Hydrodynamic behavior near dynamical criticality of a facilitated conservative lattice gas

Abstract

We investigate a $2d$-conservative lattice gas exhibiting a dynamical active-absorbing phase transition with critical density $ρ_c$. We derive the hydrodynamic equation for this model, showing that all critical exponents governing the large scale behavior near criticality can be obtained from two independent ones. We show that as the supercritical density approaches criticality, distinct length scales naturally appear. Remarkably, this behavior is different from the subcritical one. Numerical simulations support the critical relations and the scale separation.

Figures (5)

• Figure 1: Blue circle particles are active, red square particles are frozen. As an example, the active particle highlighted with $\square$ can jump to one of its three neighbours indicated with X. The dashed region corresponds to the frozen phase.
• Figure 2: Median absorption time in a closed box, of size changing from 5 to 100. In the subcritical phase (a) the absorption time grows sub-linearly with the system size. In the supercritical phase (b) it grows exponentially fast at large $L$. In fact, the figure illustrates nicely that at $\rho = 0.334$ the geometric correlation length, which separates the absorbing regime from the quasi-stationary regime, is roughly $\xi \approx 20$, which is the approximative point where the exponential growth begins.
• Figure 3: $N_t$ as a function of $t$ for different reservoir densities. See equation \ref{['eq:current_between_reservoirs']}.
• Figure 4: Simulating $\rho(x),\rho_a(x)$ and $a(x)$ in a system with reservoirs $\lambda_l=0,\lambda_r=1$.
• Figure 5: We show here a collapse of $S_\rho(k)$ for different values of $\rho$. The parameters $C_1=0.03, C_2 = 0.71, \rho_c=0.3361, \nu_\times = 1.77, \gamma=1.07$ are adjusted to best fit equation \ref{['eq:S_k']}. That is, we find that $\chi(\rho) = 0.61 (\rho - 0.3361)^{-1.07}$ and $\xi_\times = 0.03 (\rho-0.3361)^{-1.77}$. Indeed, after this rescaling the curves $S_\rho(k)$ collapse as expressed in equation \ref{['eq:S_k']}. The fit (black curve) is given by $S_\rho(k) = 0.61(\rho-0.3361)^{1.07}+0.07 |k|^{0.60}$.