Chapter 1 Critical Phenomena
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Qualitative Picture of Scaling in the Entropy Formalism
Entropy 2008, 10, 224-239; DOI: 10.3390/e10030224 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.org/entropy Article Qualitative Picture of Scaling in the Entropy Formalism Hans Behringer Faculty of Physics, University of Bielefeld, D-33615 Bielefeld, Germany E-mail: [email protected] Received: 30 June 2008; in revised form: 27 August 2008 / Accepted: 2 September 2008 / Published: 5 September 2008 Abstract: The properties of an infinite system at a continuous phase transition are charac- terised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems are deduced starting from the well-established scaling re- lations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling are suppressed in the microcanonical ensemble compared to the corresponding corrections in the canonical ensemble. Keywords: Critical phenomena, microcanonical entropy, scaling relations, Gaussian and spherical model 1. Introduction In a mathematically idealised way continuous phase transitions are described in infinite systems al- though physical systems in nature or in a computer experiment are finite. The singular behaviour of physical quantities such as the response functions is related to scaling properties of the thermodynamic potential which is used for the formulation [1, 2]. Normally the Gibbs free energy is chosen for the description although other potentials also contain the full information about an equilibrium phase transi- tion. The Hankey-Stanley theorem relates the scaling properties of the Gibbs free energy to the scaling properties of any other energy-like potential that is obtained from the Gibbs free energy by a Legendre transformation [3]. -
International Centre for Theoretical Physics
REFi IC/90/116 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS FIXED SCALE TRANSFORMATION FOR ISING AND POTTS CLUSTERS A. Erzan and L. Pietronero INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1990 MIRAMARE- TRIESTE IC/90/116 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS FIXED SCALE TRANSFORMATION FOR ISING AND POTTS CLUSTERS A. Erzan International Centre for Theoretical Physics, Trieste, Italy and L. Pietronero Dipartimento di Fisica, Universita di Roma, Piazzale Aldo Moro, 00185 Roma, Italy. ABSTRACT The fractal dimension of Ising and Potts clusters are determined via the Fixed Scale Trans- formation approach, which exploits both the self-similarity and the dynamical invariance of these systems at criticality. The results are easily extended to droplets. A discussion of inter-relations be- tween the present approach and renormalization group methods as well as Glauber—type dynamics is provided. MIRAMARE - TRIESTE May 1990 To be submitted for publication. T 1. INTRODUCTION The Fixed Scale Transformation, a novel technique introduced 1)i2) for computing the fractal dimension of Laplacian growth clusters, is applied here to the equilibrium problem of Ising clusters at criticality, in two dimensions. The method yields veiy good quantitative agreement with the known exact results. The same method has also recendy been applied to the problem of percolation clusters 3* and to invasion percolation 4), also with very good results. The fractal dimension D of the Ising clusters, i.e., the connected clusters of sites with identical spins, has been a controversial issue for a long time ^ (see extensive references in Refs.5 and 6). -
Statistical Field Theory University of Cambridge Part III Mathematical Tripos
Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] –1– Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very di↵erent perspectives. They have titles like “Critical Phenomena”, “Phase Transitions”, “Renormalisation Group” or, less helpfully, “Advanced Statistical Mechanics”. Here are some that I particularly like Nigel Goldenfeld, Phase Transitions and the Renormalization Group • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. Mehran Kardar, Statistical Physics of Fields • The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. John Cardy, Scaling and Renormalisation in Statistical Physics • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent perspective than these lectures, with more focus on renormalisation in real space. Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter • Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html. -
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 -
Phase Transitions and Critical Phenomena: an Essay in Natural Philosophy (Thales to Onsager)
Phase Transitions and Critical Phenomena: An Essay in Natural Philosophy (Thales to Onsager) Prof. David A. Edwards Department of Mathematics University of Georgia Athens, Georgia 30602 http://www.math.uga.edu/~davide/ http://davidaedwards.tumblr.com/ [email protected] §1. Introduction. In this essay we will present a detailed analysis of the concepts and mathematics that underlie statistical mechanics. For a similar discussion of classical and quantum mechanics the reader is referred to [E-1] and [E-4]. For a similar discussion of quantum field theory the reader is referred to [E-2]. For a discussion of quantum geometrodynamics the reader is referred to [E- 3] Scientific theories go through many stages in their development, some eventually reaching the stage at which one might say that "the only work left to be done is the computing of the next decimal." We shall call such theories climax theories in analogy with the notion of a climax forest (other analogies are also appropriate). A climax theory has two parts: one theoretical, which has become part of pure mathematics; and the other empirical, which somehow relates the mathematics to experience. The earliest example of a climax theory is Euclidean geometry. That such a development of geometry is even possible is not obvious. One can easily imagine an Egyptian geometer explaining to a Babylonian number theorist why geometry could never become a precise science like number theory because pyramids and other such bodies are intrinsically irregular and fuzzy (similar discussions occur often today between biologists and physicists). Archimedes' fundamental work showing that the four fundamental constants related to circles and spheres (C =αr, A = βr2 , S = ϒr 2, V= δr3 ) are all simply related (1/2α=β = 1/4ϒ= 3/4δ= π), together with his estimate that 3+10/71< π < 3+1/7 will serve for us as the paradigm of what science should be all about. -
Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point
entropy Article Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point Juan Carlos Obeso-Jureidini , Daniela Olascoaga and Victor Romero-Rochín * Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico; [email protected] (J.C.O.-J.); [email protected] (D.O.) * Correspondence: romero@fisica.unam.mx; Tel.: +52-55-5622-5096 Abstract: With the use of thermodynamics and general equilibrium conditions only, we study the entropy of a fluid in the vicinity of the critical point of the liquid–vapor phase transition. By assuming a general form for the coexistence curve in the vicinity of the critical point, we show that the functional dependence of the entropy as a function of energy and particle densities necessarily obeys the scaling form hypothesized by Widom. Our analysis allows for a discussion of the properties of the corresponding scaling function, with the interesting prediction that the critical isotherm has the same functional dependence, between the energy and the number of particles densities, as the coexistence curve. In addition to the derivation of the expected equalities of the critical exponents, the conditions that lead to scaling also imply that, while the specific heat at constant volume can diverge at the critical point, the isothermal compressibility must do so. Keywords: liquid–vapor critical point; scaling hypothesis; critical exponents Citation: Obeso-Jureidini, J.C.; 1. Introduction Olascoaga, D.; Romero-Rochín, V. The full thermodynamic description of critical phenomena in the liquid–vapor phase Thermodynamic Derivation of transition of pure substances has remained as a theoretical challenge for a long time [1]. -
Three-Dimensional Phase Transitions in Multiflavor Lattice Scalar SO (Nc) Gauge Theories
Three-dimensional phase transitions in multiflavor lattice scalar SO(Nc) gauge theories Claudio Bonati,1 Andrea Pelissetto,2 and Ettore Vicari1 1Dipartimento di Fisica dell’Universit`adi Pisa and INFN Largo Pontecorvo 3, I-56127 Pisa, Italy 2Dipartimento di Fisica dell’Universit`adi Roma Sapienza and INFN Sezione di Roma I, I-00185 Roma, Italy (Dated: January 1, 2021) We investigate the phase diagram and finite-temperature transitions of three-dimensional scalar SO(Nc) gauge theories with Nf ≥ 2 scalar flavors. These models are constructed starting from a maximally O(N)-symmetric multicomponent scalar model (N = NcNf ), whose symmetry is par- tially gauged to obtain an SO(Nc) gauge theory, with O(Nf ) or U(Nf ) global symmetry for Nc ≥ 3 or Nc = 2, respectively. These systems undergo finite-temperature transitions, where the global sym- metry is broken. Their nature is discussed using the Landau-Ginzburg-Wilson (LGW) approach, based on a gauge-invariant order parameter, and the continuum scalar SO(Nc) gauge theory. The LGW approach predicts that the transition is of first order for Nf ≥ 3. For Nf = 2 the transition is predicted to be continuous: it belongs to the O(3) vector universality class for Nc = 2 and to the XY universality class for any Nc ≥ 3. We perform numerical simulations for Nc = 3 and Nf = 2, 3. The numerical results are in agreement with the LGW predictions. I. INTRODUCTION The global O(N) symmetry is partially gauged, obtain- ing a nonabelian gauge model, in which the fields belong to the coset SN /SO(N ), where SN = SO(N)/SO(N 1) Global and local gauge symmetries play a crucial role c − in theories describing fundamental interactions [1] and is the N-dimensional sphere. -
Universal Scaling Behavior of Non-Equilibrium Phase Transitions
Universal scaling behavior of non-equilibrium phase transitions Sven L¨ubeck Theoretische Physik, Univerit¨at Duisburg-Essen, 47048 Duisburg, Germany, [email protected] December 2004 arXiv:cond-mat/0501259v1 [cond-mat.stat-mech] 11 Jan 2005 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 different uni- versality 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. 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 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. -
Classifying Potts Critical Lines
Classifying Potts critical lines Gesualdo Delfino1,2 and Elena Tartaglia1,2 1SISSA – Via Bonomea 265, 34136 Trieste, Italy 2INFN sezione di Trieste Abstract We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points invariant under the permutational symmetry Sq in two dimensions, and show how one of these scattering solutions describes the ferromagnetic and square lattice antiferromagnetic critical lines of the q-state Potts model. Other solutions we determine should correspond to new critical lines. In particular, we obtain that a Sq-invariant fixed point can be found up to the maximal value q = (7+ √17)/2. This is larger than the usually assumed maximal value 4 and leaves room for a second order antiferromagnetic transition at q = 5. arXiv:1707.00998v2 [cond-mat.stat-mech] 16 Oct 2017 1 Introduction Symmetry plays a prominent role within the theory of critical phenomena. The circumstance is usually illustrated referring to ferromagnetism, for which systems with different microscopic real- izations but sharing invariance under transformations of the same group G of internal symmetry fall within the same universality class of critical behavior. In the language of the renormalization group (see e.g. [1]) this amounts to say that the critical behavior of these ferromagnets is ruled by the same G-invariant fixed point. In general, however, there are several G-invariant fixed points of the renormalization group in a given dimensionality. Even staying within ferromag- netism, a system with several tunable parameters may exhibit multicriticality corresponding to fixed points with the same symmetry but different field content. -
RANDOM WALKS from STATISTICAL PHYSICS II Two Dimensons and Conformal Invariance Wald Lectures 2011 Joint Statistical Meetings
RANDOM WALKS FROM STATISTICAL PHYSICS II Two Dimensons and Conformal Invariance Wald Lectures 2011 Joint Statistical Meetings Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago August 3, 2011 1 / 32 CRITICAL PHENOMENA IN STATISTICAL PHYSICS I Study systems at or near parameters at which a phase transition occurs I Parameter ¯ = C=T where T = temperature I Large ¯ (low temperature) | long range correlation. I Small ¯ (high temperature) | short range correlation I Critical value ¯c at which sharp transition occurs I Belief: systems at criticality \in the scaling limit" exhibit fractal-like behavior (power-law correlations) with nontrivial critical exponents. I The exponents depend on dimension. 2 / 32 TWO DIMENSIONS I Belavin, Polyakov, Zamolodchikov (1984) | critical systems in two dimensions in the scaling limit exhibit some kind of \conformal invariance". I A number of theoretical physicists (Nienhuis, Cardy, Duplantier, Saleur, ...) made predictions about critical exponents using nonrigorous methods | conformal ¯eld theory and Coulomb gas techniques. I Exact rational values for critical exponents | predictions strongly supported by numerical simulations I While much of the mathematical framework of conformal ¯eld theory was precise and rigorous (or rigorizable), the nature of the limit and the relation of the ¯eld theory to the lattice models was not well understood. 3 / 32 SELF-AVOIDING WALK (SAW) I Model for polymer chains | polymers are formed by monomers that are attached randomly except for a self-avoidance constraint. 2 ! = [!0;:::;!n];!j 2 Z ; j!j = n j!j ¡ !j¡1j = 1; j = 1;:::; n !j 6= !k ; 0 · j < k · n: I Critical exponent º: a typical SAW has diameter about j!jº. -
Self-Organization, Critical Phenomena, Entropy Decrease in Isolated Systems and Its Tests
International Journal of Modern Theoretical Physics, 2019, 8(1): 17-32 International Journal of Modern Theoretical Physics ISSN: 2169-7426 Journal homepage:www.ModernScientificPress.com/Journals/ijmtp.aspx Florida, USA Article Self-Organization, Critical Phenomena, Entropy Decrease in Isolated Systems and Its Tests Yi-Fang Chang Department of Physics, Yunnan University, Kunming, 650091, China Email: [email protected] Article history: Received 15 January 2019, Revised 1 May 2019, Accepted 1 May 2019, Published 6 May 2019. Abstract: We proposed possible entropy decrease due to fluctuation magnification and internal interactions in isolated systems, and these systems are only approximation. This possibility exists in transformations of internal energy, complex systems, and some new systems with lower entropy, nonlinear interactions, etc. Self-organization and self- assembly must be entropy decrease. Generally, much structures of Nature all are various self-assembly and self-organizations. We research some possible tests and predictions on entropy decrease in isolated systems. They include various self-organizations and self- assembly, some critical phenomena, physical surfaces and films, chemical reactions and catalysts, various biological membranes and enzymes, and new synthetic life, etc. They exist possibly in some microscopic regions, nonlinear phenomena, many evolutional processes and complex systems, etc. As long as we break through the bondage of the second law of thermodynamics, the rich and complex world is full of examples of entropy decrease. Keywords: entropy decrease, self-assembly, self-organizations, complex system, critical phenomena, nonlinearity, test. DOI ·10.13140/RG.2.2.36755.53282 1. Introduction Copyright © 2019 by Modern Scientific Press Company, Florida, USA Int. J. Modern Theo. -
Indicators of Conformal Field Theory: Entanglement Entropy and Multiple
Indicators of Conformal Field Theory: entanglement entropy and multiple-point correlators Pranay Patil, Ying Tang, Emanuel Katz, and Anders W. Sandvik Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA The entanglement entropy (EE) of quantum systems is often used as a test of low-energy de- scriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state expressed in the valence-bond basis (where the amplitudes decay as a power law of the bond length and the wave function is the product of all bond amplitudes), we show that even though the EE exhibits the expected CFT behavior, there is no CFT description of this state. We provide numerical tests of the behavior predicted by CFT for the correlation functions in the critical transverse-field Ising chain and the J-Q spin chain, where the conformal structure is well understood. That behavior is not reproduced in the amplitude-product state. I. INTRODUCTION tuned to criticality for particular weightings of the sin- glet covers which contribute to it, and one may then ask whether a state similar to the ground state of the Heisen- The emergence of conformal invariance at a quantum berg Hamiltonian could be constructed this way.