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 $\stackrel{\ensuremath{\rightarrow}}{A}A+A,$ $A+\stackrel{\ensuremath{\rightarrow}}{A}A$, and $\stackrel{\ensuremath{\rightarrow}}{A}0$. We study a hierarchy of such DP processes for particle species
A,B,…, unidirectionally coupled via the reactions $\stackrel{\ensuremath{\rightarrow}}{A}B,\dots{}$ (with rates ${\ensuremath{\mu}}_{\mathrm{AB}},\dots{})$. When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents ${\ensuremath{\beta}}_{i}$ 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 ${\ensuremath{\beta}}_{i}{=1/2}^{i\ensuremath{-}1}$ at the multicritical point. Using field-theoretic renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we identify a new crossover exponent $\ensuremath{\varphi}$, and compute $\ensuremath{\varphi}=1+O({\ensuremath{\epsilon}}^{2})$ in the multicritical regime (for small ${\ensuremath{\mu}}_{\mathrm{AB}})$ of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, ${\ensuremath{\beta}}_{2}=1/2\ensuremath{-}\ensuremath{\epsilon}/8+O({\ensuremath{\epsilon}}^{2})$. Outside the multicritial region, we discuss the crossover to ordinary DP behavior, with the density exponent ${\ensuremath{\beta}}_{1}=1\ensuremath{-}\ensuremath{\epsilon}/6+O({\ensuremath{\epsilon}}^{2})$. 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.
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.
We study a one-dimensional, nonequilibrium Potts-like model that has q symmetric absorbing states. For q=2, as expected, the model belongs to the parity-conserving universality class. For q=3 the critical behavior depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity-conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behavior.
At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.
Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.
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.