We present the mapping relation between a one-dimensional conserved lattice gas model and a two species reaction $A+B\ensuremath{\rightarrow}0$. From the kinetics of $A+B\ensuremath{\rightarrow}0$, we show that the anomalous critical decay of an order parameter in the conserved lattice gas model results from finite-size effects induced by the domain structure for random initial conditions where the scaling relation $z={\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ is violated. A simple initial condition satisfying the scaling relation is also discussed.
We present the mapping relation between a one-dimensional conserved lattice gas model and a two species reaction A+B\ensuremath→0. From the kinetics of A+B\ensuremath→0, we show that the anomalous critical decay of an order parameter in the conserved lattice gas model results from finite-size effects induced by the domain structure for random initial conditions where the scaling relation z=\ensuremathν\ensuremath∥/\ensuremathν\ensuremath⊥ is violated. A simple initial condition satisfying the scaling relation is also discussed.
There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.
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.
The conserved lattice gas model in one dimension, generated from various initial states, was intensively studied. The dynamic critical exponents α and ν ║ associated with the decay of activeparticle densities and the correlation time, respectively, were found to depend drastically on the initial states. The perfectly-ordered initial state prepared by repeatedly placing the block ‘0011’ yielded a standard critical behavior, with the critical exponents satisfying all the scaling relations. On the other hand, the natural initial states and the ordered initial states of blocks of odd numbers of particles yielded an exponent α slightly larger than that of an ordered state of ‘0011’, but the data for active-particle densities did not satisfy scaling function. The exponents α and ν ║ were also calculated from, respectively, the autocorrelation function and the persistence distribution of the active-particle densities, both in the steady states. The value of α was found to be close to the value for an ordered state of ‘0011’ whereas the value of ν ║ was consistent with that for the random initial states.
We extend standard finite-size scaling methods to study the dynamical evolution of an unstable state far from equilibrium as the system undergoes a first order phase transition. We suggest that the nonequilibrium structure factor S(q, t, L), at late times and for large enough lattice sizes, scales as S(q, t, L)=LdF(qL, t1/x/L). L is the linear dimension of the system and 1/x is the domain growth exponent. We obtain x=2 in the case of the kinetic Ising model with a nonconserved order parameter. For a critical quench in a system with conserved order parameter, scaling of the peak of the structure factor gives 1/x≈0.27. Higher wavenumbers, however, are more consistent with x=3.
We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to Tc from random initial states. As in the case of subcritical quenches, we observe dynamical scaling of the equal time order parameter correlation function G(r,t,t), but here the nonequilibrium correlation length grows as L(t)\ensuremath{\sim}t1/z, where z=4/15 is the usual dynamic critical exponent for this system. Moreover, we find dynamical scaling for correlations with the initial condition G(r,0,t). The temporal decay of autocorrelations at Tc yields a critical exponent ${\ensuremath{\lambda}}_{\mathit{c}}$=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}${\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$(t)G\ifmmode \tilde{}\else \~{}\fi{}(r/L(t)).
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.