Stationary distribution of finite-size systems with absorbing states

We introduce a procedure that allows us to obtain nontrivial stationary distributions of finite-size models with absorbing states. Two models are studied: the contact process and the sandpile model with height restriction. To avoid the permanence of the system in the absorbing state we create a small perturbation that drives the system out of the absorbing state. In the former model a particle is created, in the latter an active site is created. The stationary distributions around the critical point are analyzed by the use of finite-size scaling.

Figure 1

Density of particles $\rho$ for the contact process as a function of the creation rate $\lambda$ for several values of the system size $L$.

Figure 2

Number of particles $n=L\rho$ for the contact process as a function of the creation rate $\lambda$ for several values of the system size $L$. The continuous line represents the expansion of $n$ up to ${\lambda }^2$ as given by Eq. (20).

Figure 3

Reduced second-order cumulant $u$ for the contact process as a function of the creation rate $\lambda$ for several values of the system size $L$. The continuous line represents the expansion of $u$ up to ${\lambda }^2$ as given by Eq. (22).

Figure 4

Data collapse of probability density $P(L,0,x)$ at the critical point obtained for several values of $L$.

Figure 5

Density of active sites $\rho$ for the sandpile model with height restriction as a function of the density of particles $\zeta$ for several values of the system size $L$.

Figure 6

Reduced second-order cumulant $u$ for the sandpile model with height restriction as a function of the density of particles $\zeta$ for several values of the system size $L$.

Figure 7

Data collapse of probability density $P(L,0,x)$ at the critical point obtained for several values of $L$ for the sandpile with height restriction.