Self-organisation has many forms, many of which have been studied for systems in equilibrium or metastable equilibrium, as in crystal formation or in gelation. The striking feature of these phenomena is the emergence of complex patterns of aggregation just from elementary interactions among the constituents. These are driven by an imbalance in the thermodynamic potentials for example the chemical potential. However, in recent years it has been shown that this spontaneous organisation is not the prerogative of equilibrium for (passive) systems: completely out of equilibrium systems such as bacterial colonies or self-propelled colloids present a similar behaviour, even if thermodynamic concepts such as a chemical potential are difficult to generalise to these systems. An interesting example is the amoeba-like crawling crystals that Abraham Mauleon-Amieva, here in Bristol has studied in his PhD and which present an interesting competition between electrostatic and active forces, with multiple mesh-phases, see Phys. Rev. E 102, 032609.
An important question in this area is to understand whether the phase diagrams of similar active systems can be understood exclusively invoking effective equilibrium concepts. A possible route is, for example, to think that the active forces lead to collisions and these can be effectively coarse-grained into suitable attractive effective two-body interactions. It would such an effective attraction to favour aggregation and hence explain (in an effective picture) the observed motility-induced self organisation.
In a recent article published in Physical Review letters with Nigel B Wilding we explore these ideas for an elementary model for active matter in three dimensions, active Brownian particles. Through the characterisation of the phase diagram, its phase separations and single phase fluid region we show that the system indeed shares many similarities with passive systems with short ranged interactions in 3d: a metastable liquid-gas phase separation, a crystalline phase, several pre-critical lines. However, a quantitative analysis of the effective interactions shows that it is not possible to explain the motility-induced phase separation only in terms of effective twobody interactions: multibody effects involving large numbers of particles are important, and can be quantified using information-theoretic tools.
For more details, see