Relating Fisher information to order parameters

Mikhail Prokopenko*, Joseph T. Lizier, Oliver Obst, X. Rosalind Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

70 Citations (Scopus)


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.

Original languageEnglish
Article number041116
Pages (from-to)1-10
Number of pages10
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number4
Publication statusPublished - 13 Oct 2011
Externally publishedYes


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