Generalized thermodynamics of phase equilibria in scalar active matter

Motility-induced phase separation (MIPS) arises generically in fluids of self-propelled particles when interactions lead to a kinetic slowdown at high densities. Starting from a continuum description of scalar active matter akin to a generalized Cahn-Hilliard equation, we give a general prescription for the mean densities of coexisting phases in flux-free steady states that amounts, at a hydrodynamics scale, to extremizing an effective free energy. We illustrate our approach on two well-known models: self-propelled particles interacting either through a density-dependent propulsion speed or via direct pairwise forces. Our theory accounts quantitatively for their phase diagrams, providing a unified description of MIPS.

Figure 1

(a) Phase diagrams of QSAPs. The solid lines correspond to common tangent constructions on $\varphi \left(R\right)$ (red) or $f\left(\rho \right)$ (black). Dashed lines correspond to the spinodals $d^2\varphi /d{R}^2=0$. Data points are from simulations of run-and-tumble particles (RTPs; $\alpha =1,D_r=0$) or active Brownian particles (ABPs; $\alpha =0,D_r=1$) in one dimension on lattice or two dimensions in continuous space. Black triangles correspond to Supplemental Material movies showing nucleation or spinodal decomposition [46]. For all plots, $v\left(\rho \right)=v_0+\frac{v_1-v_0}{2}[1+tanh\left(\frac{\rho -{\rho }_m}{L_f}\right)], K\left(r\right)=exp(-\frac{1}{1-r^2})/Z$ with $Z$ a normalization constant, ${\rho }_m=200,v_1=5,L_f=100$. (b) Common tangent construction on $\varphi \left(R\right)$ for $v_0=20$.

Figure 2

(a) Mechanical pressure of PFAPs. Semiempirical EOS $g_0\left(\nu \right)$ (line) vs numerical measurements (symbols) for various $\mathrm{Pe}$. Open symbols correspond to binodals and horizontal lines to the pressure $\overline{g}$ predicted by Eq. (13). (b) Corresponding phase diagrams obtained via the modified (red; see text) and the equal-area (blue) Maxwell constructions, compared with numerically measured binodals (black). Dashed lines correspond to the spinodals $g_0^{\prime }\left(\rho \right)=0$. Black triangles correspond to Supplemental Material movies showing nucleation or spinodal decomposition [46].