Transitions driven by multibody interactions in an effective model of active matter

TL;DR

The paper investigates how multibody interactions shape phase behavior in an active-matter model under the Fox-UCNA approximation. It shows that the stationary distribution contains multibody terms and that a purely two-body effective potential cannot explain Motility-Induced Phase Separation (MIPS); in particular, a saturating activity-induced attraction $W_{\rm eff}$ does not suffice to drive phase separation. By mapping the grand-canonical measure to a replicated, pairwise-interacting system and applying a Mayer expansion in the infinite-dimensional limit, the authors derive a free-energy functional whose minimization yields a phase diagram with a first-order MIPS-like transition at large persistence and a continuous liquid–liquid transition between a paramagnetic and a spin-glass–like orientational phase. This work highlights the crucial role of multibody effects in active-matter thermodynamics and suggests a spin-glass interpretation for the orientational degrees of freedom encoded by replicated auxiliary variables. The findings clarify when pairwise reductions are adequate and reveal new nonequilibrium transitions arising from multibody interactions.

Abstract

When out-of-equilibrium particles interact by means of pairwise forces, their stationary distribution in general exhibits many-body interactions. In the particular case of active particles, it has been shown numerically that the Motility Induced Phase Separation cannot be explained by the effective attraction emerging from two isolated particles, thereby highlighting the role of multibody interactions. In this work, we study the thermodynamics of the Fox-UCNA approximation for active particles interacting by means of pairwise repulsive forces. Working at large space dimension we establish that multibody interactions up to infinite order are instrumental in giving rise to such collective phenomena as phase transitions. We recover a MIPS-like first order transition, but also find a liquid-liquid transition at somewhat lower persistence times. This new transition is connected to a spin glass phase of orientational-like degrees of freedom with disordered interactions set by the particle positions themselves.

Figures (6)

• Figure 1: Effective pair interaction for different values of $\hat{\tau}$ for $\beta = 1$ and $\sigma = 1$. (a) The pair potential is taken of the form $\widehat{U}(h) = u_0 \, e^{- \lambda h}$ with $u_0 = 1$ and $\lambda = 6$. (b) The pair potential is taken of the form $\widehat{U}(h) = u_0 \, h^2 \Theta(-h)/2$ with $u_0 = 1$.
• Figure 2: Effective pair potential for different values of $\hat{\tau}$ for $\beta = 1$ and $\sigma = 1$ and $\widehat{U}(h) = u_0 \, e^{- \lambda h}$ with $u_0 = 1$ and $\lambda = 6$. The depth of the attractive well saturates as $\hat{\tau}$ increases.
• Figure 3: Numerical solution of the saddle-point equations at $c_2 = 0.1$ and $c_1 = 0.7$. At any density, there exist two branches of solutions: One corresponds to a paramagnetic phase with $q - p = 0$ and one to a spin-glass phase with $q - p \neq 0$. They intersect at a low and a high density. (a) The stable branch (continuous, black) is the paramagnetic one at low and high densities and the spin-glass one at intermediate densities. The red dashed line corresponds to the unstable branch of solution. (b) The free energy of the unstable branch $g_{\rm un}$ is always higher than the free energy of the stable one $g_{\rm st}$.
• Figure 4: Phase transitions in the large-$d$ UCNA steady state. (a) Free energy as a function of the density for $c_2 = 0.1$ and $c_1 = 2.9$. The blue part of the curve corresponds to a paramagnetic state and the orange one to a spin-glass state. The dashed red line indicates the convex envelop of the free energy from which one can read the binodals. (b) Phase diagram of the UCNA. The blue line indicates the liquid-liquid continuous phase transition between the paramagnetic and the spin-glass phases (the dots were obtained from the numerical solution of the saddle point equations and the continuous line is the theoretical prediction of \ref{['app:analytical_transition']}). The orange dots form the binodals of the phase separation. The critical point of the MIPS-like phase transition lies well within the spin-glass phase.
• Figure 5: Number fluctuations $\kappa$ as a function of the reduced density $\hat{\varphi}$ for $c_1 = 1.9$ and $c_2 = 0.1$. They are continuous with discontinuous derivative upon crossing the paramagnet-to-spin-glass transition marked by the dashed vertical line. Note that for the ideal gas, $\kappa = 1$ so that $d\kappa$ diverges in the limit of infinite dimension when $\hat{\varphi} \rightarrow 0$. The secondary peak signals the proximity of the MIPS critical point.