Non-equilibrium phase separation in active matter
Active matter, characterised by the ability to inject energy into the environment locally, forms an important class of non-equilibrium systems. Recently there has been a surge of interest in active systems with chemical reactions, fuelled in particular by studies in biomolecular condensates, or ‘membraneless organelles’, within cells. In contrast to their passive counterparts, such systems have conserved and non-conserved dynamics that do not, in general, derive from a shared free energy. This mismatch breaks time- reversal symmetry (TRS) and leads to new types of dynamical competition that are absent in or near equilibrium. We construct a canonical scalar field theory to describe such systems, with conserved and non-conserved dynamics obeying Model B and Model A respectively (in the Hohenberg-Halperin classification), chosen such that the two free energies involved are incompatible. The resulting minimal model is shown to capture the various phenomenologies reported previously for more complicated models within this class, including microphase separation, limit cycles and droplet splitting. To expand upon the systematic study of non-equilibrium reaction-diffusion systems, we also consider that the diffusive dynamics can break time reversal symmetry in its own right. This happens only at higher order in the gradient expansion, but is the leading behaviour without reactions present. We incorporate the higher gradient terms into Model AB and show that for slow reaction rates the system can undergo a new type of hierarchical microphase separation, which we call ‘bubbly microphase separation’. In this state, small droplets of one fluid are continuously created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and diffusive dynamics. Motivated by the distinct non-equilibrium nature of active systems, we introduce the entropy production rate (EPR) as a quantitative measure of time reversal symmetry breaking. It can be defined either at the particle level or at the level of coarse-grained fields such as the density; the EPR for the latter quantifies the extent to which these coarse-grained fields behave irreversibly. In this work, we first develop a general method to compute the EPR of scalar Langevin field theories with additive noise (this large class includes the aforementioned Model AB). Treating the scalar field φ as the sole observable, we arrive at an expression for the EPR that is non-negative for every field configuration and is quadratic in the time-antisymmetric component of the dynamics. Our general expression is a function of the quasipotential, which determines the full probability distribution for field configurations, and is not generally calculable. To alleviate this difficulty, we present a small-noise expansion of the EPR, that only requires knowledge of the deterministic (mean-field) solution for the scalar field in steady state, which generally is calculable, at least numerically. We demonstrate this calculation for the case of Model AB. We then present a similar EPR calculation for Model AB with the conservative and non-conservative contributions viewed as separate observable quantities. The results are qualitatively different, confirming that the field-level EPR depends on the choice of coarse-grained information retained within the dynamical description.
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