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Physical Review E

Validity of scaling relations in absorbing phase transitions with a conserved field

Sang Bub Lee, Sang-Gui Lee

Published Oct 10, 2008• 22 citations
Publisher

Abstract

The two scaling relations in absorbing phase transitions, ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=\ensuremath{\beta}∕\ensuremath{\theta}$ and $z={\ensuremath{\nu}}_{\ensuremath{\parallel}}∕{\ensuremath{\nu}}_{\ensuremath{\perp}}$, are studied for a conserved lattice gas model. The critical indices calculated elaborately from the all-sample average density of active particles appear to satisfy both relations. However, the exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ calculated from the surviving samples does not appear to be consistent with the value in the thermodynamic limit. This is in contrast with earlier observations [M. Rossi et al., Phys. Rev. Lett. 85, 1803 (2000); S. L\"ubeck and P. C. Heger, Phys. Rev. E. 68, 056102 (2003)], in that the former scaling relation was claimed to be violated.

DOI
10.1103/physreve.78.040103

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The two scaling relations in absorbing phase transitions, \ensuremathν\ensuremath∥=\ensuremathβ∕\ensuremathθ{\ensuremath{\nu}}_{\ensuremath{\parallel}}=\ensuremath{\beta}∕\ensuremath{\theta}\ensuremathν\ensuremath∥​=\ensuremathβ∕\ensuremathθ and z=\ensuremathν\ensuremath∥∕\ensuremathν\ensuremath⊥z={\ensuremath{\nu}}_{\ensuremath{\parallel}}∕{\ensuremath{\nu}}_{\ensuremath{\perp}}z=\ensuremathν\ensuremath∥​∕\ensuremathν\ensuremath⊥​, are studied for a conserved lattice gas model. The critical indices calculated elaborately from the all-sample average density of active particles appear to satisfy both relations. However, the exponent \ensuremathν\ensuremath⊥{\ensuremath{\nu}}_{\ensuremath{\perp}}\ensuremathν\ensuremath⊥​ calculated from the surviving samples does not appear to be consistent with the value in the thermodynamic limit. This is in contrast with earlier observations [M. Rossi et al., Phys. Rev. Lett. 85, 1803 (2000); S. L"ubeck and P. C. Heger, Phys. Rev. E. 68, 056102 (2003)], in that the former scaling relation was claimed to be violated.
The critical behavior of absorbing phase transitions for two typical models in the Manna universality class, the conserved Manna model and the conserved lattice gas model, both on a square lattice, was investigated using the natural initial states. Various critical exponents were estimated using the static and dynamic simulations. The exponents characterizing dynamics of active particles differ considerably from the known exponents obtained using the random initial states, whereas those associated with the steady-state quantities remain the same. The critical exponents for both models were consistent with errors of less than 1% and satisfied the known scaling relations; thus, the known violation of scaling relations for models with a conserved field was resolved using the natural initial states. The results differed by 7%∼12% from the directed percolation values.
We analyze numerically the critical behavior of a conserved lattice gas that was recently introduced as an example of the new universality class of absorbing phase transitions with a conserved field [Phys. Rev. Lett. 85, 1803 (2000)]. We determine the critical exponent of the order parameter as well as the critical exponent of the order parameter fluctuations in D=2,3,4,5 dimensions. A comparison of our results and those obtained from a mean-field approach and a field theory suggests that the upper critical dimension of the absorbing phase transition is four.
A one-dimensional conserved lattice-gas model is known to undergo continuous absorbing phase transitions where some of the critical exponents are exactly known. In one dimension, we recently showed that the model is mapped onto a two species reaction A+B→0 with diffusion rate of D_{A}>0 and D_{B}=0. In this work, it is explicitly shown from the scaling theory for A+B→0 that the observed scaling behavior of the conserved lattice-gas model is not associated with the absorbing phase transitions. Instead, the model indeed undergoes a crossover between two different scaling behaviors of A+B→0, the scaling behaviors of equal and unequal initial densities of two species. The crossover is similar to the absorbing transitions in many respects but some important features of continuous transitions such as the diverging fluctuations of an order parameter are absent.
Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.