On phase transitions in Schlögl's second model

AbstractWe study Schlögl's second model, characterized by chemical reactions $$\begin{array}{*{20}c} {2X\underset{{k_2 }}{\overset{{k_1 }}{\longleftrightarrow}}3X,} & {X\underset{{k_4 }}{\overset{{k_3 }}{\longleftrightarrow}}0,} \\ \end{array} $$ ind-dimensional space. The reactions are assumed to be local; local fluctuations are fully taken into account, and particle transport occurs via diffusion.In contrast to previous investigations, we find no phase transition whenk4≠0 andd<4. Fork4=0,k3≠0, and 1≦d<4, we find a second-order phase transition which is in the same universality class as the transition in Schlögl's first model. Only ford≧4 we do find the first-order transition found also by previous authors.These claims are supported by extensive Monte Carlo calculations for various realizations of this process on discrete space-time lattices.

[1]  H. Abarbanel,et al.  Structure of the Pomeranchuk singularity in Reggeon field theory , 1974 .

[2]  C. Gardiner,et al.  Stochastic models of firstorder nonequilibrium phase transitions in chemical reactions , 1975 .

[3]  Richard C. Brower,et al.  Critical Exponents for the Reggeon Quantum Spin Model , 1978 .

[4]  J. Hammersley,et al.  Percolation processes , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  F. Schlögl.  Chemical reaction models for non-equilibrium phase transitions , 1972 .

[6]  H. Janssen,et al.  On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state , 1981 .

[7]  P. Grassberger,et al.  Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour , 1979 .

[8]  G. Nicolis,et al.  A master equation description of local fluctuations , 1975 .

[9]  K. McNeil,et al.  Nonequilibrium phase transitions in chemical reactions , 1974 .

[10]  P. Richter,et al.  Fluctuations in spatially homogeneous chemical steady states , 1976 .

[11]  P. Grassberger,et al.  Reggeon field theory and markov processes , 1978 .

[12]  W. Kinzel,et al.  Directed percolation: a finite-size renormalisation group approach , 1981 .

[13]  G. Nicolis,et al.  Stochastic analysis of a nonequilibrium phase transition: Some exact results , 1977 .

[14]  M. Furman,et al.  Quantum spin model for Reggeon field theory , 1977 .

[15]  David Griffeath,et al.  Additive and Cancellative Interacting Particle Systems , 1979 .

[16]  P. Grassberger,et al.  Fock Space Methods for Identical Classical Objects , 1980 .

[17]  E. Tirapegui,et al.  On the critical behaviour of the Schlögl model , 1981 .

[18]  Grégoire Nicolis,et al.  EFFECT OF FLUCTUATIONS ON BIFURCATION PHENOMENA , 1979 .

[19]  M. Moshe.  Recent developments in Reggeon field theory , 1978 .

[20]  M. Bellac,et al.  Reggeon field theory for α(0) > 1 , 1976 .

[21]  J. Cardy,et al.  Directed percolation and Reggeon field theory , 1980 .

[22]  K. McNeil,et al.  Correlations in stochastic theories of chemical reactions , 1976 .

[23]  M. Doi.  Second quantization representation for classical many-particle system , 1976 .

[24]  G. Nicolis,et al.  Systematic analysis of the multivariate master equation for a reaction-diffusion system , 1980 .

[25]  K. Kaplan.  H. Haken, Synergetics. An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology (2nd Edition). XI + 355 S., 152 Abb. Berlin—Heidelberg—New York 1978. Springer-Verlag. DM 66,00 , 1980 .

[26]  Grégoire Nicolis,et al.  Self-Organization in nonequilibrium systems , 1977 .

[27]  H. Janssen.  Stochastisches Reaktionsmodell für einen Nichtgleichgewichts-Phasenübergang , 1974 .

[28]  T. E. Harris.  Contact Interactions on a Lattice , 1974 .