Finite-size scaling of directed percolation in the steady state

Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a $d$-dimensional hypercube of finite edge length $L$ with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.

Figure 1

Scaling regions (schematically) above $d_c$ where corrections to the lowest mode approximation resulting from higher modes are essential (I and III) or negligible (II).

Figure 2

The universal order parameter scaling function $M_1$ (inset) and the universal fourth order ratio scaling function $Q$ as a function of the rescaled source.

Figure 3

The ratio $U$ as a function of $\ln b$ with $a$ as a parameter. For $a=0$, $U$ lies on the abscissa, $U\equiv 1∕2$. For $a$ decreasing form zero, the values of $U$ deviate increasingly from $1∕2$ for $b⪡1$.

Figure 4

(Color online) The ratio $U$ as a function of $L$. The meaning of the various curves is explained in the text.

Figure 5

The universal ratio $U$ at the bulk critical point as a function of the scaled source $h∕L^{\Delta ∕\nu }$ in dimensions $d=1,2,3$.

Figure 6

(Color online) The universal ratio $U$ versus system size $L$ for $d=4$. The blue lower and the red middle curves represent our analytical results (5.12, 8.10) with $L_0=2.4$, $K=0$ and $L_0=1.5$, $K=-0.5$, respectively. For comparison, we included the dashed upper curve, where we have disregarded any two-loop contributions and where we have fitted $L_0$ to the data points for larger $L$, $L_0=5.6$. The solid dots stem from our Monte-Carlo simulations of critical sDP on bcc lattices of linear sizes ranging from $L=4$ to $L=64$.