Lamellar to Micellar Phases and Beyond: When Tactic Active Systems Admit Free Energy Functionals

We consider microscopic models of active particles whose velocities, rotational diffusivities, and tumbling rates depend on the gradient of a local field that is either externally imposed or depends on all particle positions. Despite the fundamental differences between active and passive dynamics at the microscopic scale, we show that a large class of such tactic active systems admit fluctuating hydrodynamics equivalent to those of interacting Brownian colloids in equilibrium. We exploit this mapping to show how taxis may lead to the lamellar and micellar phases observed for soft repulsive colloids. In the context of chemotaxis, we show how the competition between chemoattractant and chemorepellent may lead to a bona fide equilibrium liquid-gas phase separation in which a loss of thermodynamic stability of the fluid signals the onset of a chemotactic collapse.

Figure 1

Simulations of dynamics, Eqs. (1)–(2), (top) and of the equilibrium dynamics, Eq. (14), (bottom) under the conditions of the mapping, Eq. (15), with $K(\mathit{r})$ defined by Eq. (16). Color encodes the local density. Parameters: $v_0=1$, $v_1=0.2$, ${\alpha }_0=50$, ${\sigma }_0=0.3$, ${\sigma }_1=1$, $ϵ=5$, $A=10$, $D_t={\alpha }_1={\mathrm{\Gamma }}_i=0$, $dt={10}^{-3}$. Snapshots taken at $t=20000$ in a system of size $20\times 20$. From left to right, ${\rho }_0\equiv N/L^2$ is equal to 2, 4, 8, and 10.

Figure 2

Simulations of the active dynamics, Eqs. (1)–(3), top, and the equilibrium dynamics, Eq. (14), bottom, using the pair potential, Eq. (19), under the conditions of the mapping, Eq. (15). Color encodes the local density. From left to right ${\ell }_r=1/6,1/8,1/10$. Chemotactic collapse is predicted when $\overline{V}<0$, i.e., below ${\ell }_r=1/8.2$. Parameters: $L=40$, ${\rho }_0=7$, $a_a/2\pi {\nu }_a=0.05$, ${\ell }_a=0.25$, $a_r/2\pi {\nu }_r=0.2$, ${\alpha }_0=100$, ${\alpha }_1^{r,a}=200$, $v_0=1$, $T^{-1}=200$, $\gamma =1$, $v_1={\mathrm{\Gamma }}_i=D_t=0$, $dt={10}^{-3}$. The snapshots were taken after a time $t={10}^4$.