The introduction of a metric onto the space of parameters in models in
Statistical Mechanics and beyond gives an alternative perspective on their
phase structure. In such a geometrization, the scalar curvature, R, plays a
central role. A non-interacting model has a flat geometry (R=0), while R
diverges at the critical point of an interacting one. Here, the information
geometry is studied for a number of solvable statistical-mechanical models.Comment: 6 pages with 1 figur

D.A. Johnston

Dolan

Fisher

Janke

Janyszek

R. Kenna

W. Janke

English

arXiv

Crossref

Information geometry and phase transitions

The introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrisation, the scalar curvature, Full-size image 

Janke, W.

Johnston, D. A.

Kenna, Ralph

Coventry University Pure Portal

CURVE is the Institutional Repository for Coventry University   Information geometry and phase transitions  Janke, W. , Johnston, D. A. and Kenna, R.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Janke, W. , Johnston, D. A. and Kenna, R. (2004) Information geometry and phase transitions. Physica A: Statistical Mechanics and its Applications, volume 336 (1-2): 181-186 http://dx.doi.org/10.1016/j.physa.2004.01.023  DOI 10.1016/j.physa.2004.01.023 ISSN 0378-4371  Publisher: Elsevier  NOTICE: This is the author’s version of a work that was accepted for publication in Physica A: Statistical Mechanics and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica A: Statistical Mechanics and its Applications, [336, 1-2, 2004] DOI: 10.1016/j.physa.2004.01.023   © 2004, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/  Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.   This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.        arXiv:cond-mat/0401092v1  [cond-mat.stat-mech]  7 Jan 2004Information Geometry and Phase TransitionsW. Janke a, D.A. Johnston b and R. Kenna caInstitut fu¨r Theoretische Physik, Universita¨t Leipzig, Augustusplatz 10/11,04109 Leipzig, Germany.bDepartment of Mathematics, Heriot-Watt University, Riccarton, Edinburgh,EH14 4AS, Scotland.cSchool of Mathematical and Information Sciences, Coventry University,Coventry, CV1 5FB, England.AbstractThe introduction of a metric onto the space of parameters in models in StatisticalMechanics and beyond gives an alternative perspective on their phase structure.In such a geometrization, the scalar curvature, R, plays a central role. A non-interacting model has a ﬂat geometry (R = 0), while R diverges at the criticalpoint of an interacting one. Here, the information geometry is studied for a numberof solvable statistical-mechanical models.Key words: Information geometry, Phase transitionsPACS: 02.50.-r, 05.70.Fh1 IntroductionAs the application of statistical-physics techniques becomes more widespreadand acceptable outside the traditional physics community, it is clear that thephenomena of phase transitions have important roles to play there. Indeed,phase transitions are common to a very wide range of disciplines, from physicsto biology, economics and even to sociology. As statistical physics finds perti-nence in these areas, so too can the field draw on concepts outside the confinesof pure physics.In all of these disciplines, models are characterised by certain sets of param-eters. The idea of endowing the space of such parameters with a metric andgeometrical structure has been borrowed from parametric statistics [1]. Givena probability distribution p(x|θ), and a sample x1, . . . , xn, the objective is toPreprint submitted to Elsevier Science 14 October 2015estimate the parameter θ. This may be done by maximizing the so-called like-lihood function, L(θ) =∏ni=1 p(xi|θ), or its logarithm (called the log-likelihoodfunction),lnL(θ) =n∑i=1ln p(xi|θ) . (1)The gradient of this quantity is the score function:U(θ) =d lnL(θ)dθ, (2)and the expectation of this random variable is zero (E[U(θ)] = 0). Its varianceisVar[U(θ)] = E[−d2 lnL(θ)dθ2]. (3)This quantity is called the expected or Fisher information. Taylor expandingthe log-likelihood function, one arrives atlnL(θ + δθ)− lnL(θ) = δθd lnL(θ)dθ∣∣∣∣∣θ+(δθ)22d2 lnL(θ)dθ2∣∣∣∣∣θ+ . . . . (4)The first term on the right-hand side is zero at the true θ-value. Therefore,the closeness of two probability distributions characterised by θ and θ+ δθ, isgiven by the second term, or the Fisher information.For higher-dimensional distributions (which may be continuous), where, in-stead of θ, one has a set of parameters, θ1, θ2, . . ., the Fisher information isdefined asGij(θ) = −E[∂2 ln p(x|θ)∂θi∂θj]= −∫p(x|θ)∂2 ln p(x|θ)∂θi∂θjdx . (5)C.R. Rao suggested this is a metric. It is, in fact, the only suitable metric inparametric statistics and is called the Fisher-Rao metric [1].In generic statistical-physics models, we have two parameters, β, which wemay think of as the inverse temperature, and h, the external field. In this casethe Fisher-Rao metric is simply given byGij = ∂i∂jf , (6)2where f is the reduced free energy per site and ∂i = (∂/∂β, ∂/∂h).For such a metric the scalar curvature may be calculated asR = −12G2∣∣∣∣∣∣∣∣∣∣∣∂2βf ∂β∂hf ∂2hf∂3βf ∂2β∂hf ∂β∂2hf∂2β∂hf ∂β∂2hf ∂3hf∣∣∣∣∣∣∣∣∣∣∣, (7)where G = det(Gij) is the determinant of the metric itself. The scalar curva-ture plays a central role in any attempt to look at phase transitions from ageometrical perspective. Indeed, R measures the complexity of the system. Aflat metric implies that the system is not interacting. Conversely, and for allthe models that have been considered so far, the curvature diverges at (andonly at) a phase transition point for physical ranges of the parameter values.Using standard scaling assumptions, we can anticipate the behaviour of Rnear a second-order critical point. With t = 1− β/βc,f(β, h) = λ−1f (tλat , hλah) = t1atψ(ht−ahat), (8)whereat =1νd, ah =βδνd, (9)are the scaling dimensions for the energy and spin operators and d is thespacial dimensionality. One finds for the scalar curvature,R = −12G2∣∣∣∣∣∣∣∣∣∣∣t1at−20 t1at−2ahatt1at−30 t1at−2ahat−10 t1at−2ahat−1t1at−3ahat∣∣∣∣∣∣∣∣∣∣∣, (10)andG ∼ t2at+2ahat−2, (11)yieldingR ∼ ξd ∼ |β − βc|α−2 , (12)3where hyperscaling (νd = 2− α) is assumed and ξ is the correlation length.The systems hitherto analysed from the information-geometry perspective in-clude the Ising model in one dimension [2], the Bethe lattice Ising and mean-field models [3]. The main results hitherto established are that R is positivedefinite and diverges (as ξd ) only at the critical point.In an effort to see how generic these features are, and to discover new ones, weanalyse the Potts model in one dimension, the Ising model in two dimensionscoupled to quantum gravity and the spherical model in three dimensions.2 Information Geometry in Specific ModelsThe Potts Model in One Dimension: The one-dimensional q-state Pottsmodel, like its Ising counterpart (which corresponds to q = 2) is exactly solv-able. Although it has no true phase transition, thermodynamic quantities di-verge at zero temperature. In the Ising case, explicit calculations showed thatRIsing = 1 +cosh h√sinh2 h+ exp (−4β). (13)The correlation length is known to behave as ξ ∼ −1/ ln (tanh β), and there-fore RIsing ∼ ξ as β →∞.For plotting purposes, it is convenient to introducey = exp (β) and z = exp (h) . (14)The curvature is plotted in Fig. 1. It is clear that RIsing is positive definite andexhibits a h → −h (or z → 1/z) symmetry. Furthermore, given a particularvalue for y, RIsing is maximum along the zero-field ridge z = 1.The one-dimensional q-state Potts model is also exactly solvable, so it is sen-sible to exploit it in an attempt to decide which of the previous features aregeneric. An explicit calculation [4] shows thatRPotts = A(q, y, z) +B(q, y, z)√η(q, y, z)∼ y ∼ ξ , (15)for h = 0 and y →∞. Here, η(q, y, z) is the Potts analogue of the Ising term,sinh2 h + exp (−4β), and A and B are smooth functions of y and z. So theexpected scaling (12) holds with d = 1 or α = 1.4Figure 1 also shows that, in the Potts case, RPotts is no longer positive definite.Furthermore, the z → 1/z symmetry of the Ising model is no longer present.For a given y, RPotts no longer peaks along z = 1. Finally, an analysis ofthe Lee-Yang (complex h) zeroes of the model reveals that the curvature alsodiverges as the locus of zeroes, z0(y), is approached. In fact,R ∼ (z − z0(y))− 12 , (16)so that this divergence is characterised by an edge exponent σ = 1/2.246810y12345z246810246810y12345z–2–1012Fig. 1. The scalar curvature for the Ising model (left) and the 10-state Potts model(right) in one dimension.The Three-Dimensional Spherical Model (and the Ising Model onPlanar Random Graphs): The spherical model is given byZspherical =∫ds1 . . . dsLdδ(∑is2i − Ld)expβ∑〈ij〉sisj + h∑isi , (17)and can be solved by exponentiating the constraint and using steepest descent.Allowing the lattice extent, L, to diverge reveals no transition for d = 1, 2.The transition for d = 3 has α = −1, β = 1/2 and γ = 2. This is thesame set of critical exponents as for the Ising model on planar random graphs(matter coupled to 2D gravity). This is remarkable, because there are noobvious similarities between the two models.Explicit calculations (in both models) give [5,6]R ∼ |β − βc|−2 , (18)5which does not accord with the prediction of (12). That prediction isR ∼ |β−βc|α−2, with α = −1. The source of the discrepancy can be traced back to thetop left term in the determinant in (10), which is t−α. In both current models,α is, in fact, negative and this term vanishes as criticality is approached. It isreplaced by a constant term coming from the regular part of the free energy.Both models then yield the same result (18).3 ConclusionsIn statistical physics, and in related fields – from the bio-sciences to economics– phase transitions play a central role. Thus new insights into the character-isation of critical phenomena are of paramount importance. Here, geometricideas from the field of parametric statistics are “borrowed” and explored. Itis found that the curvature associated with the Fisher-Rao metric is, indeed,a useful quantity in the characterisation of phase transitions. Some featuresfound in the Ising model are found not to be generic. In particular, R can benegative and there is no symmetry nor ridge along h = 0. More surprisingly,the naively expected scaling behaviour (12) fails in both the three-dimensionalspherical model and in the Ising model on planar random graphs. Reasons forthis are given. Once again, it is curious that all critical exponents for the lattertwo models coincide, although there is no obvious physical relation betweenthem.Acknowledgements: W.J. and D.J. were partially supported by EC IHPnetwork “Discrete Random Geometries: From Solid State Physics to QuantumGravity” HPRN-CT-1999-000161.References[1] R.A. Fisher, Phil. Trans. R. Soc. Lond. A 222 (1922) 309;C.R. Rao, Bull. Calcutta Math. Soc. 37 (1945) 81.[2] H. Janyszek, Rep. Math. Phys. 24 (1986) 1; ibid. 11;H. Janyszek and R. Mruga la, Phys. Rev. A 39 (1989) 6515.[3] B.P. Dolan, Proc. Roy. Soc. London A 454 (1998) 2655.[4] B.P. Dolan, D.A. Johnston, and R. Kenna, J. Phys. A 35 (2002) 9025.[5] W. Janke, D.A. Johnston, and Ranasinghe P.K.C. Malmini, Phys. Rev. E 66(2002) 056119.[6] W. Janke, D.A. Johnston, and R. Kenna, Phys. Rev. E 67 (2003) 046106.6

The introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrisation, the scalar curvature, R, plays a central role. A non-interacting model has a flat geometry (R=0), while R diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statistical-mechanical models. © 2004 Elsevier B.V. All rights reserved.</p

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Information Geometry and Phase Transitions

Information Geometry and Phase Transitions

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