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Universal Scaling Behavior of Non-Equilibrium Phase Transitions Universal scaling behavior of non-equilibrium phase transitions HABILITATIONSSCHRIFT der FakultÄat furÄ Naturwissenschaften der UniversitÄat Duisburg-Essen vorgelegt von Sven LubÄ eck aus Duisburg Duisburg, im Juni 2004 Zusammenfassung Kritische PhÄanomene im Nichtgleichgewicht sind seit Jahrzehnten Gegenstand inten- siver Forschungen. In Analogie zur Gleichgewichtsthermodynamik erlaubt das Konzept der UniversalitÄat, die verschiedenen NichtgleichgewichtsphasenubÄ ergÄange in unterschied- liche Klassen einzuordnen. Alle Systeme einer UniversalitÄatsklasse sind durch die glei- chen kritischen Exponenten gekennzeichnet, und die entsprechenden Skalenfunktionen werden in der NÄahe des kritischen Punktes identisch. WÄahrend aber die Exponenten zwischen verschiedenen UniversalitÄatsklassen sich hÄau¯g nur geringfugigÄ unterscheiden, weisen die Skalenfunktionen signi¯kante Unterschiede auf. Daher ermÄoglichen die uni- versellen Skalenfunktionen einerseits einen emp¯ndlichen und genauen Nachweis der UniversalitÄatsklasse eines Systems, demonstrieren aber andererseits in ubÄ erzeugender- weise die UniversalitÄat selbst. Bedauerlicherweise wird in der Literatur die Betrachtung der universellen Skalenfunktionen gegenubÄ er der Bestimmung der kritischen Exponen- ten hÄau¯g vernachlÄassigt. Im Mittelpunkt dieser Arbeit steht eine bestimmte Klasse von Nichtgleichgewichts- phasenubÄ ergÄangen, die sogenannten absorbierenden PhasenubÄ ergÄange. Absorbierende PhasenubÄ ergÄange beschreiben das kritische Verhalten von physikalischen, chemischen sowie biologischen Systemen. Zu Beginn wird eine EinfuhrungÄ in absorbierende Phasen- ubÄ ergÄange und in das Konzept der UniversalitÄat geboten. Verschiedene universelle Skalenfunktionen werden am Beispiel der UniversalitÄatsklasse der gerichteten Perko- lation sowie der sogenannten Manna-UniversalitÄatsklasse untersucht. Insbesondere die gerichtete Perkolation wird als Paradigma von absorbierenden PhasenubÄ ergÄangen ange- sehen. Es werden Skalenfunktionen betrachtet, die neben den statischen auch die dy- namischen Eigenschaften beschreiben. DarubÄ er hinaus wird der Einuss einer endlichen SystemgrÄo¼e untersucht, das sogenannte ¯nite-size Skalenverhalten. Dabei ermÄoglicht die Anwendung eines zum Ordnungsparameter konjugierten, externen Feldes die Be- stimmung verschiedener Funktionen und physikalischer GrÄo¼en, wie z. B. der Zustands- gleichung und der SuszeptibilitÄat. Eine furÄ absorbierende PhasenubÄ ergÄange geeignete Form des ublicÄ hen ¯nite-size Skalenverhaltens basiert ebenfalls auf der Anwendung des konjugierten Feldes. Die erhaltenen universellen Skalenfunktionen kÄonnen mit anderen numerisch oder experimentell gewonnenen Kurven verglichen werden und so eindeutig zur Identi¯zierung des jeweiligen kritischen Verhaltens beitragen. Universelle Skalenfunktionen eigenen sich neben der ubÄ erzeugenden Demonstration der UniversalitÄat auch zur UbÄ erprufungÄ von Ergebnissen der Renormierungsgruppen- theorie. Dies ist von fundamentalem Interesse, da unser VerstÄandnis von kritischen PhÄa- nomenen auf der Renormierungsgruppentheorie beruht. Die zunehmende VerfugbarkÄ eit von prÄazisen numerischen Ergebnissen fuhrtÄ dabei zu einem fruchtbaren Wechselspiel zwischen Simulationen und Renormierungsgruppenanalysen. iv Summary 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 di®erent uni- versality classes than the exponents. Thus, universal scaling functions o®er 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. Unfor- tunately, 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 introduc- tion is presented. The universal scaling behavior of two di®erent 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 ¯nite size scaling functions. In particular, the e®ect of an external ¯eld conjugated to the order parameter is investigated. Incorporating the conjugated ¯eld, it is possible to determine the equation of state, the suscepti- bility, and to perform a modi¯ed ¯nite-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. vi Contents vii Contents 1 Introduction 1 1.1 Opening remarks and outline 1 1.2 Scaling theory 6 1.3 Universality 10 1.4 Remarks on renormalization group theory 16 2 Absorbing phase transitions 25 2.1 De¯nitions and the contact process at ¯rst glance 25 2.2 Numerical simulation methods 32 2.2.1 Steady state scaling behavior 33 2.2.2 Dynamical scaling behavior 37 3 Directed percolation 43 3.1 Lattice models of directed percolation 48 3.1.1 Contact process 48 3.1.2 Domany-Kinzel automaton 49 3.1.3 Pair contact process 54 3.1.4 Other models 56 3.2 Steady state scaling behavior 58 3.2.1 Below the upper critical dimension 59 3.2.2 Above the upper critical dimension 64 3.2.3 At the upper critical dimension 67 3.2.4 Universal amplitude combinations 70 3.2.5 Finite-size scaling behavior 74 3.3 Dynamical scaling behavior 80 3.3.1 Homogeneous particle source 80 3.3.2 Localized particle source 82 3.4 Conclusions and comments 87 viii Contents 4 Absorbing phase transitions with a conserved ¯eld 89 4.1 Lattice models of absorbing phase transitions with a conserved ¯eld 89 4.1.1 Manna model 90 4.1.2 Conserved threshold transfer process 92 4.1.3 Conserved lattice gas 94 4.2 Steady state scaling behavior 95 4.2.1 Equation of state and uctuations 95 4.2.2 Universal amplitude combinations 99 4.2.3 Finite-size scaling behavior 100 4.3 Dynamical scaling behavior and hyperscaling relations 101 4.3.1 Homogenous and localized particle source 102 4.3.2 Hyperscaling relations 102 4.4 Self-organized criticality 106 4.4.1 Sandpile models and absorbing phase transitions 108 4.4.2 Avalanche exponents 111 5 Universal crossover scaling 115 5.1 Crossover to mean ¯eld scaling behavior 116 5.1.1 Control parameter driven crossover 121 5.1.2 Field driven crossover: Violation of Widom scaling law 124 6 Concluding remarks and outlook 129 7 Appendix 133 A.1 Branching process 133 A.2 Scaling laws 133 A.3 Critical exponents, universal amplitude combinations and critical parameters 135 A.3.1 Directed percolation 135 A.3.2 Compact directed percolation 135 A.3.3 Manna universality class 136 A.3.4 Critical parameters 137 A.3.5 Crossover parameters 138 References 139 List of Figures 158 Introduction 1 Introduction 1.1 Opening remarks and outline One of the most impressive features of continuous phase transitions is the concept of universality, that allows to group the great variety of di®erent critical phenomena into a small number of universality classes (see [1] for a recent review). All systems belonging to a given universality class have the same critical exponents, and certain scaling func- tions (equation of state, correlation functions, etc.) become identical near the critical point. The universality has its origin in the long range character of the uctuations. Close to the transition point, the corresponding correlation length becomes much larger than the typical range of molecular interactions. Then the behavior of the cooperative phenomena becomes independent of the microscopic details of the considered system. The concept of universality is well established for equilibrium phase transitions where a unifying theoretical framework exists. In that case, the universal behavior of short- range interacting systems depends only on few fundamental parameters, namely the dimensionality of space and the symmetry of the order parameter [2]. Classical ex- amples of such universal behavior in critical equilibrium systems are for instance the coexistence curve of liquid-vapor systems [3] and the equation of state in ferromagnetic systems (e.g. [1, 4]). A complete understanding
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