Simple Model for Active Nematics: Quasi-Long-Range Order and Giant Fluctuations

We propose a simple microscopic model for active nematic particles similar in spirit to the Vicsek model for self-propelled polar particles. In two dimensions, we show that this model exhibits a Kosterlitz-Thouless-like transition to quasi-long-range orientational order and that in this nonequilibrium context, the ordered phase is characterized by giant density fluctuations, in agreement with the predictions of Ramaswamy et al. [Europhys. Lett. 62, 196 (2003)].

Figure 1

Transition to nematic order at density $\rho =N/L^2=\frac{1}{2}$. (a) (Time-averaged) scalar order parameter $⟨S⟩$ vs noise strength $\sigma$ for various sizes. (b) $⟨S⟩$ vs $N$ at $\sigma =0.02,0.04,0.06,0.08,0.1$ from top to bottom; the lines are fitted power laws. Each point represents an average over ${10}^6–{10}^7$ time steps after transients. (c) Same as (b) but for larger noise values: $\sigma =0.11,0.12,0.13,0.14,0.15,0.16,0.17,0.18$. For large $\sigma$ values these curves cross over to $1/\sqrt{N}$ decay. (d) Exponents $\zeta$ extracted from the power laws shown in (b) and (c) vs $\sigma$. The line linking the symbols is a (cubic) fit. For the last 2 points, the exponent is only effective, as it was estimated from small $N$ values, before signs of crossover appear. The dotted lines indicate the threshold ${\sigma }_{\mathrm{c}}=0.113(5)$ estimated from the condition $\zeta ({\sigma }_{\mathrm{c}})=\frac{1}{16}$ (see text).

Figure 2

Orientational correlation function $g_2(r)$ as calculated from the orientation field coarse grained over boxes of linear size 4 [$\rho =\frac{1}{2}$, time average over ${10}^6–{10}^7$ time steps after transients for each run, $L=256$ in (a), (d), and $L=128$ in (b)]. (a) In log-log scales at small noise values $\sigma =0.02$, 0.04, 0.06, 0.08, 0.10, 0.11, 0.12, and 0.13 from top to bottom. (b) In lin-log scales for $\sigma =0.14$ to 0.3 from top to bottom. (c) Variation of the correlation length $\xi$ extracted from the exponential tails in (b) as the transition point is approached (dashed line at the estimated threshold ${\sigma }_{\mathrm{c}}=0.113$). (d) Same as in (a), data multiplied by the expected exponent at threshold ${\eta }_{\mathrm{c}}=\frac{1}{4}$: around the estimated threshold value ${\sigma }_{\mathrm{c}}=0.113(5)$, the curves are flat and straight.

Figure 3

Root mean square density fluctuations $\Delta n$ in square boxes containing $⟨n⟩$ particles on average (linear size $\ell =\sqrt{n/\rho }$, $\rho =\frac{1}{2}$, time average over ${10}^6–{10}^7$ time steps after transients). The saturation at large $⟨n⟩$ occurs when $\ell \simeq L$. Top two curves: giant fluctuations in the ordered phase at 2 different system sizes. Bottom curve: normal fluctuations in the disordered phase.

Figure 4

Typical time series of $S$ in the ordered phase. ($\rho =\frac{1}{2}$, $L=128$, $\sigma =0.1$). Note the large excursions to small $S$ values which make time averaging difficult. (b)–(d) Snapshots taken during the run shown in (a) at the times indicated there. For each particle, a small segment (of arbitrary length) centered on its position and aligned on its orientation is drawn. (b), (d) are typical ordered states, while (c) represents the more disordered episodes when the macroscopic structure changes orientation.