interactions and hence undergo independent motions. We address three cases: sedimentation under gravity; confinement by a harmonic external potential; and rectification by a strip of 'funnel gates' which we model by a zone in which tumble rate depends on swim direction. We compare our results with recent experimental and simulation literature and highlight similarities and differences with the diffusive motion of colloidal particles.

We study the behaviour of interacting self-propelled particles, whose self-propulsion speed decreases with their local density. By combining direct simulations of the microscopic model with an analysis of the hydrodynamic equations obtained by explicitly coarse graining the model, we show that interactions lead generically to the formation of a host of patterns, including moving clumps, active lanes and asters. This general mechanism could explain many of the patterns seen in recent experiments and simulations.