Multichain models of conserved lattice gas

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents $\beta =1=\nu =\eta$, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system ($M\times L$ square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit $L\to \infty ,$ these models exhibit APT at ${\rho }_c=\frac{1}{2}(1+\frac{1}{M})$ with $\beta =1.$ On the other hand, for even-chain systems transition occurs at ${\rho }_c=\frac{1}{2}$ with $\beta =1,2$ for $M=2,4$, respectively, and $\beta =3$ for $M\ge 6.$ We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Figure 1

Schematic description of the model: Particles having two occupied nearest neighbors and one vacant nearest neighbor are active (filled circle), whereas all other particles are inactive (open circle). The active particles hop to the only vacant nearest neighbor they have.

Figure 2

(a) Plot of density $\rho$ as a function of $z:\rho$ approaches the critical value ${\rho }_c=\frac{1}{2}$ as $z\to 0.$ (b) Plot of order parameter ${\rho }_a$ (i.e., steady-state density of active particles) versus $\rho$. ${\rho }_a$ becomes nonzero above critical density ${\rho }_c=\frac{1}{2}$. For $\rho =1$, all the sites are occupied and thus ${\rho }_a=0$.

Figure 3

The order parameter ${\rho }_a$ is the sum of the steady-state averages of several three-rung configurations, of which one of the terms, which is lowest order in $z$, is ${\rho }_a^{*}$ given by Eq. (52). (a) A parametric plot of ${\rho }_a^{*}\left(z\right)$ as a function of $\rho \left(z\right)$, calculated following the transfer-matrix method (solid line) for $M=3,5,7,9,11$ (top to bottom) is compared with the same obtained for different densities (symbols) using Monte Carlo simulations of the restricted CLG dynamics (density conserving) on a $M\times L$ system, with $L=1000$ and $M=3,5,7.$ Clearly, ${\rho }_a^{*}$ vanish at ${\rho }_c=\frac{1}{M}\lfloor \frac{M+1}{2}\rfloor$. (b) Log scale plot of ${\rho }_a^{*}$ as a function of $(\rho -{\rho }_c)$, along with a line with unit slope (dashed line) indicates that ${\rho }_a^{*}\sim {(\rho -{\rho }_c)}^{\beta }$ with $\beta =1.$

Figure 4

Parametric plot of ${\rho }_a^{*}\left(z\right)$, the steady-state average of the active three-rung configuration in (58), as a function of $\rho \left(z\right)$. (a), (b), (c), and (d) correspond to even $M=2,4,6$, and 8, respectively. Solid lines are obtained from the transfer-matrix formulation and the symbols correspond to the same obtained from Monte Carlo simulation of the restricted CLG on the $M\times L$ system with $L=1000$. Clearly, ${\rho }_a^{*}$ vanishes at ${\rho }_c=\frac{1}{2}$ for all $M$.

Figure 5

(a) $\rho \left(z\right)$ as a function of $z$. (b) Log scale plot of ${\rho }_a^{*}$ shown in Fig. 4 for $M=2,4,6,8,10$ (top to bottom) as a function of $\rho -{\rho }_c$ shows that $\beta =1,2$ for $M=2,4$ and $\beta =3$ for even $M>4.$ Lines with slope 1, 2, 3 are shown in dashed lines for comparison.

Figure 6

The density of active particles ${\rho }_a\left(t\right)$ as a function $t$ for $M=2.$ The evolution from a random initial condition (IC) exhibits undershooting and long relaxation to the stationary state; both these ill effects are avoided if we use the natural IC. Here $L={10}^4$ and $\rho =0.53.$