Relating Fisher information to order parameters.
Abstract
We study phase transitions and relevant order parameters via statistical estimation theory using the Fisher information matrix. The assumptions that we make limit our analysis to order parameters representable as a negative derivative of thermodynamic potential over some thermodynamic variable. Nevertheless, the resulting representation is sufficiently general and explicitly relates elements of the Fisher information matrix to the rate of change in the corresponding order parameters. The obtained relationships allow us to identify, in particular, second-order phase transitions via divergences of individual elements of the Fisher information matrix. A computational study of random Boolean networks supports the derived relationships, illustrating that Fisher information of the magnetization bias (that is, activity level) is peaked in finite-size networks at the critical points, and the maxima increase with the network size. The framework presented here reveals the basic thermodynamic reasons behind similar empirical observations reported previously. The study highlights the generality of Fisher information as a measure that can be applied to a broad range of systems, particularly those where the determination of order parameters is cumbersome.
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Fisher information has been shown to be related to phase transitions.
“Fisher information, which has been shown to be related to phase transitions (Prokopenko et al., 2011).”
From Information Measures of Complexity, Emergence, Self-organization, Homeostasis, and Autopoiesis (2013) · cites this paper as Prokopenko et al., 2011
C is correlated with Fisher information.
“C is correlated with Fisher information, which has been shown to be related to phase transitions (Prokopenko et al., 2011).”
From Information Measures of Complexity, Emergence, Self-organization, Homeostasis, and Autopoiesis (2013) · cites this paper as Prokopenko et al., 2011
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Entropy, Economics, and Criticality
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Thermodynamic Efficiency of Interactions in Self-Organizing Systems
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Self-citationGeometry and control of thermodynamic systems described by generalized exponential families
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Critical echo state network dynamics by means of Fisher information maximization
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Advances in Quantum Metrology: Continuous Variables in Phase Space
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Physics-informed GCN-LSTM framework for long-term forecasting of 2D and 3D microstructure evolution
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Geometric quantification of cell phenotype transition manifolds with information geometry
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Exponential Families with External Parameters
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“3 Fisher Information C is correlated with Fisher information, which has been shown to be related to phase transitions (Prokopenko et al., 2011).”
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Criticality of the configuration-space statistical geometry
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- Authors
- Prokopenko, M., Lizier, J. T., Obst, O., & Wang, X. R.
- Affiliations
- Commonwealth Scientific and Industrial Research Organisation (AU)
- Max Planck Institute for Mathematics in the Sciences (DE)
- Venue
- Physical Review E
- Publisher
- American Physical Society (APS)
- Published
- 2011-10-13
- Volume
- 84
- Issue
- 4
- Pages
- 041116-041116
- ISSN
- 1539-3755 (ISSN-L)
- 1550-2376
- OpenAlex
- W1971124708
- Cited by
- 122
- References
- 34
- Citations by year
- 2026: 10 · 2025: 4 · 2024: 4 · 2023: 3 · 2022: 8 · 2021: 9 · 2020: 8 · 2019: 6 · 2018: 15 · 2017: 16 · 2016: 7 · 2015: 3 · 2014: 14 · 2013: 12 · 2012: 3
- FWCI
- 6.92
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review E (publishedVersion)
- PubMed (publishedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Keywords
- Fisher information matrix, order parameter, phase transitions, second-order phase transition, critical points, random Boolean networks, thermodynamic potential, finite-size scaling, critical phenomena
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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