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...