Geodesics in information geometry: Classical and quantum phase transitions

Kumar, Prashant; Mahapatra, Subhash; Phukon, Prabwal; Sarkar, Tapobrata

doi:10.1103/physreve.86.051117

Cited by 37 publications

(30 citation statements)

References 17 publications

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“…[103]) and a geodesic-analysis (for more details, see Ref. [111]) were considered for this specific model. The geometric parameters used in this case were the anisotropy parameter γ and the magnetic field intensity h. For this system the components of the metric tensor on the parameter manifold diverge at the quantum phase transition, while the Ricci scalar curvature does not (see Eqs.…”

Section: Quantum Frameworkmentioning

confidence: 99%

Information geometric methods for complexity

Felice

Cafaro

Mancini

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence, we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.

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Section: Quantum Frameworkmentioning

confidence: 99%

Information geometric methods for complexity

Felice

Cafaro

Mancini

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In a different context, pioneering efforts by Ruppeiner and others have shown that the equilibrium state space of a thermodynamic system can be uplifted to a Riemannian geometry via the Gaussian fluctuation moments which constitute the metric [10]. Thermodynamic geometry has been investigated for a variety of systems ranging from fluids and magnetic systems to various black hole systems, and significant information has been revealed through the invariants of the geometry, like the geodesics or the curvature scalar [10][11][12][13][14][15]. Thermodynamic geometry forms a remarkable connection from the thermodynamic description to the underlying statistical description.…”

Section: Introductionmentioning

confidence: 99%

Restricted thermodynamic fluctuations and the Ruppeiner geometry of black holes

Sahay

2017

Phys. Rev. D

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Thermodynamic fluctuation metrics in Ruppeiner's formalism are worked out for Kerr-AdS black holes in the extended state space. The implications of constraints upon the state space geometry and their correspondence with thermodynamical ensembles are explicitly worked out in the most general setting. The state space scalar curvature for a given ensemble is found to be sensitive to the instabilities or phase transitions therein. In particular, it is found that the appropriate Ruppeiner scalar curvature does encode critical phenomena in the Kerr-AdS black holes. A detailed study is undertaken of the curvature contour of the state space of the 4D Kerr-AdS black hole, and suitable inferences are drawn. In particular, thermodynamic geometry suggests an instability in the Schwarzschild-AdS limit for all the ensembles except the pressure ensemble, which is equivalent to the unextended state space of the Kerr-AdS black holes. The extrinsic geometry of the ensemble hypersurfaces is introduced, and its relevance to constrained thermodynamic fluctuations is discussed. A new interpretation for the thermodynamic curvature of black hole systems is suggested.

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“…Of particular interest in these studies is the role of the scalar (Riemannian) curvature. It was shown to diverge at critical transition points and on the spinodal curve [25,26], thus effectively preventing geodesics from crossing into the unphysical area of phase space [27]. See also [28] for a general renormalization group analysis of IG near criticality.…”

Section: Introductionmentioning

confidence: 99%

Information geometric analysis of phase transitions in complex patterns: the case of the Gray-Scott reaction–diffusion model

et al. 2016

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The Fisher-Rao metric from Information Geometry is related to phase transition phenomena in classical statistical mechanics. Several studies propose to extend the use of Information Geometry to study more general phase transitions in complex systems. However, it is unclear whether the Fisher-Rao metric does indeed detect these more general transitions, especially in the absence of a statistical model. In this paper we study the transitions between patterns in the Gray-Scott reactiondiffusion model using Fisher information. We describe the system by a probability density function that represents the size distribution of blobs in the patterns and compute its Fisher information with respect to changing the two rate parameters of the underlying model. We estimate the distribution non-parametrically so that we do not assume any statistical model. The resulting Fisher map can be interpreted as a phase-map of the different patterns. Lines with high Fisher information can be considered as boundaries between regions of parameter space where patterns with similar characteristics appear. These lines of high Fisher information can be interpreted as phase transitions between complex patterns. arXiv:1512.02077v2 [cond-mat.stat-mech]

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Geodesics in information geometry: Classical and quantum phase transitions

Cited by 37 publications

References 17 publications

Information geometric methods for complexity

Information geometric methods for complexity

Restricted thermodynamic fluctuations and the Ruppeiner geometry of black holes

Information geometric analysis of phase transitions in complex patterns: the case of the Gray-Scott reaction–diffusion model

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