We analyze numerically three different models exhibiting an absorbing phase transition. We focus on the finite-size scaling as well as the dynamical scaling behavior. An accurate determination of several critical exponents allows one to validate certain hyperscaling relations. Using these hyperscaling relations it is possible to express the avalanche exponents of a self-organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition.
We analyze numerically three different models exhibiting an absorbing phase transition. We focus on the finite-size scaling as well as the dynamical scaling behavior. An accurate determination of several critical exponents allows one to validate certain hyperscaling relations. Using these hyperscaling relations it is possible to express the avalanche exponents of a self-organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition.
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.
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.
We analyze numerically the critical behavior of a conserved lattice gas that was recently introduced as an example of the new universality class of absorbing phase transitions with a conserved field [Phys. Rev. Lett. 85, 1803 (2000)]. We determine the critical exponent of the order parameter as well as the critical exponent of the order parameter fluctuations in D=2,3,4,5 dimensions. A comparison of our results and those obtained from a mean-field approach and a field theory suggests that the upper critical dimension of the absorbing phase transition is four.
We consider two different systems exhibiting a continuous phase transition into an absorbing state. Both models belong to the same universality class; i.e., they are characterized by the same scaling functions and the same critical exponents. Varying the range of interactions, we examine the crossover from the mean-field-like to the non-mean-field scaling behavior. A phenomenological scaling form is applied in order to describe the full crossover region, which spans several decades. Our results strongly support the hypothesis that the crossover function is universal.
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago.