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arXiv:0706.1144 · New Journal of Physics

Block scaling in the directed percolation universality class

Gunnar Pruessner

Published Nov 7, 2008• 3 citations
arXivHTMLPDF

Abstract

The universal behaviour of the directed percolation universality class is well understood—both the critical scaling and the finite size scaling. This paper focuses on the block (finite size) scaling of the order parameter and its fluctuations, considering (sub-)blocks of linear size l in systems of linear size L . The scaling depends on the choice of the ensemble, as only the conditional ensemble produces the block-scaling behaviour as established in equilibrium critical phenomena. The dependence on the ensemble can be understood by an additional symmetry present in the unconditional ensemble. The unconventional scaling found in the unconditional ensemble is a reminder of the possibility that scaling functions themselves have a power-law dependence on their arguments.

DOI
10.1088/1367-2630/10/11/113003

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The universal behaviour of the directed percolation universality class is well understood—both the critical scaling and the finite size scaling. This paper focuses on the block (finite size) scaling of the order parameter and its fluctuations, considering (sub-)blocks of linear size l in systems of linear size L . The scaling depends on the choice of the ensemble, as only the conditional ensemble produces the block-scaling behaviour as established in equilibrium critical phenomena. The dependence on the ensemble can be understood by an additional symmetry present in the unconditional ensemble. The unconventional scaling found in the unconditional ensemble is a reminder of the possibility that scaling functions themselves have a power-law dependence on their arguments.
Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.
We discuss the universal scaling laws of order-parameter fluctuations in any system in which a second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with a finite-size scaling analysis. The relation between the order parameter, the criticality, and the scaling law of fluctuations has been established, and the connection between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or at-equilibrium Ising and percolation models.
Universal scaling laws of fluctuations (the Delta\\DeltaDelta-scaling laws) can be derived for equilibrium and off-equilibrium systems when combined with the finite-size scaling analysis. In any system in which the second-order critical behavior can be identified, the relation between order parameter, criticality and scaling law of fluctuations has been established and the relation between the scaling function and the critical exponents has been found.
Universal scaling laws of fluctuations (the Δ-scaling laws) occur for any system in which second-order critical behaviour can be identified. This can be established rigorously for at-equilibrium systems when combined withfinite-size scaling analysis. For dynamic critical systems, the relations between the order parameter, the criticality and the scaling law of fluctuations can be proven numerically. Moreover, one can find general relation between the form of the scaling function and the critical exponents of the corresponding infinite system. The case of the first-order phase transition is currently under investigation. There is no background theory for precise finite-size scaling analysis, and the results are numerical. Systematic data on the q-state 2-dimensional Potts model with q > 4, shows that at the critical point, the order-parameter distribution can be splitted into two peaks - characteristic of the two coexistent phases -, each of them exhibiting a particular Δ-scaling. Conjecture is done about general occurence of this feature.
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.