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Relating Fisher information to order parameters


Mikhail Prokopenko, Joseph T. Lizier, Oliver Obst, and X. Rosalind Wang. Relating Fisher information to order parameters. Physical Review E, 84(4):041116, 2011.


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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 vari- able. 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 diver- gences of individual elements of the Fisher information matrix. A computational study of random Boolean networks (RBNs) 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.

BiBTeX Entry


@article{PLOW11,
	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 vari- able. 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 diver- gences of individual elements of
		   the Fisher information matrix. A computational study of random Boolean
		   networks (RBNs) 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.},
	Author = {Mikhail Prokopenko and Joseph T. Lizier and Oliver Obst
		   and X. Rosalind Wang},
	Doi = {10.1103/PhysRevE.84.041116},
	Journal = {Physical Review E},
	Keywords = {phase transitions, Fisher information, order parameter,
		   thermodynamic potential, free entropy, random Boolean network, critical
		   points},
	Number = 4,
	Pages = {041116},
	Title = {Relating Fisher information to order parameters},
	Volume = 84,
	Year = 2011,