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Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (30 November 2020) | Viewed by 20733

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Special Issue Editor

Prof. Dr. Ralph Kenna

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Guest Editor

Statistical Physics Group, Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, UK
Interests: phase transitions and critical phenomena; high-dimensional systems; finite-size scaling; boundary conditions; complexity science; interdisciplinary applications of statistical physics; scientometrics; sociophysics; digital humanities

Special Issue Information

Dear Colleagues,

Ludwig Boltzmann and contemporaries pioneered the development of statistical physics towards the end of the 19^th century. The pillars on which the discipline rests include “bottom-up” theories of phase transitions and critical phenomena, built on other pioneering ideas and work such as that of Wilhelm Lenz and Ernst Ising at the start of the 20^th century. In the words of Stephen Hawking, we are now in the “century of complexity”, moving on from basic laws that govern matter to how everything is connected to everything else.

Although Ising’s original investigations did not deliver the desired result of a phase transition, the idea that randomness coupled with gross simplification at the micro-level could explain changes of state at the macro-level was ground-breaking. Now we know the importance of dimensionality; interaction range; symmetries; whether the model is classical or quantum, equilibrium, or non-equilibrium, etc., in understanding the physics of change.

A vast body of research covers how all sorts of variants on such systems describe increasingly complex systems, but the essential idea to apply probabilistic considerations to simplified many-body systems was borrowed from socio systems. In recent times, with the emergence of the notion of “emergence”, the statistical physics of complex systems has re-embraced its interdisciplinary birthplace, delivering rich physics in and beyond physics and contributing to our understanding of the world.

This Special Issue focuses on models that are simplified at the micro level but complex at the macro level. We are interested in negative results like Ising’s as well as positive results, and, reflecting the birthplace of statistical physics, we welcome interdisciplinary considerations as well as traditional physics. Thus, this Issue focuses on the concept of change—how the simple can deliver the complex through non-trivial mechanisms, wherever they arise.

This special issue is dedicated to the fond memory of Professor Ian Campbell who has contributed so much to our understanding of phase transitions and emergent phenomena. In particular, Ian’s discovery of extended scaling, and his research into hyperscaling and spin glasses have contributed very significantly to theories of critical phenomena and we anticipate they will contribute more in the years to come. We are honored that Ian’s last paper (co-authored with Per-Håkan Lundow) is published in this special issue.

Prof. Dr. Ralph Kenna
Guest Editor

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Keywords

Phase transitions
Critical phenomena
Universality
Scaling
Statistical physics concepts applied to other disciplines

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Published Papers (6 papers)

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Research

22 pages, 409 KiB

Open AccessArticle

Exact Recovery of Stochastic Block Model by Ising Model

by Feng Zhao, Min Ye and Shao-Lun Huang

Entropy 2021, 23(1), 65; https://doi.org/10.3390/e23010065 - 2 Jan 2021

Cited by 2 | Viewed by 2818

Abstract

In this paper, we study the phase transition property of an Ising model defined on a special random graph—the stochastic block model (SBM). Based on the Ising model, we propose a stochastic estimator to achieve the exact recovery for the SBM. The stochastic algorithm can be transformed into an optimization problem, which includes the special case of maximum likelihood and maximum modularity. Additionally, we give an unbiased convergent estimator for the model parameters of the SBM, which can be computed in constant time. Finally, we use metropolis sampling to realize the stochastic estimator and verify the phase transition phenomenon thfough experiments. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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16 pages, 1003 KiB

Open AccessArticle

(Four) Dual Plaquette 3D Ising Models

by Desmond A. Johnston and Ranasinghe P. K. C. M. Ranasinghe

Entropy 2020, 22(6), 633; https://doi.org/10.3390/e22060633 - 8 Jun 2020

Cited by 3 | Viewed by 3590

Abstract

A characteristic feature of the

3 d

plaquette Ising model is its planar subsystem symmetry. The quantum version of this model has been shown to be related via a duality to the X-Cube model, which has been paradigmatic in the new and rapidly [...] Read more.

A characteristic feature of the

3 d

3 d

plaquette Ising and the X-Cube model is similar to that between the

2 d

quantum transverse spin Ising model and the Toric Code. Gauging the global symmetry in the case of the

2 d

Ising model and considering the gauge invariant sector of the high temperature phase leads to the Toric Code, whereas gauging the subsystem symmetry of the

3 d

quantum transverse spin plaquette Ising model leads to the X-Cube model. A non-standard dual formulation of the

3 d

plaquette Ising model which utilises three flavours of spins has recently been discussed in the context of dualising the fracton-free sector of the X-Cube model. In this paper we investigate the classical spin version of this non-standard dual Hamiltonian and discuss its properties in relation to the more familiar Ashkin–Teller-like dual and further related dual formulations involving both link and vertex spins and non-Ising spins. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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16 pages, 1670 KiB

Open AccessArticle

Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)

by Maddalena Dilucca, Luca Leuzzi, Giorgio Parisi, Federico Ricci-Tersenghi and Juan J. Ruiz-Lorenzo

Entropy 2020, 22(2), 250; https://doi.org/10.3390/e22020250 - 22 Feb 2020

Cited by 4 | Viewed by 5204

Abstract

We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field,

h > 0

and

T_{c} (h) < T < T_{c} (h = 0)

, of the

d = 1

spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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10 pages, 497 KiB

Open AccessArticle

Hyperscaling Violation in Ising Spin Glasses

by Ian A. Campbell and Per H. Lundow

Entropy 2019, 21(10), 978; https://doi.org/10.3390/e21100978 - 8 Oct 2019

Cited by 3 | Viewed by 2913

Abstract

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension

D = 6

. Estimates are obtained for the “violation of hyperscaling exponent” values in the various models. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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9 pages, 306 KiB

Open AccessArticle

Kinetic Models of Discrete Opinion Dynamics on Directed Barabási–Albert Networks

by F. Welington S. Lima and J. A. Plascak

Entropy 2019, 21(10), 942; https://doi.org/10.3390/e21100942 - 26 Sep 2019

Cited by 12 | Viewed by 2554

Abstract

Kinetic models of discrete opinion dynamics are studied on directed Barabási–Albert networks by using extensive Monte Carlo simulations. A continuous phase transition has been found in this system. The critical values of the noise parameter are obtained for several values of the connectivity of these directed networks. In addition, the ratio of the critical exponents of the order parameter and the corresponding susceptibility to the correlation length have also been computed. It is noticed that the kinetic model and the majority-vote model on these directed Barabási–Albert networks are in the same universality class. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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16 pages, 291 KiB

Open AccessFeature PaperArticle

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

by Nickolay Izmailian and Ralph Kenna

Entropy 2019, 21(9), 895; https://doi.org/10.3390/e21090895 - 15 Sep 2019

Viewed by 2481

Abstract

The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be

c = - 2

) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future. Full article

(This article belongs to the Special Issue Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models)

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