Disorder-Induced Long-Ranged Correlations in Scalar Active Matter

We study the impact of quenched random potentials and torques on scalar active matter. Microscopic simulations reveal that motility-induced phase separation is replaced in two dimensions by an asymptotically homogeneous phase with anomalous long-ranged correlations and nonvanishing steady-state currents. Using a combination of phenomenological models and a field-theoretical treatment, we show the existence of a lower-critical dimension $d_c=4$, below which phase separation is only observed for systems smaller than an Imry-Ma length scale. We identify a weak-disorder regime in which the structure factor scales as $S(q)\sim 1/q^2$, which accounts for our numerics. In $d=2$, we predict that, at larger scales, the behavior should cross over to a strong-disorder regime. In $d>2$, these two regimes exist separately, depending on the strength of the potential.

Figure 1

Snapshots of simulations (a) without and (b) with disorder. Color encodes density, obtained by averaging occupancies over four neighboring sites. (c) The pair correlation functions are shown in linear scale with (black) and without (red) disorder. (d) Pair correlation with disorder using log-linear scale. The dashed lines correspond to linear (red) and logarithmic (black) decays. Parameters: (a),(b) $L=300$; (b) $\mathrm{\Delta }V=7.5$; $v=13$, $\alpha =1$, $n_M=2$, ${\rho }_0\equiv N/L^2=1$.

Figure 2

(a) In $d=1$, asymmetric potentials leads to a nonzero average force on the particles, indicated by a bold arrow. This is accompanied by a nonvanishing ratchet current. (b) In $d=2$, a random potential leads to a steady-state field of random forces exerted on the particles. (c) Measurement of the force $f(A)$ exerted on the active particles inside a region of area $A$ in the presence of a random potential. The amplitude of the force is quantified by $|f(A)|\equiv [\overline{f^2(A)}{]}^{1/2}$, where $f(A)$ is obtained by time averaging ${\sum }_{\overrightarrow{i}\in A}{n}_{\overrightarrow{i}}(V_{\overrightarrow{i}+\overrightarrow{e}}-V_{\overrightarrow{i}-\overrightarrow{e}})/2$ with $\overrightarrow{e}$ as an arbitrary unit vector.

Figure 3

The current induced by disorder and its statistical properties. (a) Current vector map for a realization of disorder. The color code is the steady-state current normalized by the maximum value measured. (b) The variance of the sum of current $J(\mathcal{C})$ along a contour $\mathcal{C}$ as a function of the its perimeter $c$. Parameters: $v=13$, $\alpha =1$, $\mathrm{\Delta }V=6.5$ in (a).

Figure 4

Measurement of the effective self-propulsion speed $v(\rho )$ in our simulations (a) without and (b) with disorder for $v=13$ and $\alpha =1$. Both systems exhibit a similar decay that, in the absence of disorder, would lead to MIPS. Note that the kink observed for $v(\rho )$ in (a) stems from the occurrence of phase separation.