Critical behavior of a two-species reaction-diffusion problem

We present a Monte Carlo study in dimension $d=1\mathrm{}$ of the two-species reaction-diffusion process $A+\overrightarrow{B}2B$ and $\overrightarrow{B}A.\mathrm{}$ Below a critical value ${\rho }_c$ of the conserved total density $\rho$ the system falls into an absorbing state without B particles. Above ${\rho }_c$ the steady state B particle density ${\rho }_B^{\mathrm{st}}$ is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities $\rho$ between zero and some maximum density. From finite-size scaling we obtain the steady state exponents $\beta =0.435\mathrm{}\left(10\right),\nu =2.21\mathrm{}\left(5\right),$ and $\eta =-0.606\left(4\right)\mathrm{}$ for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value ${\theta }^{\prime }=0.30\mathrm{}\left(2\right),$ in agreement with the theoretical relation ${\theta }^{\prime }=-\eta /2\mathrm{}$ due to Van Wijland et al. [Physica A 251, 179 (1998)].