1998/09/30 by Y. Y. Goldschmidt, Yadin Y. Goldschmidt, Haye Hinrichsen +5 · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Material Dynamics and Properties #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat
paper · pdf · doi:10.1103/physreve.59.6381
published as Phys. Rev. E 59 (1999) 6381 · 29 pages, 19 figures, revtex, 2 columns, revised Jan 1995: minor changes and additions; accepted for publication in Phys. Rev. E
arxiv created 1999/03/16 · openalex publication_date 1999/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension dc=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes \stackrel\ensuremath→AA+A, A+\stackrel\ensuremath→AA, and \stackrel\ensuremath→A0. We study a hierarchy of such DP processes for particle species A,B,…, unidirectionally coupled via the reactions \stackrel\ensuremath→AB,… (with rates \ensuremathμAB,…). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents \ensuremathβi which are markedly reduced at each hierarchy level i>~2. This scenario can be understood on the basis of the mean-field rate equations, which yield \ensuremathβi=1/2^i\ensuremath-1 at the multicritical point. Using field-theoretic renormalization-group techniques in d=4\ensuremath-\ensuremathε dimensions, we identify a new crossover exponent \ensuremathφ, and compute \ensuremathφ=1+O(\ensuremathε2) in the multicritical regime (for small \ensuremathμAB) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, \ensuremathβ2=1/2\ensuremath-\ensuremathε/8+O(\ensuremathε2). Outside the multicritial region, we discuss the crossover to ordinary DP behavior, with the density exponent \ensuremathβ1=1\ensuremath-\ensuremathε/6+O(\ensuremathε2). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d<~3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.