Non-reciprocity across scales in active mixtures

Abstract

In active matter, particles typically experience mediated interactions, which are not constrained by Newton's third law and are therefore generically non-reciprocal. Non-reciprocity leads to a rich set of emerging behaviors that are hard to account for starting from the microscopic scale, due to the absence of a generic theoretical framework out of equilibrium. Here we consider bacterial mixtures that interact via mediated, non-reciprocal interactions (NRI) like quorum-sensing and chemotaxis. By explicitly relating microscopic and macroscopic dynamics, we show that, under conditions that we derive explicitly, non-reciprocity may fade upon coarse-graining, leading to large-scale equilibrium descriptions. In turn, this allows us to account quantitatively, and without fitting parameters, for the rich behaviors observed in microscopic simulations including phase separation, demixing, and multi-phase coexistence. We also derive the condition under which non-reciprocity survives coarse-graining, leading to a wealth of dynamical patterns. Again, our analytical approach allows us to predict the phase diagram of the system starting from its microscopic description. All in all, our work demonstrates that the fate of non-reciprocity across scales is a subtle and important question.

Fig. 1. Simulations of two species of RTPs with self-inhibition and global activation of the motility.

The self-propulsion speed is regulated through Eq. (11). The figures report simulation results for different average densities ${\rho }_{\alpha ,\beta }^0$. Normalized densities are defined as: ${\phi }_{\mu }={\rho }_{\mu }/\left({\rho }_{\alpha }^0+{\rho }_{\beta }^0\right)$. a For ${\rho }_{\alpha }^0=15$ and ${\rho }_{\beta }^0=50$, species β undergoes phase separation (bottom left), accompanied by a slight modulation of the density of α (top left). The total density field is inhomogeneous (top right). The local difference in composition simply recapitulates the variations of species β (bottom right). b For ${\rho }_{\alpha }^0={\rho }_{\beta }^0=55$, the two species demix: the total density field is homogeneous (top right) while each species experiences phase separation (left column). The dense phase of one species is co-localized with the dilute phase of the other species (bottom right). c For ${\rho }_{\alpha }^0={\rho }_{\beta }^0=75$, we observe a triple-coexistence regime. Each species exhibits dense, dilute, and intermediate phases (left column). The intermediate phases of both species are co-localized, whereas their dense and dilute phases are demixed (bottom right). The overall density field shows the existence of a well-mixed dense phase---which results from the sum of the intermediate densities of each species---and one with an intermediate density---which corresponds to the demixed regions (top right). All parameters and numerical details are given in Methods.

Fig. 2. Phase diagram and common-tangent construction in the presence of self-inhibition and global activation of motility.

a Phase diagram of two species of RTPs experiencing self-inhibition and global activation of motility according to Eq. (11). White regions correspond to homogeneous well-mixed phases. Red, green, and ochre regions indicate one-species phase separation, demixing, and triple phase coexistence, respectively. Stars correspond to snapshots shown in Fig. 1a–c. Coexistence lines (solid) and tie-lines (dashed) are predicted using a tangent plane construction on the free energy density f(ρ_α, ρ_β) as detailed in the Supplementary Information. Black squares show coexisting densities measured in simulations. b Plot of the free energy density in the triple coexistence regime from Fig. 1c. The points where the tangent plane in blue touches the surface determine the three compositions that will be observed in the coexistence region.

Fig. 3. Simulations of two species of RTPs when non-reciprocity survives coarse-graining.

Microscopic simulations of Eq. (15), with self-inhibition and non-reciprocal cross interactions. a and c Dynamical and static patterns observed in simulations, respectively; α-particles are depicted in red, β-particles in blue. The snapshots correspond to the larger symbols shown in (b). b Phase diagram as the couplings ${\kappa }_{\alpha }^c$ and ${\kappa }_{\beta }^c$ are varied, where we remind that ${\kappa }_{\mu }^c>0$ corresponds to the activation of the motility of species μ by the density of species ν, whereas ${\kappa }_{\mu }^c<0$ corresponds to an inhibition. The background colors correspond to the predictions of linear stability analysis which are confirmed by numerical simulations (small symbols). The phase diagram is symmetric with respect to the dashed line ${\kappa }_{\beta }^c={\kappa }_{\alpha }^c$ upon inverting the roles of α- and β-particles in (a, c). See Methods for other parameters and numerical details.