Equilibrium to off-equilibrium crossover in homogeneous active matter

We study the crossover between equilibrium and off-equilibrium dynamical universality classes in the Vicsek model near its ordering transition. Starting from the incompressible hydrodynamic theory of Chen et al. [Critical phenomenon of the order-disorder transition in incompressible active fluids, New J. Phys. 17, 042002 (2015)], we show that increasing the activity leads to a renormalization group (RG) crossover between the equilibrium ferromagnetic fixed point, with dynamical critical exponent $z=2$, and the off-equilibrium active fixed point, with $z=1.7$ (in $d=3$). We run simulations of the classic Vicsek model in the near-ordering regime and find that critical slowing down indeed changes with activity, displaying two exponents that are in remarkable agreement with the RG prediction. The equilibrium to off-equilibrium crossover is ruled by a characteristic length scale, beyond which active dynamics takes over. The larger the activity is, the smaller is such a length scale, suggesting the existence of a general trade-off between activity and the system's size in determining the dynamical universality class of active matter.

Figure 1

RG flow and crossover. Top: The system admits four fixed points: The free theory, ${\alpha }^{*}=0$, $u^{*}=0$ (black circle), and the incompressible stirred fluid, ${\alpha }^{*}=4/3\epsilon$, $u^{*}=0$ (purple triangle [24]), are of no interest to us. On the other hand, the equilibrium ferromagnet, ${\alpha }^{*}=0$, $u^{*}=2/17\epsilon$ (green square), and the Vicsek active-ferromagnetic fixed point, ${\alpha }^{*}=124/113\epsilon$, $u^{*}=6/113\epsilon$ (red circle), give rise to the equilibrium to off-equilibrium crossover. If the initial value of the activity, ${\alpha }_0$, is small, the flow quickly approaches the equilibrium fixed point (red dots) and remains around it for many RG iterations, before crossing over to the Vicsek fixed point. Bottom: The crossover is evident when plotting the running parameters over an RG streamline with small ${\alpha }_0$. The running critical exponent $z$ flows from its equilibrium value $z=2$ to its off-equilibrium value, $z=1.7$ (in $d=3$).

Figure 2

Vicsek dynamics in three dimensions. (a) and (c) Normalized dynamical correlation functions, $\widehat{C}(k,t)=C(k,t)/C(k,0)$, at $k=1/\xi$ for $v_0=0.05$ and $v_0=0.20$, respectively. (b) and (d) Correlation functions plotted against the scaling variable $t{k}^z$; $z=2$ in (b) and $z=1.7$ in (d). (e) Relaxation time vs correlation length for the two different values of the speed $v_0$. Lines are best fit to the RG results, $z=1.7$ (blue, high $v_0$, high activity) and $z=2$ (red, low $v_0$, low activity).

Figure 3

Numerical simulations: test of homogeneity. Analysis of homogeneity for data sets with $v_0=0.05$ (red) and $v_0=0.2$ (blue). (a) Temporal series of polarization $\varphi$ for both activities and for size $N=2048$ at the critical point. (b) The mean first-neighbor distance is computed for each size $N=128,256,384,512,1024,2048$, and it scales as $x\sim {\rho }_0^{1/3}$ as predicted for homogeneous systems. (c) Temporal series of $n_t$: the number of particles entering into a box of size $l_0=L/7$ that is centered in the middle of the bigger box of size $L$ for $N=2048$. (d) Average in time of $n_t$ for all the sizes $N$ and speed: This quantity scales linearly with $N$, as predicted for homogeneous systems $\langle n_t\rangle =N/7^3$.

Figure 4

Numerical simulations. (a) and (c) Scaling of dynamic correlation functions $\widehat{C}(k,t)$ using values of $z$ extrapolated by the linear fits of Fig. 2: (a) $v_0=0.05$ with $z=2.0$ and (b) $v_0=0.20$ with $z=1.7$. (b) and (d) Same correlation functions reversing the scaling procedure: (c) $v_0=0.05$ with $z=1.7$ and (d) $v_0=0.05$ with $z=2.0$. The scaling hypothesis is not well verified for (b) and (d).