Eur. Phys. J. B (2014) 87: 215
https://doi.org/10.1140/epjb/e2014-50142-1

Regular Article

Fisher information of Markovian decay modes

Nonequilibrium equivalence principle, dynamical phase transitions and coarse graining

Complex Systems and Statistical Mechanics, University of Luxembourg, Campus Limpertsberg, 162a avenue de la Faïencerie, 1511 Luxembourg, Luxembourg

^a e-mail: matteo.polettini@uni.lu

Received: 3 March 2014
Received in final form: 7 July 2014
Published online: 17 September 2014

Abstract

We introduce the Fisher information in the basis of decay modes of Markovian dynamics, arguing that it encodes important information about the behavior of nonequilibrium systems. In particular we generalize an orthonormality relation between decay eigenmodes of detailed balanced systems to normal generators that commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical distributions, we relate the result to the choice of a coordinate patch that makes the Fisher-Rao metric Euclidean at the steady distribution, realizing a sort of statistical equivalence principle. We then classify nonequilibrium systems according to their spectrum, showing that a degenerate Fisher matrix is the signature of the insurgence of a class of dynamical phase transitions between nonequilibrium regimes, characterized by level crossing and power-law decay in time of suitable order parameters. An important consequence is that normal systems cannot manifest critical behavior. Finally, we study the Fisher matrix of systems with time-scale separation.