Fisher information of Markovian decay modes - Nonequilibrium equivalence principle, dynamical phase transitions and coarse graining

TL;DR

This paper explores how Fisher information in decay modes reveals insights into nonequilibrium systems, classifies their phases, and links spectral properties to dynamical phase transitions, emphasizing the role of Fisher matrix degeneracy.

Contribution

It generalizes orthonormality of decay modes to normal generators, relates Fisher-Rao metric to statistical equivalence, and classifies nonequilibrium phases via spectral analysis.

Findings

Degenerate Fisher matrix indicates dynamical phase transitions.

Normal systems cannot exhibit critical behavior.

Fisher matrix analysis applies to systems with time-scale separation.

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.