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arXiv:cond-mat/9909347

Nonuniversality and critical point shift in systems with infinitely many absorbing configurations

Ronald Dickman

Published Sep 23, 1999• 1 citation
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Abstract

A detailed study of critical spreading in the one-dimensional pair contact process is performed using a recently devised reweighting method. The results confirm the validity of a generalized hyperscaling relation among the (nonuniversal) spreading exponents, and support the assertion that the critical point location does not depend on the particle density ϕ. It appears that the exponents z and delta+eta, once thought to be invariant, exhibit a systematic dependence on ϕ.

DOI
10.48550/arxiv.cond-mat/9909347

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A detailed study of critical spreading in the one-dimensional pair contact process is performed using a recently devised reweighting method. The results confirm the validity of a generalized hyperscaling relation among the (nonuniversal) spreading exponents, and support the assertion that the critical point location does not depend on the particle density ϕ. It appears that the exponents z and delta+eta, once thought to be invariant, exhibit a systematic dependence on ϕ.
Continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations. Spreading from a localized source is characterized by nonuniversal critical exponents, which vary continuously with the density \ensuremath{\varphi} in the surrounding region. The exponent \ensuremath{\delta} changes by more than an order of magnitude and \ensuremath{\eta} changes sign. The location of the critical point also depends on \ensuremath{\varphi}, which has important implications for scaling. As expected on the basis of universality, the static critical behavior belongs to the directed percolation class. \textcopyright{} 1996 The American Physical Society.
A hyperscaling relation for the critical exponents of absorbing phase transitions is tested in the bosonic pair contact process with diffusion. To this end spreading is considered, i.e. the time evolution out of an initial seed. It is shown that like in the case of spatial homogeneous initial conditions the autocorrelation function exhibits a phase transition at the critical point of the first moment. Some of the critical exponents can be determined exactly which is an unusual property of an absorbing phase transition and provides a possibility to test the hyperscaling relation. In the case of the bosonic pair contact process with diffusion three sets of exponents can be considered referring to the number of particles, number of pairs and number of active sites. It is argued that in special cases it is generally impossible to produce adequate data numerically.
I determine the critical behavior of a nonequilibrium three-dimensional lattice model exhibiting a phase transition to an absorbing state. I study the model in the vicinity of the critical point, and in the subcritical region, via time-dependent Monte Carlo simulations. The method used in the subcritical region is very efficient. The results for the directly measured critical exponents, \ensuremath{\nu}=1.11\ifmmode\pm\else\textpm\fi{}0.01, \ensuremath{\eta}=0.114\ifmmode\pm\else\textpm\fi{}0.004, and z=1.052\ifmmode\pm\else\textpm\fi{}0.003, are consistent with those of directed percolation. \ensuremath{\delta}=0.732\ifmmode\pm\else\textpm\fi{}0.004 is obtained from the hyperscaling relation 4\ensuremath{\delta}+2\ensuremath{\eta}=dz, and \ensuremath{\beta}=0.813\ifmmode\pm\else\textpm\fi{}0.011 from \ensuremath{\beta}=\ensuremath{\nu}\ensuremath{\delta}. These results are the most precise so far for a three-dimensional model with directed percolation critical behavior.
One-dimensional systems with an infinite number of absorbing states exhibit a phase transition that is not fully understood yet. Their static critical exponents are universal and belong in the Reggeon field theory (or directed percolation) universality class. However, exponents associated with the spreading of a localized seed appear to be nonuniversal depending on the nature of the initial condition. We investigate this problem by integrating numerically a non-Markovian Langevin equation proposed recently to describe such phase transitions. We find that the static critical exponents are universal, as expected. On the other hand, the Langevin equation reproduces the nonuniversal behavior observed in microscopic models for exponents associated with the spreading of an initially localized seed and satisfies the generalized hyperscaling relation proposed for those systems.
We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent δ. To obtain an accurate estimate of δ, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 62, 469 (2013)KPSJAS0374-488410.3938/jkps.62.469]. For small diffusion rate (d≤0.5), the leading corrections-to-scaling term is found to be ∼t^{-0.15}, whereas for large diffusion rate (d=0.95) it is found to be ∼t^{-0.5}. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent δ is 0.173(3) irrespective of d. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At d=0, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent ϕ=2.6(1). We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At d=1, the crossover from the mean field PCPD to the PCPD is described by ϕ=2 which is argued to be exact.