Abstract The critical behaviour of an absorbing phase transition for the conserved continuous Manna model (CCMM) was investigated in one dimension via extensive Monte Carlo simulations, using sequential updates from the uniform initial states. The discrete conserved Manna model (DCMM) was also revisited and previous data, together with newly generated data, were reanalysed and compared with those for the CCMM. All critical exponents in the two models appear to be consistent to within statistical error, whereas the off-critical scaling function takes a different form in each model. The results differed by less than 10% from those of the directed percolation (DP) universality class. To examine the existence of the Griffith phase, additional simulations were carried out on a stripe of 10 5 × 20 lattice sites, 20% of which were diluted. While the non-universal power laws were reported for the models in the DP class, there was no sign of the Griffith phase in either model. Thus, we conclude that both the DCMM and CCMM belong to a universality class that is different from the DP class. We examine the possibility of weak universality between the discrete and continuous models, whereby the critical exponents are similar, but the off-critical scaling functions are different.
Abstract The critical behaviour of an absorbing phase transition for the conserved continuous Manna model (CCMM) was investigated in one dimension via extensive Monte Carlo simulations, using sequential updates from the uniform initial states. The discrete conserved Manna model (DCMM) was also revisited and previous data, together with newly generated data, were reanalysed and compared with those for the CCMM. All critical exponents in the two models appear to be consistent to within statistical error, whereas the off-critical scaling function takes a different form in each model. The results differed by less than 10% from those of the directed percolation (DP) universality class. To examine the existence of the Griffith phase, additional simulations were carried out on a stripe of 10 5 × 20 lattice sites, 20% of which were diluted. While the non-universal power laws were reported for the models in the DP class, there was no sign of the Griffith phase in either model. Thus, we conclude that both the DCMM and CCMM belong to a universality class that is different from the DP class. We examine the possibility of weak universality between the discrete and continuous models, whereby the critical exponents are similar, but the off-critical scaling functions are different.
The nonequilibrium absorbing phase transition of the discrete conserved Manna model was studied via Monte Carlo simulations on a one-dimensional chain, using the natural initial states with a sequential update. The critical density of the particles was found to be smaller than the recently reported value, and the order-parameter exponent was considerably different from the directed percolation (DP) value. The influence of quenched disorder was also studied on a diluted strip of L_{x}×L_{y} lattice sites with L_{x}≫L_{y}, and the results were compared with those of the contact process (CP). It was found that the Manna model and the CP exhibited distinctly different behaviors; the CP exhibited nonuniversal power-law decreases of active-site densities in the Griffith phase, whereas the Manna model showed a standard critical behavior. These results consistently suggest that the Manna model belongs to a universality class that is different from the DP class.
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 applied an alternative method for measuring characteristic lengths reported recently by one of us [J. M. Kim, J. Stat. Mech. (2021) 03321310.1088/1742-5468/abe599] to the models in the Manna universality class, i.e., the stochastic Manna sandpile and conserved lattice gas models in various dimensions. The universality of the Manna model has been under long debate particularly in one dimension since the work of M. Basu et al. [Phys. Rev. Lett. 109, 015702 (2012)10.1103/PhysRevLett.109.015702], who claimed that the Manna model belongs to the directed percolation (DP) universality class and that the independent Manna universality class does not exist. We carried out Monte Carlo simulations for the stochastic Manna sandpile model in one, two, and three dimensions and the conserved lattice gas model in three dimensions, using both the natural initial states (NISs) and uniform initial states (UISs). In two and three dimensions, the results for R(t), defined by R(t)=L[〈ρ_{a}^{2}〉/〈ρ_{a}〉^{2}-1]^{1/d}, L and ρ_{a} being, respectively, the system size and activity density, yielded consistent results for the two initial states. R(t) is proportional to the correlation length following R(t)∼t^{1/z} at the critical point. In one dimension, the data of R(t) for the Manna model using NISs yielded anomalous behavior, suggesting that NISs require much longer prerun time steps to homogenize the distribution of particles and larger systems to eliminate the finite-size effect than those employed in the literature. On the other hand, data from UISs yielded a power-law behavior, and the estimated critical exponents differed from the values in the DP class.
We investigate the dynamical scaling behavior of the static diffusive epidemic process and a fixed-energy Manna sandpile model, undergoing nonequilibrium absorbing phase transitions in one dimension. These models belong to the so-called conserved directed-percolation or Manna universality class characterized by the conservation of the total particle number, activity coupled to a nondiffusive conserved field and infinitely many absorbing states. We measure the dynamical exponents of these models in one dimension by using the critical spreading simulation of a localized activity in absorbing configurations. In the spreading simulations, boundaries are never touched, so the results are free from the finite-size effects. In contrast to the scattered results for the different models from the previous finite-size scaling analyses, we obtain consistent estimates of the dynamical exponents for both models.
Absorbing phase transitions in restricted exclusion processes are characterized by simple integer exponents. We show that this critical behaviour flows to the directed percolation (DP) universality class when particle conservation is broken suitably. The same transition, when studied using the average density as the controlling parameter, yields critical exponents quite different from DP; we argue that these exponents are actually related to DP by a scaling factor 1/ β DP . These conclusions also apply to conserved lattice gas in one dimension.
Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions. In particular, the effect of an external field conjugated to the order parameter is investigated. Incorporating the conjugated field, it is possible to determine the equation of state, the susceptibility, and to perform a modified finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments. Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work. Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results. Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay between numerical investigations and renormalization group analyzes.